tag:theconversation.com,2011:/id/topics/maths-skills-16361/articlesMaths skills – The Conversation2018-08-17T12:20:34Ztag:theconversation.com,2011:article/964412018-08-17T12:20:34Z2018-08-17T12:20:34ZMaths: six ways to help your child love it<figure><img src="https://images.theconversation.com/files/230790/original/file-20180806-191028-12mefqt.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">
</span> <span class="attribution"><span class="source">shutterstock</span></span></figcaption></figure><p>There is a widespread perception that mathematics is inaccessible, and ultimately boring. Just mentioning it can cause a negative reaction in people, as many mathematicians witness at any social event when the dreaded question arrives: “what is your job?”</p>
<p>For many people, school maths lessons are the time when any interest in the subject turns into disaffection. And eventually maths becomes a topic many people don’t want to engage with <a href="http://www.bsrlm.org.uk/wp-content/uploads/2016/02/BSRLM-IP-27-1-04.pdf">for the rest of their lives</a>. A percentage of the population, at least 17% – possibly much higher depending on <a href="https://www.frontiersin.org/articles/10.3389/fpsyg.2016.00508/full">the metrics applied</a> – develops maths anxiety. This is a debilitating fear of performing any numerical task, which results in chronic underachievement in subjects involving mathematics.</p>
<p>At the opposite end of the spectrum, professional mathematicians see mathematics as <a href="https://www.lms.ac.uk/library/frames-of-mind">fun, engaging, challenging and creative</a>. And as maths fans, we are trying to address this chasm in perception of mathematics, to allow everybody to access its beauty and power. So here are our six ways you can help children fall back in love with mathematics. </p>
<h2>1. Focus on the whys</h2>
<p>The Australian teacher <a href="https://www.youtube.com/channel/UCq0EGvLTyy-LLT1oUSO_0FQ">Eddie Woo</a> has become an internet sensation for his engaging way of presenting mathematics. He starts from the ideas and, using pictures and graphs, develops the theory. </p>
<p>He does not ask his students to do repetitive exercises, but to work with him in developing intuition. And he asks the most powerful question a learner of mathematics can ask: “Why?”. It is possible to hear throughout his classes the “oohs” and “ahhs” of students in the background, when a novel concept is understood. </p>
<h2>2. Make it relevant</h2>
<p>Traditionally (and in particular in the UK) mathematics is taught in a systematic way, <a href="https://eclass.uoa.gr/modules/document/file.php/MATH103/ELENA%20NARDI/NARDI3.pdf">based on rote learning and individual study</a>. Some students thrive in such a system, others, typically more empathetic students – often female – find such an approach to mathematics isolating and disconnected from their values and their reality.</p>
<p>Connecting mathematical concepts with applications in reality can bring meaning to lessons and lectures, and motivate students to put in the necessary effort to understand. For example, derivatives – ways of calculating rates of change – can be introduced as a way to measure slopes, and slopes are experienced in everyday life – think about the skatepark or the big hill you cycle up. </p>
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<img alt="" src="https://images.theconversation.com/files/230791/original/file-20180806-191038-197vk2x.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/230791/original/file-20180806-191038-197vk2x.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/230791/original/file-20180806-191038-197vk2x.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/230791/original/file-20180806-191038-197vk2x.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/230791/original/file-20180806-191038-197vk2x.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/230791/original/file-20180806-191038-197vk2x.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/230791/original/file-20180806-191038-197vk2x.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">Make maths about real life to capture kids imaginations.</span>
<span class="attribution"><span class="source">Pexels</span></span>
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<h2>3. Recognise the challenge</h2>
<p>There is an effort component in learning mathematics. It can be challenging, and understanding it sometimes involves stress, frustration, and struggle over time. This can be an emotionally complex environment for children. But it is one where persistence and perseverance are rewarded when a new concept is understood. </p>
<p>With each success, students gain confidence that they can progress in learning more mathematics. In this way, learning mathematics can be compared to climbing a mountain: plenty of effort, but also some truly blissful moments.</p>
<h2>4. Be a maths role model</h2>
<p>Some people like to climb mountains solo, while others prefer good company to share the effort. Similarly, some people are happy to study mathematics on their own, but others need more help <a href="https://www.nature.com/articles/srep23011">navigating this challenging subject</a>. Research shows that students who are failing in maths tend to be more empathetic than systematising. These are also the students more affected by reactions of people surrounding them: parents, teachers and the media. </p>
<h2>5. Make maths matter</h2>
<p>So given that <a href="https://hpl.uchicago.edu/sites/hpl.uchicago.edu/files/uploads/Maloney%252c%20E.A.%252c%20Schaeffer%252c%20M.W.%252c%20%26%20Beilock%252c%20S.L.%252c%20%25282013%2529.%20Mathematics%20anxiety%20and%20stereotype%20threat.pdf">maths anxiety can spread from one generation</a> to another, parents clearly have a role to play in making sure their children don’t clam up at the very thought of numbers. This is important, because a parent who learns how to avoid passing on mathematical anxiety gives their child a chance to learn a beautiful subject and to access <a href="http://www.bbc.co.uk/news/education-41693230">some of the best paid, most interesting, jobs around</a>. </p>
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<img alt="" src="https://images.theconversation.com/files/230792/original/file-20180806-191035-w0uvxk.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/230792/original/file-20180806-191035-w0uvxk.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/230792/original/file-20180806-191035-w0uvxk.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/230792/original/file-20180806-191035-w0uvxk.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/230792/original/file-20180806-191035-w0uvxk.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/230792/original/file-20180806-191035-w0uvxk.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/230792/original/file-20180806-191035-w0uvxk.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">Don’t scared of maths, it could rub off on your child.</span>
<span class="attribution"><span class="source">Shutterstock</span></span>
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<h2>6. Join the dots</h2>
<p>When it comes to maths, both inside and outside the classroom, the emphasis should shift from solely the numerical aspect to include connected aspects, such as concepts and links with other subjects and everyday applications. This will allow children to see mathematics as a social practice – where discussing mathematical challenges with classmates, teachers and parents becomes the norm.</p><img src="https://counter.theconversation.com/content/96441/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 organization that would benefit from this article, and have disclosed no relevant affiliations beyond their academic appointment.</span></em></p>Make maths more fun with these tipsSue Johnston-Wilder, Associate Professor, Mathematics Education, University of WarwickDavide Penazzi, Lecturer in Mathematics, University of Central LancashireLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/973162018-06-07T20:28:49Z2018-06-07T20:28:49ZBees join an elite group of species that understands the concept of zero as a number<figure><img src="https://images.theconversation.com/files/221898/original/file-20180606-137295-1owba2y.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Bees live in complex environments, and make lots of decisions every day that are crucial for survival. </span> <span class="attribution"><a class="source" href="https://www.shutterstock.com/image-photo/honey-bees-kept-bee-box-hive-1099240694?src=73MQBwoOpxtqlXSw7mEAng-1-75">from www.shutterstock.com </a></span></figcaption></figure><p>When it comes to bees, it seems that nothing really does matter. </p>
<p>As shown in a <a href="http://science.sciencemag.org/lookup/doi/10.1126/science.aar4975">paper</a> published today, our research demonstrates that the honeybee can understand the quantitative value of nothing, and place zero in the correct position along a line of sequential numbers. </p>
<p>This is the first evidence showing that an insect brain can understand the concept of zero, and has implications for our understanding of how complex number processing evolved. More broadly, it may help us design better artificial intelligence solutions for operating in complex environments.</p>
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Read more:
<a href="http://theconversation.com/want-a-better-camera-just-copy-bees-and-their-extra-light-sensing-eyes-80385">Want a better camera? Just copy bees and their extra light-sensing eyes</a>
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<iframe width="440" height="260" src="https://www.youtube.com/embed/KQkP85I2UJM?wmode=transparent&start=0" frameborder="0" allowfullscreen></iframe>
<figcaption><span class="caption">Bee brains are tiny - but they do get that zero is a number.</span></figcaption>
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<h2>What is ‘zero’, anyway?</h2>
<p>There are <a href="https://www.sciencedirect.com/science/article/pii/S1364661316301255">four stages of understanding the concept of zero</a> in human culture, history, psychology and animal learning.</p>
<p><strong>Stage one:</strong> Understanding zero as the absence of something, such as no food on your plate. This first level is likely enabled at an early stage of visual processing. </p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/221135/original/file-20180531-69490-1ja5sg3.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/221135/original/file-20180531-69490-1ja5sg3.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=370&fit=crop&dpr=1 600w, https://images.theconversation.com/files/221135/original/file-20180531-69490-1ja5sg3.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=370&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/221135/original/file-20180531-69490-1ja5sg3.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=370&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/221135/original/file-20180531-69490-1ja5sg3.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=464&fit=crop&dpr=1 754w, https://images.theconversation.com/files/221135/original/file-20180531-69490-1ja5sg3.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=464&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/221135/original/file-20180531-69490-1ja5sg3.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=464&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">Perceiving nothing on your plate requires understanding the absence of information.</span>
