tag:theconversation.com,2011:/es/topics/mathematics-98/articlesMathematics – The Conversation2024-03-26T12:39:43Ztag:theconversation.com,2011:article/2238092024-03-26T12:39:43Z2024-03-26T12:39:43ZHow AI and a popular card game can help engineers predict catastrophic failure – by finding the absence of a pattern<figure><img src="https://images.theconversation.com/files/584159/original/file-20240325-10630-wq22k6.png?ixlib=rb-1.1.0&rect=84%2C498%2C1343%2C882&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Can you find a matching set?</span> <span class="attribution"><a class="source" href="https://en.wikipedia.org/wiki/File:Set_isomorphic_cards.svg">Cmglee/Wikimedia Commons</a>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span></figcaption></figure><p>Humans are very good at <a href="https://doi.org/10.1016/j.neuron.2018.05.013">spotting patterns</a>, or repeating features people can recognize. For instance, ancient Polynesians navigated across the Pacific by <a href="https://nla.gov.au/nla.obj-318911458/view?partId=nla.obj-318923632#page/n90/mode/1up">recognizing many patterns</a>, from the stars’ constellations to more subtle ones such as the directions and sizes of ocean swells.</p>
<p>Very recently, <a href="https://scholar.google.com/citations?user=L1lKOGsAAAAJ&hl=en">mathematicians like me</a> have started to study large collections of objects that have no patterns of a particular sort. How large can collections be before a specified pattern has to appear somewhere in the collection? Understanding such scenarios can have significant real-world implications: For example, what’s the smallest number of server failures that would lead to the severing of the internet?</p>
<p>Research from mathematician <a href="https://scholar.google.com/citations?user=b7P6YbkAAAAJ&hl=en">Jordan Ellenberg</a> at the University of Wisconsin and researchers at <a href="https://deepmind.google/">Google’s Deep Mind</a> have proposed a novel approach to this problem. Their work <a href="https://doi.org/10.1038/s41586-023-06924-6">uses artificial intelligence to find</a> large collections that don’t contain a specified pattern, which can help us understand some worst-case scenarios.</p>
<h2>Patterns in the card game Set</h2>
<p>The idea of patternless collections can be illustrated by a popular card game <a href="https://brilliant.org/wiki/set-game/">called Set</a>. In this game, players lay out 12 cards, face up. Each card has a different simple picture on it. They vary in terms of number, color, shape and shading. Each of these four features can have one of three values.</p>
<p>Players race to look for “sets,” which are groups of three cards in which every feature is either the same or different in each card. For instance, cards with one solid red diamond, two solid green diamonds and three solid purple diamonds form a set: All three have different numbers (one, two, three), the same shading (solid), different colors (red, green, purple) and the same shape (diamond).</p>
<figure>
<iframe width="440" height="260" src="https://www.youtube.com/embed/7AeEr9QtDF0?wmode=transparent&start=0" frameborder="0" allowfullscreen=""></iframe>
<figcaption><span class="caption">Marsha Falco originally created the game Set to help explain her research on population genetics.</span></figcaption>
</figure>
<p>Finding a set is usually possible – but not always. If none of the players can find a set from the 12 cards on the table, then they flip over three more cards. But they still might not be able to find a set in these 15 cards. The players continue to flip over cards, three at a time, until someone spots a set.</p>
<p>So what is the maximum number of cards you can lay out without forming a set?</p>
<p>In 1971, mathematician Giuseppe Pellegrino showed that the <a href="https://www.quantamagazine.org/set-proof-stuns-mathematicians-20160531/">largest collection of cards without a set is 20</a>. But if you chose 20 cards at random, “no set” would happen only <a href="https://www-cs-faculty.stanford.edu/%7Eknuth/programs/setset-all.w">about one in a trillion times</a>. And finding these “no set” collections is an extremely hard problem to solve.</p>
<h2>Finding ‘no set’ with AI</h2>
<p>If you wanted to find the smallest collection of cards with no set, you could in principle do an exhaustive search of every possible collection of cards chosen from the deck of 81 cards. But there are an enormous number of possibilities – on the order of 10<sup>24</sup> (that’s a “1” followed by 24 zeros). And if you increase the number of features of the cards from four to, say, eight, the complexity of the problem would overwhelm any computer doing an exhaustive search for “no set” collections.</p>
<p>Mathematicians love to think about computationally difficult problems like this. These complex problems, if approached in the right way, can become tractable. </p>
<p>It’s easier to find best-case scenarios – here, that would mean the fewest number of cards that could contain a set. But there were few known strategies that could explore bad scenarios – here, that would mean a large collection of cards that do not contain a set.</p>
<p>Ellenberg and his collaborators approached the bad scenario with a type of AI called <a href="https://theconversation.com/ai-to-z-all-the-terms-you-need-to-know-to-keep-up-in-the-ai-hype-age-203917">large language models, or LLMs</a>. The researchers first wrote computer programs that generate some examples of collections of many that contain no set. These collections typically have “cards” with more than four features.</p>
<p>Then they fed these programs to the LLM, which soon learned how to write many similar programs and choose the ones that give rise to the largest set-free collections to undergo the process again. Iterating that process by repeatedly tweaking the most successful programs enables them to find larger and larger set-free collections.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/584175/original/file-20240325-28-3w7tk3.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="Square of nine circles, four of which are colored blue, connected by grey, red, green, and yellow lines" src="https://images.theconversation.com/files/584175/original/file-20240325-28-3w7tk3.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/584175/original/file-20240325-28-3w7tk3.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=600&fit=crop&dpr=1 600w, https://images.theconversation.com/files/584175/original/file-20240325-28-3w7tk3.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=600&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/584175/original/file-20240325-28-3w7tk3.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=600&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/584175/original/file-20240325-28-3w7tk3.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=754&fit=crop&dpr=1 754w, https://images.theconversation.com/files/584175/original/file-20240325-28-3w7tk3.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=754&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/584175/original/file-20240325-28-3w7tk3.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=754&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">This is another version of a ‘no set,’ where no three components of a set are linked by a line.</span>
<span class="attribution"><a class="source" href="https://doi.org/10.1038/s41586-023-06924-6">Romera-Peredes et al./Nature</a>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span>
</figcaption>
</figure>
<p>This method allows people to explore disordered collections – in this instance, <a href="https://doi.org/10.1038/s41586-023-06924-6">collections of cards that contain no set</a> – in an entirely new way. It does not guarantee that researchers will find the absolute worst-case scenario, but they will find scenarios that are much worse than a random generation would yield.</p>
<p>Their work can help researchers understand how events might align in a way that leads to catastrophic failure. </p>
<p>For example, how vulnerable is the electrical grid to a malicious attacker who destroys select substations? Suppose that a bad collection of substations is one where they don’t form a connected grid. The worst-case scenario is now a very large number of substations that, when taken all together, still don’t yield a connected grid. The amount of substations excluded from this collection make up the smallest number a malicious actor needs to destroy to deliberately disconnect the grid.</p>
<p>The work of Ellenberg and his collaborators demonstrates yet another way that AI is a very powerful tool. But to solve very complex problems, at least for now, it still needs human ingenuity to guide it.</p><img src="https://counter.theconversation.com/content/223809/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>John Edward McCarthy is partially supported by National Science Foundation Grant
DMS 2054199. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.</span></em></p>What mathematicians call ‘disordered collections’ can help engineers explore real-world worst-case scenarios. The simple card game Set illustrates how to predict internet and electrical grid failures.John Edward McCarthy, Professor of Mathematics, Arts & Sciences at Washington University in St. LouisLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/2235092024-03-01T13:40:34Z2024-03-01T13:40:34ZThe ‘average’ revolutionized scientific research, but overreliance on it has led to discrimination and injury<figure><img src="https://images.theconversation.com/files/578352/original/file-20240227-22-rs4i9u.jpg?ixlib=rb-1.1.0&rect=0%2C0%2C5591%2C3722&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">The average can tell you a lot about a dataset, but not everything. </span> <span class="attribution"><a class="source" href="https://www.gettyimages.com/photos/bell-curve?assettype=image&alloweduse=availableforalluses&agreements=pa%3A174132&family=creative&phrase=bell%20curve&sort=best">marekuliasz/iStock via Getty Images Plus</a></span></figcaption></figure><p>When analyzing a set of data, one of the first steps many people take is to compute an average. You might compare your height against the average height of people where you live, or brag about your favorite baseball player’s batting average. But while the average can help you study a dataset, it has important limitations. </p>
<p>Uses of the average that ignore these limitations have led to serious issues, such as <a href="https://www.routledge.com/The-Disability-Studies-Reader/Davis/p/book/9781138930230">discrimination</a>, <a href="https://www.gao.gov/products/gao-23-105595">injury</a> and even life-threatening accidents. </p>
<p>For example, the U.S. Air Force used to design its planes for “the average man,” but abandoned the practice when pilots <a href="https://www.youtube.com/watch?v=4eBmyttcfU4&pp=ygURdG9kZCByb3NlIGF2ZXJhZ2U%3D">couldn’t control their aircraft</a>. The average has many uses, but it doesn’t tell you anything about the variability in a dataset.</p>
<p>I am a <a href="https://scholar.google.com/citations?user=zEYYuIcAAAAJ&hl=en">discipline-specific education researcher</a>, meaning I study how people learn, with a focus on engineering. My research includes study of how engineers use averages in their work.</p>
<h2>Using the average to summarize data</h2>
<p>The average has been around for a long time, with its use documented as early as the ninth or eighth century BCE. In an early instance, the Greek poet Homer <a href="https://www.penguinrandomhouse.com/books/292278/the-history-of-the-peloponnesian-war-by-thucydides-translated-by-rex-warner-introduction-and-notes-by-m-i-finley/">estimated the number of soldiers</a> on ships by taking an average.</p>
<p>Early astronomers wanted to predict future locations of stars. But to make these predictions, they first needed accurate measurements of the stars’ current positions. Multiple astronomers would take position measurements independently, but they often arrived at different values. Since a star has just one true position, these discrepancies were a problem.</p>
<p>Galileo in 1632 was the <a href="https://doi.org/10.1080/0025570X.2006.11953386">first to push for a systematic approach</a> to address these measurement differences. His analysis was the beginning of <a href="https://press.princeton.edu/books/paperback/9780691208428/the-rise-of-statistical-thinking-1820-1900">error theory</a>. Error theory helps scientists reduce uncertainty in their measurements.</p>
<h2>Error theory and the average</h2>
<p>Under error theory, researchers interpret a set of measurements as falling around a true value that is corrupted by error. In astronomy, a star has a true location, but early astronomers may have had unsteady hands, blurry telescope images and bad weather – all sources of error.</p>
<p>To deal with error, researchers often assume that measurements are unbiased. In statistics, this means they evenly distribute around a central value. Unbiased measurements still have error, but they can be combined to better estimate the true value.</p>
<p>Say three scientists have each taken three measurements. Viewed separately, their measurements may seem random, but when unbiased measurements are put together, they evenly distribute around a middle value: the average.</p>
<p>When measurements are unbiased, the average will tend to sit in the middle of all measurements. In fact, we can show mathematically that <a href="https://doi.org/10.1080/0025570X.2006.11953386">the average is closest</a> to all possible measurements. For this reason, the average is an excellent tool for dealing with measurement errors.</p>
<h2>Statistical thinking</h2>
<p>Error theory was, in its time, considered revolutionary. Other scientists admired the precision of astronomy and sought to bring the same approach to their disciplines. The 19th century scientist Adolphe Quetelet applied ideas from error theory to study humans and <a href="https://press.princeton.edu/books/paperback/9780691208428/the-rise-of-statistical-thinking-1820-1900">introduced the idea</a> of taking averages of human heights and weights.</p>
<p>The average helps make comparisons across groups. For instance, taking averages from a dataset of male and female heights can show that the males in the dataset are taller – on average – than the females. However, the average does not tell us everything. In the same dataset, we could likely find individual females who are taller than individual males.</p>
<p>So, you can’t consider only the average. You should also consider the spread of values by thinking statistically. <a href="https://doi.org/10.1111/j.1751-5823.1999.tb00442.x">Statistical thinking</a> is defined as thinking carefully about variation – or the tendency of measured values to be different.</p>
<p>For example, different astronomers taking measurements of the same star and recording different positions is one example of variation. The astronomers had to think carefully about where their variation came from. Since a star has one true position, they could safely assume their variation was due to error.</p>
<p>Taking the average of measurements makes sense when variation comes from sources of error. But researchers have to be careful when interpreting the average when there is real variation. For instance, in the height example, individual females can be taller than individual males, even if men are taller on average. Focusing on the average alone <a href="https://doi.org/10.1080/26939169.2024.2308119">neglects variation</a>, which has caused serious issues.</p>
<p>Quetelet did not just take the practice of computing averages from error theory. He also took the assumption of a single true value. He elevated an ideal of “the average man” and suggested that <a href="https://press.princeton.edu/books/paperback/9780691208428/the-rise-of-statistical-thinking-1820-1900">human variability was fundamentally error</a> – that is, not ideal. To Quetelet, there’s something wrong with you if you’re not exactly average height.</p>
<p>Researchers who study <a href="https://www.routledge.com/The-Disability-Studies-Reader/Davis/p/book/9781138930230">social norms</a> note that Quetelet’s ideas about “the average man” contributed the modern meaning of the word “normal” – normal height, as well as normal behavior.</p>
<p>These ideas have been used by some, such as <a href="https://theconversation.com/francis-galton-pioneered-scientific-advances-in-many-fields-but-also-founded-the-racist-pseudoscience-of-eugenics-144465">early statisticians</a>, to divide populations in two: people who are in some way superior and those who are inferior. </p>
<p>For instance, the <a href="https://www.genome.gov/about-genomics/fact-sheets/Eugenics-and-Scientific-Racism">eugenics movement</a> – a despicable effort to prevent “inferior” people from having children – <a href="https://www.routledge.com/The-Disability-Studies-Reader/Davis/p/book/9781138930230">traces its thinking</a> to these ideas about “normal” people.</p>
<p>While Quetelet’s idea of variation as error <a href="https://doi.org/10.1080/15017410600608491">supports practices of discrimination</a>, Quetelet-like uses of the average also have direct connections to modern engineering failures.</p>
<h2>Failures of the average</h2>
<p>In the 1950s, the U.S. Air Force designed its aircraft for “the average man.” It assumed that a plane designed for an average height, average arm length and the average along several other key dimensions <a href="https://www.youtube.com/watch?v=4eBmyttcfU4&pp=ygURdG9kZCByb3NlIGF2ZXJhZ2U%3D">would work for most pilots</a>.</p>
<p>This decision contributed to as many as <a href="http://www.toddrose.com/endofaverage">17 pilots crashing in a single day</a>. While “the average man” could operate the aircraft perfectly, real variation got in the way. A shorter pilot would have trouble seeing, while a pilot with longer arms and legs would have to squish themselves to fit. </p>
<p>While the Air Force assumed most of its pilots would be close to average along all key dimensions, it found that out of 4,063 pilots, <a href="https://books.google.com/books/about/The_Average_Man.html?id=NxmdHAAACAAJ">zero were average</a>.</p>
<p>The Air Force solved the problem by designing for variation – it designed adjustable seats to account for the real variation among pilots.</p>
<p>While adjustable seats might seem obvious now, this “average man” thinking still causes problems today. In the U.S., women experience <a href="https://doi.org/10.2105/AJPH.2011.300275">about 50% higher odds of severe injury</a> in automobile accidents.</p>
<p>The Government Accountability Office blames this disparity on crash-test practices, where female passengers are crudely represented using a <a href="https://www.gao.gov/products/gao-23-105595">scaled version of a male dummy</a>, much like the Air Force’s “average man.” The first female crash-test dummy <a href="https://www.npr.org/2022/11/01/1133375223/the-first-female-crash-test-dummy-has-only-now-arrived">was introduced in 2022</a> and has yet to be adopted in the U.S.</p>
<p>The average is useful, but it has limitations. For estimating true values or making comparisons across groups, the average is powerful. However, for individuals who exhibit real variability, the average simply doesn’t mean that much.</p><img src="https://counter.theconversation.com/content/223509/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Zachary del Rosario receives funding from the National Science Foundation, and has worked with Citrine Informatics and Toyota Research Institute.</span></em></p>The average might come in handy for certain data analyses, but is any one person really ‘average’?Zachary del Rosario, Assistant Professor of Engineering, Olin College of EngineeringLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/2239412024-02-28T13:11:56Z2024-02-28T13:11:56ZUnderstanding how the brain works can transform how school students learn maths<figure><img src="https://images.theconversation.com/files/578023/original/file-20240226-29-lwslum.jpg?ixlib=rb-1.1.0&rect=11%2C176%2C7337%2C4726&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">
</span> <span class="attribution"><a class="source" href="https://www.shutterstock.com/image-photo/school-kid-doing-math-exercise-homework-2090861887">SrideeStudio/Shutterstock</a></span></figcaption></figure><p>School mathematics teaching is stuck in the past. An adult revisiting the school that they attended as a child would see only superficial changes from what they experienced themselves. </p>
<p>Yes, in some schools they might see a room full of electronic tablets, or the teacher using a touch-sensitive, interactive whiteboard. But if we zoom in on the details – the tasks that students are actually being given to help them make sense of the subject – things have <a href="https://bera-journals.onlinelibrary.wiley.com/doi/10.1002/curj.159">hardly changed at all</a>.</p>
<p>We’ve learnt a huge amount in recent years about cognitive science – how our brains work and how people learn most effectively. This understanding has the potential to revolutionise what teachers <a href="https://www.deansforimpact.org/tools-and-resources/the-science-of-learning">do in classrooms</a>. But the design of mathematics teaching materials, such as textbooks, has <a href="https://www.tandfonline.com/doi/full/10.1080/14794802.2022.2158122">benefited very little from this knowledge</a>.</p>
<p>Some of this knowledge is counter-intuitive, and therefore unlikely to be applied unless done so deliberately. What learners prefer to experience, and what teachers think is likely to be most effective, often isn’t what will help the most.</p>
<p>For example, cognitive science <a href="https://link.springer.com/article/10.3758/s13423-014-0588-3">tells us</a> that practising similar kinds of tasks all together generally leads to less effective learning than mixing up tasks that require different approaches. </p>
<p>In mathematics, practising similar tasks together could be a page of questions each of which requires addition of fractions. Mixing things up might involve bringing together fractions, probability and equations in immediate succession.</p>
<p>Learners make more mistakes when doing mixed exercises, and are likely to feel frustrated by this. Grouping similar tasks together is therefore likely to be much easier for the teacher to manage. But the mixed exercises give the learner important practice at deciding what method they need to use for each question. This means that more knowledge is retained afterwards, making this what is known as a <a href="https://bjorklab.psych.ucla.edu/wp-content/uploads/sites/13/2016/04/EBjork_RBjork_2011.pdf">“desirable difficulty”</a>.</p>
<h2>Cognitive science applied</h2>
<p>We are just now beginning to apply findings like this from cognitive science to design better teaching materials and to support teachers in using them. Focusing on school mathematics makes sense because mathematics is a compulsory subject which many people find difficult to learn.</p>
<p>Typically, school teaching materials are chosen by gut reactions. A head of department looks at a new textbook scheme and, based on their experience, chooses whatever seems best to them. What else can they be expected to do? But even the best materials on offer are generally not designed with cognitive science principles such as “desirable difficulties” in mind.</p>
<p>My colleagues and I have been researching <a href="https://bera-journals.onlinelibrary.wiley.com/doi/10.1002/curj.249">educational design</a> that applies principles from <a href="https://psycnet.apa.org/record/2015-00153-003">cognitive science</a> to mathematics teaching, and are developing <a href="https://www.lboro.ac.uk/services/lumen/curriculum/">materials for schools</a>. These materials are not designed to look easy, but to include “desirable difficulties”. </p>
<p>They are not divided up into <a href="https://www.foster77.co.uk/Foster,%20Teach%20Secondary,%20Stop%20planning%20lessons.pdf">individual lessons</a>, because this pushes the teacher towards moving on when the clock says so, regardless of student needs. Being responsive to students’ developing understanding and difficulties requires materials designed according to the size of the ideas, rather than what will fit conveniently onto a double-page spread of a textbook or into a 40-minute class period.</p>
<h2>Switching things up</h2>
<p>Taking an approach led by cognitive science also means changing how mathematical concepts are explained. For instance, diagrams have always been a prominent feature of mathematics teaching, but often they are used haphazardly, based on the teacher’s personal preference. In textbooks they are highly restricted, due to space constraints. </p>
<p>Often, similar-looking diagrams are used in different topics and for very different purposes, leading to confusion. For example, three circles connected as shown below can indicate partitioning into a sum (<a href="https://www.ncetm.org.uk/classroom-resources/primm-102-introducing-whole-and-parts-part-part-whole/">the “part-whole model”</a>) or a product of prime factors. </p>
<p>These involve two very different operations, but are frequently represented by the same diagram. Using the same kind of diagram to represent conflicting operations (addition and multiplication) leads to learners muddling them up and becoming confused.</p>
<figure class="align-center ">
<img alt="Diagram of connected circles with numbers inside, as described above." src="https://images.theconversation.com/files/577980/original/file-20240226-16-w2ab9l.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/577980/original/file-20240226-16-w2ab9l.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=228&fit=crop&dpr=1 600w, https://images.theconversation.com/files/577980/original/file-20240226-16-w2ab9l.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=228&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/577980/original/file-20240226-16-w2ab9l.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=228&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/577980/original/file-20240226-16-w2ab9l.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=286&fit=crop&dpr=1 754w, https://images.theconversation.com/files/577980/original/file-20240226-16-w2ab9l.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=286&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/577980/original/file-20240226-16-w2ab9l.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=286&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Number diagrams showing numbers that add together to make six and numbers that multiply to make six.</span>
<span class="attribution"><span class="source">Colin Foster</span></span>
</figcaption>
</figure>
<p>The <a href="https://www.cambridge.org/core/books/abs/multimedia-learning/coherence-principle/4E80B70CB76E2166B76E5653EBDE7D3E">“coherence principle”</a> from cognitive science means avoiding diagrams where their drawbacks outweigh their benefits, and using diagrams and animations in a purposeful, consistent way across topics.</p>
<p>For example, <a href="https://www.foster77.co.uk/Foster,%20Using%20coherent%20representations%20of%20number%20in%20the%20school%20mathematics%20curriculum.pdf">number lines</a> can be introduced at a young age and incorporated across many topic areas to bring coherence to students’ developing understanding of number. Number lines can be used to solve equations and also to represent probabilities, for instance. </p>
<p>Unlike with the circle diagrams above, the uses of number lines shown below don’t conflict but reinforce each other. In each case, positions on the number line represent numbers, from zero on the left, increasing to the right.</p>
<figure class="align-center ">
<img alt="Number line in red and black demonstrating how to solve an equation, as described above" src="https://images.theconversation.com/files/578260/original/file-20240227-16-wz37l0.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/578260/original/file-20240227-16-wz37l0.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=133&fit=crop&dpr=1 600w, https://images.theconversation.com/files/578260/original/file-20240227-16-wz37l0.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=133&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/578260/original/file-20240227-16-wz37l0.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=133&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/578260/original/file-20240227-16-wz37l0.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=167&fit=crop&dpr=1 754w, https://images.theconversation.com/files/578260/original/file-20240227-16-wz37l0.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=167&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/578260/original/file-20240227-16-wz37l0.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=167&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">A number line used to solve an equation.</span>
<span class="attribution"><span class="source">Colin Foster</span></span>
</figcaption>
</figure>
<figure class="align-center ">
<img alt="Number line with values from left to right: 0, unlikely, even chance, likely, 1." src="https://images.theconversation.com/files/577987/original/file-20240226-17-ys3jg7.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/577987/original/file-20240226-17-ys3jg7.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=80&fit=crop&dpr=1 600w, https://images.theconversation.com/files/577987/original/file-20240226-17-ys3jg7.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=80&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/577987/original/file-20240226-17-ys3jg7.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=80&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/577987/original/file-20240226-17-ys3jg7.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=101&fit=crop&dpr=1 754w, https://images.theconversation.com/files/577987/original/file-20240226-17-ys3jg7.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=101&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/577987/original/file-20240226-17-ys3jg7.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=101&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">A number line used to show probability.</span>
<span class="attribution"><span class="source">Colin Foster</span></span>
</figcaption>
</figure>
<p>There are disturbing <a href="https://theconversation.com/gcse-and-higher-results-show-worsening-gap-between-richer-and-poorer-pupils-pandemic-assessment-shows-we-should-reconsider-exams-216409">inequalities</a> in the learning of mathematics, with students from poorer backgrounds underachieving relative to their wealthier peers. There is also a huge <a href="https://www.wisecampaign.org.uk/a-level-results-2023/">gender participation gap</a> in maths, at A-level and beyond, which is taken by far more boys than girls. </p>
<p>Socio-economically advantaged families have always been able to buy their children out of difficulties by using private tutors, but less privileged families cannot. Better-quality teaching materials, based on insights from cognitive science, mitigate the impact for students who have traditionally been disadvantaged by gender, race or financial background in the learning of mathematics.</p><img src="https://counter.theconversation.com/content/223941/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Colin Foster receives funding from Research England and UKRI Economic and Social Research Council. He works for Loughborough University and is Director of the Loughborough University Mathematics Education Network.</span></em></p>Principles from cognitive science can help help in the design of more effective teaching materials for maths.Colin Foster, Reader in Mathematics Education, Loughborough UniversityLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/2209152024-02-28T12:33:16Z2024-02-28T12:33:16ZAnyone can play Tetris, but architects, engineers and animators alike use the math concepts underlying the game<figure><img src="https://images.theconversation.com/files/572289/original/file-20240130-15-cg1jbw.jpg?ixlib=rb-1.1.0&rect=0%2C0%2C2991%2C2434&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Tetris has hooked people for decades. </span> <span class="attribution"><a class="source" href="https://newsroom.ap.org/detail/MusicofGaming/8f0f44af03b145208aa578e21b453275/photo?Query=tetris&mediaType=photo&sortBy=&dateRange=Anytime&totalCount=284&digitizationType=Digitized&currentItemNo=1&vs=true&vs=true">AP Photo/Richard Drew</a></span></figcaption></figure><p>With its bright colors, easy-to-learn rules and familiar music, the video game Tetris has endured as a pop culture icon over the last 40 years. Many people, like me, have been playing the game for decades, and it has evolved to adapt to new technologies like game systems, phones and tablets. But until January 2024, nobody had ever been able to beat it.</p>