<span class="attribution"><span class="source">'Adrian Dyer/RMIT University'</span></span>
</figcaption>
</figure>
<p><strong>Stage two:</strong> Understanding zero as “nothing” vs. “something”, such as the presence or absence of light in a room. “Nothing” is thus treated as a meaningful behavioural category.</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/221138/original/file-20180531-69497-2kt6a1.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/221138/original/file-20180531-69497-2kt6a1.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=218&fit=crop&dpr=1 600w, https://images.theconversation.com/files/221138/original/file-20180531-69497-2kt6a1.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=218&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/221138/original/file-20180531-69497-2kt6a1.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=218&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/221138/original/file-20180531-69497-2kt6a1.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=274&fit=crop&dpr=1 754w, https://images.theconversation.com/files/221138/original/file-20180531-69497-2kt6a1.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=274&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/221138/original/file-20180531-69497-2kt6a1.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=274&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Perceiving the absence of information relative to a stimulus like a light is the second stage of understanding zero.</span>
<span class="attribution"><span class="source">'Adrian Dyer/RMIT University'</span></span>
</figcaption>
</figure>
<p><strong>Stage three:</strong> Understanding that zero can have a numeric value and belongs at the low end of the positive number line. For example: 0 < 1 < 2 < 3 etc. (where < means “less than”). </p>
<figure class="align-right ">
<img alt="" src="https://images.theconversation.com/files/221133/original/file-20180531-69484-zv83tx.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/221133/original/file-20180531-69484-zv83tx.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=751&fit=crop&dpr=1 600w, https://images.theconversation.com/files/221133/original/file-20180531-69484-zv83tx.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=751&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/221133/original/file-20180531-69484-zv83tx.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=751&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/221133/original/file-20180531-69484-zv83tx.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=944&fit=crop&dpr=1 754w, https://images.theconversation.com/files/221133/original/file-20180531-69484-zv83tx.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=944&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/221133/original/file-20180531-69484-zv83tx.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=944&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">By learning to choose less than options, bees learnt over one day to be able to transfer information and place zero at the lower end of all previously experienced numbers.</span>
<span class="attribution"><span class="source">Scarlett Howard, Adrian Dyer and Jair Garcia/RMIT University</span></span>
</figcaption>
</figure>
<p><strong>Stage four:</strong> Understanding that zero can be assigned a symbolic representation which can be used in modern mathematics and calculations, for example: 1 – 1 = 0.</p>
<p>Our new study shows honeybees have achieved stage three of understanding the concept of zero. </p>
<p>The honeybee now joins the elite few species which have demonstrated an understanding of zero to this advanced level. While <a href="https://www.sciencedirect.com/science/article/pii/S0010027700001128">rhesus monkeys</a>, <a href="https://www.hindawi.com/journals/ijz/2011/806589/">vervet monkeys</a>, a single <a href="https://link.springer.com/article/10.1007/s100710100086">chimpanzee</a>, and one <a href="https://homepages.uni-tuebingen.de/andreas.nieder/Nieder%20(2016)%20TICS.pdf">African grey parrot</a> have demonstrated the ability to learn or spontaneously understand the concept of zero, this is the first time that such a high level of cognitive number processing has been observed in an insect.</p>
<h2>Why care about zero?</h2>
<p>The <a href="https://homepages.uni-tuebingen.de/andreas.nieder/Nieder%20(2016)%20TICS.pdf">importance of zero</a> throughout human history is not to be underestimated. </p>
<p><a href="http://www-groups.dcs.st-and.ac.uk/history/HistTopics/Chinese_numerals.html">Chinese counting rods used a blank space</a> to help represent a place holder in values, however zero went unnoticed as a number with a quantitative value for centuries. For example, Roman numerals do not have a symbol for zero. </p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/221140/original/file-20180531-69501-196wsew.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/221140/original/file-20180531-69501-196wsew.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=218&fit=crop&dpr=1 600w, https://images.theconversation.com/files/221140/original/file-20180531-69501-196wsew.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=218&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/221140/original/file-20180531-69501-196wsew.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=218&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/221140/original/file-20180531-69501-196wsew.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=274&fit=crop&dpr=1 754w, https://images.theconversation.com/files/221140/original/file-20180531-69501-196wsew.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=274&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/221140/original/file-20180531-69501-196wsew.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=274&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">Zero in Chinese Rod Calculus around 4th Century B.C.</span>
<span class="attribution"><a class="source" href="https://en.wikipedia.org/wiki/0#/media/File:Zero_in_Rod_Calculus.png">Wikimedia Commons</a></span>
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<p>It was not until <a href="https://archive.org/details/Brahmasphutasiddhanta_Vol_1">628AD that zero had a written record</a> which noted it as a number in its own right by Indian mathematician Brahma Gupta in his book Brahmasputha Siddhanta. This is the first written record to provide rules to use when doing calculations with zero. </p>
<p>The earliest record of the symbolic zero (0) we are familiar with today is from an <a href="https://www.livehistoryindia.com/amazing-india/2017/04/29/zero-number-one">Indian inscription on the wall</a> of a temple in Gwalior, India (AD 876). Arabic numerals, along with the modern idea of zero, did not reach the West <a href="https://www.britannica.com/topic/Hindu-Arabic-numerals">until 1200 AD</a>. </p>
<p>Interestingly, while it took centuries for the concept of zero to be fully understood and utilised in human culture, honeybees have learnt to apply previous number knowledge to demonstrate an understanding of zero <em>within a day</em> when presented with training to promote numerical cognition.</p>
<h2>How we asked bees about zero</h2>
<p>Bees often forage in <a href="https://theconversation.com/plants-use-advertising-like-strategies-to-attract-bees-with-colour-and-scent-92673">complex environments</a> and have evolved <a href="https://theconversation.com/which-square-is-bigger-honeybees-see-visual-illusions-like-humans-do-87673">visual processing solutions</a> adapted to this life. </p>
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Read more:
<a href="http://theconversation.com/which-square-is-bigger-honeybees-see-visual-illusions-like-humans-do-87673">Which square is bigger? Honeybees see visual illusions like humans do</a>
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<p>In our research, we tested number processing in bees by individually training them with special apparatus to collect a sugary reward, and learn the rules of “less than” considering the numbers 1 – 6. </p>
<p>An individual bee would need to choose between two numbers each time it returned to the experiment. For example, a bee would be presented with two new numbers (3 vs. 4; 1 vs. 2; 2 vs. 5, etc.) until it had reached, over many learning events, at least 80% accuracy for landing on and thus choosing the lowest number.</p>
<p>Once the bee achieved this, it would be presented with the previously unseen stimulus of “an empty set” representing zero. </p>
<figure class="align-right ">
<img alt="" src="https://images.theconversation.com/files/221707/original/file-20180605-119885-1mp78sb.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/221707/original/file-20180605-119885-1mp78sb.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=837&fit=crop&dpr=1 600w, https://images.theconversation.com/files/221707/original/file-20180605-119885-1mp78sb.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=837&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/221707/original/file-20180605-119885-1mp78sb.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=837&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/221707/original/file-20180605-119885-1mp78sb.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=1052&fit=crop&dpr=1 754w, https://images.theconversation.com/files/221707/original/file-20180605-119885-1mp78sb.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=1052&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/221707/original/file-20180605-119885-1mp78sb.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=1052&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">A honeybee chooses to land on the zero option.</span>
<span class="attribution"><span class="source">Scarlett Howard/RMIT University</span></span>
</figcaption>
</figure>
<p>Surprisingly, bees trained to the “less than” rule preferred to visit the empty set rather than any other higher value number. This means bees understood an empty set was lower in number than a set containing actual elements.</p>
<p>In further experiments, other bees were able to place zero at the low end of the numerical continuum and demonstrated numerical distance effects. Numerical distance effects are demonstrated when accuracy increases as the difference between two numbers increases. The study showed that while bees could differentiate between zero and one, they performed better when the numbers were further apart, such as in the case of zero vs. six.</p>
<p>The next step for research on the processing of zero is to understand how small and seemingly simple brains (like those of bees) represent zero in a neurological sense. </p>
<hr>
<p>
<em>
<strong>
Read more:
<a href="http://theconversation.com/plants-use-advertising-like-strategies-to-attract-bees-with-colour-and-scent-92673">Plants use advertising-like strategies to attract bees with colour and scent</a>
</strong>
</em>
</p>
<hr>
<p>Andreas Nieder, an expert in numerical competency in animals from the University of Tübingen in Germany <a href="http://science.sciencemag.org/content/360/6393/1069">writes</a>: </p>
<blockquote>
<p>The advanced numerical skills of bees and other animals raise the question of how their brains transform “nothing” into an abstract concept of zero. </p>
</blockquote>
<p>This new research on bees has created many new questions in the field and also makes it clear that brain size and complexity does not fully determine intelligence and, in particular, numerical ability.</p><img src="https://counter.theconversation.com/content/97316/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Scarlett Howard received funding from The Company of Biologists (JEB Travelling Fellowship) and has an Australian Government Research Training Program Scholarship.</span></em></p><p class="fine-print"><em><span>Adrian Dyer receives funding from The Australian Research Council.</span></em></p><p class="fine-print"><em><span>Aurore Avarguès-Weber a reçu des financements de la fondation Fyssen. </span></em></p>The Romans may not have had a symbol for zero, but bees understand what it means beyond just the simple assumption "there's nothing there".Scarlett Howard, PhD candidate, RMIT UniversityAdrian Dyer, Associate Professor, RMIT UniversityAurore Avarguès-Weber, Researcher , Université de Toulouse III - Paul SabatierLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/928582018-03-15T11:20:15Z2018-03-15T11:20:15ZHigh number of adults unable to do basic mathematical tasks<figure><img src="https://images.theconversation.com/files/210274/original/file-20180314-113469-1dxoeka.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">'I don't even know how much this is going to cost.'</span> <span class="attribution"><a class="source" href="https://www.shutterstock.com/download/success?src=_8uSGJoRNieQEMUSqROWJA-1-36">shutterstock</a></span></figcaption></figure><p>Suppose, a litre of cola costs US$3.15. If you buy one third of a litre of cola, how much would you pay?</p>
<p>The above may seem like a rather basic question. Something that you would perhaps expect the vast majority of adults to be able to answer? Particularly if they are allowed to use a calculator. </p>
<p>Unfortunately, the reality is that a large number of adults across the world struggle with even such basic financial tasks (the correct answer is US$1.05, by the way).</p>
<p>Using Organisation for Economic Cooperation and Development (OECD) Programme for International Assessment of Adult Competencies (PIAAC) <a href="http://www.google.co.uk/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&ved=0ahUKEwjQqYO009fZAhXLDMAKHRKQBF8QFggnMAA&url=http%3A%2F%2Fwww.oecd.org%2Fskills%2Fpiaac%2F&usg=AOvVaw0doGOG0MDqeIWDq46xfSIa">data</a>, my co-authors and I have looked at how adults from 31 countries answer four <a href="https://johnjerrim.com/piaac/">relatively simple financial questions</a>. </p>
<figure class="align-right zoomable">
<a href="https://images.theconversation.com/files/209099/original/file-20180306-146661-16wex9g.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/209099/original/file-20180306-146661-16wex9g.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/209099/original/file-20180306-146661-16wex9g.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=964&fit=crop&dpr=1 600w, https://images.theconversation.com/files/209099/original/file-20180306-146661-16wex9g.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=964&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/209099/original/file-20180306-146661-16wex9g.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=964&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/209099/original/file-20180306-146661-16wex9g.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=1211&fit=crop&dpr=1 754w, https://images.theconversation.com/files/209099/original/file-20180306-146661-16wex9g.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=1211&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/209099/original/file-20180306-146661-16wex9g.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=1211&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 estimated proportion of adults who could answer the question correctly.</span>
<span class="attribution"><span class="license">Author provided</span></span>
</figcaption>
</figure>
<p>As well as the question above, participants were asked questions such as: “Suppose, upon your trip to the grocery store you purchase four types of tea packs: Chamomile Tea (US$4.60), Green Tea (US$4.15), Black Tea (US$3.35) and Lemon Tea (US$1.80). If you paid for all these items with a US$20 bill, how much change would you get?”</p>
<p>The results (as seen in the table) allowed us to create an estimated range for the percentage of the adult population who would be able to answer the cola question correctly. These results are based upon a random sample of adults from each country.</p>
<p>We found that Lithuania, Austria and Slovakia were the most successful, but even in these countries, one in four adults failed to give the correct answer. </p>
<p>In many other countries, the situation is even worse. Four in every ten adults in places like England, Canada, Spain and the US can’t make this straightforward calculation – even when they had a calculator to hand. Similarly, less than half of adults in places like Chile, Turkey and South Korea can get the right answer. </p>
<h2>Basic calculations</h2>
<p>Of course, not all groups within each country perform quite so poorly, and there are notable differences in financial literacy skills between different demographic groups. </p>
<p>Across the four financial questions adults were asked, in most countries, men tended to perform slightly better than women. The young (particularly 25- to 34-year-olds) were also found to perform better than the over-55s. </p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/210276/original/file-20180314-113469-bv14d3.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/210276/original/file-20180314-113469-bv14d3.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/210276/original/file-20180314-113469-bv14d3.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/210276/original/file-20180314-113469-bv14d3.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/210276/original/file-20180314-113469-bv14d3.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/210276/original/file-20180314-113469-bv14d3.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/210276/original/file-20180314-113469-bv14d3.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">Many adults struggle with basic financial tasks, like working out what’s better value at a supermarket.</span>
<span class="attribution"><a class="source" href="https://www.shutterstock.com/download/success?src=_8uSGJoRNieQEMUSqROWJA-1-53">Shutterstock</a></span>
</figcaption>
</figure>
<p>The starkest differences were seen by education group. Returning to the first question given above, in many countries adults with a “low” level of education (the equivalent of completing secondary school) had less than a 50% chance of getting the question correct. In places like Canada and United States, this fell to as low as 25%. </p>
<h2>Financial headache</h2>
<p>Our results clearly highlight how many adults are ill equipped to make key financial decisions. And how in fact, many struggle to cope with even very simple financial tasks. </p>
<p>In the long term, this highlights the critical need for financial literacy to be taught in schools, to ensure young people are equipped for the complex financial decisions they will face in the real world. </p>
<p>More immediately, though, given the low level of financial skills among many adults, it is vital that the information provided with financial products is as simple and straightforward to interpret as possible. And in the age of payday loans, and high interest credit cards, adequate advice and guidance must also be available where needed. Because otherwise, there is a real danger that a large proportion of the population is at risk of making serious financial mistakes.</p><img src="https://counter.theconversation.com/content/92858/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>The team who wrote the paper received funding from the OECD. This work was commissioned by the Organisation for Economic Co-operation and Development (OECD) who has granted Cambridge University Technical Services Ltd the right to publish it for non-commercial purposes only. </span></em></p>New research shows that many adults across the world are financially illiterate and unable to complete even basic mathematical calculations.John Jerrim, Lecturer in Economics and Social Statistics, UCLLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/917572018-02-27T15:22:11Z2018-02-27T15:22:11ZMathematics: forget simplicity, the abstract is beautiful - and important<figure><img src="https://images.theconversation.com/files/207820/original/file-20180226-140213-yox11e.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">
</span> <span class="attribution"><span class="source">Shutterstock</span></span></figcaption></figure><p>Why is mathematics so complicated? It’s a question many students will ask while grappling with a particularly complex calculus problem – and their teachers will probably echo while setting or marking tests.</p>
<p>It wasn’t always this way. Many fields of mathematics germinated from the study of real world problems, before the underlying rules and concepts were identified. These rules and concepts were then defined as abstract structures. For instance, algebra, the part of mathematics in which letters and other general symbols are used to represent numbers and quantities in formulas and equations was born from solving problems in arithmetic. Geometry emerged as people worked to solve problems dealing with distances and area in the real world. </p>
<p>That process of moving from the concrete to the abstract scenario is known, appropriately enough, as <a href="https://betterexplained.com/articles/learning-to-learn-math-abstraction/">abstraction</a>. Through abstraction, the underlying essence of a mathematical concept can be extracted. People no longer have to depend on real world objects, as was once the case, to solve a mathematical puzzle. They can now generalise to have wider applications or by matching it to other structures can illuminate similar phenomena. An example is the adding of integers, fractions, complex numbers, vectors and matrices. The concept is the same, but the applications are different. </p>
<p>This evolution was necessary for the development of mathematics, and important for other scientific disciplines too. </p>
<p>Why is this important? Because the growth of abstraction in maths gave disciplines like chemistry, physics, astronomy, geology, meteorology the ability to explain a wide variety of complex physical phenomena that occur in nature. If you grasp the process of abstraction in mathematics, it will equip you to better understand abstraction occurring in other tough science subjects like chemistry or physics.</p>
<h2>From the real world to the abstract</h2>
<p>The earliest example of abstraction was when humans counted before symbols existed. A sheep herder, for instance, needed to keep track of his flock of sheep without having any sort of symbolic system akin to numbers. So how did he do this to ensure that none of his sheep wandered away or got stolen?</p>
<p>One solution is to obtain a big supply of stones. He then moved the sheep one-by-one into an enclosed area. Each time a sheep passed, he placed a stone in a pile. Once all the sheep had passed, he got rid of the extra stones and was left with a pile of stones representing his flock. </p>