<p>A teen from Oklahoma holds the Tetris title after he crashed the game on Level 157 and <a href="https://www.wired.com/story/why-everyone-is-obsessed-with-the-kid-who-beat-tetris/">beat the game</a>. Beating it means the player moved the tiles too fast for the game to keep up with the score, causing the game to crash. Artificial intelligence can <a href="https://theconversation.com/from-besting-tetris-ai-to-epic-speedruns-inside-gamings-most-thrilling-feats-220620">suggest strategies</a> that allow players to more effectively control the game tiles and slot them into place faster – these strategies helped crown the game’s first winner. </p>
<figure>
<iframe width="440" height="260" src="https://www.youtube.com/embed/aSXxa64WrEA?wmode=transparent&start=0" frameborder="0" allowfullscreen=""></iframe>
<figcaption><span class="caption">The Las Vegas sphere lit up with a massive game of Tetris in January 2024. The game’s appeal spans generations.</span></figcaption>
</figure>
<p>But there’s far more to Tetris than the elusive promise of winning. As a <a href="https://education.wfu.edu/about-the-department/faculty-and-staff-profiles/dr-leah-mccoy/">mathematician and mathematics educator</a>, I recognize that the game is based on a fundamental element of geometry, called dynamic spatial reasoning. The player uses these geometric skills to manipulate the game pieces, and playing can both test and improve a player’s dynamic spatial reasoning.</p>
<h2>Playing the game</h2>
<p>A Russian computer scientist named <a href="https://tetris.com/history-of-tetris">Alexey Pajitnov invented Tetris</a> in 1984. The game itself is very simple: The Tetris screen is composed of a rectangular game board with dropping geometric figures. These figures are called <a href="https://mathworld.wolfram.com/Tetromino.html">tetrominoes</a>, made up of four squares connected on their sides in seven different configurations. </p>
<p>The game pieces drop from the top, one at a time, stacking up from the bottom. <a href="https://tetris.com/article/33/tetris-tips-for-beginners">The player can manipulate each one</a> as it falls by turning or sliding it and then dropping it to the bottom. When a row completely fills up, it disappears and the player earns points. </p>
<p>As the game progresses, the pieces appear at the top more quickly, and the game ends when the stack reaches the top of the board.</p>
<h2>Dynamic spatial reasoning</h2>
<figure class="align-right zoomable">
<a href="https://images.theconversation.com/files/572281/original/file-20240130-25-8puxh9.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="A Tetris board, which has blocks made up of four squares arranged in different formations, stacked atop each other." src="https://images.theconversation.com/files/572281/original/file-20240130-25-8puxh9.png?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/572281/original/file-20240130-25-8puxh9.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=1100&fit=crop&dpr=1 600w, https://images.theconversation.com/files/572281/original/file-20240130-25-8puxh9.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=1100&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/572281/original/file-20240130-25-8puxh9.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=1100&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/572281/original/file-20240130-25-8puxh9.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=1382&fit=crop&dpr=1 754w, https://images.theconversation.com/files/572281/original/file-20240130-25-8puxh9.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=1382&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/572281/original/file-20240130-25-8puxh9.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=1382&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">A Tetris board.</span>
<span class="attribution"><a class="source" href="https://commons.wikimedia.org/wiki/File:Typical_Tetris_Game.svg">Brandenads/Wikimedia Commons</a></span>
</figcaption>
</figure>
<p>Manipulating the game pieces gives the player an exercise in dynamic spatial reasoning. Spatial reasoning is the ability to visualize geometric figures and how they will move in space. So, dynamic spatial reasoning is the ability to visualize actively moving figures. </p>
<p>The Tetris player must quickly decide where the currently dropping game piece will best fit and then move it there. This movement involves both translation, or moving a shape right and left, and rotation, or twirling the shape in increments of 90 degrees on its axis.</p>
<p>Spatial visualization is partly inherent ability, but partly learned expertise. <a href="https://doi.org/10.1037/a0016127">Some researchers</a> identify spatial skill as necessary for successful problem solving, and it’s often used alongside mathematics skills and verbal skills. </p>
<p>Spatial visualization is a key component of a mathematics discipline called transformational geometry, which is usually first taught in middle school. In a typical transformational geometry exercise, students might be asked to represent a figure by its x and y coordinates on a coordinate graph and then <a href="https://www.cuemath.com/geometry/transformations/">identify the transformations</a>, like translation and rotation, necessary to move it from one position to another while keeping the piece the same shape and size.</p>
<p>Reflection and dilation are the two other basic mathematical transformations, though they’re not used in Tetris. Reflection flips the image across any line while maintaining the same size and shape, and dilation changes the size of the shape, producing a similar figure. </p>
<figure>
<iframe width="440" height="260" src="https://www.youtube.com/embed/XiAoUDfrar0?wmode=transparent&start=0" frameborder="0" allowfullscreen=""></iframe>
<figcaption><span class="caption">Transformations may seem simple, but they underlie lots of more complex math concepts.</span></figcaption>
</figure>
<p>For many students, these exercises are tedious, as they involve plotting many points on graphs to move a figure’s position. But games like Tetris can help students grasp these concepts in a dynamic and engaging way.</p>
<h2>Transformational geometry beyond Tetris</h2>
<p>While it may seem simple, transformational geometry is the foundation for several advanced topics in mathematics. Architects and engineers both use transformations to draw up blueprints, which represent the real world in <a href="https://www.encyclopedia.com/education/news-wires-white-papers-and-books/scale-drawings-and-models">scale drawings</a>. </p>
<p>Animators and computer graphic designers use concepts of transformations as well. <a href="https://www.khanacademy.org/computing/pixar/animate">Animation</a> involves representing a figure’s coordinates in a matrix array and then creating a sequence to change its position, which moves it across the screen. While animators today use computer programs that automatically move figures around, they are all based on translation.</p>
<p>Calculus and differential geometry also use transformation. <a href="https://www.researchgate.net/publication/371605360_Some_optimization_problems_with_calculus">The concept of optimization</a> involves representing a situation <a href="https://www.britannica.com/science/function-mathematics">as a function</a> and then finding the maximum or minimum value of that function. Optimization problems often involve graphic representations where the student uses transformations to manipulate one or more of the variables.</p>
<p>Lots of real-world applications use optimization – for example, businesses might want to find out the minimum cost of distributing a product. Another example is figuring out the size of a theoretical box with the largest possible volume.</p>
<p>All of these advanced topics use the same concepts as the simple moves of Tetris.</p>
<p>Tetris is an engaging and entertaining video game, and players with transformational geometry skills might find success playing it. <a href="https://doi.org/10.1007/s11199-008-9498-z">Research has found</a> that manipulating rotations and translations within the game can provide a solid conceptual foundation for advanced mathematics in numerous science fields.</p>
<p>Playing Tetris <a href="https://doi.org/10.1111/j.1467-9280.2007.01990.x">may lead students</a> to <a href="https://doi.org/10.1080/00221320209598683">a future aptitude</a> in business analytics, engineering or computer science – and it’s fun. As a mathematics educator, I encourage students and friends to play on.</p><img src="https://counter.theconversation.com/content/220915/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Leah McCoy does not work for, consult, own shares in or receive funding from any company or organization that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>People young and old love the classic video game Tetris. A working knowledge of the spatial reasoning concepts underlying Tetris can set students up for success in mathematics.Leah McCoy, Professor of Education, Wake Forest UniversityLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/2236892024-02-28T12:32:11Z2024-02-28T12:32:11ZGOP primary elections use flawed math to pick nominees<figure><img src="https://images.theconversation.com/files/578350/original/file-20240227-18-rw2ozs.jpg?ixlib=rb-1.1.0&rect=7%2C31%2C5311%2C3357&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">How people vote isn't always reflected in how elections are decided.</span> <span class="attribution"><a class="source" href="https://www.gettyimages.com/detail/illustration/map-made-of-stickman-figure-with-patriotic-royalty-free-illustration/1281610356">bamlou/DigitalVision Vectors via Getty Images</a></span></figcaption></figure><p>Republicans around the country are picking a nominee to run for president. However, their process – designed and run by the party, not government officials – is a mess of flawed mathematics that can end up delivering a result that’s in conflict with the person most voters actually support.</p>
<p>As a <a href="https://ivolic.wellesley.edu/">mathematics professor</a> and co-founder of the <a href="https://mathematics-democracy-institute.org/">Institute for Mathematics and Democracy</a>, I watched this contradictory process play out in 2016, shaping the political landscape ever since. Elements of it are visible again in 2024.</p>
<p>There are many ways bad mathematics interferes with our democracy, as I explain in my forthcoming book, “<a href="https://press.princeton.edu/books/hardcover/9780691248806/making-democracy-count">Making Democracy Count: How Mathematics Improves Voting, Electoral Maps, and Representation</a>.” Here’s how the Republican primaries can manage to defy democratic ideals and deliver a nominee even though most voters prefer someone else:</p>
<h2>Splitting votes among many candidates</h2>
<p>In 2016, former President Donald Trump became the Republican choice, having <a href="https://www.thegreenpapers.com/P16/R">won 44.9% of the votes</a> cast in primaries. That was nearly twice the share of votes won by the runner-up, Ted Cruz, who had 25.1% of the primary votes.</p>
<p>But during primary season, <a href="https://fairvote.org/new_polls_show_that_gop_split_vote_problem_continues/">polls suggested</a> that in head-to-head primaries, Trump would have lost not only to Cruz, but also to <a href="https://www.thegreenpapers.com/P16/R">third-place Republican finisher John Kasich</a> and Marco Rubio, who placed fourth.</p>
<p>In other words, a majority of Republican voters preferred Cruz, Kasich and Rubio to Trump. But none of the three took the lead because of the party’s nomination system, which assigned <a href="https://www.thegreenpapers.com/P16/R">Trump 58.3% of the delegates</a> at the Republican National Convention – the formal process by which the nominee is selected.</p>
<h2>An attempt at proportional representation</h2>
<p>The Republican Party says its primaries are meant to <a href="https://prod-cdn-static.gop.com/docs/Rules_Of_The_Republican_Party.pdf">encourage proportional assignment</a> of delegates to candidates. So if a candidate wins, say, 40% of the votes, she should win as close to 40% of the delegates as possible. </p>
<p>This sounds reasonable, and it aligns with most people’s notion of fairness. For primaries taking place before March 15, the Republican Party mandates proportional allocation, but with lots of exceptions that can effectively turn the election into winner-take-all or winner-take-most. After March 15, the exceptions become the norm, pulling the outcome further from proportional representation.</p>
<p>The Democratic Party has a more centralized system and <a href="https://ballotpedia.org/Democratic_delegate_rules,_2024">mandates proportionality</a> for all its primaries.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/578351/original/file-20240227-24-2cce3e.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="People stand at tables and raise their arms." src="https://images.theconversation.com/files/578351/original/file-20240227-24-2cce3e.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/578351/original/file-20240227-24-2cce3e.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/578351/original/file-20240227-24-2cce3e.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/578351/original/file-20240227-24-2cce3e.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/578351/original/file-20240227-24-2cce3e.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/578351/original/file-20240227-24-2cce3e.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/578351/original/file-20240227-24-2cce3e.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=503&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Crowd members cheer at the 2020 Republican National Convention.</span>
<span class="attribution"><a class="source" href="https://newsroom.ap.org/detail/Election2020RNC/b1f38c3935c94ade99728f71b15da1fd/photo">Travis Dove/The New York Times via AP, Pool</a></span>
</figcaption>
</figure>
<h2>Allocation of delegates</h2>
<p>The process begins with the <a href="https://prod-cdn-static.gop.com/media/documents/2020_RNC_Call_of_the_Convention_1575665975.pdf">states each receiving a number of delegates</a> that will later be assigned to candidates.</p>
<p>Each state gets 10 at-large delegates, and three delegates for each congressional district it contains. A state can also get additional delegates based on how Republican it is – depending on whether its people voted for a Republican presidential candidate in the previous general election, and on how much of its legislature is Republican.</p>
<p>These allocations can result in inequities. For instance, Massachusetts and Utah, two of the states voting on Super Tuesday, both get 40 delegates. That’s because Massachusetts has more congressional districts, while Utah is more Republican. </p>
<p>But Utah has roughly 960,000 <a href="https://ballotpedia.org/Partisan_affiliations_of_registered_voters">registered Republican voters</a>, and Massachusetts has about 440,000. That means for any candidate to get a Utah delegate would require support from at least twice as many voters as that candidate would need to get a Massachusetts delegate.</p>
<h2>Assigning delegates to candidates</h2>
<p>There are as many as <a href="https://doi.org/10.1007/978-3-031-24954-9">seven different proportional methods</a> used to assign states’ at-large delegates, each with its own mathematical problems. And in 21 states, the delegates allocated because of each congressional district are also assigned by the same methods as the at-large delegates.</p>
<p>In other states, the three delegates in each congressional district are all allocated to the winner in that district. And in still other states, the district delegates are allocated with a 2-1 split: The top vote-getter in the district receives two delegates and the runner-up receives one.</p>
<p>Math makes clear that these methods are not proportional representation: Imagine that three candidates in a close race get 33.5%, 33.3% and 33.2% of the votes, respectively. The winner-take-all method would give all three delegates to the top scorer. And in the 2-1 split, the last-place candidate would get nothing. </p>
<p>In some states, the party’s rules also allow the method of counting to vary depending on how dominant a candidate’s win is. For instance, <a href="https://www.desertsun.com/story/news/politics/elections/2024/02/08/california-primary-new-state-party-rules-could-accelerate-trump-nomination/72400693007/">California is the latest state to adopt the practice</a> in which a candidate who wins more than half the statewide vote gets all of the state’s delegates.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/578358/original/file-20240227-24-2mcqmn.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="Four people stand behind lecterns on a stage." src="https://images.theconversation.com/files/578358/original/file-20240227-24-2mcqmn.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/578358/original/file-20240227-24-2mcqmn.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/578358/original/file-20240227-24-2mcqmn.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/578358/original/file-20240227-24-2mcqmn.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/578358/original/file-20240227-24-2mcqmn.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/578358/original/file-20240227-24-2mcqmn.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/578358/original/file-20240227-24-2mcqmn.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=503&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Republican candidates for president appear at a debate in Milwaukee in August 2023.</span>
<span class="attribution"><a class="source" href="https://newsroom.ap.org/detail/Election2024Debate/876ea8c85d5048fea034a652dd1348bc/photo">AP Photo/Morry Gash</a></span>
</figcaption>
</figure>
<h2>Two candidates doesn’t make the math clearer</h2>
<p>The GOP’s system offers other significant advantages to winners as well. </p>
<p>Suppose a state has eight districts with three delegates apiece and in each, Candidate Alice gets 51% of the votes and Candidate Bob gets 49%. If the allocation was 2-1, Alice would get 16 delegates and Bob would get eight.</p>
<p>Then there are the 10 at-large delegates the party assigns to each state. Most proportional methods would split these delegates evenly, with five given to each candidate. That would deliver a grand total for Alice of 21 delegates, and 13 for Bob.</p>
<p>In that situation, Alice would get 51% of the votes but 62% of the delegates. This “winner’s bonus” was evident in many states Trump won in the 2016 primary, <a href="https://ballotpedia.org/Republican_delegates_from_Alabama,_2016">such as Alabama</a>, where his vote share was 43% but he collected 72% of the delegates. In the 2020 Democratic primary races, <a href="https://en.wikipedia.org/wiki/Results_of_the_2020_Democratic_Party_presidential_primaries#Major_candidates">Joe Biden won 51.6% of the votes</a> and 68% of the delegates overall.</p>
<p>Winner-take-all is problematic too. Consider Utah and Massachusetts again. If a candidate won Utah by a landslide, and another narrowly won in Massachusetts, they would both get 40 delegates – based on vastly different numbers of actual votes cast by supporters.</p>
<h2>An additional barrier</h2>
<p>Most states also require candidates to get a certain percentage of voter support before being assigned any delegates at all. Under the Republican rules, some states set this bar as high as 20%. The Democratic Party mandates a 15% threshold for every state. </p>
<p>These thresholds are biased toward more popular candidates and can even cause <a href="https://doi.org/10.1007/s11127-018-0608-3">mathematically counterintuitive delegate allocations</a>.</p>
<p>The combination of winner-take-all and complicated threshold structures is where all hope of proportionality and fairness vanishes. For example, in 2016, Trump won all of South Carolina’s 50 delegates by <a href="https://ballotpedia.org/Republican_delegates_from_South_Carolina,_2016#South_Carolina_primary_results">garnering 33% of the votes</a> and <a href="https://ballotpedia.org/Republican_delegates_from_Florida,_2016#Florida_primary_results">all of Florida’s 99 delegates</a> with 46% of the votes.</p>
<p>This phenomenon is occurring again in this cycle: In the 2024 South Carolina primary, <a href="https://www.nytimes.com/interactive/2024/02/24/us/elections/results-south-carolina-republican-primary.html">Trump won 60% of the vote</a> but landed 94% of the state’s delegates. </p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/578360/original/file-20240227-22-p8xqbc.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="A woman stands on a stage holding a microphone in front of a U.S. flag and the South Carolina state flag." src="https://images.theconversation.com/files/578360/original/file-20240227-22-p8xqbc.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/578360/original/file-20240227-22-p8xqbc.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/578360/original/file-20240227-22-p8xqbc.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/578360/original/file-20240227-22-p8xqbc.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/578360/original/file-20240227-22-p8xqbc.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/578360/original/file-20240227-22-p8xqbc.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/578360/original/file-20240227-22-p8xqbc.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=503&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Nikki Haley got 40% of the primary vote in her home state of South Carolina, but only 4% of the state’s Republican delegates.</span>
<span class="attribution"><a class="source" href="https://newsroom.ap.org/detail/Election2024MichiganWhattoWatch/9fc716699bf2449aa70ab90ee7956350/photo">AP Photo/Andrew Harnik</a></span>
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<h2>Picking a single winner</h2>
<p>Ultimately, the party delegate system has to arrive at a single winner. Somehow, one candidate must win a majority of the delegate votes that are cast at the summer convention. For this year’s Republican nomination, this is 1,215 of the 2,429 delegates. </p>
<p>Even if the delegate apportionment reflected Republican voters’ preferences in perfect proportion, the system has yet another inherent flaw. Suppose the process gave 35% of the delegates to one candidate, 30% to another, 20% to a third, and then split the remaining 15% between several others. Who should win the nomination?</p>
<p>In a sequential process often called a “<a href="https://ballotpedia.org/Brokered_conventions">brokered convention</a>,” various candidates who recognize they cannot win the nomination free their delegates to vote for others. As its name suggests, this method more closely resembles a business deal than a fair election – and it’s very far from the eyes of the voters and even more distant from the rigor of mathematics.</p>
<p>There is no unbiased way to pick a single nominee using the GOP’s current primary structure. Voters are reluctant to risk wasting their votes by supporting less popular candidates. Candidates who appear weaker exit races earlier because they don’t think they can clear the hurdles in enough states. As a result, candidates with small but committed followings can rise to the top – even if most people prefer someone else.</p>
<h2>Some alternatives</h2>
<p>Math does offer some options for possible solutions that eliminate the flaws of winner-take-all, reduce divisiveness, ensure that each voter has an equal say, and enact the will of a majority.</p>
<p>One way could be using <a href="https://theconversation.com/what-is-ranked-choice-voting-a-political-scientist-explains-165055">ranked-choice voting</a>, in which people rank all the candidates in their order of preference. A system that would be mathematically most representative and inclusive would involve nonpartisan primaries with some number of top vote-getters advancing to the general election. Both would be held with ranked-choice voting. Alaska and <a href="https://www.ncsl.org/elections-and-campaigns/state-primary-election-types">several other states</a> use this method in state elections, but not for the presidential race.</p><img src="https://counter.theconversation.com/content/223689/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Ismar Volić does not work for, consult, own shares in or receive funding from any company or organization that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>There are many ways bad mathematics interferes with our democracy. Assigning delegates is just one example.Ismar Volić, Professor of Mathematics, Director of Institute for Mathematics and Democracy, Wellesley CollegeLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/2185162024-02-13T01:16:56Z2024-02-13T01:16:56ZWhat are ‘multiplication facts’? Why are they essential to your child’s success in maths?<figure><img src="https://images.theconversation.com/files/574826/original/file-20240212-25-kdfbw0.jpg?ixlib=rb-1.1.0&rect=0%2C110%2C6689%2C4325&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">
</span> <span class="attribution"><a class="source" href="https://www.pexels.com/photo/colorful-plastic-numbers-for-kids-to-learn-from-5412109/">Karolina Grabowska/ Pexels</a>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span></figcaption></figure><p>One of the essential skills students need to master in primary school mathematics are “multiplication facts”. </p>
<p>What are they? What are they so important? And how can you help your child master them? </p>
<h2>What are multiplication facts?</h2>
<p>Multiplication facts typically describe the answers to multiplication sums up to 10x10. Sums up to 10x10 are called “facts” as it is expected they can be easily and quickly recalled. You may recall learning multiplication facts in school from a list of times tables. </p>
<p>The shift from “times tables” to “multiplication facts” is not just about language. It stems from teachers wanting children to see how multiplication facts can be used to solve a variety of problems beyond the finite times table format. </p>
<p>For example, if you learned your times tables in school (which typically went up to 12x12 and no further), you might be stumped by being asked to solve 15x8 off the top of your head. In contrast, we hope today’s students can use their multiplication facts knowledge to quickly see how 15x8 is equivalent to 10x8 plus 5x8. </p>
<p>The shift in terminology also means we are encouraging students to think about the connections between facts. For example, when presented only in separate tables, it is tricky to see how 4x3 and 3x4 are directly connected. </p>
<h2>Maths education has changed</h2>
<p>In a previous piece, we talked about how <a href="https://theconversation.com/why-would-they-change-maths-how-your-childs-maths-education-might-be-very-different-from-yours-207030">mathematics education has changed</a> over the past 30 years. </p>
<p>In today’s mathematics classrooms, <a href="https://www.australiancurriculum.edu.au/resources/mathematics-proficiencies">teachers still focus</a> on developing students’ mathematical accuracy and fast recall of essential facts, including multiplication facts. </p>
<p>But we also focus on developing essential problem-solving skills. This helps students form connections between concepts, and learn how to reason through a variety of real-world mathematical tasks. </p>
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<strong>
Read more:
<a href="https://theconversation.com/why-would-they-change-maths-how-your-childs-maths-education-might-be-very-different-from-yours-207030">'Why would they change maths?' How your child's maths education might be very different from yours</a>
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<h2>Why are multiplication facts so important?</h2>
<p>By the end of primary school, it is expected students will know multiplication facts up to 10x10 and can recall the related division fact (for example, 10x9=90, therefore 90÷10=9). </p>
<p>Learning multiplication facts is also essential for developing “multiplicative thinking”. This is an understanding of the relationships between quantities, and is something we need to know how to do on a daily basis. </p>
<p>When we are deciding whether it is better to purchase a 100g product for $3 or a 200g product for $4.50, we use multiplicative thinking to consider that 100g for $3 is equivalent to 200g for $6 – not the best deal! </p>
<p>Multiplicative thinking is needed in nearly all maths topics in high school and beyond. It is used in many topics across algebra, geometry, statistics and probability.</p>
<p>This kind of thinking is profoundly important. Research <a href="https://www.education.vic.gov.au/Documents/school/teachers/teachingresources/discipline/maths/teaching-with-the-big-ideas-in-mathematics.pdf">shows</a> students who are more proficient in multiplicative thinking perform significantly better in mathematics overall.</p>
<p>In 2001, an <a href="https://www.education.vic.gov.au/Documents/school/teachers/teachingresources/discipline/maths/mynumfreport.pdf">extensive RMIT study</a> found there can be as much as a seven-year difference in student ability within one mathematics class due to differences in students’ ability to access multiplicative thinking. </p>
<p>These findings have been confirmed in more recent studies, including <a href="https://www.sciencedirect.com/science/article/pii/S0732312320301012?casa_token=k2Q1Fm2-7ekAAAAA:QWRso-ioRmtsx45VlrncevZd8fX5jc0V8nDbbiPCBWAjE_2OGaAC0O_VBStlGxalI1fbgPFV#sec0025">a 2021 paper</a>. </p>
<p>So, supporting your child to develop their confidence and proficiency with multiplication is key to their success in high school mathematics. How can you help? </p>
<p>Below are three <a href="https://eric.ed.gov/?id=ED520252">research-based</a> tips to help support children from Year 2 and beyond to learn their multiplication facts. </p>
<h2>1. Discuss strategies</h2>
<p>One way to help your child’s confidence is to discuss strategies for when they encounter new multiplication facts. </p>
<p>Prompt them to think of facts they already and how they can be used for the new fact. </p>
<p>For example, once your child has mastered the x2 multiplication facts, you can discuss how 3x6 (3 sixes) can be calculated by doubling 6 (2x6) and adding one more 6. We’ve now realised that x3 facts are just x2 facts “and one more”!</p>
<figure class="align-center ">
<img alt="Dots representing 3 times 5, 3 times 6 and 3 times 7" src="https://images.theconversation.com/files/574915/original/file-20240212-24-3jkw1c.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/574915/original/file-20240212-24-3jkw1c.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=188&fit=crop&dpr=1 600w, https://images.theconversation.com/files/574915/original/file-20240212-24-3jkw1c.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=188&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/574915/original/file-20240212-24-3jkw1c.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=188&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/574915/original/file-20240212-24-3jkw1c.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=237&fit=crop&dpr=1 754w, https://images.theconversation.com/files/574915/original/file-20240212-24-3jkw1c.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=237&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/574915/original/file-20240212-24-3jkw1c.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=237&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
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<span class="attribution"><span class="source">The Conversation</span>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span>
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<p>Strategies can be individual: students should be using the strategy that makes the most sense to them. So you could ask a questions such as “if you’ve forgotten 6x7, how could you work it out?” (we might personally think of 6x6=36 and add one more 6, but your child might do something different and equally valid). </p>
<p>This is a great activity for any quiet car trip. It can also be a great drawing activity where you both have a go at drawing your strategy and then compare. Identifying multiple strategies develops flexible thinking. </p>
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<p>
<em>
<strong>
Read more:
<a href="https://theconversation.com/maths-anxiety-is-a-real-thing-here-are-3-ways-to-help-your-child-cope-200822">'Maths anxiety' is a real thing. Here are 3 ways to help your child cope</a>
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<h2>2. Help them practise</h2>
<p>Practising recalling facts under a friendly time crunch can be helpful in achieving what teachers call “fluency” (that is, answering quickly and easily). </p>
<p>A great game you could play with your children is “<a href="https://www.pta.org/docs/default-source/uploadedfiles/required-handout-activity-sheet-math-night-multiplication-heads-up#:%7E:text=One%20card%20is%20placed%20face%20up%20so%20that%20everyone%20can%20see%20it.&text=The%20player%20places%20a%20card%20on%20their%20forehead.&text=The%20facilitator%20says%20out%20loud,card%20and%20the%20forehead%20card.&text=The%20player%20identifies%20the%20card%20on%20their%20forehead%20as%20quickly%20as%20possible">multiplication heads up</a>” .