<p>Every time he needed to count the sheep, he removed the stones from his pile; one for each sheep. If he had stones left over, it means some sheep had wandered away or perhaps been stolen. This one-to-one correspondence helped the shepherd to keep track of his flock. </p>
<p>Today, we use the Arabic numbers (also known as the <a href="https://www.britannica.com/topic/Hindu-Arabic-numerals">Hindu-Arabic numerals</a>): 0,1,2,3,4,5,6,7,8,9 to represent any integer, that is any whole number. </p>
<p>This is another example of abstraction, and it’s powerful. It means we’re able to handle any amount of sheep, regardless of how many stones we have. We’ve moved from real-world objects – stones, sheep – to the abstract. There is real strength in this: we’ve created a space where the rules are minimalistic, yet the games that can be played are endless.</p>
<p>Another advantage of abstraction is that it reveals a deeper connection between different fields of mathematics. Results in one field can suggest concepts and ideas to be explored in a related field. Occasionally, methods and techniques developed in one field can be directly applied to another field to create similar results. </p>
<h2>Tough concepts, better teaching</h2>
<p>Of course, abstraction also has its disadvantages. Some of the mathematical subjects taught at university level – Calculus, Real Analysis, Linear Algebra, Topology, Category Theory, Functional Analysis and Set Theory among them – are very advanced examples of abstraction. </p>
<p>These concepts can be quite difficult to learn. They’re often tough to visualise and their rules rather unintuitive to manipulate or reason with. This means students need a degree of mathematical maturity to process the shift from the concrete to the abstract. </p>
<p>Many high school kids, particularly from developing countries, come to university with an <a href="https://link.springer.com/chapter/10.1007/978-3-319-12688-3_18">undeveloped level</a> of intellectual maturity to handle abstraction. This is because of the way mathematics was taught at high school. I have seen many students struggling, giving up or not even attempting to study mathematics because they weren’t given the right tools at school level and they think that they just “can’t do maths”. </p>
<p>Teachers and lecturers can improve this abstract thinking by being aware of abstractions in their subject and learning to demonstrate abstract concepts through concrete examples. Experiments are also helpful to familiarise and assure students of an abstract concept’s solidity.</p>
<p>This teaching principle is applied in some school systems, such as <a href="http://montessoritraining.blogspot.co.za/2008/07/montessori-philosophy-moving-from.html">Montessori</a>, to help children improve their abstract thinking. Not only does this guide them better through the maze of mathematical abstractions but it can be applied to other sciences as well.</p><img src="https://counter.theconversation.com/content/91757/count.gif" alt="La Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Harry Zandberg Wiggins ne travaille pas, ne conseille pas, ne possède pas de parts, ne reçoit pas de fonds d'une organisation qui pourrait tirer profit de cet article, et n'a déclaré aucune autre affiliation que son poste universitaire.</span></em></p>Through abstraction, the underlying essence of a mathematical concept can be extracted.Harry Zandberg Wiggins, Lecturer, University of PretoriaLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/886782017-12-15T11:14:26Z2017-12-15T11:14:26ZMaths challenge: England has one of the biggest gaps between high and low performing pupils in the developed world<figure><img src="https://images.theconversation.com/files/199038/original/file-20171213-27555-xl6o5f.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">England’s top performing maths pupils achieve a very high standard but the bottom performers lag far behind.</span> <span class="attribution"><span class="source">shutterstock</span></span></figcaption></figure><p>When it comes to maths, many primary school children in the UK are struggling to achieve their potential, <a href="https://epi.org.uk/report/world-class-primary/">according to new research</a>. </p>
<p><a href="https://epi.org.uk/report/world-class-primary/">The recent report</a> from the <a href="https://epi.org.uk/report/world-class-primary/">Education Policy Institute</a> and UCL’s Institute of Education shows that England has one of the biggest gaps between high and low performing students in the developed world. Only New Zealand and Turkey have a bigger disparity.</p>
<p>So while England’s top performing maths pupils achieve a very high standard, the bottom performers lag far behind – with this gap well established before pupils reach secondary school.</p>
<p>It’s not surprising then that “mastery” has become something of a buzzword in the UK in the last five years. <a href="http://www.nama.org.uk/Downloads/Five%20Myths%20about%20Mathematics%20Mastery.pdf">It’s a word with lots of different meanings</a>, but it’s usually linked to how mathematics is taught in East Asia – particularly in Shanghai and Singapore. Both of which are very successful in <a href="https://theconversation.com/pisa-results-four-reasons-why-east-asia-continues-to-top-the-leaderboard-69951">international league tables such as PISA</a>.</p>
<p>In Shanghai and Singapore the mastery method involves whole class <a href="http://beyondlevels.website/wp-content/uploads/2016/05/Mastery-differentiation-and-fixed-ability-thinking-M-Boylan-Learning-first-Sheffield.pdf">interactive teaching as the main approach</a>. The idea is that by using teacher questions, step by step progression, diagrams and carefully designed practice exercises, all pupils progress together. And daily intervention is also used to support those pupils who need extra tuition.</p>
<h2>The maths gap</h2>
<p>Interest in adopting East Asian approaches to maths have recently been made an educational priority by the UK government. In the recent budget, the chancellor Philip Hammond announced <a href="https://schoolsweek.co.uk/budget-2017-hammond-announces-cash-for-computing-and-maths-mastery/">plans to invest £27m in the expansion</a> of the Teaching for Mastery maths programme to a further 3,000 schools. </p>
<p>The government has previously issued funding for a teacher exchange with Shanghai along with a whole raft of other measures to boost the uptake of the mastery approach to British schools. As part of my research, colleagues and I have been <a href="https://www.gov.uk/government/uploads/system/uploads/attachment_data/file/666450/MTE_third_interim_report_121217.pdf">evaluating the teacher exchange programme with Shanghai</a>. </p>
<p>We discovered after the initial exchange, responses were quite varied in the 48 schools that sent and hosted teachers in 2014 and 2015. Some teachers decided the method wasn’t for them, others adopted some practices, while some completely changed their teaching methods. </p>
<figure>
<iframe width="440" height="260" src="https://www.youtube.com/embed/CHP6Gc7CqW0?wmode=transparent&start=0" frameborder="0" allowfullscreen></iframe>
</figure>
<p>Schools adopting a mastery approach often follow the official guidance of the <a href="https://www.ncetm.org.uk/resources/47230">National Centre for Teaching Mathematics</a> – but there are lots of other influences. Not only the <a href="https://www.ncetm.org.uk/resources/49739">Shanghai exchange</a> but also independent curriculum and professional development organisations – such as <a href="https://mathsnoproblem.com/">Maths No Problem </a> and <a href="https://www.mathematicsmastery.org/">Mathematics Mastery</a>. On top of all this, “mastery” is also the latest catchphrase for suppliers of resources and educational consultants. </p>
<p>The outcome is often lots of experimentation and variation in schools – and ultimately, lots of different versions of mastery. But overall, the types of changes we saw schools make after the exchange are ones mathematics educators have advocated for a <a href="https://educationendowmentfoundation.org.uk/tools/guidance-reports/maths-ks-two-three/">long time and are backed by research</a>.</p>
<h2>Mastering mastery</h2>
<p>In our evaluation, we found a number of ways schools are implementing the mastery method into their classrooms. For some schools, this has meant using an <a href="https://mathsnoproblem.com/">East Asian inspired textbook</a>. For others, they have continued to select from a wide range of resources, but think more about mastery methods in everyday teaching. </p>
<p>We have seen how schools involved in the exchange have made much greater use of concrete and visual models to support abstract thinking. Equipment that previously might only have been used with young children or those struggling with mathematics, has been dusted off and integrated into teaching more generally – <a href="https://educationendowmentfoundation.org.uk/tools/guidance-reports/maths-ks-two-three/">which has been shown to help children’s understanding</a>.</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/199042/original/file-20171213-27555-1m0n63s.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/199042/original/file-20171213-27555-1m0n63s.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/199042/original/file-20171213-27555-1m0n63s.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/199042/original/file-20171213-27555-1m0n63s.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/199042/original/file-20171213-27555-1m0n63s.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/199042/original/file-20171213-27555-1m0n63s.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/199042/original/file-20171213-27555-1m0n63s.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">If England is to be considered world class at primary in maths, the performance of pupils at the bottom must be improved.</span>
<span class="attribution"><span class="source">Shutterstock</span></span>
</figcaption>
</figure>
<p>We have also seen more schools <a href="https://www.tes.com/news/school-news/breaking-views/how-make-mixed-ability-work-let-children-take-control-lesson">teaching in “mixed ability” sessions</a> (the Shanghai way) with teaching at a pace whereby children don’t get left behind – if children don’t “get it” in the lesson, they are given daily catch up support.</p>
<p>But we have found that many of the schools involved in the exchange have been less keen to adopt the mastery way of seating pupils in rows and daily homework. In these schools, pupils are more likely to face the front only some of the time, and instead of daily homework, time is used each day for maths practice within school. </p>
<h2>Making it add up</h2>
<p>But despite many British schools now adopting some form of mastery approach to mathematics, there still tends to be scepticism around the approach with lots of <a href="http://www.nama.org.uk/Downloads/Five%20Myths%20about%20Mathematics%20Mastery.pdf">myths about mastery emerging</a>. </p>
<p>Mastery sceptics point to the <a href="https://www.theguardian.com/world/2014/oct/09/east-asian-school-success-culture-curriculum-teaching">cultural factors</a> and <a href="https://schoolsweek.co.uk/schools-minister-admits-shanghai-maths-teachers-only-do-two-lessons-a-day/">very different working conditions of East Asian teachers</a> as underpinning their success. And there is also concern over <a href="https://schoolsweek.co.uk/what-is-teaching-for-mastery-in-maths/">whether the fastest learners will be held back</a>.</p>
<p>The sceptics might be proved right, but the way maths is taught in many English schools is changing. So if benefits for pupils are shown, then conversations may well shift from “whether” to implement mastery, to “which” aspects of East Asian teaching are most useful when adapted for the Western classroom.</p><img src="https://counter.theconversation.com/content/88678/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Mark Boylan receives funding from the Department for Education, England for evaluating the Mathematics Teacher Exchange: China-England. </span></em></p>But could the influences of Shanghai and Singapore help?Mark Boylan, Professor of Education, Sheffield Hallam UniversityLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/786602017-06-21T10:31:01Z2017-06-21T10:31:01ZChallenging the status quo in mathematics: Teaching for understanding<figure><img src="https://images.theconversation.com/files/174303/original/file-20170618-28772-1vhqkpw.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">How can we change math instruction to meet the needs of today's kids?</span> <span class="attribution"><a class="source" href="https://flic.kr/p/97aGY8">World Bank Photo Collection / flickr</a>, <a class="license" href="http://creativecommons.org/licenses/by-nc-nd/4.0/">CC BY-NC-ND</a></span></figcaption></figure><p>Despite decades of <a href="http://files.eric.ed.gov/fulltext/ED372969.pdf">reform efforts</a>, mathematics teaching in the U.S. <a href="http://www.jstor.org/stable/20405948">has changed little</a> in the last century. As a result, it seems, American students have been left behind, now ranking <a href="https://nces.ed.gov/pubs2017/2017048.pdf#page=31">40th in the world</a> in math literacy. </p>
<p>Several state and national reform efforts have tried to improve things. The most recent <a href="http://www.corestandards.org/Math/">Common Core standards</a> had a great deal of promise with their focus on how to teach mathematics, but after several years, <a href="http://journals.sagepub.com/doi/full/10.3102/0013189X17711899">changes in teaching practices</a> have been minimal. </p>
<p><iframe id="Grc6N" class="tc-infographic-datawrapper" src="https://datawrapper.dwcdn.net/Grc6N/1/" height="400px" width="100%" style="border: none" frameborder="0"></iframe></p>
<p>As an education researcher, I’ve observed teachers trying to implement reforms – often with limited success. They sometimes make changes that are more cosmetic than substantive (e.g., more student discussion and group activity), while failing to get at the heart of the matter: What does it truly mean to teach and learn mathematics?</p>
<h2>Traditional mathematics teaching</h2>
<p>Traditional middle or high school mathematics teaching in the U.S. <a href="http://www.jstor.org/stable/20405948">typically follows this pattern</a>: The teacher demonstrates a set of procedures that can be used to solve a particular kind of problem. A similar problem is then introduced for the class to solve together. Then, the students get a number of exercises to practice on their own.</p>
<figure class="align-right ">
<img alt="" src="https://images.theconversation.com/files/174300/original/file-20170618-28759-1jyothn.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/174300/original/file-20170618-28759-1jyothn.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=686&fit=crop&dpr=1 600w, https://images.theconversation.com/files/174300/original/file-20170618-28759-1jyothn.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=686&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/174300/original/file-20170618-28759-1jyothn.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=686&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/174300/original/file-20170618-28759-1jyothn.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=862&fit=crop&dpr=1 754w, https://images.theconversation.com/files/174300/original/file-20170618-28759-1jyothn.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=862&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/174300/original/file-20170618-28759-1jyothn.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=862&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">The basics of math instruction have changed little since George Eaton taught at Phillips Academy (1880-1930).</span>
<span class="attribution"><a class="source" href="https://flic.kr/p/jKrzFZ">Phillips Academy Archives and Special Collections / flickr</a></span>
</figcaption>
</figure>
<p>For example, when students learn about the area of shapes, they’re given a set of formulas. They put numbers into the correct formula and compute a solution. More complex questions might give the students the area and have them work backwards to find a missing dimension. Students will often learn a different set of formulas each day: perhaps squares and rectangles one day, triangles the next. </p>
<p>Students in these kinds of lessons are learning to follow a rote process to arrive at a solution. This kind of instruction is so common that it’s seldom even questioned. After all, within a particular lesson, it makes the math seem easier, and students who are successful at getting the right answers find this kind of teaching to be very satisfying.</p>
<p>But it turns out that teaching mathematics this way can actually <a href="http://www.jstor.org/stable/3696735">hinder learning</a>. Children can become dependent on <a href="http://www.jstor.org/stable/10.5951/teacchilmath.21.1.0018">tricks and rules</a> that don’t hold true in all situations, making it harder to adapt their knowledge to new situations.</p>
<p>For example, in traditional teaching, children learn that they should distribute a number by multiplying across parentheses and will practice doing so with numerous examples. When they begin learning how to solve equations, they often have trouble realizing that it’s not always needed. To illustrate, take the equation 3(x + 5) = 30. Children are likely to multiply the 3 across the parentheses to make 3x + 15 = 30. They might just as easily have divided both sides by 3 to make x + 5 = 10, but a child who learned the distribution method might have great difficulty recognizing the alternate method – or even that both procedures are equally correct.</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/174582/original/file-20170619-22075-1mmjc2g.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/174582/original/file-20170619-22075-1mmjc2g.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=320&fit=crop&dpr=1 600w, https://images.theconversation.com/files/174582/original/file-20170619-22075-1mmjc2g.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=320&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/174582/original/file-20170619-22075-1mmjc2g.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=320&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/174582/original/file-20170619-22075-1mmjc2g.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=402&fit=crop&dpr=1 754w, https://images.theconversation.com/files/174582/original/file-20170619-22075-1mmjc2g.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=402&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/174582/original/file-20170619-22075-1mmjc2g.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=402&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Students who learn by rote drilling often have trouble realizing that there are equally valid alternative methods for solving a problem.</span>
<span class="attribution"><span class="source">Kaitlyn Chantry</span></span>
</figcaption>
</figure>
<h2>More than a right answer</h2>
<p>A key missing ingredient in these traditional lessons is conceptual understanding. </p>
<p>Concepts are ideas, meaning and relationships. It’s not just about knowing the procedure (like how to compute the area of a triangle) but also the significance behind the procedure (like what area means). How concepts and procedures are related is important as well, such as how the area of a triangle can be considered half the area of a rectangle and how that relationship can be seen in their area formulas. </p>
<p>Teaching for conceptual understanding has <a href="http://math.coe.uga.edu/Olive/EMAT3500f08/instrumental-relational.pdf">several benefits</a>. Less information has to be memorized, and students can translate their knowledge to new situations more easily. For example, understanding what area means and how areas of different shapes are related can help students understand the concept of volume better. And learning the relationship between area and volume can help students understand how to interpret what the volume means once it’s been calculated.</p>
<p>In short, building relationships between <a href="https://doi.org/10.1007/s10648-015-9302-x">how to solve a problem and why it’s solved that way</a> helps students <a href="https://doi.org/10.1037//0022-0663.91.1.175">use what they already know</a> to solve new problems that they face. Students with a truly conceptual understanding can see how methods emerged from <a href="https://doi.org/10.1037/0022-0663.91.1.175">multiple interconnected ideas</a>; their relationship to the solution goes deeper than rote drilling.</p>
<p>Teaching this way is a critical first step if students are to begin recognizing mathematics as meaningful. Conceptual understanding is a key ingredient to helping people think mathematically and use mathematics outside of a classroom.</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/174193/original/file-20170616-537-p8ad2j.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/174193/original/file-20170616-537-p8ad2j.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=412&fit=crop&dpr=1 600w, https://images.theconversation.com/files/174193/original/file-20170616-537-p8ad2j.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=412&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/174193/original/file-20170616-537-p8ad2j.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=412&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/174193/original/file-20170616-537-p8ad2j.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=517&fit=crop&dpr=1 754w, https://images.theconversation.com/files/174193/original/file-20170616-537-p8ad2j.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=517&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/174193/original/file-20170616-537-p8ad2j.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=517&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Procedural learning promotes memorization instead of critical thinking and problem solving.</span>
<span class="attribution"><a class="source" href="https://www.shutterstock.com/image-photo/math-study-exam-set-book-pencil-250606378">m.jrn/shutterstock.com</a></span>
</figcaption>
</figure>
<h2>The will to change</h2>
<p>Conceptual understanding in mathematics has been recognized as important for <a href="http://www.nctm.org/uploadedFiles/About/President,_Board_and_Committees/Board_Materials/MLarson-SF-NCTM-4-16.pdf">over a century</a> and widely discussed for decades. So why has it not been incorporated into the curriculum, and why does traditional teaching abound? </p>
<p>Learning conceptually can take longer and be more difficult than just presenting formulas. Teaching this way may require additional time commitments both in and outside the classroom. Students may have never been asked to think this way before.</p>