Using a deck of cards, your child places a card to their forehead where you can see but they cannot. You then flip over the top card on the deck and reveal it to your child. Using the revealed card and the card on your child’s head you tell them the result of the multiplication (for example, if you flip a 2 and they have a 3 card, then you tell them “6!”). </p>
<p>Based on knowing the result, your child then guesses what their card was. </p>
<p>If it is challenging to organise time to pull out cards, you can make an easier game by simply quizzing your child. Try to mix it up and ask questions that include a range of things they know well with and ones they are learning.</p>
<p>Repetition and rehearsal will mean things become stored in long-term memory. </p>
<h2>3. Find patterns</h2>
<p>Another great activity to do at home is print some multiplication grids and explore patterns with your child. </p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/575154/original/file-20240212-16-92wlnn.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="Multiplication tables for 0 to 10, with colour columns to show connections between numbers" src="https://images.theconversation.com/files/575154/original/file-20240212-16-92wlnn.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/575154/original/file-20240212-16-92wlnn.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=416&fit=crop&dpr=1 600w, https://images.theconversation.com/files/575154/original/file-20240212-16-92wlnn.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=416&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/575154/original/file-20240212-16-92wlnn.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=416&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/575154/original/file-20240212-16-92wlnn.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=523&fit=crop&dpr=1 754w, https://images.theconversation.com/files/575154/original/file-20240212-16-92wlnn.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=523&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/575154/original/file-20240212-16-92wlnn.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=523&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
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<span class="attribution"><span class="source">The Conversation</span>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span>
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<p>A first start might be to give your child <a href="https://www.scholastic.com/content/dam/teachers/blogs/alycia-zimmerman/migrated-files/blank_multiplication_table.pdf">a blank</a> or <a href="https://www.math-salamanders.com/times-table-grid.html">partially blank</a> multiplication grid which they can practise completing. </p>
<p>Then, using coloured pencils, they can colour in patterns they notice. For example, the x6 column is always double the answer in the x3 column. Another pattern they might see is all the even answers are products of 2, 4, 6, 8, 10. They can also notice half of the grid is repeated along the diagonal. </p>
<p>This also helps your child become a mathematical thinker, not just a calculator. </p>
<p>The importance of multiplication for developing your child’s success and confidence in mathematics cannot be understated. We believe these ideas will give you the tools you need to help your child develop these essential skills.</p><img src="https://counter.theconversation.com/content/218516/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>The authors do not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and have disclosed no relevant affiliations beyond their academic appointment.</span></em></p>Multiplication facts typically describe the answers to multiplication sums up to 10x10. They are called “facts” as it is expected they can be easily and quickly recalled.Bronwyn Reid O'Connor, Lecturer in Mathematics Education, University of SydneyBen Zunica, Lecturer in Secondary Maths Education, University of SydneyLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/2213522024-02-05T13:34:48Z2024-02-05T13:34:48ZHow bats ‘leapfrog’ their way home at night – new research<figure><img src="https://images.theconversation.com/files/572171/original/file-20240130-27-o1vlrb.jpg?ixlib=rb-1.1.0&rect=0%2C10%2C3413%2C2539&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">The greater horseshoe bat is one of the UK's 18 bat species. </span> <span class="attribution"><a class="source" href="https://www.shutterstock.com/image-photo/flying-bat-hunting-forest-greater-horseshoe-1494098204">Rudmer Zwerver/Shutterstock</a></span></figcaption></figure><p>A silent ballet takes place above our heads at night as Britain’s bat populations leave their roosts to forage for food. Although their initial movement away from roosts is fairly well understood, until recently little was known about how they returned home. </p>
<p>But our <a href="https://link.springer.com/article/10.1007/s11538-023-01233-5">new research</a> shows how bats may use a “leap-frogging” motion to make their way home, something which could help conservationists in future.</p>
<p>As they flit through the darkness, bats play a crucial role in the health of our ecosystems. From keeping insect populations in check to dispersing seeds and pollinating plants, they provide a multitude of benefits. </p>
<p>In the UK alone, the 18 bat species devour agricultural pests such as cockchafers with impressive efficiency. So, it is imperative that we not only understand and appreciate bats, but also actively support and safeguard their populations for the wellbeing of our planet.</p>
<p>But bat populations are vulnerable to pollution, climate change and loss of roosting locations. Habitat fragmentation and light pollution can also interrupt how bats feed. This is particularly important during the maternity season <strong>in early summer</strong>, when bats gather together to have and raise their young.</p>
<p>An integral aspect of effective bat conservation lies in unravelling the mysteries of how bats move. This not only helps us understand how bats navigate and use their environment, but also helps in identifying and protecting their roosts. </p>
<h2>Radio-tracking</h2>
<p>Conventional methods for pinpointing bat roosts primarily hinge on radio-tracking surveys. This arduous process involves capturing bats, attaching small radio transmitters to them before releasing them and following the signals throughout the night. </p>
<p>Our team conducted a radio-tracking survey in Devon which monitored 12 greater horseshoe bats over 24 nights. The trajectories of seven of those bats over 14 nights were extracted from the data for analysis, ensuring that in each case, a bat’s beginning and ending roost were the same.</p>
<p>Using this data, we measured the population’s average distance from the roost. We found two distinctive patterns in the data we analysed: an initial spread of bats within the first one to two hours after sunset and a gradual return to the roost afterwards.</p>
<p>The initial spread reflects the expected random dispersal of bats leaving their roosts to forage after sunset. The return to the roost, occurring two to eight hours after sunset, is more complicated. </p>
<p>This prompted us to explore two potential mechanisms influencing the bats’ return. First, a “pull mechanism”, where the roost attracts the bats home, and second, a mechanism pushing the bats who range furthest away back to the roost.</p>
<p>We modelled the pushing mechanism as a leapfrog process. Imagine this as a cascade effect, where the outermost bats begin their return. Once the “outer” bats have passed or “leapt over” bats that are closer to the roost, the “inner” bats become the furthest out causing them to return too.</p>
<p>This motion unfolds systematically, like a synchronised dance, as each bat from the periphery of the foraging range follows suit in returning to the roost after being “leapfrogged”.</p>
<p>But what causes the bats to return in this manner? One plausible explanation underscores how bats rely on each other for effective navigation, like tiny radar signals. If a bat experiences prolonged silence or predominantly hears calls from one direction, it might decide to move closer to the roost, anticipating the presence of other colony members. </p>
<p>But a bat might return more slowly, prolonging foraging, if it perceives the presence of bats beyond its current location. So, it is the outer bats that would drive the return as they would not be surrounded by calls.</p>
<h2>How does this research help bats?</h2>
<p>The significance of these findings extends beyond just describing the movements of bats. They have laid the foundation for work that promises easier discovery of new bat roosts, potentially reducing the need for labour-intensive bat tracking surveys in the future. </p>
<p>One of the immediate effects of our research includes informing a measurement of the “core sustenance zone” for greater horseshoe bats. This is where most of their foraging occurs, so it’s important in bat ecology, conservation and construction planning.</p>
<p>The leapfrogging mechanism also allows us to ascribe intention to bat movements. Namely, through using surrounding bat calls they can identify where the population is relative to their position, suggesting whether or not they are on the periphery of the group, which is an indicator of their vulnerability. Should they be furthest from the roost they move back towards the bulk of the population and closer to the roost.</p>
<p>While these interpretations hold promise, further rigorous testing is essential. And we need to think about the safety and wellbeing of the bat population.</p>
<p>Our observations are also specific to greater horseshoe bats during the summer months. Different bat species have distinct flight patterns and habitat preferences, with the same species displaying diverse behaviours at different times of the year. </p>
<p>So, while we have taken some crucial first steps, we still have a lot of work to do in unravelling the characteristics of bat motions in general.</p><img src="https://counter.theconversation.com/content/221352/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Fiona Mathews receives funding from Devon Area of Outstanding Natural Beauty, Devon County Council and the Natural Environment Research Council. She is affiliated with the UK Mammal Society, Mammal Conservation Europe, Ecotype Genetics and Ecology Search Services Ltd. </span></em></p><p class="fine-print"><em><span>Thomas Woolley does not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>Maths plays a crucial role in new research which finds that bats “leapfrog” their way home at night.Thomas Woolley, Senior Lecturer in Applied Mathematics, Cardiff UniversityFiona Mathews, Professor of Environmental Biology, University of SussexLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/2190542024-02-02T13:17:20Z2024-02-02T13:17:20ZOrbital resonance − the striking gravitational dance done by planets with aligning orbits<figure><img src="https://images.theconversation.com/files/571696/original/file-20240126-29-xc09zr.jpg?ixlib=rb-1.1.0&rect=19%2C4%2C1578%2C792&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Planets can gravitationally affect each other when their orbits line up. </span> <span class="attribution"><a class="source" href="https://exoplanets.nasa.gov/trappist1/">NASA/JPL-Caltech</a></span></figcaption></figure><p>Planets orbit their parent stars while separated by enormous distances – in our solar system, planets are like <a href="https://science.nasa.gov/learning-resources/how-big-is-the-solar-system/">grains of sand</a> in a region the size of a football field. The time that planets take to orbit their suns have no specific relationship to each other. </p>
<p>But sometimes, their orbits display striking patterns. For example, astronomers studying <a href="https://exoplanets.nasa.gov/news/1771/discovery-alert-watch-the-synchronized-dance-of-a-6-planet-system/">six planets orbiting a star</a> 100 light years away have just found that they orbit their star with an almost rhythmic beat, in perfect synchrony. Each pair of planets completes their orbits in times that are the ratios of whole numbers, allowing the planets to align and exert a gravitational push and pull on the other during their orbit.</p>
<p>This type of gravitational alignment is called <a href="https://www.aanda.org/glossary/175-orbital-resonance">orbital resonance</a>, and it’s like a harmony between distant planets.</p>
<p>I’m an <a href="https://scholar.google.com/citations?user=OrRLRQ4AAAAJ&hl=en">astronomer</a> who studies and writes about <a href="https://wwnorton.com/books/9780393343861">cosmology</a>. Researchers have discovered <a href="https://exoplanets.nasa.gov/">over 5,600 exoplanets</a> in the past 30 years, and their extraordinary diversity continues to surprise astronomers.</p>
<h2>Harmony of the spheres</h2>
<p>Greek mathematician <a href="https://www.auroraorchestra.com/2019/05/pythagoras-the-music-of-the-spheres/">Pythagoras</a> discovered the principles of musical harmony 2,500 years ago by analyzing the sounds of blacksmiths’ hammers and plucked strings. </p>
<p>He believed mathematics was at the heart of the natural world and proposed that the Sun, Moon and planets each emit unique hums based on their orbital properties. He thought this “music of the spheres” would be imperceptible to the human ear.</p>
<p>Four hundred years ago, <a href="https://www.skyscript.co.uk/kepler.html">Johannes Kepler</a> picked up this idea. He proposed that musical intervals and harmonies described the motions of the six known planets at the time. </p>
<p>To Kepler, the solar system had two basses, Jupiter and Saturn; a tenor, Mars; two altos, Venus and Earth; and a soprano, Mercury. These roles reflected how long it took each planet to orbit the Sun, lower speeds for the outer planets and higher speeds for the inner planets. </p>
<p>He called the book he wrote on these mathematical relationships “<a href="https://archive.org/details/ioanniskepplerih00kepl">The Harmony of the World</a>.” While these ideas have some similarities to the concept of orbital resonance, planets don’t actually make sounds, since <a href="https://theconversation.com/why-isnt-there-any-sound-in-space-an-astronomer-explains-why-in-space-no-one-can-hear-you-scream-217885">sound can’t travel through the vacuum of space</a>.</p>
<h2>Orbital resonance</h2>
<p><a href="https://www.aanda.org/glossary/175-orbital-resonance">Resonance happens when</a> planets or moons have orbital periods that are <a href="https://www.youtube.com/watch?v=qDHKveVSc0Y">ratios of whole numbers</a>. The orbital period is the time taken for a planet to make one complete circuit of the star. So, for example, two planets orbiting a star would be in a 2:1 resonance when one planet takes twice as long as the other to orbit the star. Resonance is seen in only <a href="https://arxiv.org/abs/1703.03634">5% of planetary systems</a>.</p>
<figure class="align-center ">
<img alt="A simple animated diagram showing a planet, as a dot, with three smaller dots making circles around it, and occasionally flashing when two of the three line up." src="https://images.theconversation.com/files/571674/original/file-20240126-17-ofefj2.gif?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/571674/original/file-20240126-17-ofefj2.gif?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=403&fit=crop&dpr=1 600w, https://images.theconversation.com/files/571674/original/file-20240126-17-ofefj2.gif?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=403&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/571674/original/file-20240126-17-ofefj2.gif?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=403&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/571674/original/file-20240126-17-ofefj2.gif?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=506&fit=crop&dpr=1 754w, https://images.theconversation.com/files/571674/original/file-20240126-17-ofefj2.gif?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=506&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/571674/original/file-20240126-17-ofefj2.gif?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=506&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Orbital resonance, as seen with Jupiter’s moons, happens when planetary bodies’ orbits line up – for example, Io orbits Jupiter four times in the time it takes Europa to orbit twice and Ganymede to orbit once.</span>
<span class="attribution"><a class="source" href="https://en.wikipedia.org/wiki/File:Galilean_moon_Laplace_resonance_animation_2.gif">WolfmanSF/Wikimedia Commons</a></span>
</figcaption>
</figure>
<p>In the solar system, Neptune and Pluto are in a 3:2 resonance. There’s also a <a href="https://www.planetary.org/space-images/orbital-resonances-of-galilean-moons">triple resonance</a>, 4:2:1, among Jupiter’s three moons: Ganymede, Europa and Io. In the time it takes Ganymede to orbit Jupiter, Europa orbits twice and Io orbits four times. Resonances occur naturally, when planets happen to have orbital periods that are the ratio of whole numbers. </p>
<p>Musical intervals describe the relationship between two musical notes. In the musical analogy, important <a href="http://hyperphysics.phy-astr.gsu.edu/hbase/Music/mussca.html">musical intervals</a> based on ratios of frequencies are the fourth, 4:3, the fifth, 3:2, and the octave, 2:1. Anyone who plays the <a href="https://globalguitarnetwork.com/perfect-intervals/">guitar or the piano</a> might recognize these intervals.</p>
<figure>
<iframe width="440" height="260" src="https://www.youtube.com/embed/2V3bvZu2Xqo?wmode=transparent&start=0" frameborder="0" allowfullscreen=""></iframe>
<figcaption><span class="caption">Musical intervals can be used to create scales and harmony.</span></figcaption>
</figure>
<p>Orbital resonances can change <a href="https://theconversation.com/earth-isnt-the-only-planet-with-seasons-but-they-can-look-wildly-different-on-other-worlds-216874">how gravity influences</a> two bodies, causing them to speed up, slow down, stabilize on their orbital path and sometimes have their orbits disrupted.</p>
<p>Think of pushing a <a href="https://astrobites.org/2018/07/05/small-black-hole-meets-big-black-hole/">child on a swing</a>. A planet and a swing both have a natural frequency. Give the child a push that matches the swing motion and they’ll get a boost. They’ll also get a boost if you push them every other time they’re in that position, or every third time. But push them at random times, sometimes with the motion of the swing and sometimes against, and they get no boost. </p>
<figure>
<iframe width="440" height="260" src="https://www.youtube.com/embed/qDHKveVSc0Y?wmode=transparent&start=0" frameborder="0" allowfullscreen=""></iframe>
<figcaption><span class="caption">Orbital resonance can cause planets or asteroids to speed up or start to wobble.</span></figcaption>
</figure>
<p>For planets, the boost can keep them continuing on their orbital paths, but it’s much more likely to disrupt their orbits.</p>
<h2>Exoplanet resonance</h2>
<p>Exoplanets, or planets outside the solar system, show striking examples of resonance, not just between two objects but also between resonant “chains” involving three or more objects. </p>
<hr>
<figure class="align-right zoomable">
<a href="https://images.theconversation.com/files/567788/original/file-20240103-23-yg479z.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="A square box with the words 'Art & Science Collide' and a drawing of a lightbulb with its wire filament in the shape of a brain, surrounded by a circle." src="https://images.theconversation.com/files/567788/original/file-20240103-23-yg479z.png?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/567788/original/file-20240103-23-yg479z.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=600&fit=crop&dpr=1 600w, https://images.theconversation.com/files/567788/original/file-20240103-23-yg479z.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=600&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/567788/original/file-20240103-23-yg479z.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=600&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/567788/original/file-20240103-23-yg479z.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=754&fit=crop&dpr=1 754w, https://images.theconversation.com/files/567788/original/file-20240103-23-yg479z.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=754&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/567788/original/file-20240103-23-yg479z.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=754&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Art & Science Collide series.</span>
</figcaption>
</figure>
<p><em><strong><a href="https://theconversation.com/us/topics/art-in-science-series-2024-149583">This article is part of Art & Science Collide</a></strong>, a series examining the intersections between art and science.</em></p>
<p><em>You may be interested in:</em></p>
<p><a href="https://theconversation.com/literature-inspired-my-medical-career-why-the-humanities-are-needed-in-health-care-217357">Literature inspired my medical career: Why the humanities are needed in health care</a></p>
<p><a href="https://theconversation.com/i-wrote-a-play-for-children-about-integrating-the-arts-into-stem-fields-heres-what-i-learned-about-encouraging-creative-interdisciplinary-thinking-218001">I wrote a play for children about integrating the arts into STEM fields – here’s what I learned about interdisciplinary thinking</a> </p>
<p><a href="https://theconversation.com/pictures-have-been-teaching-doctors-medicine-for-centuries-a-medical-illustrator-explains-how-218998">Pictures have been teaching doctors medicine for centuries − a medical illustrator explains how</a></p>
<hr>
<p>The star <a href="http://oklo.org/2010/06/23/a-second-laplace-resonance/">Gliese 876</a> has three planets with orbit period ratios of 4:2:1, just like Jupiter’s three moons. <a href="https://skyandtelescope.org/astronomy-news/kepler-finds-planets-in-tight-dance/">Kepler 223</a> has four planets with ratios of 8:6:4:3. </p>
<p>The red dwarf <a href="https://arxiv.org/abs/2212.08695">Kepler 80</a> has five planets with ratios of 9:6:4:3:2, and <a href="https://www.esa.int/ESA_Multimedia/Images/2021/01/Infographic_of_the_TOI-178_planetary_system">TOI 178</a> has six planets, of which five are in a resonant chain with ratios of 18:9:6:4:3. </p>
<p><a href="https://www.sci.news/astronomy/trappist-1-planetary-harmonies-04851.html">TRAPPIST-1</a> is the record holder. It has seven Earth-like planets, two of which <a href="https://theconversation.com/ultracool-dwarf-star-hosts-three-potentially-habitable-earth-sized-planets-just-40-light-years-away-58695">might be habitable</a>, with orbit ratios of 24:15:9:6:4:3:2. </p>
<p>The newest example of a resonant chain is the <a href="https://www.digitaltrends.com/space/exoplanet-orbital-resonance-chain/">HD 110067</a> system. It’s about 100 light years away and has six sub-Neptune planets, a common type of exoplanet, with orbit ratios of 54:36:24:16:12:9. The discovery is interesting because most resonance chains are unstable and disappear over time. </p>
<p>Despite these examples, resonant chains are rare, and <a href="https://www.astronomy.com/science/astronomers-find-six-planets-orbiting-in-resonance/">only 1% of all planetary systems display them</a>. Astronomers think that planets form in resonance, but small gravitational nudges from passing stars and wandering planets erase the resonance over time. With HD 110067, the resonant chain has survived for billions of years, offering a rare and pristine view of the system as it was when it formed.</p>
<h2>Orbit sonification</h2>
<p>Astronomers use <a href="https://science.howstuffworks.com/sonification.htm">a technique called sonification</a> to translate complex visual data into sound. It gives people a different way to appreciate the beautiful images from the <a href="https://science.nasa.gov/mission/hubble/multimedia/sonifications/">Hubble Space Telescope</a>, and it has been applied to <a href="https://www.system-sounds.com/">X-ray data and gravitational waves</a>.</p>
<p>With exoplanets, sonification can convey the mathematical relationships of their orbits. Astronomers at the European Southern Observatory created what they call “<a href="https://www.youtube.com/watch?v=-WevvRG9ysY">music of the spheres</a>” for the TOI 178 system by associating a sound on a pentatonic scale to each of the five planets. </p>
<figure>
<iframe width="440" height="260" src="https://www.youtube.com/embed/-WevvRG9ysY?wmode=transparent&start=0" frameborder="0" allowfullscreen=""></iframe>
<figcaption><span class="caption">Music from planetary orbits, created by astronomers at the European Southern Observatory.</span></figcaption>
</figure>
<p>A <a href="https://www.youtube.com/watch?v=WS5UxLHbUKc">similar musical translation</a> has been done for the TRAPPIST-1 system, with the orbital frequencies scaled up by a factor of 212 million to bring them into audible range. </p>
<p>Astronomers have also <a href="https://www.youtube.com/watch?v=2rrODAG7nmI&t=3s">created a sonification</a> for the HD 110067 system. People may not agree on whether these renditions sound like actual music, but it’s inspiring to see Pythagoras’ ideas realized after 2,500 years.</p><img src="https://counter.theconversation.com/content/219054/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Chris Impey receives funding from the National Science Foundation. </span></em></p>Orbital resonance is kind of like musical harmony, but systems that display it are far more rare than songs with harmonic melodies.Chris Impey, University Distinguished Professor of Astronomy, University of ArizonaLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/2193562024-01-25T16:01:30Z2024-01-25T16:01:30ZSpreadsheet errors can have disastrous consequences – yet we keep making the same mistakes<figure><img src="https://images.theconversation.com/files/570338/original/file-20240119-21-5frvd3.jpg?ixlib=rb-1.1.0&rect=75%2C0%2C8386%2C5573&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Making mistakes with spreadsheets can not only cause us personal frustration but can also lead to some very serious consequences. </span> <span class="attribution"><a class="source" href="https://www.shutterstock.com/image-photo/sad-tired-medical-coding-bill-spreadsheets-2197496803">Andrey_Popov/Shutterstock</a></span></figcaption></figure><p>Spreadsheet blunders aren’t just frustrating personal inconveniences. They can have serious consequences. And in the last few years alone, there have been a myriad of spreadsheet horror stories. </p>
<p>In August 2023, the Police Service of Northern Ireland <a href="https://www.bbc.co.uk/news/uk-northern-ireland-66445452">apologised</a> for a data leak of “monumental proportions” when a spreadsheet that contained statistics on the number of officers it had and their rank was shared online in response to a freedom of information request. </p>
<p>There was a second overlooked tab on the spreadsheet that contained the personal details of 10,000 serving police officers. </p>
<p>A <a href="https://anro.wm.hee.nhs.uk/Portals/3/Anaesthetics%20Recruitment%20-%20Significant%20Incident%20Report%20-%20Dec%2021.pdf?ver=hqDrm_-syzeLmBcfbigWJA%3D%3D">series of spreadsheet errors</a> disrupted the recruitment of trainee anaesthetists in Wales in late 2021. The Anaesthetic National Recruitment Office (ANRO), the body responsible for their selection and recruitment, told all the candidates for positions in Wales they were “unappointable”, despite some of them achieving the highest interview scores.</p>
<p>The blame fell on the process of consolidating interview data. Spreadsheets from different areas lacked standardisation in formatting, naming conventions and overall structure. To make matters worse, data was manually copied and pasted between various spreadsheets, a time-consuming and error-prone process.</p>
<p>ANRO only discovered the blunder when rejected applicants questioned their dismissal letters. The fact that not a single candidate seemed acceptable for Welsh positions should have been a red flag. No testing or validation was apparently applied to the crucial spreadsheet, a simple step that could have prevented this critical error.</p>
<p>In 2021, Crypto.com, an online provider of cryptocurrency, <a href="https://www.theguardian.com/australia-news/2023/sep/24/a-crypto-firm-sent-a-disability-worker-10m-by-mistake-months-later-she-was-arrested-at-an-australian-airport">accidentally transferred</a> US$10.5 million (£8.3 million) instead of US$100 into the account of an Australian customer due to an incorrect number being entered on a spreadsheet. </p>
<p>The clerk who processed the refund for the Australian customer had wrongly entered her bank account number in the refund field in a spreadsheet. It was seven months before the mistake was spotted. The recipient attempted to flee to Malaysia but was stopped at an Australian airport carrying a large amount of cash.</p>
<p>In 2022, Íslandsbanki, a state-owned Icelandic bank, sold a portion of shares that were badly undervalued due to a <a href="https://www.bloomberg.com/news/articles/2022-11-14/bungled-excel-sheet-hurts-profits-from-islandsbanki-sale">spreadsheet error</a>. When consolidating assets from different spreadsheets, the spreadsheet data was not “cleaned” and formatted properly. The bank’s shares were subsequently undervalued by as much as £16 million. </p>
<h2>The dark matter of corporate IT</h2>
<p>The above is just a fraction of the spreadsheet errors that are regularly made by various organisations. </p>
<p>Spreadsheets represent unknown risks in the form of errors, privacy violations, trade secrets and compliance violations. Yet they are also critical for the way many organisations make their decisions. For this reason, they have been <a href="https://www.igi-global.com/article/end-user-computing/81295">described</a> by experts as the “dark matter” of corporate IT. </p>
<p>Industry <a href="https://www.igi-global.com/article/know-spreadsheet-errors/55750">studies</a> show that 90% of spreadsheets containing more than 150 rows have at least one major mistake. </p>
<p>This is understandable because spreadsheet errors are easy to make but difficult to spot. My <a href="https://aisel.aisnet.org/cais/vol25/iss1/34/">own research</a> has shown that inspecting the spreadsheet’s code is the most effective way of debugging them, but this approach still only catches between 60% and 80% of all errors. </p>
<figure class="align-center ">
<img alt="A close up of Microsoft Excel spreadsheet." src="https://images.theconversation.com/files/570385/original/file-20240119-15-gebegy.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/570385/original/file-20240119-15-gebegy.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=401&fit=crop&dpr=1 600w, https://images.theconversation.com/files/570385/original/file-20240119-15-gebegy.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=401&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/570385/original/file-20240119-15-gebegy.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=401&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/570385/original/file-20240119-15-gebegy.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/570385/original/file-20240119-15-gebegy.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/570385/original/file-20240119-15-gebegy.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=503&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">As many as 9 out of 10 spreadsheets are estimated to contain errors.</span>
<span class="attribution"><a class="source" href="https://www.shutterstock.com/image-photo/new-york-usa-august-18-2017-699112366">PixieMe/Shutterstock</a></span>
</figcaption>
</figure>
<p>Spreadsheets’ appeal doesn’t just exist in the financial world. They are indispensable in <a href="https://eusprig.org/wp-content/uploads/1801.10231.pdf">engineering</a>, <a href="https://ijcis.net/index.php/ijcis/article/view/79">data science</a> and even in <a href="https://ntrs.nasa.gov/citations/20150008644">sending robots</a> to Mars. The key to their success is their flexibility. </p>
<p>Spreadsheet software is constantly evolving, with more features becoming available that increase their appeal. For instance, you can now automate many tasks in Excel (the most popular spreadsheet software) using Python scripting.</p>
<p>But given all of the aforementioned problems, isn’t it time for Excel and other spreadsheet software to be sidelined in favour of something more reliable? </p>
<h2>Human error</h2>
<p>The underlying cause of these spreadsheet problems is not the software but human error. </p>
<p>The issue is that most users don’t see the need to plan or test their work. Most users <a href="https://www.igi-global.com/article/errors-operational-spreadsheets/4145">describe</a> their first step in creating a new spreadsheet as merely jumping straight in and entering numbers or code directly. </p>
<p>Many of us don’t consider spreadsheets to warrant serious consideration. This means we become <a href="https://eusprig.org/wp-content/uploads/0804.0941.pdf">complacent</a> and assume there is no need to test, validate or verify our work.</p>
<p><a href="https://www.igi-global.com/gateway/article/3762">Research</a> on “cognitive load”, the amount of mental effort required for a task, shows that building complex spreadsheets demands as much concentration as a GP making a diagnosis. This intense mental strain makes mistakes more likely. But GPs study their profession for many years before becoming qualified while most spreadsheet users are <a href="https://eusprig.org/wp-content/uploads/0803.1862.pdf">self-taught</a>. </p>
<p>To break the cycle of repeated spreadsheet errors, there are several things organisations can do. First, introducing standardisation would help to minimise confusion and mistakes. For example, this would mean consistent formatting, naming conventions and data structures across spreadsheets.</p>
<p>Second, improving training is crucial. Equipping users with the knowledge and skills to build robust and accurate spreadsheets could help them identify and avoid pitfalls. </p>
<p>Finally, fostering a culture of critical thinking towards spreadsheets is vital. This would mean encouraging users to continually question calculations, validate their data sources and double-check their work.</p><img src="https://counter.theconversation.com/content/219356/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Simon Thorne is affiliated with The European Spreadsheets Risks Interest Group</span></em></p>Spreadsheet-related errors can have serious consequences in the private and public sector. But what can we do to overcome them?Simon Thorne, Senior Lecturer in Computing and Information Systems, Cardiff Metropolitan UniversityLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/2180012024-01-12T13:28:40Z2024-01-12T13:28:40ZI wrote a play for children about integrating the arts into STEM fields − here’s what I learned about encouraging creative, interdisciplinary thinking<figure><img src="https://images.theconversation.com/files/562522/original/file-20231129-27-a3te04.jpeg?ixlib=rb-1.1.0&rect=18%2C18%2C4007%2C2999&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Scenes from 'The STEAM Plays,' performed in Michigan schools. </span> <span class="attribution"><span class="source">Thalia Lara</span></span></figcaption></figure><p>Often, science and art are described as starkly different things. That narrative can start early on, with kids encouraged to pursue a STEM – short for science, technology, engineering and math – education that may or may not include an arts education. </p>
<p>As a <a href="https://people.cal.msu.edu/roznows5/">professor of acting</a>, I’d never thought much about the STEM fields until I received a <a href="https://grad.msu.edu/news/steampower-facultystaff-fellows">fellowship to integrate the arts</a> into STEM educational models. I used the opportunity to write and direct a play for elementary schoolers that showed how the arts can improve upon and extend work in STEM fields when properly integrated – but it wasn’t an easy process. </p>
<h2>STEM or STEAM?</h2>
<p>Whether <a href="https://theconversation.com/explainer-whats-the-difference-between-stem-and-steam-95713">STEM should be augmented to STEAM</a> – science, technology, engineering, arts and math – with the <a href="https://doi.org/10.3390/educsci11070331">addition of the arts</a> remains <a href="https://www.washingtonpost.com/news/innovations/wp/2018/06/12/why-liberal-arts-and-the-humanities-are-as-important-as-engineering/">something of a debate</a>. </p>
<p>The origins of STEM education can be traced to as early as the <a href="https://www.archives.gov/milestone-documents/morrill-act">Morrill Act</a> of 1862, which promoted agricultural science and later engineering at land grant universities. In 2001, the National Science Foundation pushed a focus on STEM education in order to <a href="https://www.stemschool.com/articles/rich-history-of-stem-education-in-the-united-states">make the U.S. more competitive globally</a>. </p>
<p>A Biden-Harris initiative launched in December 2022 called <a href="https://www.ed.gov/news/press-releases/us-department-education-launches-new-initiative-enhance-stem-education-all-students">You Belong in STEM</a> offers support of more than US$120 billion for K-12 STEM education until the year 2025. But, starting in 2012, the United States Research Council has explored the idea of a <a href="https://www.frontiersin.org/articles/10.3389/feduc.2021.709560/">STEAM education</a>. </p>
<p>Researchers have found that when integrated into a STEM education, <a href="https://doi.org/10.1016/j.procs.2013.09.317">the arts make space for curiosity and innovation</a>. So why the lack of agreement and consistency around whether it should be STEM or STEAM? </p>
<figure>
<iframe width="440" height="260" src="https://www.youtube.com/embed/GOYN70wszoo?wmode=transparent&start=0" frameborder="0" allowfullscreen=""></iframe>
<figcaption><span class="caption">Lots of careers bridge both science and arts, from game design to photography and engineering.</span></figcaption>
</figure>
<p>The bias toward emphasizing a STEM education could be driven by the <a href="https://www.cnbc.com/2023/10/20/more-students-pursue-stem-degrees-because-of-high-paying-careers.html">higher future salaries</a> of STEM majors or the significant funding that is connected more to <a href="https://doi.org/10.1057/s41599-021-00891-x">STEM-based research</a> and grants than to the arts. A STEAM education takes more time and <a href="https://theconversation.com/improving-science-literacy-means-changing-science-education-178291">is more complex</a> than a traditional STEM educational model. </p>
<p>Or it could simply be that many academics in STEM fields lack the incentive for interdisciplinary work that brings in the arts, and vice versa. In fact, that was exactly the position I was in as an arts-based researcher asked to create something about STEM disciplines that I knew very little about.</p>
<h2>Putting on the play</h2>
<p>It took me several tries and lots of research to get the script of my STEAM-centered play to its current form. </p>
<p>At first, I made basic discoveries. I learned that <a href="https://www.invent.org/blog/trends-stem/stem-steam-defined">there is a debate</a> about whether the arts should be included in a STEM education. I learned that “<a href="https://stemeducationguide.com/is-psychology-stem/">soft sciences” like psychology</a> are <a href="https://doi.org/10.1007/s12124-020-09545-0">not included</a> in many STEM educational models. I lacked a background in most of the disciplines included in STEM. And I struggled to find a project that inspired me.</p>
<p>But eventually I began work on five one-act plays, called “The STEAM Plays: Using the Arts to Talk about STEM.” Each focused on a category of STEAM education. I wrote the first draft of the show with a chip on my shoulder, trying to prove that the arts did indeed belong in STEM education.</p>
<p>The tone was defensive and provocative – and not entirely appropriate for the elementary age range I was focused on. </p>
<p>The new, revised version that toured Michigan elementary schools in the Fall of 2023 contains 20 bite-sized comedic scenes and songs that dramatize how the arts are integral to many STEM fields. These include how engineering skills go into designing a celebrity’s evening gown, how bakers need to know some basic chemistry, and how the mathematical algorithms of TikTok find new videos for each user.</p>