<p>There are systemic obstacles to face as well. A new teacher may face pressure from fellow teachers who teach in traditional ways. The <a href="https://www.thoughtco.com/high-stakes-testing-overtesting-in-americas-public-schools-3194591">culture of overtesting</a> in the last two decades means that students face more pressure than ever to get right answers on tests. </p>
<p>The results of these tests are also being <a href="https://tcta.org/node/13251-issues_with_test_based_value_added_models_of_teacher_assessment">tied to teacher evaluation systems</a>. Many teachers feel pressure to teach to the test, drilling students so that they can regurgitate information accurately.</p>
<p>If we really want to improve America’s mathematics education, we need to rethink both our education system and our teaching methods, and perhaps to <a href="http://www.nea.org/home/40991.htm">consider how other countries approach mathematics instruction</a>. Research has provided evidence that teaching conceptually has <a href="http://www.ascd.org/publications/educational-leadership/feb04/vol61/num05/Improving-Mathematics-Teaching.aspx">benefits</a> not offered by traditional teaching. And students who learn conceptually typically do <a href="https://doi.org/10.3102/0034654310374880">as well or better</a> on achievement tests. </p>
<p>Renowned education expert <a href="https://pasisahlberg.com/">Pasi Sahlberg</a> is a former mathematics and physics teacher from Finland, which is renowned for its world-class education. He <a href="http://www.smithsonianmag.com/innovation/why-are-finlands-schools-successful-49859555/">sums it up</a> well:</p>
<blockquote>
<p>We prepare children to learn how to learn, not how to take a test.</p>
</blockquote><img src="https://counter.theconversation.com/content/78660/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Christopher Rakes receives funding from the National Science Foundation. </span></em></p>Math instruction is stuck in the last century. How can we change teaching methods to move past rote memorization and help students develop a more meaningful understanding – and be better at math?Christopher Rakes, Assistant Professor of Mathematics Education, University of Maryland, Baltimore CountyLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/753032017-06-18T19:52:30Z2017-06-18T19:52:30ZYour guide to solving the next online viral maths problem<figure><img src="https://images.theconversation.com/files/174130/original/file-20170616-565-1v5w26n.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">You need to think like a mathematician to solve those viral maths problems.</span> <span class="attribution"><span class="source">Shutterstock/Dean Drobot</span></span></figcaption></figure><p>How many times have you seen a post online or part of your social media feed that says something like “<a href="https://mic.com/articles/143062/this-math-problem-is-stumping-the-whole-internet-can-you-solve-it#.zfJUBEV19">This Math Problem Is Stumping the Whole Internet. Can You Solve It?</a>” or “<a href="https://www.facebook.com/tara.haelle/posts/568740469816867">Apparently 9 out of 10 people get this wrong. Do you know the answer?</a>”</p>
<p>At the heart of the post is usually a problem involving numbers and symbols, such as this one:</p>
<div data-react-class="Tweet" data-react-props='{"tweetId":"631405920078073856"}'></div>
<p>They’re usually followed by plenty of <a href="https://www.reddit.com/r/AskReddit/comments/1b9n6u/i_am_really_confused_why_do_some_people_say_the/">discussion online</a> about the <a href="https://community.spiceworks.com/topic/137542-what-is-the-answer-to-the-equation-6-2-1-2-x-is-it-9-or-1">various possible answers</a> to the problem, including questions about why some people think even <a href="https://productforums.google.com/forum/#!topic/websearch/kZkTv_WTSxA">Google’s online calculator gets the wrong answer</a>.</p>
<div data-react-class="Tweet" data-react-props='{"tweetId":"499506580258168832"}'></div>
<p>This is just one of many examples of similar problems that have been doing the rounds for years but <a href="https://www.reddit.com/r/AskReddit/comments/6gb0hb/what_the_heck_is_the_answer_to_6221/">still continue to baffle some people</a>. Here’s another:</p>
<div data-react-class="Tweet" data-react-props='{"tweetId":"770686417857024000"}'></div>
<p>No matter how hard you try, it’s impossible to resist that challenge. You give it a go and then look at the comments section only to find some people agree with your answer while others have something completely different. </p>
<p>So let me outline the correct way to approach these online equations with the minimum of fuss. I’ll explain why in some cases there may be more than one possible correct answer.</p>
<h2>The language of mathematics</h2>
<p>In the English language we read from left to right. It therefore seems very natural to look at mathematical equations in the same way. </p>
<p>But you wouldn’t try to read Mandarin or Arabic like this, and nor should you attempt to do so with the distinct language of mathematics. </p>
<p>To be maths-literate, it is important to understand the relevant rules about “spelling” and “grammar” in mathematics.</p>
<p>A strict set of rules known as the <a href="https://www.mathsisfun.com/operation-order-bodmas.html">order of operations</a> defines the correct arithmetical grammar. These rules tell us the order in which we must perform mathematical operations such as addition and multiplication when both appear in an equation.</p>
<p>In Australia, the mnemonic BODMAS (Brackets, Order, Division, Multiplication, Addition, Subtraction) is typically taught to students to help them remember the correct order. Here, the ‘Order’ in BODMAS refers to mathematical powers such as squared, cubed or square root. </p>
<p>In other countries, this may be taught as <a href="http://www.mathsisfun.com/operation-order-pemdas.html">PEMDAS</a>, <a href="http://www.mathsisfun.com/operation-order-bodmas.html">BEDMAS or BIDMAS</a>, but these all boil down to exactly the same thing.</p>
<p>This means that if, for example, we have an equation that contains both addition and multiplication, we always carry out multiplication first regardless of the order in which they are written.</p>
<p>Consider the following equations:</p>
<blockquote>
<p>(a) 3×4+2</p>
<p>(b) 2+3×4 </p>
</blockquote>
<p>When we apply BODMAS, we can see that these equations are exactly the same (or equivalent) – in both cases we begin by calculating 3×4=12, then compute 12+2=14.</p>
<p>But some people are likely to get the wrong answer for the second equation because they will try to solve it from left to right. They will do the addition first (2+3=5) and then multiplication (5×4) to obtain an incorrect answer of 20.</p>
<h2>Brackets can make a difference</h2>
<p>This is where brackets (or parentheses) can be a very useful part of arithmetical punctuation. In English, a well-placed comma can be the difference between saying “Let’s eat, John” and “Let’s eat John”.</p>
<p>The same applies in maths, where a well-placed bracket can completely change our calculation. Brackets are used to give priority to a particular part of an equation – we always carry out the calculation inside the bracket before dealing with what is outside.</p>
<p>If we introduce brackets around the addition in equations (a) and (b) above, then we have two new equations:</p>
<blockquote>
<p>(c) 3×(4+2)</p>
<p>(d) (2+3)×4</p>
</blockquote>
<p>These equations are no longer equivalent to each other. In both cases, the brackets tell us to do the addition before we do the multiplication. This means we have to calculate 3×6 for (c) and 5×4 for (d). We now get different answers, (c) is 18 and (d) is 20. </p>
<p>Note that for equations (a) and (b), brackets were not necessary because BODMAS tells us to carry out multiplication before addition anyway. However, adding brackets that reinforce the BODMAS rules can help to avoid any confusion.</p>
<h2>More rules</h2>
<p>Understanding BODMAS gets us most of the way there in terms of solving these problems, but it also helps to be aware of the commutative and associative properties of mathematics.</p>
<p>A mathematical operation is commutative if it does not matter which order the operands (numbers) are written in. Addition is commutative, since a+b=b+a.</p>
<p>But subtraction is not, because a-b is not the same as b-a. It is also straightforward to show that multiplication is commutative, but division is not. </p>
<p>Such distinctions exist in the English language too. Ordering “vodka and orange juice” is the same as ordering “orange juice and vodka”, but “shaken not stirred” is not the same as “stirred not shaken”.</p>
<p>An operation is associative if, when we have multiple consecutive occurrences of this operation, it does not matter which order we carry them out in. </p>
<p>Again, addition and multiplication have this property, while subtraction and division do not. If we have the equation a+b+c, then it does not matter whether we solve it as (a+b)+c or a+(b+c).</p>
<p>But if we have a-b-c then the order is important, as (a-b)-c is not the same as a-(b-c) and we should always work from left to right. See for youself:</p>
<blockquote>
<p> (3-2)-1=0</p>
<p> 3-(2-1)=2</p>
</blockquote>
<p>Again, English language implicitly has such concepts; “rum and coke and lime” is the same product regardless of whether rum is added to (coke and lime), or lime is added to a (rum and coke).</p>
<p>But we cannot rearrange any of these operations in “order then drink then leave” – a successful trip to the pub relies on these actions being carried out in exactly that order.</p>
<p>Once we understand the correct order of operations and the associative and commutative properties, we have the toolbox to solve any simple, well-defined arithmetical equation.</p>
<h2>So do you know the answer?</h2>
<p>So let’s return to the original problem:</p>
<blockquote>
<p> 6÷2(1+2)=?</p>
</blockquote>
<p>The equation has more than one legitimate meaning. Some might believe the answer is 1, others might think the answer is 9. And neither answer is really wrong.</p>
<p>After carrying out the addition inside the brackets, we are left with 6÷2(3). Some people will argue that we should work from left to right, calculating 6÷2=3 and then multiplying 3×3=9, which is the answer given by Google’s calculator.</p>
<p>Others, and I consider myself part of this camp, would argue that 2(1+2) should be computed in its entirety first, since the juxtaposition of these terms without a × sign implies that it consists of a single element.</p>
<p>A mathematician would more normally express the equation as follows:</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/173965/original/file-20170615-3453-n98wx9.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/173965/original/file-20170615-3453-n98wx9.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=100&fit=crop&dpr=1 600w, https://images.theconversation.com/files/173965/original/file-20170615-3453-n98wx9.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=100&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/173965/original/file-20170615-3453-n98wx9.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=100&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/173965/original/file-20170615-3453-n98wx9.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=126&fit=crop&dpr=1 754w, https://images.theconversation.com/files/173965/original/file-20170615-3453-n98wx9.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=126&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/173965/original/file-20170615-3453-n98wx9.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=126&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption"></span>