<p>In each of the scenes, students can see how artistic imagination and creative thinking expand STEM education.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/563453/original/file-20231204-23-wjyepx.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="A group of people performing on a stage, wearing brightly colored costumes. The background is a screen projecting blue, green and yellow geometric shapes. The two performers on the left have their arms crossed and stand back to back, same on the right." src="https://images.theconversation.com/files/563453/original/file-20231204-23-wjyepx.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/563453/original/file-20231204-23-wjyepx.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=450&fit=crop&dpr=1 600w, https://images.theconversation.com/files/563453/original/file-20231204-23-wjyepx.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=450&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/563453/original/file-20231204-23-wjyepx.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=450&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/563453/original/file-20231204-23-wjyepx.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=566&fit=crop&dpr=1 754w, https://images.theconversation.com/files/563453/original/file-20231204-23-wjyepx.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=566&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/563453/original/file-20231204-23-wjyepx.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=566&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">‘The STEAM Plays’ in action. Performers, from left: Alex Spevetz, Marcus Pennington, Zoe Dorst, Cassidy Williams and Olivia Hagar.</span>
<span class="attribution"><span class="source">Rob Roznowski</span></span>
</figcaption>
</figure>
<h2>Beyond the stage</h2>
<p>These themes emerge from a wider scholarly understanding that STEM isn’t done in a creativity vacuum, and <a href="https://doi.org/10.1016/j.procs.2013.09.317">stimulating students’ artistic thinking</a> will help them both in the science classroom and the art studio.</p>
<p>One plot point of the show is about an evil genius who views the arts as less important trying to keep the arts out of STEM. He swaps the bodies of a scientist and an actor, as well as an engineer and a creative writer. In each body swap, the STEM professional and the artist recognize how similar their work is. In the final scene, the evil genius tries to switch the bodies of Pythagoras and Taylor Swift, only to realize that music is all about math.</p>
<hr>
<figure class="align-right ">
<img alt="A square box with the words 'Art & Science Collide' and a drawing of a lightbulb with its wire filament in the shape of a brain, surrounded by a circle." src="https://images.theconversation.com/files/567788/original/file-20240103-23-yg479z.png?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/567788/original/file-20240103-23-yg479z.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=600&fit=crop&dpr=1 600w, https://images.theconversation.com/files/567788/original/file-20240103-23-yg479z.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=600&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/567788/original/file-20240103-23-yg479z.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=600&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/567788/original/file-20240103-23-yg479z.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=754&fit=crop&dpr=1 754w, https://images.theconversation.com/files/567788/original/file-20240103-23-yg479z.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=754&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/567788/original/file-20240103-23-yg479z.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=754&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Art & Science Collide series.</span>
</figcaption>
</figure>
<p><em><strong><a href="https://theconversation.com/us/topics/art-in-science-series-2024-149583">This article is part of Art & Science Collide</a></strong>, a series examining the intersections between art and science.</em></p>
<p><em>You may be interested in:</em></p>
<p><a href="https://theconversation.com/literature-inspired-my-medical-career-why-the-humanities-are-needed-in-health-care-217357">Literature inspired my medical career: Why the humanities are needed in health care</a></p>
<p><a href="https://theconversation.com/art-and-science-entwined-this-course-explores-the-long-interrelated-history-of-two-ways-of-seeing-the-world-210250">Art and science entwined: This course explores the long, interrelated history of two ways of seeing the world </a></p>
<p><a href="https://theconversation.com/art-illuminates-the-beauty-of-science-and-could-inspire-the-next-generation-of-scientists-young-and-old-168925">Art illuminates the beauty of science – and could inspire the next generation of scientists young and old</a> </p>
<hr>
<p>Many teachers have provided rave reviews. “The plays did an excellent job of highlighting the importance and value of arts in our educational system,” one noted. “Students walked away enjoying and having a deeper understanding of how all of the different aspects of STEAM were able to work together collaboratively.</p>
<p>A STEAM education in which <a href="https://www.ucf.edu/online/engineering/news/comparing-stem-vs-steam-why-the-arts-make-a-difference/">students learn soft skills</a> like empathy, collaboration, emotional intelligence and creativity through the arts helps prepare students for the job market. And these discussions aren’t confined only to K-12 education – many research grants <a href="https://new.nsf.gov/funding/learn/research-types/learn-about-interdisciplinary-research">encourage interdisciplinary work</a>.</p>
<p>My understanding of the STEM and STEAM debate and my experience writing, producing and watching how people respond to my show have helped me understand how the arts are necessary to every student’s education. I learned that without artistic imagination, STEM students’ big-picture thinking skills can get stifled. </p>
<p>It only took writing a play for children for me to get it myself.</p><img src="https://counter.theconversation.com/content/218001/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Rob Roznowski received funding from Michigan State University from two places. As part of the STEAMpower Fellowship <a href="https://grad.msu.edu/news/steampower-facultystaff-fellows">https://grad.msu.edu/news/steampower-facultystaff-fellows</a> $10,000
and the Humanities And Arts Grant Proposal System. <a href="https://research.msu.edu/humanities-and-arts-research-program">https://research.msu.edu/humanities-and-arts-research-program</a>
The first fellowship covered the writing and research. The HARPwas awarded to tour and design the play. $7000</span></em></p>Is it a STEM education or a STEAM education? Integrating arts into science programming and vice versa can pique kids’ curiosity − a play touring Michigan aims to do just that.Rob Roznowski, Professor of Acting, Michigan State UniversityLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/2145762024-01-07T19:04:19Z2024-01-07T19:04:19ZHere’s why you should (almost) never use a pie chart for your data<figure><img src="https://images.theconversation.com/files/558554/original/file-20231109-25-j7ehuz.jpg?ixlib=rb-1.1.0&rect=810%2C436%2C4761%2C3377&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">
</span> <span class="attribution"><a class="source" href="https://www.shutterstock.com/image-photo/lemon-pie-flat-lay-on-blue-1663719415">YesPhotographers/Shutterstock</a></span></figcaption></figure><p>Our lives are becoming increasingly data driven. Our phones monitor our time and internet usage and online surveys discern our opinions and likes. These data harvests are used for telling us how well we’ve slept or what we might like to buy. </p>
<p>Numbers are becoming more important for everyday life, yet people’s numerical skills are falling behind. For example, the percentage of Year 12 schoolchildren in Australia taking higher and intermediate mathematics <a href="https://amsi.org.au/?publications=year-12-participation-in-calculus-based-mathematics-subjects-takes-a-dive-2">has been declining for decades</a>. </p>
<p>To help the average person understand big data and numbers, we often use visual summaries, such as pie charts. But while non-numerate folk will avoid numbers, most numerate folk will avoid pie charts. Here’s why.</p>
<h2>What is a pie chart?</h2>
<p>A pie chart is a circular diagram that represents numerical percentages. The circle is divided into slices, with the size of each slice proportional to the category it represents. It is named because it resembles a sliced pie and can be “served” in many different ways. </p>
<p>An example pie chart below shows Australia’s two-party preferred vote before the last election, with Labor on 55% and the the Coalition on 45%. The two near semi-circles show the relatively tight race – this is a useful example of a pie chart. </p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/560670/original/file-20231121-23-sgp640.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="55% for labor, 45% for coalition on a red and blue pie chart" src="https://images.theconversation.com/files/560670/original/file-20231121-23-sgp640.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/560670/original/file-20231121-23-sgp640.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=360&fit=crop&dpr=1 600w, https://images.theconversation.com/files/560670/original/file-20231121-23-sgp640.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=360&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/560670/original/file-20231121-23-sgp640.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=360&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/560670/original/file-20231121-23-sgp640.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=452&fit=crop&dpr=1 754w, https://images.theconversation.com/files/560670/original/file-20231121-23-sgp640.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=452&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/560670/original/file-20231121-23-sgp640.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=452&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">A simple pie chart showing the percentages for the two major Australian parties in an opinion poll.</span>
<span class="attribution"><span class="source">Victor Oguoma</span></span>
</figcaption>
</figure>
<h2>What’s wrong with pie charts?</h2>
<p>Once we have more than two categories, pie charts can easily misrepresent percentages and become hard to read.</p>
<p>The three charts below are a good example – it is very hard to work out which of the five areas is the largest. The pie chart’s circularity means the areas lack a common reference point. </p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/556782/original/file-20231031-27-3dz8ta.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/556782/original/file-20231031-27-3dz8ta.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/556782/original/file-20231031-27-3dz8ta.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=208&fit=crop&dpr=1 600w, https://images.theconversation.com/files/556782/original/file-20231031-27-3dz8ta.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=208&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/556782/original/file-20231031-27-3dz8ta.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=208&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/556782/original/file-20231031-27-3dz8ta.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=261&fit=crop&dpr=1 754w, https://images.theconversation.com/files/556782/original/file-20231031-27-3dz8ta.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=261&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/556782/original/file-20231031-27-3dz8ta.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=261&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Three example pie charts, each with five similar categories. Can you quickly tell which colour is the largest in each pie?</span>
<span class="attribution"><a class="source" href="https://commons.wikimedia.org/wiki/File:Piecharts.svg">Schutz/Wikimedia Commons</a>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span>
</figcaption>
</figure>
<p>Pie charts also do badly when there are lots of categories. For example, this chart from a study on data sources used for COVID data visualisation shows hundreds of categories in one pie. </p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/556770/original/file-20231031-19-uurqzu.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/556770/original/file-20231031-19-uurqzu.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/556770/original/file-20231031-19-uurqzu.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=340&fit=crop&dpr=1 600w, https://images.theconversation.com/files/556770/original/file-20231031-19-uurqzu.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=340&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/556770/original/file-20231031-19-uurqzu.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=340&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/556770/original/file-20231031-19-uurqzu.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=428&fit=crop&dpr=1 754w, https://images.theconversation.com/files/556770/original/file-20231031-19-uurqzu.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=428&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/556770/original/file-20231031-19-uurqzu.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=428&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">A pie chart with dozens of categories. Not every category has a label, it’s not clear what the total number of categories is and what the unlabelled slices refer to.</span>
<span class="attribution"><a class="source" href="https://doi.org/10.3390/informatics7030035">Trajkova et al., Informatics (2020)</a>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span>
</figcaption>
</figure>
<p>The tiny slices, lack of clear labelling and the kaleidoscope of colours make interpretation difficult for anyone.</p>
<p>It’s even harder for a colour blind person. For example, this is a simulation of what the above chart would look like to a person with deuteranomaly or reduced sensitivity to green light. This is the most common type of colour blindness, affecting roughly <a href="https://wearecolorblind.com/articles/a-quick-introduction-to-color-blindness/">4.6% of the population</a>. </p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/558560/original/file-20231109-27-4714o0.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/558560/original/file-20231109-27-4714o0.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/558560/original/file-20231109-27-4714o0.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=449&fit=crop&dpr=1 600w, https://images.theconversation.com/files/558560/original/file-20231109-27-4714o0.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=449&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/558560/original/file-20231109-27-4714o0.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=449&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/558560/original/file-20231109-27-4714o0.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=564&fit=crop&dpr=1 754w, https://images.theconversation.com/files/558560/original/file-20231109-27-4714o0.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=564&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/558560/original/file-20231109-27-4714o0.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=564&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">The same data chart as above, but run through a simulation filter to demonstrate what it would look like for someone with a common type of colour blindness.</span>
<span class="attribution"><a class="source" href="https://doi.org/10.3390/informatics7030035">Trajkova et al., Informatics (2020); modified.</a>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span>
</figcaption>
</figure>
<p>It can get even worse if we take pie charts and make them three-dimensional. This can lead to egregious misrepresentations of data.</p>
<p>Below, the yellow, red and green areas are all the same size (one-third), but appear to be different based on the angle and which slice is placed at the bottom of the pie.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/556772/original/file-20231031-25-bdpq56.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/556772/original/file-20231031-25-bdpq56.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/556772/original/file-20231031-25-bdpq56.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=196&fit=crop&dpr=1 600w, https://images.theconversation.com/files/556772/original/file-20231031-25-bdpq56.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=196&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/556772/original/file-20231031-25-bdpq56.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=196&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/556772/original/file-20231031-25-bdpq56.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=246&fit=crop&dpr=1 754w, https://images.theconversation.com/files/556772/original/file-20231031-25-bdpq56.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=246&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/556772/original/file-20231031-25-bdpq56.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=246&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">A standard two-dimensional pie chart and two three-dimensional pie charts. In every chart the proportions are one-third but there appear to be differences between states in the three-dimensional versions.</span>
<span class="attribution"><span class="source">Victor Oguoma</span>, <a class="license" href="http://creativecommons.org/licenses/by-nd/4.0/">CC BY-ND</a></span>
</figcaption>
</figure>
<h2>So why are pie charts everywhere?</h2>
<p>Despite the well known problems with pie charts, they are everywhere. They are in journal articles, PhD theses, political polling, books, newspapers and government reports. They’ve even been used by the Australian Bureau of Statistics.</p>
<p>While statisticians have criticised them for decades, it’s hard to argue with this logic: “if pie charts are so bad, why are there so many of them?”</p>
<p>Possibly they are popular because they are popular, which is a circular argument that suits a pie chart.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/556781/original/file-20231031-17-hfvpgr.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/556781/original/file-20231031-17-hfvpgr.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/556781/original/file-20231031-17-hfvpgr.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=395&fit=crop&dpr=1 600w, https://images.theconversation.com/files/556781/original/file-20231031-17-hfvpgr.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=395&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/556781/original/file-20231031-17-hfvpgr.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=395&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/556781/original/file-20231031-17-hfvpgr.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=496&fit=crop&dpr=1 754w, https://images.theconversation.com/files/556781/original/file-20231031-17-hfvpgr.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=496&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/556781/original/file-20231031-17-hfvpgr.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=496&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">A collection of terrible pie charts gathered from various open access sources, including ‘exploded’ pie charts and 3D pie charts.</span>
<span class="attribution"><span class="source">Adrian Barnett and Victor Oguoma</span>, <a class="license" href="http://creativecommons.org/licenses/by-nd/4.0/">CC BY-ND</a></span>
</figcaption>
</figure>
<h2>What’s a good alternative to pie charts?</h2>
<p>There’s a simple fix that can effectively summarise big data in a small space and still allow creative colour schemes. </p>
<p>It’s the humble bar chart. Remember the brain-aching pie chart example above with the five categories? Here’s the same example using bars – we can now instantly see which category is the largest.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/556773/original/file-20231031-25-9vdsm4.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/556773/original/file-20231031-25-9vdsm4.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/556773/original/file-20231031-25-9vdsm4.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=430&fit=crop&dpr=1 600w, https://images.theconversation.com/files/556773/original/file-20231031-25-9vdsm4.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=430&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/556773/original/file-20231031-25-9vdsm4.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=430&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/556773/original/file-20231031-25-9vdsm4.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=541&fit=crop&dpr=1 754w, https://images.theconversation.com/files/556773/original/file-20231031-25-9vdsm4.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=541&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/556773/original/file-20231031-25-9vdsm4.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=541&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Three pie charts, each with five similar categories, and the same data presented using bar charts.</span>
<span class="attribution"><a class="source" href="https://commons.wikimedia.org/wiki/File:Piecharts.svg">Schutz/Wikimedia Commons</a>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span>
</figcaption>
</figure>
<p>Linear bars are easier on the eye than the non-linear segments of a pie chart. But beware the temptation to make a humble bar chart look more interesting by adding a 3D effect. As you already saw, 3D charts distort perception and make it harder to find a reference point.</p>
<p>Below is a standard bar chart and a 3D alternative of the number of voters in the 1992 US presidential election split by family income (from under US$15K to over $75k). Using the 3D version, can you tell the number of voters for each candidate in the highest income category? Not easily. </p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/556775/original/file-20231031-17-dscfue.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/556775/original/file-20231031-17-dscfue.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/556775/original/file-20231031-17-dscfue.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=157&fit=crop&dpr=1 600w, https://images.theconversation.com/files/556775/original/file-20231031-17-dscfue.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=157&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/556775/original/file-20231031-17-dscfue.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=157&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/556775/original/file-20231031-17-dscfue.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=197&fit=crop&dpr=1 754w, https://images.theconversation.com/files/556775/original/file-20231031-17-dscfue.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=197&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/556775/original/file-20231031-17-dscfue.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=197&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">The same voter data presented as a standard two-dimensional bar chart and an unhelpful three-dimensional version.</span>
<span class="attribution"><span class="source">Victor Oguoma</span>, <a class="license" href="http://creativecommons.org/licenses/by-nd/4.0/">CC BY-ND</a></span>
</figcaption>
</figure>
<h2>Is it ever okay to use a pie chart?</h2>
<p>We’ve shown some of the worst examples of pie charts to make a point. Pie charts can be okay when there are just a few categories and the percentages are dissimilar, for example with one large and one small category.</p>
<p>Overall, it is best to use pie charts sparingly, especially when there is a more “digestible” alternative – the bar chart.</p>
<p>Whenever we see pie charts, we think one of two things: their creators don’t know what they’re doing, or they know what they are doing and are deliberately trying to mislead.</p>
<p>A graphical summary aims to easily and quickly communicate the data. If you feel the need to spruce it up, you’re likely reducing understanding without meaning to do so.</p>
<hr>
<p>
<em>
<strong>
Read more:
<a href="https://theconversation.com/3-questions-to-ask-yourself-next-time-you-see-a-graph-chart-or-map-141348">3 questions to ask yourself next time you see a graph, chart or map</a>
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</em>
</p>
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<img src="https://counter.theconversation.com/content/214576/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Adrian Barnett is a member of the Statistical Society of Australia.</span></em></p><p class="fine-print"><em><span>Victor Oguoma is a member of the Statistical Society of Australia.</span></em></p>They are popular because they are popular, which is a circular argument that suits a pie chart. But there are some serious downsides to using the humble pie.Adrian Barnett, Professor of Statistics, Queensland University of TechnologyVictor Oguoma, Senior Research Fellow, Poche Centre for Indigenous Health, The University of QueenslandLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/2193202024-01-03T20:26:58Z2024-01-03T20:26:58ZAI is our ‘Promethean fire’: using it wisely means knowing its true nature – and our own minds<p>Future historians may well regard 2023 as a landmark in the advent of artificial intelligence (AI). But whether that future will prove <a href="https://a16z.com/ai-will-save-the-world/">utopian</a>, <a href="https://www.toolify.ai/ai-news/ais-apocalyptic-vision-24116">apocalyptic</a> or <a href="https://www.mckinsey.com/featured-insights/mckinsey-explainers/whats-the-future-of-generative-ai-an-early-view-in-15-charts">somewhere in between</a> is anyone’s guess. </p>
<p>In February, ChatGPT set the record as the fastest app to reach <a href="https://www.reuters.com/technology/chatgpt-sets-record-fastest-growing-user-base-analyst-note-2023-02-01/">100 million users</a>. It was followed by similar “large language” AI models from Google, Amazon, Meta and other big tech firms, which collectively look poised to transform education, healthcare and many other knowledge-intensive fields. </p>
<p>However, AI’s potential for harm was underscored in May by an <a href="https://www.safe.ai/statement-on-ai-risk">ominous statement</a> signed by leading researchers: </p>
<blockquote>
<p>Mitigating the risk of extinction from AI should be a global priority alongside other societal-scale risks such as pandemics and nuclear war. </p>
</blockquote>
<p>In November, responding to the growing concern about AI risk, 27 nations (including the UK, US, India, China and the European Union) pledged cooperation at an inaugural AI Safety Summit at Bletchley Park in England, to ensure the safe development of AI for the <a href="https://www.gov.uk/government/publications/ai-safety-summit-2023-the-bletchley-declaration/the-bletchley-declaration-by-countries-attending-the-ai-safety-summit-1-2-november-2023">benefit of all</a>. </p>
<p>To achieve this, researchers focus on <a href="https://en.wikipedia.org/wiki/AI_alignment">AI alignment</a> – that is, how to make sure AI models are consistent with human values, preferences and goals. But there’s a problem – AI’s so-called “<a href="https://www.technologyreview.com/2017/04/11/5113/the-dark-secret-at-the-heart-of-ai/">dark secret</a>”: large-scale models are so complex they are like a black box, impossible for anyone to fully understand. </p>
<h2>AI’s black box problem</h2>
<p>Although the transparency and explainability of AI systems are <a href="https://www.sciencedirect.com/science/article/pii/S1566253519308103?casa_token=eMCns9rVBmoAAAAA:ZozMhIZEA-Sd4IWnBBWRC6KmXV3THV4lqMYkWKf8-NrwaTxEKHqU2EAw4B-RZP0sCg0wazbml3o">important research goals</a>, such efforts seem unlikely to keep up with the frenetic pace of innovation. </p>
<p>The black box metaphor explains why people’s beliefs about AI are all over the map. Predictions range from utopia to extinction, and many even believe an artificial general intelligence (AGI) will soon <a href="https://www.science.org/content/article/if-ai-becomes-conscious-how-will-we-know">achieve sentience</a>. </p>
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<p>
<em>
<strong>
Read more:
<a href="https://theconversation.com/a-year-of-chatgpt-5-ways-the-ai-marvel-has-changed-the-world-218805">A year of ChatGPT: 5 ways the AI marvel has changed the world</a>
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<p>But this uncertainty compounds the problem. AI alignment should be a two-way street: we must not only ensure AI models are consistent with human intentions, but also that our beliefs about AI are accurate. </p>
<p>This is because we are remarkably adept at creating futures that accord with those beliefs, even if we are unaware of them. </p>
<p>So-called “<a href="https://journals.sagepub.com/doi/abs/10.1111/1467-8721.ep10770698">expectancy effects</a>”, or self-fulfilling prophecies, are well known in psychology. And research has shown that manipulating users’ beliefs influences not just how they <a href="https://dl.acm.org/doi/pdf/10.1145/3529225">interact with AI</a>, but how AI <a href="https://dspace.mit.edu/bitstream/handle/1721.1/152316/NMI_AI_beholder_Final-Unformatted%5B85%5D.pdf?sequence=1&isAllowed=y">adapts to the user</a>. </p>
<p>In other words, how our beliefs (conscious or unconscious) affect AI can potentially increase the likelihood of any outcome, including catastrophic ones. </p>
<hr>
<p>
<em>
<strong>
Read more:
<a href="https://theconversation.com/how-ai-sees-the-world-what-happened-when-we-trained-a-deep-learning-model-to-identify-poverty-217586">How AI 'sees' the world – what happened when we trained a deep learning model to identify poverty</a>
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<hr>
<h2>AI, computation, logic and arithmetic</h2>
<p>We need to probe more deeply to understand the basis of AI – like Alice in Wonderland, head down the rabbit hole and see where it takes us. </p>
<p>Firstly, what is AI? It runs on computers, and so is automated computation. From its origin as the “<a href="https://idp.springer.com/authorize/casa?redirect_uri=https://link.springer.com/content/pdf/10.1007/BF02478259.pdf&casa_token=Joxu9OnlEd4AAAAA:uCl-_FASTbvXSWBtZAt5bS24ZSRvOMsdufe1PG6PXY1TSdNoU0gL8a5j6I7lGmk4rqrSCbIqE0CQoxd9BnA">perceptron</a>” – an artificial neuron defined mathematically in 1943 by neurophysiologist <a href="https://en.wikipedia.org/wiki/Warren_Sturgis_McCulloch">Warren McCulloch</a> and logician <a href="https://en.wikipedia.org/wiki/Walter_Pitts">Walter Pitts</a> – AI has been intertwined with the cognitive sciences, neuroscience and computer science. </p>
<p>This convergence of <a href="https://www.cs.cmu.edu/afs/cs/academic/class/15883-f21/readings/churchland-1992-ch3.pdf">minds</a>, <a href="https://www.cell.com/fulltext/S0896-6273(17)30509-3">brains</a> and <a href="https://www.nature.com/articles/s41583-020-00395-8">machines</a> has led to the widely-held belief that, because AI is computation by machine, then natural intelligence (the mind) must be computation by the brain.</p>
<p>But what is computation? In the late 19th century, mathematicians <a href="https://en.wikipedia.org/wiki/Richard_Dedekind">Richard Dedekind</a> and <a href="https://en.wikipedia.org/wiki/Giuseppe_Peano">Giuseppe Peano</a> proposed a set of axioms which <a href="https://en.wikipedia.org/wiki/Peano_axioms">defined arithmetic in terms of logic</a>, and inspired attempts to ground all mathematics on a secure <a href="https://en.wikipedia.org/wiki/Hilbert%27s_program">formal basis</a>. </p>
<p>Although the logician <a href="https://en.wikipedia.org/wiki/Kurt_G%C3%B6del">Kurt Gödel</a> later proved this goal was <a href="https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems">unachievable</a>, his work was the starting point for mathematician (and code-breaker) <a href="https://en.wikipedia.org/wiki/Alan_Turing">Alan Turing</a>. His “<a href="https://en.wikipedia.org/wiki/Turing_machine">Turing machine</a>”, an abstract device capable of <a href="https://en.wikipedia.org/wiki/Church%E2%80%93Turing_thesis">universal computation</a>, is the foundation of computer science. </p>
<hr>
<p>
<em>
<strong>
Read more:
<a href="https://theconversation.com/forget-dystopian-scenarios-ai-is-pervasive-today-and-the-risks-are-often-hidden-218222">Forget dystopian scenarios – AI is pervasive today, and the risks are often hidden</a>
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<h2>Deep structure of perception</h2>
<p>So, computation is based on mathematical ideas that trace back to efforts to define arithmetic in logic. But our knowledge of arithmetic exists <a href="https://academic.oup.com/pq/article-abstract/68/273/717/4969397?redirectedFrom=PDF&casa_token=Z-7sIkFvtL0AAAAA:UOuTtFoVh9mpu6guxajdbe44O93oCe6PANK-Uz9yWL_0iX8lo-Lla-pPatTGINKxrAqB-MBpCtmts4cz">prior to logic</a>. If we want to understand the basis of AI, we need to go further and ask where arithmetic itself comes from. </p>
<p>My colleagues and I have recently shown that arithmetic is based on the “<a href="https://theconversation.com/arithmetic-has-a-biological-origin-its-an-expression-in-symbols-of-the-deep-structure-of-our-perception-211337">deep structure</a>” of perception. This structure is like coloured glasses that shape our perception in particular ways, so that our experience of the world is ordered and manageable. </p>
<p>Arithmetic consists of a set of elements (numbers) and operations (addition, multiplication) that combine pairs of elements to give another element. We asked: of all possibilities, why are numbers the elements, and addition and multiplication the operations? </p>
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<p>
<em>
<strong>
Read more:
<a href="https://theconversation.com/arithmetic-has-a-biological-origin-its-an-expression-in-symbols-of-the-deep-structure-of-our-perception-211337">Arithmetic has a biological origin – it's an expression in symbols of the 'deep structure' of our perception</a>
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<p>We showed by <a href="https://psycnet.apa.org/fulltext/2023-84614-001.pdf?sr=1">mathematical proof</a> that when the deep structure of perception was assumed to limit the possibilities, arithmetic was the result. In other words, when our mind views the abstract world through the same “coloured glasses” that shape our experience of the physical world, it “sees” numbers and arithmetic. </p>
<p>Because arithmetic is the foundation for mathematics, the implication is that mathematics is a reflection of the mind – an expression in symbols of its fundamental nature and creativity. </p>
<p>Although the deep structure of perception is shared with other animals and so a product of evolution, only humans have invented mathematics. It is our most intimate creation – and by enabling the development of AI, perhaps our most consequential. </p>
<h2>A Copernican revolution of the mind</h2>
<p>Our account of <a href="https://theconversation.com/arithmetic-has-a-biological-origin-its-an-expression-in-symbols-of-the-deep-structure-of-our-perception-211337">arithmetic’s origin</a> is consistent with views of the 18th century philosopher Immanuel Kant. According to him, our knowledge of the world is structured by “pure intuitions” of space and time that exist prior to sense experience – analogous to the coloured glasses we can never remove.</p>
<p>Kant claimed his <a href="https://plato.stanford.edu/entries/kant/?rid=903123293s840c38">philosophy</a> was a “Copernican revolution of the mind”. In the same way ancient astronomers believed the Sun revolved around the Earth because they were unaware of the Earth’s motion, Kant argued, philosophers who believed all knowledge is derived from <a href="https://en.wikipedia.org/wiki/Empiricism">sense experience</a> (John Locke and David Hume, for example) overlooked how the mind shapes perception. </p>
<hr>
<p>
<em>
<strong>
Read more:
<a href="https://theconversation.com/ai-is-closer-than-ever-to-passing-the-turing-test-for-intelligence-what-happens-when-it-does-214721">AI is closer than ever to passing the Turing test for ‘intelligence’. What happens when it does?</a>
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<p>Although Kant’s views were shaped by the <a href="https://books.google.com/books?hl=en&lr=&id=cBtOFUg4tHAC&oi=fnd&pg=PR11&dq=Kant+and+the+exact+sciences&ots=2Uu_qCDZT-&sig=WKqrfULN9w5qL6TfVc63PEyP5RQ">natural sciences of his day</a>, they have proved <a href="https://www.cell.com/trends/cognitive-sciences/fulltext/S1364-6613(10)00216-0">influential in contemporary psychology</a>. </p>
<p>The recognition that arithmetic is a <a href="https://theconversation.com/arithmetic-has-a-biological-origin-its-an-expression-in-symbols-of-the-deep-structure-of-our-perception-211337">natural consequence of our perception</a>, and thus biologically based, suggests a similar Kantian shift in our understanding of computation.</p>
<p>Computation is not “outside” or separate from us in an abstract realm of mathematical truth, but inherent in our mind’s nature. The mind is more than computation; the brain is not a computer. Rather, computation – the basis for AI – is, like mathematics, a symbolic expression of the mind’s nature and creativity. </p>
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<p>
<em>
<strong>
Read more:
<a href="https://theconversation.com/computer-says-no-more-employers-are-using-ai-to-recruit-increasing-the-risk-of-discrimination-218598">'Computer says no’: more employers are using AI to recruit, increasing the risk of discrimination</a>
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<h2>Promethean fire</h2>
<p>What are the implications for AI? Firstly, AI is not a mind and will never become sentient. The idea we can transcend our biological nature and achieve immortality by uploading our minds to the cloud is only <a href="https://medium.com/iva-to/on-achieving-immortality-3ed1d567f7a2">fantasy</a>. </p>
<p>Yet if the principles of mind on which AI is based are shared by all humanity (and likely other living creatures as well), it may be possible to transcend the limitations of our individual minds.</p>
<p>Because computation is universal, we are free to simulate and create any outcome we choose in our increasingly connected virtual and physical worlds. In this way, AI is truly our <a href="https://www.japantimes.co.jp/editorials/2023/11/10/ai-global-governance/">Promethean fire</a>, a gift to humanity stolen from the gods as in <a href="https://en.wikipedia.org/wiki/Prometheus">Greek mythology</a>. </p>
<p>As a global civilisation, we are likely at a turning point. AI will not become sentient and decide to <a href="https://newatlas.com/technology/ai-danger-kill-everyone/">kill us all</a>. But we are very capable of “apocalypsing” ourselves with it – expectation can create reality. </p>
<p>Efforts to ensure AI alignment, safety and security are vitally important, but may not be enough if we lack awareness and collective wisdom. Like Alice, we need to wake up from the dream and recognise the reality and power of our minds.</p><img src="https://counter.theconversation.com/content/219320/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Randolph Grace receives funding from the Royal Society Te Apārangi, Marsden Fund</span></em></p>AI will not become sentient and decide to kill us all. But our own conscious or unconscious beliefs about AI can potentially increase the likelihood of any outcome, including catastrophic ones.Randolph Grace, Professor of Psychology, University of CanterburyLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/2172472023-12-26T17:15:34Z2023-12-26T17:15:34ZHow counting by 10 helps children learn about the meaning of numbers<figure><img src="https://images.theconversation.com/files/564574/original/file-20231208-15-3eojg4.jpg?ixlib=rb-1.1.0&rect=0%2C242%2C4383%2C3017&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Using concrete tools or objects matters for fostering mathematical development – but how can children best learn to count by 10?