</figcaption>
</figure>
<p>That leaves us with 6÷6=1.</p>
<p>The problem here is a poorly constructed equation, and the ÷ is the main culprit. Mathematicians rarely use this sign (or the multiplication sign ×); in practice, they prefer to use clear, unambiguous notation such as fractions.</p>
<p>If we want to convey the first meaning above, it would be more common to write the equation with extra brackets as (6/2)(1+2), which gives the answer 9. To convey the second meaning, we would write 6/(2(1+2)) as shown in the equation above, which gives the answer 1.</p>
<p>By writing things this way, we can eliminate the whole debate and save everyone a lot of time and energy.</p>
<p>Online puzzles can be a great way to refresh your mathematical skills, but it’s important to watch out for the deliberately confusing ones. </p>
<p>The next time one pops up on your timeline, remember BODMAS and you should be fine. </p>
<p>But if the answer is still not clear, then it’s best to avoid the debate and instead step back, take a deep breath and say: “They haven’t spelled that correctly!”</p><img src="https://counter.theconversation.com/content/75303/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Craig Anderson 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>There's a reason why some people get different answers to those frustrating viral maths problems. You need to learn how to "read" the maths.Craig Anderson, Postdoctoral Research Fellow in Statistics, University of Technology SydneyLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/709252017-01-09T20:26:48Z2017-01-09T20:26:48ZSouth Africa can't compete globally without fixing its attitude to maths<figure><img src="https://images.theconversation.com/files/152080/original/image-20170109-23482-uehzdp.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Investing in pupils' maths skills is an investment in a country's economy.</span> <span class="attribution"><span class="source">Global Partnership for Education/Flickr</span></span></figcaption></figure><p>South Africa is not producing enough school leavers who are competent in maths and science. This is a fact borne out by international assessments such as the <a href="http://timssandpirls.bc.edu/publications/timss/2015-methods.html">Trends in International Mathematics and Science Study</a> (TIMMS) and the World Economic Forum’s <a href="https://www.weforum.org/reports/the-global-competitiveness-report-2016-2017-1">Global Competitiveness Report</a>. These show that South Africa is not making much headway when it comes to maths and science. </p>
<p>The 2016 Global Competitiveness Report ranked South Africa last among 140 countries for maths and science. This places it behind poorer African countries like Mozambique and Malawi.</p>
<p>In 2016 there was a <a href="https://businesstech.co.za/news/government/148875/matric-results-2016-maths-and-science-suffers/">marginal improvement</a> in the maths pass rate, from 49.1% the previous year to 51.1%. The country is moving at a glacial pace in an area that demands urgent attention. After all, science and maths are key to any country’s economic development and its competitiveness in the global economy. </p>
<p>The TIMMS study ranks Singapore, Hong Kong, South Korea and Japan among its top maths and science performers. It’s no coincidence that these countries feature among the <a href="http://www.wipo.int/pressroom/en/articles/2016/article_0008.html">top 20</a> on the Global Innovation Index. Good quality education fuels an economy. South Africa needs to increase its supply of science and technology university graduates, which at the moment constitute the bulk of <a href="http://www.dhet.gov.za/Gazette/Government%20Gazette%20No%2039604,%2019%20January%202016.%20List%20of%20Occupations%20in%20High%20Demand%202015.pdf">scarce skills</a> outlined by the Department of Higher Education and Training. </p>
<p>But instead of chasing improved results the government is lowering the bar for maths at school level. At the end of 2016 it set <a href="http://www.education.gov.za/Portals/0/Documents/Publications/Circular%20A3%20of%202016.pdf?ver=2016-12-07-154005-723">20% as a passing mark</a> for pupils in grades 7, 8 and 9. This lends credence to the common view of maths as a subject only the “gifted” can comprehend. </p>
<p>It’s time to place a premium on maths and to ensure that pupils – especially those from poorer backgrounds – receive the necessary support to excel at maths. This is critical if South Africa is to produce the human capital needed to drive economic growth and create new industries in the future. </p>
<h2>How maths and science boost economies</h2>
<p>Maths and science are a gateway to new industries. Mastery of them endows an economy with the human capital needed to ride the technological wave. In his work on the industries of the future Alec Ross, who advised Hillary Clinton on innovation during her term as US Secretary of State, <a href="http://www.simonandschuster.com/books/The-Industries-of-the-Future/Alec-Ross/9781476753652">points out</a> that sectors such as robotics, advanced life sciences, codification of money, big data and cybersecurity – all of which require mastery of technology and mathematical skills – are the pillars of the <a href="https://www.weforum.org/agenda/2016/01/the-fourth-industrial-revolution-what-it-means-and-how-to-respond/">fourth industrial revolution</a>. </p>
<p>Simply put, this “revolution” is the age of technology that’s already upon us.</p>
<p>More importantly, a grasp of maths and science boosts confidence and expands career possibilities for pupils. This ultimately gives them an edge in the labour market. </p>
<p>Many students drop out of maths not by choice but because they’re frustrated by a lack of adequate support. I speak from experience: I dropped the subject when I was 14 at the end of what’s now Grade 9 but used to be called Standard 7. Our maths teacher didn’t encourage those he called “slow learners” to continue with the subject and I was one of many intimidated into giving up on maths.</p>
<p>But succeeding in maths, or in any area of skill, isn’t entirely a matter of genetic endowment. Psychologist Anders Ericsson, <a href="http://www.goodreads.com/book/show/26312997-peak">in his book Peak</a>, draws on three decades of research to prove why natural talent and other innate factors have less of an impact than what he calls deliberate or purposeful practice.</p>
<p>He contends that</p>
<blockquote>
<p>a number of successful efforts have shown that pretty much any child can learn math if it is taught in the right way.</p>
</blockquote>
<p>South Africa should be focusing on how to teach maths in the right way rather than buying into the myth that it is an impossible subject. The current approach is robbing the economy of critical human capital.</p>
<h2>Radical interventions</h2>
<p>Some may argue, though, that any improvement or shift is impossible in an education system that’s plagued by weak infrastructure, a lack of teacher development and support and too few qualified maths and science teachers. Even if the numbers of teachers in these subjects were to increase, it’s crucial that the quality of education rises too.</p>
<p>Radical interventions are needed, now – or South Africa will never become a global player in the fourth industrial revolution. </p>
<p>The country must develop new teacher training methods and nurture a supportive environment for teachers. Innovative teaching tools should be introduced in the early phases to demystify maths and science for young pupils. If these subjects are more fun to learn, more pupils may be drawn to them as future career options.</p>
<p>Taking these steps will give South Africa a better chance in the future to harness the talent of its youth to powering the economy, and improve its global competitiveness.</p><img src="https://counter.theconversation.com/content/70925/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Mzukisi Qobo 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>Good quality education fuels an economy. South Africa needs to increase its supply of science and technology university graduates. But instead it's lowering the bar, especially when it comes to maths.Mzukisi Qobo, Associate Professor at the Institute for Pan African Thought and Conversation, University of JohannesburgLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/705962016-12-21T19:52:22Z2016-12-21T19:52:22Z20% maths decree sets a dangerous precedent for schooling in South Africa<figure><img src="https://images.theconversation.com/files/150744/original/image-20161219-24307-1e5x8xg.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Learning the fundamentals of maths can equip children with critical thinking and reasoning skills.</span> <span class="attribution"><span class="source">Reuters/Mohamed Nureldin Abdallah</span></span></figcaption></figure><p>The department of basic education in South Africa has reacted to pupils’ exceedingly low attainment rates in maths education in a controversial fashion. It has issued <a href="http://www.education.gov.za/Portals/0/Documents/Publications/Circular%20A3%20of%202016.pdf?ver=2016-12-07-154005-723">an urgent circular</a> to its heads of departments, principals, managers, directors and exam and curriculum heads outlining a “special condonation dispensation”. This applies to pupils completing grades 7, 8 and 9 in the 2016 academic year which has just ended.</p>
<p>Pupils who did not meet the 40% criteria in mathematics will now be able to progress to the next grade provided they met all other pass requirements and obtained more than 20% in mathematics. Only those who passed Grade 9 maths with 30% or more will be allowed to continue with the subject.</p>
<p>Those who achieved in the 20% band will have to take mathematical literacy in the last years of their school careers. This is a somewhat different and far less demanding subject.</p>
<p>The move has been widely condemned from most quarters. For instance, the education department in the Western Cape province, <a href="http://www.gov.za/af/node/759324">warned</a> that if no “drastic action” is taken, “we will be sitting in the same position next year”. Indeed. The national department claims that its directive constitutes “<a href="http://www.iol.co.za/news/south-africa/no-such-thing-as-a-20-percent-pass-mark-for-maths-7160039">an interim measure</a>”. But how does it hope to address the crisis in maths education in future years? What can be done to instil pupils with the valuable, relevant skills developed through good maths teaching?</p>
<h2>Building critical thinking skills</h2>
<p>South African pupils’ chronic under-performance in maths is not a one-off event. It has <a href="http://www.telegraph.co.uk/education/2016/11/29/revealed-world-pupil-rankings-science-maths-timss-results/">become entrenched</a>. An earlier <a href="http://www.amesa.org.za/TIMSSR.htm">study shows</a> that pupils’ basic maths abilities – calculating fractions, simple number sense, analysis and probability – have steadily declined. </p>