</span> <span class="attribution"><span class="source">(Shutterstock)</span></span></figcaption></figure><iframe style="width: 100%; height: 100px; border: none; position: relative; z-index: 1;" allowtransparency="" allow="clipboard-read; clipboard-write" src="https://narrations.ad-auris.com/widget/the-conversation-canada/how-counting-by-10-helps-children-learn-about-the-meaning-of-numbers" width="100%" height="400"></iframe>
<p>When children start school, they learn how to recite their numbers (“one, two, three…”) and how to write them (1, 2, 3…). Learning about what those numbers mean is even more challenging, and this becomes trickier yet when numbers have more than one digit — such as 42 and 608. </p>
<p>It turns out that the meaning of such “multidigit” numbers cannot be gleaned from simply looking at them or by performing calculations with them. Our number system has many hidden meanings that are not transparent, making it <a href="https://doi.org/10.1037/dev0001145">difficult for children</a> to comprehend it. </p>
<p>In collaboration with elementary teachers, the Mathematics Teaching and Learning Lab at <a href="https://www.concordia.ca/">Concordia University</a> explores tools that can support young children’s understanding of multidigit numbers.</p>
<p>We investigate the impact of using concrete objects (like bundling straws into groups of 10). We also investigate the use of visual tools, such as number lines and charts, or words to represent numbers (the word for 40 is “forty”) and written notation (for example, 42). </p>
<p>Our recent research examined whether the “hundreds chart” — 10 by 10 grids containing numbers from one to 100, with each row in the chart containing numbers in groups of 10 — could be useful for teaching children about counting by 10, something foundational for understanding how numbers work. </p>
<h2>What’s in a number?</h2>
<p>Most adults know that the placement of the “4” and “2” in 42 means four tens and two ones, respectively. </p>
<p>But when young children start learning about numbers, they do not naturally see 10s and ones in a number like 42. They think the number represents 42 things counted from one to 42 without distinguishing between the meaning of the digits “4” and “2.” Over time, through counting and other activities, children see the four as a <a href="https://doi.org/10.1177/1053451221994827">collection of 40 ones</a>. </p>
<p>This realization is not sufficient, however, for <a href="https://doi.org/10.1111/ssm.12258">learning more advanced topics</a> in math. </p>
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Read more:
<a href="https://theconversation.com/mathematical-thinking-begins-in-the-early-years-with-dialogue-and-real-world-exploration-128282">Mathematical thinking begins in the early years with dialogue and real-world exploration</a>
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<p>An important next step is to see that 42 is made up of four distinct groups of 10 and two ones, and that the four 10s can be counted as if they were ones (for example, 42 is one, two, three, four 10s and one, two, “ones”). </p>
<p>Ultimately, one of the most challenging aspects of understanding numbers is that groups of ten and ones are <a href="https://doi.org/10.17226/12519">different kinds of units</a>. </p>
<h2>Numbers can be arranged in different ways</h2>
<p>The numbers in hundreds charts can be arranged in different ways. A top-down hundreds chart has the digit “1” in the top-left corner and 100 in the bottom-right corner. </p>
<p>The numbers increase by 10 moving downward one row at a time, like going from 24 to 34 using one hop down, for instance. A second type of chart is the “bottom-up” chart, which has the numbers increasing in the opposite direction. </p>
<h2>Counting by 10s</h2>
<p>Children can move from one number to another in the chart to <a href="https://doi.org/10.5951/teacchilmath.24.3.00e1">solve problems</a>. Considering 24 + 20, for example, children could start on 24 and move 20 spaces to land on 44. </p>
<p>Another way would be to move up (or down, depending on the chart) two rows (for example, counting “one,” “two”) until they land on 44. This second method shows a developing understanding of multidigit numbers being composed of distinct groups of 10, which is critical for an advanced knowledge of the number system. </p>
<p>For her master’s research at Concordia University, Vera Wagner, one of the authors of this story, thought children might find it more intuitive to solve problems with the bottom-up chart, where the numbers get larger with upward movement. </p>
<p>After all, plants grow taller and liquid rises in a glass as it is filled. Because of such <a href="https://doi.org/10.1111/tops.12278">familiar experiences</a>, she thought children would move by tens more frequently in the bottom-up chart than in the top-down chart. </p>
<h2>Study with kindergarteners, Grade 1 students</h2>
<p>To examine this hypothesis, we worked with 47 kindergarten and first grade students in Canada and the United States. All the children but one spoke English at home. In addition to English, 14 also spoke French, four spoke Spanish, one spoke Russian, one spoke Arabic, one spoke Mandarin and one communicated to some extent in ASL at home. </p>
<p>We assigned all child participants in the study an online version of <a href="http://mathchart.ca/chart.html#nt">either a top-down</a> or <a href="https://mathchart.ca/chart.html#reversednt">bottom-up</a> hundreds chart, programmed by research assistant André Loiselle, to solve arithmetic word problems. </p>
<p><a href="https://doi.org/10.1111/ssm.12593">What we found surprised us</a>: children counted by tens more often with the top-down chart than the bottom-up one. This was the exact opposite of what we thought they might do!</p>
<p>This finding suggests that the top-down chart fosters children’s counting by tens as if they were ones (that is, up or down one row at a time), an important step in their mathematical development. Children using the bottom-up chart were more likely to confuse the digits and move in the wrong direction. </p>
<h2>Tools can impact learning</h2>
<p>Our research suggests that the types of tools used in the math classroom can impact children’s learning in <a href="https://doi.org/10.1016/j.learninstruc.2008.03.005">different ways</a>. </p>
<p>One advantage of the top-down chart could be the corresponding left-to-right and downward movement that matches the direction in which children learn to read in English and French, the official languages of instruction in the schools in our study. Children who learn to read in a different direction (for example, <a href="https://escholarship.org/uc/item/4tt0k00j">from right to left, as in Arabic</a>) may interact with some math tools differently from children whose first language is English or French. </p>
<p>The role of cultural experiences in math learning opens up questions about the design of teaching tools for the classroom, and the relevance <a href="https://theconversation.com/culturally-responsive-teaching-in-a-globalized-world-109881">of culturally responsive</a> mathematics teaching. Future research could seek to directly examine the relation between reading direction and the use of the hundreds chart.</p><img src="https://counter.theconversation.com/content/217247/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Helena Osana received funding from the Social Sciences and Humanities Research Council of Canada for this research. </span></em></p><p class="fine-print"><em><span>Jairo A. Navarrete-Ulloa receives funding from the National Agency for Research and Development (ANID) in Chile. </span></em></p><p class="fine-print"><em><span>Vera Wagner does not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>Findings of a study suggest using a ‘hundreds chart’ showing numbers one through 100, beginning with one in the top-left corner, fosters children’s counting by 10s.Helena Osana, Professor, Principal Investigator of the Mathematics Teaching and Learning Lab, Concordia UniversityJairo A. Navarrete-Ulloa, Adjunct assistant professor, Institute of Education Sciences, Universidad de O’Higgins (Chile)Vera Wagner, Research Assistant, Mathematics Teaching and Learning Lab, Concordia UniversityLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/2182502023-12-20T19:05:47Z2023-12-20T19:05:47ZHow many people need to be in a room for two to share a birthday? It’s fewer than you think. Here’s why<figure><img src="https://images.theconversation.com/files/560653/original/file-20231121-4807-e2cxhf.jpg?ixlib=rb-1.1.0&rect=0%2C0%2C3994%2C2000&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">A veridical paradox with practical uses. Cottonbro Studio/Pexels</span> <span class="attribution"><a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span></figcaption></figure><p>Have you ever bumped into someone with the same birthday as you? What about someone sharing a birthday in your workplace? How common is a shared birthday, anyway?</p>
<p>The <a href="https://www.scientificamerican.com/article/bring-science-home-probability-birthday-paradox/">birthday problem</a>, as it’s called by mathematicians, reveals problems with our understanding of number theory, probabilities and our assumptions of how the world works. It comes back to how counter-intuitive maths is for a lot of people.</p>
<p>In the birthday problem you are asked “what’s the minimum number of people in a room to get better than 50% chance of two people having the same birthday?” A simple question with a puzzling answer.</p>
<p>To get a more intuitive understanding of this problem we’ve created an interactive simulation, below. It arranges birthdays along a line, with January on the left and December on the right.</p>
<p><iframe id="tc-infographic-982" class="tc-infographic" height="400px" src="https://cdn.theconversation.com/infographics/982/2b47006750b0481a7781fede8a7655f3bf3b387a/site/index.html" width="100%" style="border: none" frameborder="0"></iframe></p>
<h2>The intuitive answer is the wrong one</h2>
<p>When the birthday problem is described to maths students for the first time, the majority of responses are that a group of 183 people is needed to have a better than even chance of two people having the same birthday.</p>
<p>The thinking here is: 183 is half of 365 (number of days in the year). Students assume they only need to compare others against a single person – themselves, and then <a href="https://betterexplained.com/articles/understanding-the-birthday-paradox/">try to match their birthday</a> with other people.</p>
<p>If you use this assumption, you need to find 183 people to have an even chance of finding a person matching with you. However, when students understand that not every combination has to be with yourself – for example, person 2 and person 5 might be the right combination – it becomes clearer the number needed is lower than 183.</p>
<h2>Combinations do not scale linearly</h2>
<p>If you’ve been playing with the interactive above you may have come across the answer to the birthday problem: only about 23 people are needed for a greater than 50% chance of a shared birthday. But how can this be if there are 15 times more days in the year?</p>
<p>We’ve created another interactive below to visualise how the connections between people in a room do not scale linearly as you add more people. Play around with adding a node and see if you can guess how many connections should be added.</p>
<p>It should give you a better grasp of factorial growth through multiplication, which is the area of number theory that underpins the birthday problem.</p>
<p><iframe id="tc-infographic-995" class="tc-infographic" height="400px" src="https://cdn.theconversation.com/infographics/995/9c3076a412616df132bae72eb5e4448ad50520ed/site/index.html" width="100%" style="border: none" frameborder="0"></iframe></p>
<h2>Large numbers are hard to comprehend</h2>
<p>The COVID-19 pandemic showed the world most of us have a limited understanding of exponential growth when presented with models of what could occur if the pandemic were left unchecked. </p>
<p>There are many great examples of the ill-understood power of exponential growth, but one I often use is asking this question: would you take $1 million on the first day of the month only, or one cent on the first day of the month, doubled each day until the end of the month (30 days)?</p>
<p>Nearly all people choose the $1 million lump sum. However, if you choose the one cent option, you end the month with approximately 10 times more money due to the effect of exponential growth.</p>
<p><div data-react-class="TiktokEmbed" data-react-props="{"url":"https://www.tiktok.com/@axiomalpha/video/7056421079149202734"}"></div></p>
<p>In a <a href="https://www.forbes.com/sites/alexknapp/2011/11/17/the-seduction-of-the-exponential-curve/?sh=52448f7d2480">similar tale</a>, the supposed inventor of chess requested to sell their game to a king for some rice. They proposed placing a single grain on the first tile and doubling it each tile. The amount of rice on the final tile would be the sale price.</p>
<p>As you can see below, that number can very quickly become larger than the entire world’s rice supply.</p>
<p><iframe id="tc-infographic-1004" class="tc-infographic" height="400px" src="https://cdn.theconversation.com/infographics/1004/bc0ac6dafe6f7260f7180b9cfb39a15b1c270f70/site/index.html" width="100%" style="border: none" frameborder="0"></iframe></p>
<h2>Asking ‘what comes next’ is seldom simple</h2>
<p>Another real world example of counter-intuitive mathematics was on the <a href="https://www.elderresearch.com/blog/gamblers-fallacy/">roulette wheels in Monte Carlo in 1913</a>. There was a run of 26 straight black results, which is improbable but not impossible. </p>
<p>One striking aspect of the story is that gamblers increasingly bet on red as the run of blacks continued, thinking red was “due”. However, the mathematics says otherwise. Each spin of the roulette wheel has no memory of what happened before, so the chance of red appearing does not increase as time goes on. In short, lots of people lost money that night!</p>
<p>However, some situations do rely on the probabilities of what came before. For example, in the Monty Hall problem you are trying to win a car, which is hidden behind one of three doors (the others have goats). You are given the option to pick a door, then shown a different door with a goat.</p>
<p>The question is: should you stay with your first pick or switch doors? Try it out with the game below.</p>
<p><iframe id="tc-infographic-1000" class="tc-infographic" height="400px" src="https://cdn.theconversation.com/infographics/1000/c55c43cd1b856a9237b1add3d624f68fad7bdcc1/site/index.html" width="100%" style="border: none" frameborder="0"></iframe></p>
<p>The assumption here is that after Monty shows you the goat then the odds of winning a car if you switch or stay are 50/50 – there are only two doors left and one of them has a car.</p>
<p>But that doesn’t take into account the likelihood that you originally picked a goat to begin with. Thus the chances of your next choice are informed by your previous choice.</p>
<h2>Becoming numerate is important</h2>
<p>These examples demonstrate the importance of being “numerate”, defined as <a href="http://www.ncesd.org/wp-content/uploads/2017/03/WaKids.preschool-box-powerpoint.pdf">being able to reason with numbers</a> and being able to apply this reasoning in a range of contexts. The importance of developed numeracy skills cannot be understated, with correlations to <a href="https://www.acer.org/au/discover/article/the-importance-of-measuring-adult-literacy-and-numeracy">better overall life outcomes</a> such as employment, income, health and well being.</p>
<p>If people are highly numerate, they can understand how our world works at a deeper level, even if it doesn’t feel like it should work that way. Also, they are likely to have a better idea of what actions will yield the desired results in certain situations.</p>
<p>So, keep turning your mind to mathematical and numerical problems, they may just come in handy one day.</p><img src="https://counter.theconversation.com/content/218250/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Ben Zunica does not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>The logic of shared birthdays is stranger than you might think, but learning to understand counter-intuitive mathematics is key to seeing the world clearly.Ben Zunica, Lecturer in Secondary Maths Education, University of SydneyLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/2186522023-12-18T12:08:43Z2023-12-18T12:08:43ZRishi Sunak wants more maths at school – but finding the teachers will be hard when university departments are closing<figure><img src="https://images.theconversation.com/files/565256/original/file-20231212-29-i8mtw0.jpg?ixlib=rb-1.1.0&rect=0%2C0%2C6359%2C5774&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">
</span> <span class="attribution"><a class="source" href="https://www.shutterstock.com/image-photo/close-female-student-writing-equation-on-1131701174">Jacob Lund/Shutterstock</a></span></figcaption></figure><p>Oxford Brookes University recently <a href="https://www.timeshighereducation.com/news/music-and-maths-courses-close-part-oxford-brookes-cuts">announced</a> it will no longer be offering mathematics degrees. This follows reported <a href="https://committees.parliament.uk/writtenevidence/120416/pdf/">reductions or proposed cuts</a> at other universities. </p>
<p>This is a problem for Prime Minister Rishi Sunak’s vision for improved maths skills across the nation. Sunak has laid out a vision for young people to study <a href="https://www.gov.uk/government/news/prime-minister-outlines-his-vision-for-maths-to-18">maths to age 18</a>. The goal is to ensure that “every young person has the maths skills they need to succeed”.</p>
<p>This focus on maths was also evident in chancellor Jeremy Hunt’s 2023 autumn statement. This included funding for a <a href="https://rss.org.uk/news-publication/news-publications/2023/general-news/government-funding-announced-for-an-academy-for-ma/">national academy of mathematical sciences</a> to build links between mathematicians in education, academia, industry and government.</p>
<p>But the success of the prime minister’s vision, and the health of mathematics education more generally, rests largely on specialist maths teachers with a mathematics degree. These teachers are needed to educate young people in maths up to the age of 18. They teach the maths skills young people need to go on to study the subject further and use it in their future jobs. </p>
<p>But there is a <a href="https://www.theguardian.com/education/2023/apr/17/shortage-of-teachers-will-be-a-big-maths-problem-for-rishi-sunak">chronic shortage</a> of specialist mathematics teachers in schools – and the university maths education that trains these teachers is under threat. </p>
<h2>University maths under threat</h2>
<p>The cuts in maths teaching at universities has resulted in a new phenomenon: <a href="https://www.protectpuremaths.uk/news/broken-promise-on-maths-puts-science-plans-in-peril">maths deserts</a>. The closing of maths departments at universities that often serve their local population means that many, not just aspiring maths teachers, can no longer study mathematics beyond A-level in their local region. </p>
<p>Students with lower A-level results or from low-income families who are more likely to be living at home while studying at their local university, are disproportionately affected by maths deserts. And it creates a negative feedback loop that sees diminishing numbers going into maths teaching. This further erodes secondary schools’ ability to provide high-quality mathematics education.</p>
<p>Past president of the London Mathematical Society Ulrike Tillmann <a href="https://www.timeshighereducation.com/blog/uk-cannot-afford-cut-m-stem">warned that maths deserts</a> will turn post A-level mathematics education into “an almost exclusively high-tariff, big-city degree” essentially concentrated at large Russell Group universities. Moreover, graduates with degrees like this are less likely to pursue <a href="https://theconversation.com/why-are-so-many-graduates-shunning-teaching-pay-but-not-bonuses-could-be-the-answer-216963">teaching as a career</a>. </p>
<h2>Protecting the discipline of mathematics</h2>
<p>The importance of mathematics cannot be overstated, both now and for the future. Mathematics underpins almost all technological development in society, from cryptography and information security through to artificial intelligence (AI) and quantum computing. </p>
<p>There is a <a href="https://www.mirror.co.uk/news/uk-news/fight-save-pure-maths-only-24697612">trend for universities</a> across the world to offer degrees in data science and AI. But a rigorous grounding in mathematics is required to ensure graduates are best equipped to meet future challenges in areas such as quantum computing or AI. This means that mathematics still needs to be taught as a discipline on its own, rather than being subsumed into seemingly more “job-ready” disciplines.</p>
<figure class="align-center ">
<img alt="Workbook showing equations to solve" src="https://images.theconversation.com/files/565266/original/file-20231212-15-deb5vi.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/565266/original/file-20231212-15-deb5vi.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/565266/original/file-20231212-15-deb5vi.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/565266/original/file-20231212-15-deb5vi.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/565266/original/file-20231212-15-deb5vi.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/565266/original/file-20231212-15-deb5vi.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/565266/original/file-20231212-15-deb5vi.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=503&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Good teaching of maths at school needs specialist maths teachers.</span>
<span class="attribution"><a class="source" href="https://www.shutterstock.com/image-photo/handwriting-mathematics-quadratic-equation-on-examination-2154251415">Mehaniq/Shutterstock</a></span>
</figcaption>
</figure>
<p>The same point applies to training future maths teachers. There is a world of difference between having a specialist teacher who loves the discipline of mathematics and is passionate about communicating it to their students, and one simply teaching out of a textbook. The former is crucial in ensuring a new generation of students go on to become excellent maths teachers and inspire future generations.</p>
<p>We need to equip our young people to manage the challenges of a rapidly shifting world. If we are to tackle challenges ranging from climate change to the explosion of AI in society and environmental resource management, then a rigorous education in mathematics – the subject <a href="https://www.theguardian.com/science/2021/jul/11/pure-folly-turing-family-join-fight-to-save-blue-skies-maths-from-neglect">Alan Turing considered</a> a combination of intuition and ingenuity – is essential.</p><img src="https://counter.theconversation.com/content/218652/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Neil Saunders does not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>University maths departments are shrinking or closing.Neil Saunders, Senior Lecturer in Mathematics, University of GreenwichLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/2196462023-12-13T13:31:09Z2023-12-13T13:31:09ZWales’s Pisa school test results have declined – but it’s not a true reflection of an education system<p>Every three years, an early Christmas gift arrives for the global education community from the Organisation for Economic Co-operation and Development (OECD). The Programme for International Student Assessments (<a href="https://www.oecd.org/pisa/">Pisa</a>) is an international test in which 15 year olds are tested on their knowledge and skills. </p>
<p>It relegates those far below the Pisa top ten as poor performers in desperate need of improvement, which this time includes Wales.</p>
<p>The Pisa scores for participating education systems around the world are unquestionably significant. But since its inception in 2000, Pisa has sparked much debate, especially among experts and policymakers, with many viewing it as a <a href="https://ffteducationdatalab.org.uk/2019/11/is-pisa-fundamentally-flawed-because-of-the-scaling-methodology-used/">flawed</a> assessment of educational outcomes. In <a href="https://www.oecd.org/pisa/Combined_Executive_Summaries_PISA_2018.pdf">2018</a>, around 600,000 students took part in the standardised Pisa tests, which measured their performance in maths, science and reading, and also looked at wellbeing.</p>
<p>Predictably, the 2023 Pisa results captured the negative impact of COVID on learners and learning, with some downward trends in performance visible across the data set. </p>
<p>The results signalled mixed fortunes for the <a href="https://www.education.ox.ac.uk/pisa-results-reveal-students-in-the-uk-have-higher-than-average-levels-of-maths-reading-and-science/">UK</a>. The <a href="https://www.bbc.co.uk/news/uk-wales-67616536">BBC headline</a>, reporting the results starkly stated: “Wales slumps to worst school test results.” Such sweeping statements are by now an anticipated byproduct of Pisa that ignore how the tests are often highly contested and <a href="https://revisesociology.com/2020/05/15/the-pisa-global-education-tests-arguments-for-and-against/">controversial</a>. </p>
<h2>Pisa in Wales</h2>
<p>Every three years, Pisa measures the ability of 15 year olds to use their reading, mathematics and science knowledge and skills to meet real-life challenges. In Wales, 2,568 pupils from 89 schools took a two-hour computer-based exercise. To put this in context, there are <a href="https://statswales.gov.wales/Catalogue/Education-and-Skills/Schools-and-Teachers/Schools-Census/Pupil-Level-Annual-School-Census/Pupils/pupils-by-yeargroup-sex">approximately</a> 33,000 pupils in Year 11 in 178 secondary schools and 27 middle schools in Wales.</p>
<p>The subsequent OECD report <a href="https://www.gov.wales/achievement-15-year-olds-program-international-student-assessment-pisa-national-report-2022">acknowledges</a> that “the sample for Wales, and for many other countries, did not meet some of the Pisa standards”. It is important to reflect on how a test taken by a sample of 15-year-old students, every three years for two hours, can possibly be a valid and reliable measure of a system’s performance even in a relative sense. </p>
<p>Pisa’s statistics show that Wales’s average score for mathematics in 2022 was significantly lower than the average across OECD countries. Wales’s average scores for mathematics, reading and science have all declined significantly since 2018. This was also the case, on average, across OECD countries for mathematics and reading. Although for science, the difference between the OECD average in 2022 was not significantly different to that in 2018. </p>
<p>It also noted that the gap in performance between pupils from the most disadvantaged backgrounds and the least disadvantaged backgrounds was smaller in Wales than it was on average across OECD countries for all subjects. </p>
<p>The important thing to observe is that Pisa deals in averages. In the latest results, those averages are derived from the 81 countries that took part, which is a huge range. The report notes a relative fall in Welsh performance against an aggregated average of OECD countries. </p>
<p>It then highlights that this decline was also the case on average across OECD countries for mathematics and reading. In other words, this is a trend. It also suggests that Wales has been more successful in closing the achievement gap between the most disadvantaged and least disadvantaged pupils than most other OECD countries. Yet this important indicator of success has been overshadowed by the blanket headlines of abject educational failure.</p>
<p>If all countries participating in Pisa now recover fully after COVID and improve their educational performance across the board, it is highly likely that Wales will “underperform” on Pisa yet again, whatever it does. If all countries in Pisa continue a steady trajectory of improvement, the country differentials will remain largely the same. Some countries may move up or down, but that movement will be marginal. </p>
<h2>A game of relatives</h2>
<p>There were no real surprises in the latest results. Countries like Singapore, Taiwan and Japan have retained their comparative advantage and will probably continue to do so, because Pisa is a game of relatives. The complexity and dynamic of any education system cannot (and should not) be at the mercy of a single measure of assessment however compelling or lucrative. </p>
<p>But what does that mean for Wales? First, it should encourage us to look at Pisa as one data set only and to not be obsessed by its findings. It is important to put Pisa in perspective by looking far beyond the simple headlines and delving into the detail of the report. </p>
<p>Second, it should be a reminder that Pisa is a snapshot of performance at a particular moment in time. It takes no account of the possibilities and potential of ongoing reform – deep contextual detail is not on its global radar. Hence, the danger is that every three years, Pisa fuels doubt, dissent and concern, when education systems need certainty, confidence and consensus about the reforms they are putting in place.</p>
<p>Third, no education system is perfect. Getting great Pisa scores is certainly no guarantee that the wellbeing and mental health of children and young people is not compromised or sidestepped along the way. The potential of human collateral damage in achieving high Pisa performance over two decades, unsurprisingly, does not feature in the OECD reports. </p>
<p>Wales has a choice, to either let this global compass direct its educational pathway, accepting that every three years it will derail and disrupt the reform agenda, or to hold its nerve. We cannot ignore Pisa, but we can put it in perspective and continue to focus on the learning and wellbeing of all children and young people in Wales. This is what matters most.</p><img src="https://counter.theconversation.com/content/219646/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Alma Harris does not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>Pisa measures 15-year-olds in reading, maths and science every three years - but is that the best way to test an education system?Alma Harris, Professor of Leadership in Education, Cardiff Metropolitan UniversityLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/2188022023-12-13T01:25:02Z2023-12-13T01:25:02Z20 people, 2.4 quintillion possibilities: the baffling statistics of Secret Santa<figure><img src="https://images.theconversation.com/files/565106/original/file-20231212-29-ba551n.jpeg?ixlib=rb-1.1.0&rect=87%2C21%2C7196%2C4739&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">
</span> <span class="attribution"><a class="source" href="https://www.shutterstock.com/image-photo/secret-santa-smiling-man-unpacks-christmas-2378052273">Harbucks / Shutterstock</a></span></figcaption></figure><p>Christmas, we’re told, is the <a href="https://www.youtube.com/watch?v=AN_R4pR1hck&ab_channel=AndyWilliamsVEVO">most wonderful time of the year</a>. For many of us, however, it is preceded by one of the least wonderful times: the awkward social spectacle of the office <a href="https://en.wikipedia.org/wiki/Secret_Santa">Secret Santa</a> or Kris Kringle, where employees agree to purchase a gift for a randomly allocated colleague.</p>
<p>As you watch your co-workers unwrap their often wildly inappropriate gifts, each chosen by a office mate they barely know, cast your mind to the sheer statistical improbability of what you’re seeing. The odds of such a combination of these cheaply re-gifted photograph frames, inexplicably scented candles or unwanted <a href="https://metro.co.uk/2020/12/25/how-do-the-makers-of-lynx-feel-about-being-a-christmas-meme-13776264/">Lynx Africa gift sets</a> being passed around your office is, in its own way, truly a Christmas miracle.</p>
<h2>The 12! ways of Christmas?</h2>
<p>To work out how many possible pairings of buyers and recipients there are, you need to calculate the <a href="https://brilliant.org/wiki/permutations/">number of permutations</a> of the people involved.</p>
<p>Consider a workplace with four employees. If there is no rule to prevent people selecting their own names, there are four people who could be selected to buy the first person’s gift. </p>
<p>Once this is decided, there are three remaining choices for the second person, then two choices for the third person. Finally, there is one choice for the last person’s workplace Santa.</p>
<p>This means there are 4 × 3 × 2 × 1 = 24 possible permutations. Mathematicians write this as 4!, which is pronounced “four factorial”.</p>
<p><iframe id="tc-infographic-1006" class="tc-infographic" height="400px" src="https://cdn.theconversation.com/infographics/1006/695db8d722e320a72096762eabdfe31099bb09c6/site/index.html" width="100%" style="border: none" frameborder="0"></iframe></p>
<p>However, factorials soon get out of hand. Spare a thought for poor Santa himself. With <a href="https://www.saturdayeveningpost.com/2019/12/in-a-word-eight-er-nine-tiny-reindeer/">nine reindeer</a>, there are 9! = 362,880 ways these could be arranged, although perhaps <a href="https://en.wikipedia.org/wiki/Rudolph_the_Red-Nosed_Reindeer">on one foggy Christmas Eve</a>, this number is reduced by the requirement to have a red nose leading his sleigh.</p>
<p>Once the office workforce swells to 20, there are more than 2.4 quintillion permutations. To put this mind-boggling 20! figure into context, that’s more than three times current estimates of <a href="https://www.oklahoman.com/story/lifestyle/2019/02/05/more-stars-than-grains-of-sand-on-earth-you-bet/60474645007/">the number of grains of sand on Earth</a>.</p>
<h2>Yule buy for someone else</h2>
<p>Of course, nobody wants to draw themselves in a Secret Santa.</p>
<p>What a Secret Santa really wants is not a permutation of all employees, but instead what mathematicians call a <a href="https://brilliant.org/wiki/derangements/">derangement</a>. This is simply a permutation where no element remains in its original position, which means no employee has to buy their own gift.</p>
<p>The calculation is <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Hassani/hassani5.pdf">far from simple</a>, but the number of ways <em>n</em> employees can be assigned another unique co-worker is called the <em>n</em> th <a href="https://www.sciencedirect.com/science/article/abs/pii/S0096300322004155">de Montmort number</a>.</p>
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Read more:
<a href="https://theconversation.com/the-mathematics-of-christmas-a-review-of-the-indisputable-existence-of-santa-claus-88508">The mathematics of Christmas: A review of the Indisputable Existence of Santa Claus</a>
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<p>Amazingly, this is equal to <em>n</em>!/<em>e</em> , rounded to the nearest whole number. The <em>e</em> here is one of the most famous numbers in mathematics, <a href="https://www.investopedia.com/terms/e/eulers-constant.asp">Euler’s number, approximately equal to 2.71828</a>, and the bane of anyone whose schooldays involved logarithm tables.</p>
<p>In the 24 permutations of four employees illustrated, there are 9 derangements, which is equal to 24/<em>e</em> rounded to the nearest whole number. For large numbers, approximately 63.2% of possible permutations are not derangements and so would be excluded.</p>
<p>For a 20-employee situation, this cuts the over 2.4 quintillion permutations to a mere 895 quadrillion or so. (This is still more than 100 million times the current global population.)</p>
<h2>Uniquely self-Santa-ed?</h2>
<p>Another surprising feature of a Secret Santa arises from the number of people who will, on average, be assigned their own name in a random draw.
It doesn’t matter if you have one person (although that is a terribly un-secret and desperately sad Secret Santa) or a billion people, the expected number of people to be allocated to buy their own gift is the same – just one person.</p>
<p>A full proof is a little more complicated than this, but think what happens if you double the number of employees. With twice as many gifts to buy, everybody’s chance of selecting themselves is halved. Twice as many people, each with half the chance of matching, then gives an unchanged average.</p>
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Read more:
<a href="https://theconversation.com/how-to-play-and-win-the-gift-stealing-game-bad-santa-according-to-a-mathematician-196483">How to play and win the gift-stealing game Bad Santa, according to a mathematician</a>
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<p>For example, of the 24 permutations of four people illustrated, one involves four self-matches, none involve three self-matches, six involve two self matches and eight involve a single self-match. In total, this gives 24 possible self-matches in the 24 permutations, so an average of one each.</p>
<h2>Ho Ho Hope for the best</h2>
<p>If you do find yourself trapped in the dystopian office whodunit of guessing which of your co-workers gifted a <a href="https://www.usmagazine.com/celebrity-news/pictures/australia-firefighters-pose-with-cute-animals-for-sexy-2024-calendar/">hunky shirtless firefighter calendar</a> to an elderly colleague from human resources, at least hope that the one-in-a-billion or one-in-trillion permutation that was drawn in your office lands you something useful.</p>
<p>Mariah Carey may have <a href="https://en.wikipedia.org/wiki/All_I_Want_for_Christmas_Is_You">assured us of more specific requests</a>, but all I want for Christmas is avoid getting dragged into a workplace Secret Santa in the first place.</p>
<p><a href="https://www.historyextra.com/period/victorian/why-we-say-bah-humbug-christmas-carol-scrooge-dickens-what-mean/">Bah humbug</a> indeed.</p><img src="https://counter.theconversation.com/content/218802/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Stephen Woodcock does not receive funding from Santa Claus or any other relevant external party. He has remained off Santa's Naughty List for over 40 years.</span></em></p>The annual ritual of buying a gift for a random recipient opens a window onto some fascinating mathematics.Stephen Woodcock, Associate Professor of Mathematical Sciences, University of Technology SydneyLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/2191672023-12-05T17:24:38Z2023-12-05T17:24:38ZLow PISA math scores post-pandemic: Policies need to consider both academic excellence and equity<iframe style="width: 100%; height: 100px; border: none; position: relative; z-index: 1;" allowtransparency="" allow="clipboard-read; clipboard-write" src="https://narrations.ad-auris.com/widget/the-conversation-canada/low-pisa-math-scores-post-pandemic-policies-need-to-consider-both-academic-excellence-and-equity" width="100%" height="400"></iframe>
<p>There will be a collective gasp as governments and policymakers <a href="https://www.theglobeandmail.com/canada/article-math-scores-of-canadian-students-are-declining-raising-concern-about/?login=true">reckon with drops in 15 year olds’ math skills</a> in the wake of the pandemic, as captured by <a href="https://www.oecd.org/pisa/">the OECD’s release</a> of the Program for International Student Assessment (PISA) 2022 results.</p>
<p>The global results capture mathematics skills based on 2022 testing. PISA typically tests math (as well as science and literacy skills) every three years, but a <a href="https://www.oecd.org/publications/new-pisa-results-strengthening-education-systems-in-the-wake-of-the-pandemic-62fc50a3-en.htm#">2021 test was postponed</a> because of the pandemic. </p>
<p>Governments <a href="https://www.theguardian.com/education/2021/jan/28/primary-schools-pandemic-causing-significant-learning-loss-england">anticipated there would be</a> a drop in test scores due <a href="https://www.cbc.ca/news/canada/toronto/new-supports-students-math-reading-writing-1.6812345">to COVID-19 disruption</a>. But few would have predicted such significant learning losses. </p>
<p>Canada was not immune from the learning challenges caused by the pandemic. Canada’s results indicate a significant drop of <a href="https://gpseducation.oecd.org/CountryProfile?primaryCountry=CAN&treshold=10&topic=PI#:%7E:text=In%20reading%20literacy%2C%20the%20main,30%20points%20higher%20for%20girls">15 score points</a> in mathematics — a score of 497, down from <a href="https://gpseducation.oecd.org/CountryProfile?primaryCountry=CAN&treshold=10&topic=PI#">512 in 2018</a>. Since PISA was first administered in 2000, Canada has <a href="https://gpseducation.oecd.org/CountryProfile?primaryCountry=CAN&treshold=10&topic=PI#:%7E:text=In%20reading%20literacy%2C%20the%20main,30%20points%20higher%20for%20girls">never experienced a drop of 15 score points in any area</a> as has happened this year.</p>
<p>As provinces across Canada take stock of test scores, and likely face weaponization of these scores by those seeking <a href="https://doi.org/10.1093/oxfordhb/9780197570685.013.9">to gain political points</a>, both policymakers and the public need to know there’s no quick fix. To address academic learning, a multi-dimensional approach is needed that promotes the success of the whole child: academic, physical and socio-emotional.</p>
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Read more:
<a href="https://theconversation.com/student-achievement-depends-on-reducing-poverty-now-and-after-covid-19-153523">Student achievement depends on reducing poverty now and after COVID-19</a>
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<h2>Significant drop in mathematics scores</h2>
<p>Canada’s math scores on the <a href="https://www.oecd.org/pisa/publications/PISA2018_CN_CAN.pdf">last three administrations of PISA</a> have been fairly stable, <a href="https://oecdch.art/a40de1dbaf/C903">albeit showing</a> a slight <a href="https://www.oecd.org/publication/pisa-2022-results/country-notes/canada-901942bb#chapter-d1e11">downward performance trend</a>.</p>
<p>Still, Canadian students have consistently performed <a href="https://www.cmec.ca/Publications/Lists/Publications/Attachments/396/PISA2018_PublicReport_EN.pdf">above the OECD average as the top English speaking jurisdiction in mathematics, science and reading achievement</a>. While Canada is still a top-performing nation and these drops closely align with <a href="https://www.oecd.org/pisa/PISA%202022%20Insights%20and%20Interpretations.pdf">average OECD performance declines</a>, they are sure to provoke calls for system reform, given the trajectory of the decline.</p>
<p>What should governments do (and not do) to address significant performance declines?</p>
<figure class="align-center ">
<img alt="Students seated at desks writing tests." src="https://images.theconversation.com/files/563470/original/file-20231204-21-n52i6y.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/563470/original/file-20231204-21-n52i6y.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/563470/original/file-20231204-21-n52i6y.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/563470/original/file-20231204-21-n52i6y.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/563470/original/file-20231204-21-n52i6y.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/563470/original/file-20231204-21-n52i6y.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/563470/original/file-20231204-21-n52i6y.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=503&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
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<span class="caption">Making up two years of lost learning is a daunting challenge.</span>
<span class="attribution"><span class="source">(Shutterstock)</span></span>
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<h2>Education policies and academic learning loss</h2>
<p><a href="https://orcid.org/0000-0002-8723-6774">My research with colleagues</a> suggests educational policymakers in Canadian provinces have taken a number of important steps to address challenges created by the pandemic. </p>
<p>No less than 62 policy documents and related supports were developed and issued across Canada’s 10 provinces from January 2020 to December 2021. Academic supports tended to focus on maintaining continuity of learning, synchronous learning during school closures and, finally, recovery catch-up strategies.</p>
<p><a href="https://www.edcan.ca/articles/covid-19-and-the-learning-loss-dilemma/">Catch-up policies</a> need to consider students’ mental and physical health — domains largely untested by large-scale assessment programs, but equally important.</p>
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Read more:
<a href="https://theconversation.com/pause-pisa-international-standardized-student-testing-its-been-two-years-of-pandemic-schooling-stress-179945">Pause PISA international standardized student testing — it's been two years of pandemic schooling stress</a>
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<h2>Resist temptation to narrow curriculum</h2>
<p>Similarly, provincial policymakers must resist the temptation to narrow the curriculum to focus on the mathematics domain at the expense of other subjects. Narrowing the curriculum often accompanies significant test score drops, which unfortunately contribute to <a href="https://theconversation.com/high-school-dropouts-cost-countries-a-staggering-amount-of-money-115396">school failure</a> and negatively impact countries’ future economic prosperity.</p>
<p>Clearly, a generational challenge like COVID-19 requires a multi-year approach that takes a long-view based on the best available evidence. And yet, PISA results have routinely been <a href="https://doi.org/10.1093/oxfordhb/9780197570685.013.9">politicized by policymakers</a> globally, including in Canada. </p>
<p>Yes, it is early days, but if the past is a good predictor, a host of education reforms such as greater privatization and <a href="https://www.ourkids.net/blog/public-schools-fraser-institute-9075">school choice will likely be offered as a remedy for our “failing schools.</a>”</p>
<p>Collectively, these types of reforms, <a href="https://theconversation.com/what-exactly-is-neoliberalism-84755">modelled on notions that the state should promote markets and competition</a> to meet social needs, have consistently shown their negative effects around the world.</p>
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Read more:
<a href="https://theconversation.com/school-choice-policies-are-associated-with-increased-separation-of-students-by-social-class-149902">'School choice' policies are associated with increased separation of students by social class</a>
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<h2>Moving forward in a post-COVID world</h2>
<p>Education is a social science concerned <a href="https://www.britannica.com/topic/social-science">with human behaviour in its social and cultural aspects</a>. What works in one context isn’t guaranteed to work in another. Looking to countries that perform high in PISA math performance like Singapore (and other Asian nations) to <a href="https://doi.org/10.1007/s10833-019-09367-x">borrow policies</a> and strategies in the hopes of emulating their success, is naïve. </p>
<p>Success in a post-COVID world will depend on local innovation and an ability to address the unique challenges of Canada’s <a href="https://www150.statcan.gc.ca/n1/daily-quotidien/221026/dq221026b-eng.htm">ethnically and linguistically diverse population</a>. Indeed, the share of immigrant students in Canada increased <a href="https://gpseducation.oecd.org/CountryProfile?primaryCountry=CAN&treshold=10&topic=PI#:%7E:text=In%20reading%20literacy%2C%20the%20main,30%20points%20higher%20for%20girls">to 34 per cent in 2022</a>.</p>
<p>Although Canada can and should consider the efficacy of education policies in similar international education contexts, it also possesses unique educational governance structures. Provincial autonomy in administering education means provinces can learn from each other as much as from other countries. </p>
<p>In many instances, performance variations are larger <a href="https://www.cmec.ca/Publications/Lists/Publications/Attachments/396/PISA2018_PublicReport_EN.pdf">across Canadian provinces</a>, than between Canada and other high-performing nations.</p>
<p>For example, the difference between Saskachewan and Québec’s PISA math scores is 46 points, approximately one-and-a-half grade levels, with the former at <a href="https://a.storyblok.com/f/81332/x/ad9bf10292/oecd_pisa_2022_results-volume-i_53f23881-en.pdf">the lower and the latter at the higher end</a>. A range of education policy and other factors, including the <a href="https://policyalternatives.ca/sites/default/files/uploads/publications/Saskatchewan%20Office/2019/02/Failing%20Report%20Card%20on%20Child%20Care%20(01-23-19).pdf">availability of early childhood education</a>, have been and should continue to be explored to account for such differences.</p>
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Read more:
<a href="https://theconversation.com/generation-c-why-investing-in-early-childhood-is-critical-after-covid-19-157095">'Generation C': Why investing in early childhood is critical after COVID-19</a>
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<h2>Push for online learning</h2>
<p>It would also be naive to expect technology to offer easy solutions to enhancing students’ access to quality education. </p>
<p>For example, in Ontario, a shift to online learning and less in-class hours, undoubtedly <a href="https://covid19-sciencetable.ca/sciencebrief/covid-19-and-education-disruption-in-ontario-emerging-evidence-on-impacts/">contributed to pandemic learning losses</a>. Yet online learning is now <a href="https://www.ontario.ca/document/education-ontario-policy-and-program-direction/policyprogram-memorandum-167">being offered as a strategy to improve education</a> — <a href="https://www.cbc.ca/news/canada/toronto/high-school-students-mandatory-online-courses-graduation-1.5368305">as it was also prior</a> to the pandemic. </p>
<p>This is despite the fact <a href="https://www.cbc.ca/news/canada/learning-curve-transitions-1.6482719">that students</a> and <a href="https://www.cbc.ca/news/canada/toronto/fed-up-parents-call-for-boycott-of-online-classes-as-province-shifts-to-virtual-learning-to-fight-covid-19-1.6304352">parents have</a> both voiced their concerns with online education. </p>
<p>Equally important is <a href="https://doi.org/10.1080/10494820.2020.1789672">that research</a> highlights the social challenges associated with a reliance on online education.</p>
<h2>Two pillars of success: excellence and equity</h2>
<p>Ultimately, Canadian policymakers should be judged on their ability to promote academic resilience in a post-COVID world that is both <a href="https://doi.org/10.1177/14782103211069002">multi-dimensional</a> and also attentive to marginalized student groups, particularly <a href="https://link.springer.com/book/10.1007/978-981-13-9863-6">those from lower socio-economic backgrounds</a>. </p>
<p>Policies designed to promote high achievement must also carefully consider the success of students living in poverty. The <a href="https://www.routledge.com/Equity-and-Excellence-in-Education-Towards-Maximal-Learning-Opportunities/Branden-Avermaet-Houtte/p/book/9780415847452">best-performing education systems</a> embody both of these characteristics, and PISA scores represent only one piece of a complex puzzle to help spur system reform. </p>
<p>The countries that equally attend to these two pillars, excellence and equity, will be the leaders of the future.</p><img src="https://counter.theconversation.com/content/219167/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Louis Volante receives funding from the Social Sciences and Humanities Research Council of Canada (SSHRC).</span></em></p>To boost post-pandemic math learning, a multi-dimensional approach is needed that promotes the success of the whole child: academic, physical and socio-emotional.Louis Volante, Professor, Brock UniversityLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/2188142023-12-05T10:01:35Z2023-12-05T10:01:35ZAustralian teenagers record steady results in international tests, but about half are not meeting proficiency standards<p>Australian high school students have achieved steady results in a new round of international tests. </p>
<p>The latest Programme for International Student Assessment (PISA) results, <a href="https://www.acer.org/au/pisa?utm_source=acer%20homepage&utm_medium=web&utm_campaign=feature%20box">released on Tuesday night</a>, show 15-year-olds have recorded similar results to 2015 and 2018 in mathematics, science and reading. </p>
<p>But when looked at over the past 20 years, Australian students’ performance has dropped significantly. PISA also shows about one in eight Australian students is a “high performer”, while one in every four or five is a “low performer”. </p>
<h2>What is PISA?</h2>
<p>PISA is an international test of 15-year-olds’ knowledge and skills as they near the end of compulsory education.</p>
<p>It looks at maths, science and reading. In 2022, for the first time, it also assessed creative thinking. The creative thinking results will be released in 2024. </p>
<p>Since 2000, PISA has been conducted every three years but the assessment planned for 2021 was postponed until 2022 because of COVID. About 690,000 students across 81 countries participated in the test. Almost 13,500 students from 743 schools did the test in Australia. </p>
<p>Students complete a computer-based test and a background questionnaire. In the test, students are presented with stimulus material, such as a brief text, sometimes accompanied by a table, graph or diagram, and a series of questions. Students have to select the correct response or provide a written response, ranging from a word or a number, to an explanation. </p>
<p>In the questionnaire, students are asked about their family background, school life and attitudes about learning.</p>
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Read more:
<a href="https://theconversation.com/australias-year-4-students-have-not-lost-ground-on-reading-despite-pandemic-disruptions-205644">Australia’s Year 4 students have not lost ground on reading, despite pandemic disruptions</a>
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<h2>Why is PISA important?</h2>
<p>While other national and international assessments (such as NAPLAN) assess what students have learned in school, PISA assesses how students apply what they have learned to real-world situations.</p>
<p>It is also one of three international assessments in which Australia participates, along with the <a href="https://www.acer.org/au/pirls">Progress in International Reading Literacy Study</a> (PIRLS), which looks at Year 4 students’ reading comprehension skills, and <a href="https://www.acer.org/au/timss">Trends in International Mathematics and Science Study</a> (TIMSS), which covers maths and science in Year 4 and Year 8.</p>
<h2>How did Australia go?</h2>
<p>The PISA 2022 results show Australia was equal tenth in maths, and equal ninth in science and reading.</p>
<p>Australian students’ performance in maths and reading has not changed significantly over the past seven years, and their performance in science has not changed significantly over four years. </p>
<p>However, Australian students’ performance has declined significantly since PISA results were first reported. There has been a decrease of 37 points in maths, 20 points in science and 30 points in reading. </p>
<p>The test does not tell us the reasons for this drop. Other countries whose performance in maths, reading and science have also declined significantly include Canada, Finland, Greece, New Zealand and Sweden.</p>
<h2>Other countries drop</h2>
<p>Australia’s standing compared with other countries has improved since the last PISA test because the performance of other countries has declined. </p>
<p>In maths, 11 countries (Austria, Belgium, Czech Republic, Denmark, Finland, Ireland, the Netherlands, Poland, Slovenia, Sweden and the United Kingdom) that outperformed Australia in 2018 are now on par with Australia. We are now outperforming six countries that were on par with Australia in 2018 (France, Iceland, Italy, New Zealand, Portugal and the Slovak Republic). </p>
<p>In Australia, male students performed significantly higher in maths (with an average score of 493 compared to the female average of 481). Female students performed significantly higher in reading (with an average of 509 compared to the male average of 487). Male and female students performed at similar levels for science. </p>
<p>Students from higher socioeconomic backgrounds performed significantly higher than students from lower socioeconomic backgrounds and students in major cities performed significantly higher than students in regional or remote areas.</p>
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<img alt="Australia sits around tenth in maths compared to OECD countries." src="https://images.theconversation.com/files/562902/original/file-20231201-17-oxq3f0.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/562902/original/file-20231201-17-oxq3f0.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=600&fit=crop&dpr=1 600w, https://images.theconversation.com/files/562902/original/file-20231201-17-oxq3f0.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=600&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/562902/original/file-20231201-17-oxq3f0.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=600&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/562902/original/file-20231201-17-oxq3f0.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=754&fit=crop&dpr=1 754w, https://images.theconversation.com/files/562902/original/file-20231201-17-oxq3f0.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=754&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/562902/original/file-20231201-17-oxq3f0.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=754&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
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<span class="caption">Australia sits around tenth in maths compared to OECD countries. The Conversation.</span>
<span class="attribution"><a class="source" href="https://www.oecd.org/pisa/">OECD, PISA</a>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span>
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<h2>What about high and low performers?</h2>
<p>About one in every eight Australian students is a “high performer” according to the test. This means they show high levels of skills and knowledge in reading, mathematics and science.</p>
<p>In reading and science, about one in every five of Australian students is a low performer, showing limited skills and knowledge in the relevant subject area, while in maths one in every four students is a low performer.</p>
<p>More than half of Australian students attained the (Australian-set) <a href="https://acara.edu.au/reporting/measurement-framework-for-schooling-in-australia">National Proficient Standard</a>. This meet this, students must “demonstrate more than elementary skills expected at that year level”. In maths, 51% attained the proficient standard, 58% attained it for science and 57% for reading.</p>
<p>Between 2018 and 2022 there was no significant change in the proportion of students who attained the National Proficient Standard. But there has been a significant decline since PISA results were first reported. This includes a 16 percentage point drop in mathematics, nine percentage points in science and 12 percentage points in reading.</p>
<p><iframe id="wssbJ" class="tc-infographic-datawrapper" src="https://datawrapper.dwcdn.net/wssbJ/1/" height="400px" width="100%" style="border: none" frameborder="0"></iframe></p>
<h2>Next steps</h2>
<p>While it is encouraging to see Australia’s results remain steady, we need to look at the bigger picture.</p>
<p>This includes a long-term decline in results and the reality that a significant proportion of students still aren’t meeting national standards.</p>
<p>Clearly, we are failing some of our 15-year-old students – because they lack basic literacy and numeracy skills and the ability to apply them to real-world situations.</p>
<p>To move forward, we need to ask how our education system can lift their performance. We also cannot forget our high performers – how can our education system support them to extend their learning further?</p>
<p>PISA not only provides us with an opportunity to compare how Australia’s education system fares against other countries. We can also to look at high-performing countries and learn how their curriculum and teaching practices could improve the education of young Australians on the cusp of adulthood.</p>
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<strong>
Read more:
<a href="https://theconversation.com/yes-australias-pisa-test-results-may-be-slipping-but-new-findings-show-most-students-didnt-try-very-hard-172050">Yes, Australia's PISA test results may be slipping, but new findings show most students didn't try very hard</a>
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<img src="https://counter.theconversation.com/content/218814/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Lisa De Bortoli does not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>The latest Programme for International Student Assessment (PISA) show Australian 15-year-olds have recorded similar results to 2015 and 2018 in maths, science and reading.Lisa De Bortoli, Senior Research Fellow, Australian Council for Educational ResearchLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/2144592023-11-14T19:06:44Z2023-11-14T19:06:44ZNuclear bombs, artificial intelligence and the madness of reason – in The Maniac, Benjamin Labatut examines the troubling dawn of the digital age<figure><img src="https://images.theconversation.com/files/559028/original/file-20231113-19-y77vl9.jpg?ixlib=rb-1.1.0&rect=6%2C6%2C4109%2C2733&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">
</span> <span class="attribution"><span class="source">GM Pictures/Shutterstock</span></span></figcaption></figure><p>A little over 100 years ago, the great German sociologist <a href="https://en.wikipedia.org/wiki/Max_Weber">Max Weber</a> warned that the growth of modern science would result in the “<a href="https://hscif.org/wp-content/uploads/2018/04/Max-Weber-Science-as-a-Vocation.pdf">disenchantment</a>” of the world. </p>
<p>He meant a world without mystery, without the unknown or the transcendent, and therefore without meaning – a world governed by the grim law of what he called “<a href="https://en.wikipedia.org/wiki/Instrumental_and_value_rationality">instrumental rationality</a>”, where everything is a means to an end and nothing an end in itself. </p>
<p>Weber feared that science and technology might reduce human existence to cold calculation and utilitarian practicality, and destroy any pursuits that do not have immediate, measurable, pragmatic effects.</p>
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<p><em>Review: The Maniac – Banjamin Labatut (Pushkin Press)</em></p>
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<p>Benjamin Labatut’s <a href="https://www.allenandunwin.com/browse/book/Benjamin-Labatut-MANIAC-9781805330677/">The Maniac</a> is a barely fictionalised account of scientific developments in the century since Weber issued his warning. And, in a strange way, it is about how wrong Weber proved to be.</p>
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<a href="https://images.theconversation.com/files/554432/original/file-20231017-23-r5mfan.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/554432/original/file-20231017-23-r5mfan.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/554432/original/file-20231017-23-r5mfan.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=917&fit=crop&dpr=1 600w, https://images.theconversation.com/files/554432/original/file-20231017-23-r5mfan.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=917&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/554432/original/file-20231017-23-r5mfan.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=917&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/554432/original/file-20231017-23-r5mfan.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=1152&fit=crop&dpr=1 754w, https://images.theconversation.com/files/554432/original/file-20231017-23-r5mfan.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=1152&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/554432/original/file-20231017-23-r5mfan.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=1152&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
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<p>The science that has spawned everything from nuclear weapons and artificial intelligence to Silicon Valley and neoliberal economics is anything but practical and mundane. It takes shape on a level of mathematical abstraction and philosophical speculation that only a tiny handful of humans can claim to understand. It operates by breaking all rules of common sense and everything that might seem useful in the workaday world, thriving on its inconsistencies and irrationalities. </p>
<p>In fact, and as Labatut’s title hints, it exists on the thin boundary between the rational and the irrational – that place where thinking tips over into madness, where the world does not lose all meaning, as Weber imagined, but becomes replete with infinite meanings, teeming with messages that only a paranoid mind could ever discern.</p>
<p>If science shuts the gates of heaven, we might say, it throws open the gates of hell.</p>
<h2>The limits of logic</h2>
<p>Labatut’s novel invites us to consider a number of figures in the history of 20th-century science whose personal lives mirrored the madness of the truths they were uncovering.</p>
<p>The Austrian physicist <a href="https://thereader.mitpress.mit.edu/paul-ehrenfest-forgotten-physicist/">Paul Ehrenfest</a> could not help but compare the irrationality of the new science with the irrationality of the emerging Nazi regime. His descent into insanity led him, in 1933, to murder his disabled son before killing himself.</p>
<p>In 1931, the mathematician and logician <a href="https://en.wikipedia.org/wiki/Kurt_G%C3%B6del">Kurt Gödel</a> developed incompleteness theorems that installed an inconsistency at the foundation of all mathematics. His debilitating psychosis is sometimes said to be not the effect but the cause of his insight.</p>
<p>The self-taught engineer and computer scientist <a href="https://discover.lanl.gov/news/0323-von-neumanns-letters/">Klára Dan</a> was behind some of the most important technological advances of the 20th century. In 1963, at the age of 52, she drove from her home in La Jolla, California, to the beach, where she walked into the surf and drowned.</p>
<p>But for Labatut, by far the most compelling of these figures (so much so that most of The Maniac consists of an elaborate character sketch of him) is the Hungarian mathematical genius Neumann János Lajos, or, as he came to be called after moving to the United States, <a href="https://www.ias.edu/von-neumann">John von Neumann</a>.</p>
<p>Labatut introduces von Neumann as “the smartest human being of the twentieth century”. And his evidence for this assertion follows not far behind. </p>
<p>Von Neumann <a href="https://en.wikipedia.org/wiki/MANIAC_I">invented the modern computer</a>, provided the mathematical foundations for <a href="https://en.wikipedia.org/wiki/Quantum_mechanics">quantum mechanics</a>, and completed <a href="https://ahf.nuclearmuseum.org/ahf/profile/john-von-neumann/#:%7E:text=Von%20Neumann's%20principal%20contribution%20to,targets%20for%20the%20atomic%20bomb.">equations required for the atomic bomb</a>. </p>
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<a href="https://images.theconversation.com/files/554467/original/file-20231018-23-fobh3c.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/554467/original/file-20231018-23-fobh3c.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/554467/original/file-20231018-23-fobh3c.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=778&fit=crop&dpr=1 600w, https://images.theconversation.com/files/554467/original/file-20231018-23-fobh3c.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=778&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/554467/original/file-20231018-23-fobh3c.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=778&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/554467/original/file-20231018-23-fobh3c.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=978&fit=crop&dpr=1 754w, https://images.theconversation.com/files/554467/original/file-20231018-23-fobh3c.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=978&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/554467/original/file-20231018-23-fobh3c.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=978&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
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<span class="caption">John von Neumann (1903-57).</span>
<span class="attribution"><a class="source" href="https://about.lanl.gov/lanl-resources/">Los Alamos National Library, via Wikimedia Commons</a>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span>
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<p>He was also the father of <a href="https://en.wikipedia.org/wiki/Game_theory">Game Theory</a>, which is key to neoliberal economics, but which he initially used to justify the Cold War strategy of <a href="https://en.wikipedia.org/wiki/Mutual_assured_destruction">Mutually Assured Destruction, or MAD</a>. This proposed that the only way to prevent the annihilation of all human existence was to arm two superpowers with the capacity to do so many times over.</p>
<p>Von Neumann both predicted and helped advance the arrival of the digital age. He foresaw self-reproducing machines, artificial intelligence, and what he was the first to call the <a href="https://en.wikipedia.org/wiki/Technological_singularity">Singularity</a> – that mythical moment when technology finally absorbs and subordinates humanity.</p>
<p>It is hard to imagine that an individual human mind could be behind so much of the world in which we all now live. As Labatut tells the story, almost everyone who met von Neumann immediately thought of him as a different species, a higher stage of human evolution, an alien being, even a god.</p>
<p>“There are two kinds of people in this world,” Labatut has von Neumann’s collaborator <a href="https://en.wikipedia.org/wiki/Eugene_Wigner">Eugene Wigner</a> say early on in the novel: “Jansci von Neumann and the rest of us.”</p>
<p>“Most mathematicians prove what they can,” Wigner declares a little later. “Von Neumann proves what he wants.”</p>
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Read more:
<a href="https://theconversation.com/kurt-godel-from-loopholes-and-dictators-to-the-incompleteness-theorems-72376">Kurt Gödel: from loopholes and dictators to the incompleteness theorems</a>
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<h2>An Inhuman Mind</h2>
<p>As if to reinforce this image of von Neumann as a god, Labatut never writes from von Neumann’s perspective or pretends to have access to the inner workings of his mind. Instead, he structures his novel as a series of almost breathless first-person accounts of those who knew or encountered him, like the testimony of witnesses to a miracle – or a catastrophe.</p>
<p>Thus, along with Wigner, we hear from von Neumann’s mother Margit Kann von Neumann, his brother Nicholas Augustus von Neumann, his first wife Mariette Kövesi, his early teacher <a href="https://en.wikipedia.org/wiki/George_P%C3%B3lya">George Pólya</a>, the mathematician and engineer <a href="https://en.wikipedia.org/wiki/Theodore_von_K%C3%A1rm%C3%A1n">Theodore von Kármán</a>, the US physicist <a href="https://en.wikipedia.org/wiki/Richard_Feynman">Richard Feynman</a>, the economist <a href="https://nl.wikipedia.org/wiki/Oskar_Morgenstern">Oskar Morgenstern</a>, and many more.</p>
<p>They all seem to have the same basic impression of the man: a bafflingly great genius, whose singular intellectual powers appeared to place him beyond good and evil, and led him to look down on mere human morality with callous indifference.</p>
<p>That explains the glee with which von Neumann threw himself into the military applications of his ideas, and the shamelessness with which he became, as Labatut puts it, “a mind for hire”, willing to “charge exorbitant fees to sit with people from IBM, RCA, the CIA, or the RAND Corporation, sometimes for no longer than a couple of minutes”. </p>
<p>If von Neumann was a god, then, he was by no means a benevolent Christian one. He was more like the Greek gods of Mount Olympus or the angry Yahweh of the Old Testament. Or perhaps he was simply a demon – wilful, arbitrary and capable of horrific acts of destruction.</p>
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<span class="caption">Richard Feynman.</span>
<span class="attribution"><a class="source" href="https://commons.wikimedia.org/wiki/File:Richard_Feynman_1959.png">Public domain, via Wikimedia Commons</a></span>
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<h2>Mechanical gods</h2>
<p>On the other hand, and unlike most gods, von Neumann was not immortal. Like so many of those around him, he died tragically young, a victim of a virulent cancer that infiltrated his otherwise insuperable brain shortly after his 53rd birthday.</p>
<p>But towards the end of The Maniac, Labatut seems to suggest that the algorithms that now dominate so much of our lives could be seen as von Neumann’s offspring. They are mechanical gods not subject to the limits of flesh and blood.</p>
<p>The novel thus has a kind of second act, in which Labatut leaves von Neumann behind and considers instead the history of <a href="https://en.wikipedia.org/wiki/Go_(game)">the ancient game Go</a>, and the moment when machines became capable of beating the best human players in the world, <a href="https://en.wikipedia.org/wiki/AlphaGo_versus_Lee_Sedol">Lee Sedol</a> and <a href="https://en.wikipedia.org/wiki/AlphaGo_versus_Ke_Jie">Ke Jie</a>.</p>
<p>The idea here is that playing games is the most discretely human activity we can imagine, for it involves not simply the application of rules or the calculation of probabilities, but a kind of creativity and prevarication, cunning and intimidation that a machine should not be able to replicate.</p>
<p>That machines now consistently beat humans at the most complex games we can dream up appears to Labatut as a fundamental tipping point. We can only tremble before these new gods the same way our ancestors did before the old ones – in fear and awe.</p>
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Read more:
<a href="https://theconversation.com/economic-theories-that-have-changed-us-game-theory-43633">Economic theories that have changed us: game theory</a>
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<hr>
<h2>The gift of fiction</h2>
<p>Such apocalyptic fantasies notwithstanding, I would like to suggest that something else is going on here as well, just beneath the surface.</p>
<p>The Maniac is deceptively presented as a collection of facts and an account of events that really happened. But that is not at all what it is. It is a novel: the invention of another great mind, namely that of Labatut. </p>
<figure class="align-left zoomable">
<a href="https://images.theconversation.com/files/555209/original/file-20231023-15-i9mwwx.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/555209/original/file-20231023-15-i9mwwx.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/555209/original/file-20231023-15-i9mwwx.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=1083&fit=crop&dpr=1 600w, https://images.theconversation.com/files/555209/original/file-20231023-15-i9mwwx.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=1083&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/555209/original/file-20231023-15-i9mwwx.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=1083&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/555209/original/file-20231023-15-i9mwwx.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=1361&fit=crop&dpr=1 754w, https://images.theconversation.com/files/555209/original/file-20231023-15-i9mwwx.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=1361&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/555209/original/file-20231023-15-i9mwwx.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=1361&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Benjamin Labatut.</span>
<span class="attribution"><a class="source" href="https://commons.wikimedia.org/wiki/File:Julieta-Labatut_COLOR.jpg">AloysusAcker, via Wikimedia Commons</a>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span>
</figcaption>
</figure>
<p>It places the facts of human experience within a fictional container. Profound scientific and technological discoveries are encompassed by something more profound: a story.</p>
<p>In that sense, the fictional form of The Maniac belies its apocalyptic content. Machines might be able to dominate the real world, but as Labatut’s novel attests, humans can invent the fiction that dominates that domination.</p>
<p>To put the same point differently, I have been writing here as if The Maniac tells us something about what the author takes to be true. But precisely because it is a novel and not a series of articles or a doctoral dissertation, there is no reason to make that assumption. What Labatut really believes is something I cannot know. The same will never be true of <a href="https://theconversation.com/irony-machine-why-are-ai-researchers-teaching-computers-to-recognise-irony-185904">the productions of a computer or an algorithm</a>.</p>
<p>“You insist that there is something that a machine cannot do,” von Neumann once declared, with characteristic arrogance. “If you tell me precisely what it is that a machine cannot do, then I can always make a machine which will do just that.”</p>
<p>“Well,” we might have retorted, “what it cannot do is anything without being told what to do.” What it cannot do is what Labatut’s novel, indeed all novels, all fictions, all stories do – tell us things that can never be verified, create truths that have no stable foundation, or weave entire worlds out of nothing at all.</p><img src="https://counter.theconversation.com/content/214459/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Charles Barbour does not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>In The Manic, Benjamin Labatut tells the story of the ‘smartest man of the 20th century’.Charles Barbour, Associate Professor, Philosophy, Western Sydney UniversityLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/2174962023-11-14T17:06:57Z2023-11-14T17:06:57ZAstraZeneca sued over jab: could it be down to a misunderstanding of how risk is calculated?<p>A multi-million-pound landmark “vaccine damage” case is set to take place <a href="https://www.bbc.co.uk/news/health-67370454">in London’s High Court</a>. The test case is being pursued by Jamie Scott who suffered a severe brain injury in April 2021 after receiving the Oxford-AstraZeneca vaccine. </p>
<p>The case being brought under the Consumer Protections Act argues that the Oxford-AstraZeneca vaccine was less safe than consumers were entitled to expect. A key part of the argument is over the efficacy of the vaccine, which claimants argue was “vastly overstated”.</p>
<p>AstraZeneca is defending the case. </p>
<p>To get into the nitty-gritty of the claim, we need to understand something about risks. There are two ways to present the change in risk brought about by a treatment or an intervention: absolute risk and relative risk. Let’s have a look at an example from my book – <a href="https://www.amazon.co.uk/Maths-Life-Death-Kit-Yates/dp/1787475425">The Maths of Life and Death</a> - to explain the difference.</p>
<p>In 2009, under the headline <a href="https://www.tandfonline.com/doi/abs/10.1080/13698575.2010.540645">Careless Pork Costs Lives</a>, the Sun reported just one of many hundreds of results from a <a href="http://discovery.ucl.ac.uk/4841/1/4841.pdf">500-page study by the World Cancer Research Fund</a> on the effect of consuming 50 grams of processed meat per day. The newspaper told readers that eating a bacon sandwich every day would increase their risk of colorectal cancer by 20%.</p>
<p>The Sun chose to focus on the “relative risk” – the risk of a particular outcome (developing cancer) for people exposed to a given risk factor (eating bacon sandwiches) as a proportion of the risk for those not exposed. If this relative risk ratio is above one, then an exposed person is more likely to develop the disease compared with someone without the exposure. If it is below one then the risk is decreased.</p>
<p>On the right-hand side of the image below (figure 1), the increase in the relative risk (relative risk ratio of 6/5 or, equivalently, 1.2) of 20% paints a dramatic picture by neglecting the people who are not affected by the disease</p>
<p>The absolute risk can sometimes be a more helpful measure. Absolute risk is the proportion of people exposed or unexposed to a particular treatment or risk factor (for example, eating or not eating bacon sandwiches) who are expected to develop a given outcome (for example, cancer). Eating 50g of processed meat per day increases the absolute lifetime risk of developing colorectal cancer from 5% to 6%. </p>
<p>On the left of figure 1, we consider the fates of two groups of 100 people. Of 100 people who eat a bacon sandwich every day, only one more of them will develop colorectal cancer than in a group of 100 people who abstain.</p>
<p>So it is true that the relative risk for those eating 50g of processed meat per day increases by 20%, but the absolute risk increases by only 1%.</p>
<p>The claimants in the court case <a href="https://www.astrazeneca.com/media-centre/press-releases/2021/covid-19-vaccine-astrazeneca-confirms-protection-against-severe-disease-hospitalisation-and-death-in-the-primary-analysis-of-phase-iii-trials.html#!">argue that</a> AstraZeneca point to <a href="https://pubmed.ncbi.nlm.nih.gov/33617777/">studies showing around 70% efficacy</a> at preventing symptomatic COVID. In the court documents, the legal claim states: “In fact, the absolute risk reduction concerning Covid-19 prevention was only 1.2 per cent.” </p>
<p>The basis of the case that the effectiveness of the vaccine was overstated seems to be, like the Sun’s “save our bacon” campaign, that AstraZeneca has inflated the apparent effectiveness of their vaccine by using a relative measure rather than an absolute measure.</p>
<h2>Good reasons for using relative risk</h2>
<p>There are good reasons why AstraZeneca would have used the relative risk reduction rather than the absolute for describing vaccine efficacy. For one, the prevalence of COVID varied significantly throughout the acute phase of the pandemic. When cases were low, the absolute reduction in risk would necessarily be lower than when the prevalence was high. </p>
<p>Imagine a hypothetical vaccine that blocked half of all infections in two prevalence scenarios: one at 10% and another at 1% prevalence. In the first scenario, the absolute risk reduction from taking the vaccine would be 5%, but in the second it would be 0.5%, whereas the relative risk reduction would always be 50%. In the face of changing prevalence, it makes sense to use the unchanging relative risk reduction to demonstrate how much safer the vaccine makes you.</p>
<p>Another good reason for using relative risk hinges on understanding how clinical trials are often run. Typically, trial volunteers are split into a treatment group that is given the vaccine and a control group that is not.</p>
<p>When running a trial, partly for time constraints and partly because it is unethical to let lots of unvaccinated people get infected, a limit is set on the proportion of people in the unvaccinated group who are allowed to be infected. Once this limit is reached, the trial is halted and the comparison between the proportion of infections in the two groups determines the vaccine efficacy. </p>
<p>Imagine, in a hypothetical trial, this limit was set at 2% and that when the trial was halted only 0.2% of the treatment group had been infected. This would give a crude estimate of the relative efficacy of the vaccine of 90%, but an absolute efficacy of only 1.8%. Even with a vaccine which blocked all infections, it would never be possible to demonstrate an absolute efficacy greater than 2% because of the trial setup.</p>
<p>Given that the UK’s <a href="https://www.ons.gov.uk/peoplepopulationandcommunity/healthandsocialcare/conditionsanddiseases/articles/coronaviruscovid19infectionsurveytechnicalarticlecumulativeincidenceofthenumberofpeoplewhohavebeeninfectedwithcovid19byvariantandageengland/9february2023">Office for National Statistics suggests</a> that we have had enough COVID infections in the UK for everyone to have been infected at least once, using the relative effectiveness seems sensible. Is it reasonable to suggest that statements of the effectiveness of vaccines should be limited because of the trial design or fluctuating prevalence?</p>
<p>Regardless of what the court rules, however, what is not in doubt is that the Oxford-AstraZeneca vaccine <a href="https://www.airfinity.com/articles/astrazeneca-and-pfizer-biontech-saved-over-12-million-lives-in-the-first">saved millions of lives</a>.</p>
<p>A <a href="https://www.bbc.co.uk/news/health-67370454">spokesperson for AstraZeneca said</a>: “Patient safety is our highest priority and regulatory authorities have clear and stringent standards to ensure the safe use of all medicines, including vaccines. Our sympathy goes out to anyone who has lost loved ones or reported health problems.”</p><img src="https://counter.theconversation.com/content/217496/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Kit is a member of Independent SAGE.</span></em></p>A mathematician explains the important difference between absolute risk and relative risk.Christian Yates, Senior Lecturer in Mathematical Biology, University of BathLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/2156982023-11-01T12:35:53Z2023-11-01T12:35:53ZCancer has many faces − 5 counterintuitive ways scientists are approaching cancer research to improve treatment and prevention<figure><img src="https://images.theconversation.com/files/553918/original/file-20231016-15-3osk1.jpg?ixlib=rb-1.1.0&rect=0%2C0%2C2700%2C1758&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Cancer cells don't follow the typical rules that allow a multicellular collective to function.</span> <span class="attribution"><a class="source" href="https://commons.wikimedia.org/wiki/File:Cancer_cells_(1).jpg">Dr. Cecil Fox/National Cancer Institute</a></span></figcaption></figure><p>How researchers conceptualize a disease informs how they treat it. Cancer is often described as uncontrollable cell growth triggered by genetic damage. But cancer can also be seen from angles that emphasize mathematics, evolutionary game theory and physics, among others.</p>
<p>Molecular biology has brought significant advances in making it possible to live with cancer as a chronic illness rather than a fatal disease. Alternative frameworks, however, can offer scientists additional insights on how to prevent tumors from spreading throughout the body and becoming resistant to treatment.</p>
<p>Here are a few unconventional lenses through which researchers are viewing cancer with fresh eyes, drawn from The Conversation’s archives.</p>
<h2>1. Evolution and natural selection of cancer</h2>
<p>The body is far from a wonderland for cells. Each individual cell competes against trillions of others for finite space and nutrients. If they’re able to cooperate in an orderly enough fashion, sharing resources and dividing labor, the collective functions effectively. Cancer cells, however, <a href="https://theconversation.com/microbes-in-your-food-can-help-or-hinder-your-bodys-defenses-against-cancer-how-diet-influences-the-conflict-between-cell-cooperators-and-cheaters-195810">cheat the system</a>: They hog resources, take up as much space as possible and <a href="https://theconversation.com/what-are-hela-cells-a-cancer-biologist-explains-169913">refuse to die</a>.</p>
<p>In this way, cancer can be thought of as <a href="https://theconversation.com/every-cancer-is-unique-why-different-cancers-require-different-treatments-and-how-evolution-drives-drug-resistance-199249">an evolutionary disease</a> – these are cells that have developed the genetic mutations to outcompete their neighbors, and subsequent cell generations inherit this survival advantage. Cancer cells benefit at the expense of the collective until the entire organism collapses.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/554078/original/file-20231016-27-4u7mpn.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="Microscopy image of pancreas tumor with multicolored cell subgroups" src="https://images.theconversation.com/files/554078/original/file-20231016-27-4u7mpn.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/554078/original/file-20231016-27-4u7mpn.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=458&fit=crop&dpr=1 600w, https://images.theconversation.com/files/554078/original/file-20231016-27-4u7mpn.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=458&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/554078/original/file-20231016-27-4u7mpn.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=458&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/554078/original/file-20231016-27-4u7mpn.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=575&fit=crop&dpr=1 754w, https://images.theconversation.com/files/554078/original/file-20231016-27-4u7mpn.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=575&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/554078/original/file-20231016-27-4u7mpn.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=575&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Most tumors are made of many different kinds of cancer cells, as shown in this pancreatic cancer sample from a mouse.</span>
<span class="attribution"><a class="source" href="https://visualsonline.cancer.gov/details.cfm?imageid=10654">Ravikanth Maddipati/Abramson Cancer Center at the University of Pennsylvania via National Cancer Institute</a></span>
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<p>Oncologist <a href="https://cancer.psu.edu/researchers/individual/-/researcher/5B6500F63D6A38DBE0540010E056499A/monika-joshi-md-mrcp">Monika Joshi</a> and pathologists <a href="https://cancer.psu.edu/researchers/individual/-/researcher/5F6E820FF5C14A2DE0540010E056499A/joshua-warrick-md">Joshua Warrick</a> and <a href="https://scholar.google.com/citations?user=YEqQHkIAAAAJ&hl=en">David DeGraff</a> believe that understanding evolution is key to understanding cancer. Screening programs are effective, for example, because removing a nascent tumor is easier than treating one that has evolved the ability to spread. Cancer cells likewise become resistant to treatments because they’re pushed to further evolve to survive.</p>
<p>Some researchers are applying the principles of evolutionary game theory to <a href="https://theconversation.com/cancers-are-in-an-evolutionary-battle-with-treatments-evolutionary-game-theory-could-tip-the-advantage-to-medicine-17017">reduce treatment resistance</a> and optimize <a href="https://theconversation.com/cancer-in-kids-is-different-from-cancer-in-grown-ups-figuring-out-how-could-lead-to-better-pediatric-treatments-212738">therapies for children</a>.</p>
<p>“The fight against cancer is a fight against evolution, the fundamental process that has driven life on Earth since time immemorial,” they wrote. “This is not an easy fight, but medicine has made tremendous progress.”</p>
<hr>
<p>
<em>
<strong>
Read more:
<a href="https://theconversation.com/every-cancer-is-unique-why-different-cancers-require-different-treatments-and-how-evolution-drives-drug-resistance-199249">Every cancer is unique – why different cancers require different treatments, and how evolution drives drug resistance</a>
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<h2>2. Fluid mechanics of cancer</h2>
<p>As much as cancer is a disease that respects no boundaries, tumor cells are still shaped by their environment. Unlike healthy cells that take the hint when their presence isn’t wanted, however, tumor cells not only <a href="https://theconversation.com/stopping-the-cancer-cells-that-thrive-on-chemotherapy-research-into-how-pancreatic-tumors-adapt-to-stress-could-lead-to-a-new-treatment-approach-197768">survive but thrive in stressful conditions</a>. Isolated cancer cells able to adapt to harsh settings are the ones that establish metastatic colonies and become resistant to treatment.</p>
<p>While researchers have focused on how biochemical signals direct cells to move from one location to another, a cell’s physical environment also affects where it migrates. Mechanical engineer <a href="https://scholar.google.com/citations?user=nKmJNpQAAAAJ&hl=en">Yizeng Li</a> found that a cell’s “solid” and “fluid” surroundings influence its movement.</p>
<p>Cancer cells encounter varying degrees of fluid viscosity, or thickness, as they travel through the body. Li and her team found that breast cancer cells counterintuitively move faster in high viscosity environments by changing their structure. This meant that fluid viscosity serves as a <a href="https://theconversation.com/how-cancer-cells-move-and-metastasize-is-influenced-by-the-fluids-surrounding-them-understanding-how-tumors-migrate-can-help-stop-their-spread-195792">mechanobiological cue for cancer cells to metastasize</a>.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/502975/original/file-20230103-105030-c8xq8d.gif?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="Animation comparing two fluids with lower and higher viscosity." src="https://images.theconversation.com/files/502975/original/file-20230103-105030-c8xq8d.gif?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/502975/original/file-20230103-105030-c8xq8d.gif?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=463&fit=crop&dpr=1 600w, https://images.theconversation.com/files/502975/original/file-20230103-105030-c8xq8d.gif?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=463&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/502975/original/file-20230103-105030-c8xq8d.gif?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=463&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/502975/original/file-20230103-105030-c8xq8d.gif?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=582&fit=crop&dpr=1 754w, https://images.theconversation.com/files/502975/original/file-20230103-105030-c8xq8d.gif?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=582&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/502975/original/file-20230103-105030-c8xq8d.gif?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=582&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">The blue fluid on the left has a lower viscosity relative to the orange fluid on the right.</span>
<span class="attribution"><a class="source" href="https://en.wikipedia.org/wiki/File:Viscosities.gif">Synapticrelay/Wikimedia Commons</a>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span>
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<p>“Cancer patients usually don’t die from the original source of the tumor but from its spread to other parts of the body,” Li wrote. “Understanding how fluid viscosity affects the movement of tumor cells could help researchers figure out ways to better treat and detect cancer before it metastasizes.”</p>
<hr>
<p>
<em>
<strong>
Read more:
<a href="https://theconversation.com/how-cancer-cells-move-and-metastasize-is-influenced-by-the-fluids-surrounding-them-understanding-how-tumors-migrate-can-help-stop-their-spread-195792">How cancer cells move and metastasize is influenced by the fluids surrounding them – understanding how tumors migrate can help stop their spread</a>
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<h2>3. Inflammation link to cardiovascular disease</h2>
<p>Apart from being leading causes of death around the world, cardiovascular disease and cancer may not initially seem to have much in common. The many risk factors they share, however – like poor diet, smoking and chronic stress – coalesce with chronic inflammation: persistent, low-grade activation of the immune system can damage cells in ways that encourage either disease to develop. </p>
<p>For biomedical engineer <a href="https://scholar.google.com/citations?user=wD6KbXkAAAAJ&hl=en">Bryan Smith</a>, the developmental parallels between these diseases signal they could be <a href="https://theconversation.com/could-a-single-drug-treat-the-two-leading-causes-of-death-in-the-us-cancer-and-cardiovascular-disease-205461">treated at the same time</a>.</p>
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<figcaption><span class="caption">Nanoparticles can ‘eat’ the plaques that cause heart disease.</span></figcaption>
</figure>
<p><a href="https://theconversation.com/drugs-4-essential-reads-on-how-theyre-made-how-they-work-and-how-context-can-make-poison-a-medicine-192590">Drugs can be repurposed</a> to target diseases for which they weren’t originally designed. Certain drugs, for example, can direct immune cells called macrophages to consume both cancer cells and the cellular debris that contribute to cardiovascular plaques.</p>
<p>“As basic science discovers other molecular parallels between these diseases, patients will be the beneficiaries of better therapies that can treat both,” wrote Smith.</p>
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<p>
<em>
<strong>
Read more:
<a href="https://theconversation.com/could-a-single-drug-treat-the-two-leading-causes-of-death-in-the-us-cancer-and-cardiovascular-disease-205461">Could a single drug treat the two leading causes of death in the US: cancer and cardiovascular disease?</a>
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<h2>4. Mathematics of cancer</h2>
<p>In certain contexts, math has unique strengths in <a href="https://theconversation.com/big-bang-of-numbers-the-conversations-book-club-explores-how-math-alone-could-create-the-universe-with-author-manil-suri-213690">describing the natural world</a>. For instance, epigenetics – where and when genes are turned on or off – plays as much a role in cancer progression as direct changes to the genetic code. Epigenetic changes can alter healthy cells to the point of losing their normal form and function. But the randomness of these changes makes it difficult to tease out pathological from normal genetic activity.</p>
<p>A mathematical concept called stochasticity – or how the randomness of the steps of a process influences how predictable its outcome will be – lends a logical framework to the <a href="https://theconversation.com/cancer-evolution-is-mathematical-how-random-processes-and-epigenetics-can-explain-why-tumor-cells-shape-shift-metastasize-and-resist-treatments-199398">epigenetic changes contributing to cancer</a>, clarifying when healthy cells rapidly develop into tumor cells. </p>
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<iframe width="440" height="260" src="https://www.youtube.com/embed/_aAhcNjmvhc?wmode=transparent&start=0" frameborder="0" allowfullscreen=""></iframe>
<figcaption><span class="caption">Twins sharing the exact same genome can develop in completely different ways because of epigenetics.</span></figcaption>
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<p>Stochasticity is commonly used to study stock market behavior and epidemic disease spread, and researchers quantify it by examining the degree of uncertainty, or entropy, of a particular outcome. Identifying high entropy areas in the genome could offer another approach to cancer detection and drug design.</p>
<p>Cancer geneticist <a href="https://scholar.google.com/citations?user=tbj-LpcAAAAJ&hl=en">Andrew Feinberg</a> has been using entropy to quantitatively describe the epigenetics of cancer. He and his colleagues found that high entropy regions of the genome in the skin become even more entropic with sun damage, increasing the chance of developing cancer. This offers a potential explanation for why cancer risk significantly increases with age.</p>
<p>“Epigenetic entropy shows that you can’t fully understand cancer without mathematics,” Feinberg wrote. “Biology is catching up with other hard sciences in incorporating mathematical methods with biological experimentation.”</p>
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<p>
<em>
<strong>
Read more:
<a href="https://theconversation.com/cancer-evolution-is-mathematical-how-random-processes-and-epigenetics-can-explain-why-tumor-cells-shape-shift-metastasize-and-resist-treatments-199398">Cancer evolution is mathematical – how random processes and epigenetics can explain why tumor cells shape-shift, metastasize and resist treatments</a>
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<h2>5. A public health issue</h2>
<p>Cancer is a disease that develops in an individual, but its socially derived causes and societal-wide effects are hardly limited to a single person.</p>
<p>Take the case of lung cancer. It is stigmatized as a disease brought on by poor lifestyle choices – a consequence of a personal decision to use tobacco products. But as thoracic oncologist <a href="https://doctors.umiamihealth.org/provider/Estelamari+Rodriguez/1257821">Estelamari Rodriguez</a> noted, the face of lung cancer has changed.</p>
<p>“Over the past 15 years, more women, never-smokers and younger people are being diagnosed with lung cancer,” she wrote. While lung cancer rates have significantly decreased for men, they have <a href="https://theconversation.com/lung-cancer-rates-have-decreased-for-the-marlboro-man-but-have-risen-steeply-for-nonsmokers-and-young-women-an-oncologist-explains-why-197581">substantially risen for women</a> around the world. Despite being the leading cause of cancer death among women, screening rates remain low compared with other cancers.</p>
<p>More broadly, cancer symptoms are often unrecognized or misdiagnosed, not only <a href="https://theconversation.com/ovarian-cancer-is-not-a-silent-killer-recognizing-its-symptoms-could-help-reduce-misdiagnosis-and-late-detection-181415">for women</a> but also for many marginalized populations, including <a href="https://theconversation.com/biopsies-confirm-a-breast-cancer-diagnosis-after-an-abnormal-mammogram-but-structural-racism-may-lead-to-lengthy-delays-185824">people of color</a>, <a href="https://theconversation.com/doctors-often-arent-trained-on-the-preventive-health-care-needs-of-gender-diverse-people-as-a-result-many-patients-dont-get-the-care-they-need-191933">transgender patients</a> and <a href="https://theconversation.com/how-obamacare-has-helped-poor-cancer-patients-85306">the uninsured</a>.</p>
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<figcaption><span class="caption">An increasing number of lung cancer diagnoses are among people who never smoked.</span></figcaption>
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<p>These disparities are due in part to biases in medical education and <a href="https://theconversation.com/yes-black-patients-do-want-to-help-with-medical-research-here-are-ways-to-overcome-the-barriers-that-keep-clinical-trials-from-recruiting-diverse-populations-185337">clinical research</a> that fail to prepare clinicians to care for the diversity of patients they’ll encounter. <a href="https://theconversation.com/the-next-attack-on-the-affordable-care-act-may-cost-you-free-preventive-health-care-166087">Tenuous access to preventive care</a> and disproportionate <a href="https://theconversation.com/arsenic-contamination-of-food-and-water-is-a-global-public-health-concern-researchers-are-studying-how-it-causes-cancer-200689">exposure to carcinogens</a> among certain populations compound these inequities.</p>
<p>The purview of cancer goes far beyond a single discipline. It takes a village of researchers, policymakers and patient advocates to achieve effective and accessible cancer care for all.</p>
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<strong>
Read more:
<a href="https://theconversation.com/lung-cancer-rates-have-decreased-for-the-marlboro-man-but-have-risen-steeply-for-nonsmokers-and-young-women-an-oncologist-explains-why-197581">Lung cancer rates have decreased for the Marlboro Man, but have risen steeply for nonsmokers and young women – an oncologist explains why</a>
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<img src="https://counter.theconversation.com/content/215698/count.gif" alt="The Conversation" width="1" height="1" />
From math to evolutionary game theory, looking at cancer through different lenses can offer further insights on how to approach treatment resistance, metastasis and health disparities.Vivian Lam, Associate Health and Biomedicine EditorLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/2154982023-10-16T15:50:09Z2023-10-16T15:50:09ZLabour’s plan to focus on early maths is solid – gaps in achievement start even before primary school<figure><img src="https://images.theconversation.com/files/554005/original/file-20231016-17-smaari.jpg?ixlib=rb-1.1.0&rect=20%2C0%2C6689%2C4466&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">
</span> <span class="attribution"><a class="source" href="https://www.shutterstock.com/image-photo/little-girl-learning-numbers-shapes-play-1059608915">NadyaEugene/Shutterstock</a></span></figcaption></figure><p>Politicians in the UK have maths on the mind. The Conservatives intend to extend compulsory maths education for young people until 18. </p>
<p>And at the Labour party conference, shadow education secretary <a href="https://www.bridgetphillipson.com/speeches/2023/10/11/bridget-phillipsons-speech-at-the-labour-partys-2023-conference/">Bridget Phillipson announced</a> the opposition’s plans to improve maths skills across the country: a focus on primary school and pre-school education rather than post-16, with an emphasis on children learning the maths they will need for everyday life. </p>
<p>Paying attention to young children’s maths is a good idea. Evidence from the UK and beyond shows that children start primary school with <a href="https://repec-cepeo.ucl.ac.uk/cepeow/cepeowp22-06.pdf">varying levels</a> of mathematical skills – and disadvantage gaps are already evident at this point, meaning that children from poorer backgrounds may not have skills at the same level as their more well-off peers. </p>
<p>The differences between children’s maths skills then remain remarkably <a href="https://doi.org/10.1002/icd.2281">stable over time</a>. Children who start primary school with mathematical abilities behind the level of their peers will typically remain behind their peers throughout school. </p>
<p>To reduce these gaps, we need to act early. But positive change won’t be achieved simply by adding more content to the primary or early years mathematics curriculums. Neither is it helpful to push children to learn more complex mathematics earlier. These approaches might lead to children learning maths in a superficial and rote manner, rather than understanding the underlying ideas.</p>
<h2>Primary focus</h2>
<p>Labour has <a href="https://labour.org.uk/updates/press-releases/labour-to-unveil-real-world-primary-maths-teaching-to-encourage-stronger-lifelong-numeracy/">raised the prospect</a> of creating a “<a href="https://www.theguardian.com/politics/2023/oct/10/labour-announces-phonics-for-maths-scheme-in-planned-curriculum-review">phonics for maths</a>”. Phonics is a <a href="https://educationhub.blog.gov.uk/2023/10/12/everything-you-need-to-know-about-phonics-in-schools/">method of learning to read</a> that teaches children the sounds that letters and combinations of letters make. It is required in primary schools, and pupils take a phonics screening check in year one to assess their progress. </p>
<p>Although not universally supported, phonics has been linked to improvements in reading levels among children in <a href="https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/1156633/PIRLS_2021_-_national_report_for_England__May_2023.pdf">England</a>. </p>
<p>However, phonics is a specific technique for teaching word reading, while mathematics is incredibly broad. It involves multiple skills as well as different types of knowledge and understanding. </p>
<p>Even in early primary school, mathematics is <a href="https://doi.org/10.1177/17470218231175325">complex</a>. Children need to understand quantities and their relationships, to recognise digits and understand place value, to carry out arithmetic procedures, to identify patterns in numbers and shapes, and much more. It is unlikely that a single technique, as phonics is, can underpin this breadth of knowledge and understanding. </p>
<p><div data-react-class="Tweet" data-react-props="{"tweetId":"1711978621751947728"}"></div></p>
<p>But in another sense, the parallel with phonics is encouraging. The phonics revolution was informed by <a href="https://doi.org/10.1177/1529100618772271">research</a> and developed from a better understanding of how children learn to read. This can and should be emulated for mathematics. Research evidence on the early stages of learning maths can help build a solid approach to teaching mathematical skills to young children.</p>
<p>Another feature of Labour’s plans is their aim to “<a href="https://www.bridgetphillipson.com/speeches/2023/10/11/bridget-phillipsons-speech-at-the-labour-partys-2023-conference/">bring maths to life</a>” by using <a href="https://labour.org.uk/updates/press-releases/labour-to-unveil-real-world-primary-maths-teaching-to-encourage-stronger-lifelong-numeracy/">real-world examples</a>: budgeting, exchange rates, sports league tables. </p>
<p>A desire to give meaning to numbers and mathematics by building on children’s experiences is a good ambition. This can be achieved through play-based and hands-on activities, which involve children manipulating objects such as counters and cards to better understand mathematical ideas and relationships. It is also important to help children see numbers and mathematical patterns in the world around them: the number of red cars on the street or the shapes of windows and doors, for instance.</p>
<p>These approaches may provide a <a href="https://educationendowmentfoundation.org.uk/education-evidence/evidence-reviews/early-years-and-key-stage-1-mathematics-teaching">stronger foundation</a> for future learning than focusing on using written digits or learning mathematical facts (such as 2 + 3 = 5) too early.</p>
<h2>Taking care</h2>
<p>But care is needed to ensure that bringing maths to life truly reflects children’s experiences and doesn’t become a gimmick. It could even increase disadvantage gaps due to differences in children’s experiences, for example, for children from families who lack access to bank accounts or have never had the experience of travelling abroad and using different currency. </p>
<p>There are already good examples out there of how to teach in this way – such as the <a href="https://www.ncetm.org.uk/maths-hubs-projects/mastering-number-at-reception-and-ks1/">Mastering Number programme</a>. Any curriculum changes need to be properly funded and developed in collaboration with experts in the field.</p>
<p>Giving children better mathematical foundations through engaging and meaningful activities can set them up for success throughout school and beyond. This would not only positively affect children’s achievement but could also change attitudes to mathematics for the better. </p>
<p>Changing attitudes to mathematics from the foundations upwards can help children and young people feel confident and engaged with the subject and see its value in their life, leading to more wanting to study the subject.</p><img src="https://counter.theconversation.com/content/215498/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Camilla Gilmore receives funding from the Economic and Social Research Council and Research England. </span></em></p>Changing attitudes to maths from the start of education can lead to more success later on.Camilla Gilmore, Professor of Mathematical Cognition, Loughborough UniversityLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/2150542023-10-05T19:03:36Z2023-10-05T19:03:36ZIs there really a 1 in 6 chance of human extinction this century?<figure><img src="https://images.theconversation.com/files/552227/original/file-20231005-23-pvcy99.jpeg?ixlib=rb-1.1.0&rect=9%2C0%2C4815%2C3106&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">
</span> <span class="attribution"><a class="source" href="https://www.shutterstock.com/image-illustration/asteroid-hits-earth-3d-rendering-1909820308">Shutterstock</a></span></figcaption></figure><p>In 2020, Oxford-based philosopher Toby Ord published a book called <a href="https://theprecipice.com/">The Precipice</a> about the risk of human extinction. He put the chances of “existential catastrophe” for our species during the next century at one in six.</p>
<p>It’s quite a specific number, and an alarming one. The claim drew <a href="https://www.dailystar.co.uk/news/latest-news/humans-have-1-6-chance-21960557">headlines</a> at the time, and has been influential since – most recently brought up by Australian politician Andrew Leigh in a <a href="https://www.andrewleigh.com/what_s_the_worst_that_could_happen_existential_risk_and_extreme_politics_speech">speech</a> in Melbourne.</p>
<p>It’s hard to disagree with the idea we face troubling prospects over the coming decades, from climate change, nuclear weapons and bio-engineered pathogens (all big issues in my view), to rogue AI and large asteroids (which I would see as less concerning).</p>
<p>But what about that number? Where does it come from? And what does it really mean?</p>
<h2>Coin flips and weather forecasts</h2>
<p>To answer those questions, we have to answer another first: what is probability?</p>
<p>The most traditional view of probability is called frequentism, and derives its name from its heritage in games of dice and cards. On this view, we know there is a one in six chance a fair die will come up with a three (for example) by observing the frequency of threes in a large number of rolls.</p>
<p>Or consider the more complicated case of weather forecasts. What does it mean when a weatherperson tells us there is a one in six (or 17%) chance of rain tomorrow?</p>
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Read more:
<a href="https://theconversation.com/the-science-of-weather-forecasting-what-it-takes-and-why-its-so-hard-to-get-right-175740">The science of weather forecasting: what it takes and why it’s so hard to get right</a>
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<p>It’s hard to believe the weatherperson means us to imagine a large collection of “tomorrows”, of which some proportion will experience precipitation. Instead, we need to look at a large number of such predictions and see what happened after them.</p>
<p>If the forecaster is good at their job, we should see that when they said “one in six chance of rain tomorrow”, it did in fact rain on the following day one time in every six.</p>
<p>So, traditional probability depends on observations and procedure. To calculate it, we need to have a collection of repeated events on which to base our estimate.</p>
<h2>Can we learn from the Moon?</h2>
<p>So what does this mean for the probability of human extinction? Well, such an event would be a one-off: after it happened, there would be no room for repeats.</p>
<p>Instead, we might find some parallel events to learn from. Indeed, in Ord’s book, he discusses a number of potential extinction events, some of which can potentially be examined in light of a history. </p>
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<a href="https://images.theconversation.com/files/552236/original/file-20231005-29-kna72i.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="A photo of the Moon with craters highlighted." src="https://images.theconversation.com/files/552236/original/file-20231005-29-kna72i.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/552236/original/file-20231005-29-kna72i.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=600&fit=crop&dpr=1 600w, https://images.theconversation.com/files/552236/original/file-20231005-29-kna72i.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=600&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/552236/original/file-20231005-29-kna72i.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=600&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/552236/original/file-20231005-29-kna72i.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=754&fit=crop&dpr=1 754w, https://images.theconversation.com/files/552236/original/file-20231005-29-kna72i.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=754&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/552236/original/file-20231005-29-kna72i.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=754&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
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<span class="caption">Counting craters on the Moon can gives us clues about the risk of asteroid impacts on Earth.</span>
<span class="attribution"><a class="source" href="https://svs.gsfc.nasa.gov/3662#media_group_352853">NASA</a></span>
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<p>For example, we can estimate the chances of an extinction-sized asteroid hitting Earth by examining how many such space rocks have hit the Moon over its history. A French scientist named Jean-Marc Salotti <a href="https://www.sciencedirect.com/science/article/pii/S0016328722000337">did this in 2022</a>, calculating the odds of an extinction-level hit in the next century at around one in 300 million. </p>
<p>Of course, such an estimate is fraught with uncertainty, but it is backed by something approaching an appropriate frequency calculation. Ord, by contrast, estimates the risk of extinction by asteroid at one in a million, though he does note a considerable degree of uncertainty.</p>
<h2>A ranking system for outcomes</h2>
<p>There is another way to think about probability, called Bayesianism after the English statistician Thomas Bayes. It focuses less on events themselves and more on what we know, expect and believe about them.</p>
<p>In very simple terms, we can say Bayesians see probabilities as a kind of ranking system. In this view, the specific number attached to a probability shouldn’t be taken directly, but rather compared to other probabilities to understand which outcomes are more and less likely.</p>
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Read more:
<a href="https://theconversation.com/bayes-theorem-the-maths-tool-we-probably-use-every-day-but-what-is-it-76140">Bayes' Theorem: the maths tool we probably use every day, but what is it?</a>
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<p>Ord’s book, for example, contains a table of potential extinction events and his personal estimates of their probability. From a Bayesian perspective, we can view these values as relative ranks. Ord thinks extinction from an asteroid strike (one in a million) is much less likely than extinction from climate change (one in a thousand), and both are far less likely than extinction from what he calls “unaligned artificial intelligence” (one in ten).</p>
<p>The difficulty here is that initial estimates of Bayesian probabilities (often called “priors”) are rather subjective (for instance, I would rank the chance of AI-based extinction much lower). Traditional Bayesian reasoning moves from “priors” to “posteriors” by again incorporating observational evidence of relevant outcomes to “update” probability values.</p>
<p>And once again, outcomes relevant to the probability of human extinction are thin on the ground. </p>
<h2>Subjective estimates</h2>
<p>There are two ways to think about the accuracy and usefulness of probability calculations: calibration and discrimination.</p>
<p>Calibration is the correctness of the actual values of the probabilities. We can’t determine this without appropriate observational information. Discrimination, on the other hand, simply refers to the relative rankings. </p>
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Read more:
<a href="https://theconversation.com/longtermism-why-the-million-year-philosophy-cant-be-ignored-193538">Longtermism – why the million-year philosophy can't be ignored</a>
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<p>We don’t have a basis to think Ord’s values are properly calibrated. Of course, this is not likely to be his intent. He himself indicates they are mostly designed to give “order of magnitude” indications. </p>
<p>Even so, without any related observational confirmation, most of these estimates simply remain in the subjective domain of prior probabilities.</p>
<h2>Not well calibrated – but perhaps still useful</h2>
<p>So what are we to make of “one in six”? Experience suggests most people have a less than perfect understanding of probability (as evidenced by, among other things, the ongoing volume of lottery ticket sales). In this environment, if you’re making an argument in public, an estimate of “probability” doesn’t necessarily need to be well calibrated – it just needs to have the right sort of psychological impact. </p>
<p>From this perspective, I’d say “one in six” fits the bill nicely. “One in 100” might feel small enough to ignore, while “one in three” might drive panic or be dismissed as apocalyptic raving. </p>
<p>As a person concerned about the future, I hope risks like climate change and nuclear proliferation get the attention they deserve. But as a data scientist, I hope the careless use of probability gets left by the wayside and is replaced by widespread education on its true meaning and appropriate usage.</p>
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Read more:
<a href="https://theconversation.com/433-people-win-a-lottery-jackpot-impossible-probability-and-psychology-suggest-its-more-likely-than-youd-think-191946">433 people win a lottery jackpot – impossible? Probability and psychology suggest it's more likely than you’d think</a>
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<img src="https://counter.theconversation.com/content/215054/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Steven Stern does not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>What are the odds of the end of humanity? There’s no real way to know.Steven Stern, Professor of Data Science, Bond UniversityLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/2136902023-09-18T12:19:43Z2023-09-18T12:19:43Z‘Big Bang of Numbers’ – The Conversation’s book club explores how math alone could create the universe with author Manil Suri<figure><img src="https://images.theconversation.com/files/548584/original/file-20230915-21-llxf3v.jpg?ixlib=rb-1.1.0&rect=30%2C36%2C4059%2C2955&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Fractals emerge on Day 4 of Suri's playful Genesis-inspired narrative about math's role in creation.</span> <span class="attribution"><a class="source" href="https://www.gettyimages.com/detail/photo/abstract-glowing-swirl-backgrounds-royalty-free-image/1129644961">oxygen/Moment via Getty Images</a></span></figcaption></figure><p><em>The Conversation U.S. launched its new book club with a bang – talking to mathematician <a href="https://scholar.google.com/citations?user=lFWFsSkAAAAJ&hl=en&oi=ao">Manil Suri</a> about his nonfiction work “<a href="https://wwnorton.com/books/9781324007036">The Big Bang of Numbers: How to Build the Universe Using Only Math</a>.” Suri, <a href="https://theconversation.com/pi-gets-all-the-fanfare-but-other-numbers-also-deserve-their-own-math-holidays-200046">a previous</a> <a href="https://theconversation.com/want-to-fix-gerrymandering-then-the-supreme-court-needs-to-listen-to-mathematicians-114345">author in</a> <a href="https://theconversation.com/declines-in-math-readiness-underscore-the-urgency-of-math-awareness-202691">The Conversation</a>, has also written an award-winning <a href="https://www.manilsuri.com/books">fiction trilogy</a>, in addition to being a professor of mathematics and statistics at the University of Maryland, Baltimore County.</em></p>
<p><em>Below is an edited excerpt from the book club discussion. You’re welcome to keep the conversation flowing by adding your own questions for Suri to the comments.</em></p>
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<figcaption><span class="caption">Watch the full book club meeting and leave your own question in the comments at the bottom of this article.</span></figcaption>
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<p><strong>What is the Big Bang of numbers and where do you go from there in the book?</strong></p>
<p>I think the story for me started way back when I was an undergraduate in Bombay. My algebra professor told us this very famous saying by <a href="https://www.britannica.com/biography/Leopold-Kronecker">Leopold Kronecker</a>, the famous mathematician, that God gave us the integers and all the rest is the work of human beings. What he meant was that once you have the whole numbers – 1, 2, 3, 4 – which are somehow coming from heaven, then you can build up the rest of mathematics from it.</p>
<p>And then he went on and said, Hey, I can actually do better. I don’t need God. I can actually, as a mathematician, create the numbers out of nothing. And he showed us this marvelous, almost magic trick, where you start with something called the empty set and then you start building the numbers.</p>
<p>It was the closest I’ve been to a religious experience, almost like the walls just dissolved and suddenly there were numbers everywhere. </p>
<p>Once I started writing my novels, I was meeting a lot of people who were artists and writers. And they would always say, you know, we used to love math when we were in school, but afterward we never had a chance to really pursue it. And can you tell us something about your mathematics?</p>
<p>So, I started building a kind of talk, which started with this big bang, as I call it, building the numbers out of nothing. I finally decided I should write a math book, and it would be aimed at a wide audience.</p>
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<span class="caption">Patterns in nature, like the triangles on this shell, can be explained by simple mathematical rules.</span>
<span class="attribution"><a class="source" href="https://wwnorton.com/books/9781324007036">Larry Cole</a></span>
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<p>And I said, well, can you go further? You can create the numbers, but can you actually start building everything, including the whole universe from that? So that was a way to try to lay out mathematics almost as a story where one thing follows from the other and everything is embedded in one narrative.</p>
<p><strong>Who were you imagining to be your readers as you were writing the book?</strong></p>
<p>There’s just so much joy to be had out of mathematics, so many things that you don’t really see in normal courses where the emphasis is always on doing the calculations, finding the right answer. So this book is written for people who want to really engage with mathematics on the level of ideas rather than get into computations and calculations.</p>
<p><strong>After you set off your Big Bang of numbers, you dig in to some of life’s big questions. What do you see as math’s role in grappling with those big thoughts, like where the universe came from, why we even exist and so on?</strong></p>
<p>Once you start talking about the Big Bang, what comes into your mind is creation. There is a doctrine called <em>creatio ex nihilo</em>, which is basically creating everything out of nothing. </p>
<p>That’s a cornerstone of many religions where God creates the universe out of nothing. It’s also in some sense being explored by physicists, where you have some sort of singularity and from that, everything emerges in the Big Bang.</p>
<p>So my thought was, both these areas, religion and physics, are in the public’s imagination much more than mathematics is. Is there a way to posit math as the creative force of everything?</p>
<p>Physicist <a href="https://www.nobelprize.org/prizes/physics/1963/wigner/biographical/">Eugene Wigner</a>, who was a Nobel laureate, talked about the “unreasonable effectiveness” of mathematics at describing everything in our physical universe. It’s so good at modeling physics and what have you. Could it be that math is really the true driving force of the universe? Rather than us just inventing it and using it to describe the universe, could the universe really be describing mathematics? Then the universe is just a physical manifestation, an approximation, if you will, of those mathematical ideas. It’s a completely different view of math.</p>
<p><strong>There’s an ongoing debate over whether math is something that people invented or whether it’s something that exists independently of us. In the book, you say that perhaps the deepest insight that math can offer us is that it’s both of those things.</strong></p>
<p>So the glib answer to your question whether math’s invented or discovered is that you have to create a new word. Instead of discovered or invented it’s “disvented.”</p>
<p>What I mean by that is simply that there are some questions we really can’t get to any kind of logical or supportable answer. One is the question of our own existence – people might believe one thing or the other, but it always comes down to: Is there some real purpose to our lives, or is our creation just something that happened randomly – you know, molecules getting together?</p>
<figure class="align-center zoomable">
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<span class="caption">Is math something that is born from the human mind?</span>
<span class="attribution"><a class="source" href="https://wwnorton.com/books/9781324007036">'The Big Bang of Numbers'</a></span>
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<p>Now if we invent mathematics, then we’re inventing it for a purpose. If it just generates by itself, starting with emptiness, building around numbers in some strange realm that we don’t know about, then it’s just wafting around, purposeless.</p>
<p>Math has that duality that can’t be resolved. So it’s a metaphor, telling us, hey, you can’t decide for math, and you’ll never be able to decide for yourself about your own existence. </p>
<p><strong>Can you tell us a bit about your previous books, the Indian novels?</strong></p>
<p>The first one was called “<a href="https://wwnorton.com/books/9780393342826">The Death of Vishnu</a>.” I went back to visit my parents in Mumbai in around 1995, and this man Vishnu, who used to live in our building and do errands, was dying on our steps. I started writing this as a short story.</p>
<p>It started going into a more philosophical realm when a writing teacher said, you know, Vishnu is also the name of the caretaker of the universe in Hindu mythology. So if you name somebody Vishnu, you need to somehow explore that. So that’s what opened up this whole new world for me.</p>
<p>The second book was “<a href="https://wwnorton.com/books/9780393333633">The Age of Shiva</a>.” That one’s the journey of a woman right after India’s independence in 1947. She’s making her way in a very male-dominated world, and she’s not perfect.</p>
<p>Then the third one, I decided, OK, I need to put in some science and math characters. So “<a href="https://wwnorton.co.uk/books/9780393346817-the-city-of-devi-77f37252-adcc-41f0-9b53-383405f76cab">The City of Devi</a>” actually has both a physicist and a statistician. Again it’s in Mumbai, set in the future with the threat of a nuclear war with Pakistan and a love triangle unfolding in front of that. </p>
<p>It’s kind of interesting. I thought that I was done with this mythical “where do we come from?” kind of philosophy that I had in the three books, but apparently not, because now “<a href="https://wwnorton.com/books/9781324007036">The Big Bang of Numbers</a>” looks at it from a mathematical perspective.</p><img src="https://counter.theconversation.com/content/213690/count.gif" alt="The Conversation" width="1" height="1" />
A book-length thought experiment uses math to investigate some of life’s big questions.Maggie Villiger, Senior Science + Technology EditorLicensed as Creative Commons – attribution, no derivatives.