<p>All of this illustrates the debilitating burden that generations of South African children have had to endure, from the apartheid era until the present day: an education system that has failed them. It has not inducted pupils into the custom of thinking and reasoning on logical, rational and critical terms. Critical thinking is a vital skill. </p>
<p>Research <a href="http://www.doane.edu/facstaff/resources/cetl-home/31812">has shown</a> that a well-cultivated critical thinker raises vital questions and problems, formulating them clearly and precisely. They are able to gather and assess relevant information, using abstract ideas to interpret it effectively. They can reach well-reasoned conclusions and solutions, testing them against relevant criteria and standards.</p>
<p>The relationship between “mathematics education” and “more complex thinking” is typically symbiotic and mutually inclusive. Good, productive mathematics education can positively <a href="http://math-site.athabascau.ca/documents/HistoryMathematicsEducation.pdf">raise pupils’ skills</a> in diagnostic, methodical thinking.</p>
<p>Defective teaching and learning could consign them to a school life steeped in frustration, fear and failure. This is precisely the fate of those <a href="https://theconversation.com/boredom-alienation-and-anxiety-in-the-maths-classroom-heres-why-69570">perpetually struggling</a> in this crucial area of learning.</p>
<p>Primary school pupils’ gross underachievement in maths education suggests they may not have been equipped, at their relevant levels, with the skills needed to <a href="http://link.springer.com/referenceworkentry/10.1007%2F978-94-007-4978-8_35#page-1">think and reason</a> effectively and meaningfully. This is why the subject occupies such an eminent place in <a href="https://www.acer.edu.au/timss">global schooling assessment criteria</a> – not only because of its content, but for the skills that are transferred and developed alongside it.</p>
<p>The education department <a href="http://www.iol.co.za/news/south-africa/no-such-thing-as-a-20-percent-pass-mark-for-maths-7160039">argues</a> that some students are more inclined towards the arts while others are better with technical subjects. This is not well founded. David Pearson, a scholar of cognitive psychology, refers in <a href="https://theconversation.com/exploding-the-myth-of-the-scientific-vs-artistic-mind-57843">his writing</a> to the domain of neuroscience which has confirmed that “everyone uses both sides of the brain when performing any task”. Pearson argues that while certain impulses of brain activity have occasionally been associated with creative or cogent thinking,</p>
<blockquote>
<p>…it doesn’t really explain who is good at what – and why. Studies have actually revealed considerable overlap in the cognitive processes supporting both scientific and artistic creativity.</p>
</blockquote>
<p>Here’s another, fairly widespread <a href="https://theconversation.com/pressured-south-african-schools-had-no-choice-but-to-relax-maths-pass-mark-70289">fable about maths</a>: “The ability to factorise quadratic functions is not a prerequisite for an educated child”. Such standpoints devalue the subject’s more authentic meaning. Instead it’s important to ponder what happens when an education system continually fails to equip its students with the aptitudes required by so many positions or professions, even if those aptitudes are not explicitly mathematical.</p>
<p>Employers have <a href="https://www.prospects.ac.uk/careers-advice/what-can-i-do-with-my-degree/mathematics">long recognised</a> that applicants with maths credits are more inclined to succeed at jobs that call for logical reasoning, precise enquiry and careful deduction. Not only this, but a wide and protracted variety of job descriptions and professional occupations – in both the sciences and humanities – <a href="http://unesdoc.unesco.org/images/0019/001914/191425e.pdf">call</a> for <a href="https://www.gov.uk/government/news/every-pupil-needs-a-good-mathematics-education">maths education</a> to a lesser or greater degree.</p>
<h2>A dangerous precedent</h2>
<p>South Africa’s maths dilemma should not be perceived purely on narrow, technical grounds. The domain of mathematics education must be seen instead in its full complexity and for its potency to endow pupils to meet some of life’s most vital challenges. Children’s dismal failure at maths is a reflection of an education system that has continually quelled their capacities to arrive at sound answers based on accurate reasoning.</p>
<p>Drastically lowering standards – such as “condoning” a 20% mark – sets a dangerous precedent from which the country may not recover for years. The education department’s directive essentially diminishes the great and important role of maths in children’s general educational as well as their broader human development. Such a course, regrettably, will assuredly exacerbate an already dire situation.</p>
<p>Maths education can only really flourish and generate more fruitful outcomes within the context of a well-functioning national education system. It is here, arguably, that the real problem lies. South Africa’s education system is merely a reflection of its broader social system. This is generally characterised by high levels of economic and social inequality, poverty, violence and abuse, and dysfunctionality. </p>
<p>It stands to reason, then, that positive educational change is incumbent upon profound social change. In the absence of social change, however, what is the fate of the overwhelming majority of South Africa’s 12 million and more school-going children today? My own <a href="http://ruralreporters.com/research-schooling-in-rural-south-africa/">field studies</a> show the importance and usefulness of gaining a deeper understanding of how certain poor, isolated schooling communities have endeavoured to overcome the odds. </p>
<p>Such an approach presents a prospective framework – broadly defined – for others to emulate or, at the very least, to contemplate while real social change remains obscure and elusive.</p><img src="https://counter.theconversation.com/content/70596/count.gif" alt="La Conversation" width="1" height="1" />
<p class="fine-print"><em><span>In this piece Clive Kronenberg draws on his prior post-doctoral fellowship studies, funded by the Critical Thinking Group attached to CPUT's Faculty of Education. </span></em></p>Maths occupies an eminent place in global schooling assessment criteria not just because of its content, but for the skills that are taught and developed alongside it.Clive Kronenberg, NRF Accredited & Senior Researcher; Lead Coordinator of the South-South Educational Collaboration & Knowlede Interchange Initiative, Cape Peninsula University of TechnologyLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/702892016-12-13T09:47:31Z2016-12-13T09:47:31ZPressured South African schools had no choice but to relax maths pass mark<figure><img src="https://images.theconversation.com/files/149835/original/image-20161213-1615-vu7id5.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">By the time pupils who struggle with Maths reach Grade 9, there are huge bottlenecks in the system.</span> <span class="attribution"><span class="source">REUTERS/Ryan Gray</span></span></figcaption></figure><p>Starting now, South Africa’s pupils will be able to obtain as <a href="http://www.education.gov.za/Portals/0/Documents/Publications/Circular%20A3%20of%202016.pdf?ver=2016-12-07-154005-723">little as 20%</a> in mathematics in Grades 7, 8 and 9 and still progress to the next year of learning. This has been touted by many as evidence of an alleged inexorable decline in educational standards.</p>
<p>The country is already known for its <a href="http://www.telegraph.co.uk/education/2016/11/29/revealed-world-pupil-rankings-science-maths-timss-results/">poor performance</a> in international standardised assessments in mathematics. This latest move may be misconstrued as condoning such poor achievement.</p>
<p>But the truth is a little more complex.</p>
<p>For Grades 7 and 8 – when pupils should be between 14 and 15 years of age – this strategy of “pushing through” to avoid repeated student retention is not new. It has been part of standard policy. This means that by the time pupils reach Grade 9, there’s a bottleneck in the system. It was inevitable that this pressure would need to be relieved.</p>
<p>To understand why, one must consider the confluence of a number of factors, including: the over-inflated importance of mathematics; a curriculum packed too full to allow for any slip-ups or slower learning rates, and the country’s struggling maths teachers. <a href="http://mg.co.za/article/2016-12-09-00-home-is-where-the-learning-is">Maths performance correlates directly with poverty factors</a>, meaning these challenges affect more than 75% of South Africa’s schools. </p>
<h2>Inflated value of maths</h2>
<p>In the past 20 years there’s been a major shift internationally towards thinking of education in purely economic terms (as opposed to critical citizenry, creativity or self-actualization). This reduction of education to purely economic ends, coupled with the conflation between mathematical prowess and problem-solving skills for the “knowledge economy”, has resulted in mathematics being isolated as “essential knowledge”. Its proponents insist that maths is required for an education of value.</p>
<p>To fully appreciate this shift in thinking, South Africans need to suspend their collective amnesia: passing mathematics was not a requirement to move into Grade 10 a generation ago. And yet adults from this era are often economically productive, creative and academically accomplished. Many would publicly acknowledge their own struggles with numbers.</p>
<p>The vast majority of jobs of many flavours and incomes do not require the type of maths taught even in Grade 9. This is forgotten when mathematics is positioned as supremely important for the job market, or for students’ personal development.</p>
<h2>Moving targets</h2>
<p>Against the backdrop of this increased emphasis on mathematics, it’s useful to consider key features of the <a href="http://www.education.gov.za/Portals/0/Documents/Policies/PolicyProgPromReqNCS.pdf?ver=2015-02-03-154857-397">National Policy Pertaining to the Promotion Requirements of the National Curriculum Statement</a>.</p>
<p>An excessive emphasis on mathematics permeates this policy. Passing mathematics with “moderate” performance (that is, 40% or more) is now a criterion for passing in every grade. It’s a criterion many students <a href="http://www.education.gov.za/Portals/0/Documents/Reports/REPORT%20ON%20THE%20ANA%20OF%202014.pdf?ver=2014-12-04-104938-000">do not meet</a>.</p>
<p>The second issue is the “maximum four years in phase” policy. According to this, a pupil may not repeat more than one year in each three year phase of compulsory schooling. If a pupil has already repeated a year in a phase, they are “progressed” through into the next grade – whether they meet the promotion/pass criteria or not.</p>
<p>This “maximum four years in phase” policy bites at the end of Grade 9. Pushing pupils through without passing maths was a viable option in lower grades, as there was a “next grade” to progress to. But leaving Grade 9 without passing means leaving school without the <a href="http://www.saqa.org.za/docs/pol/2003/getc.pdf">General Education and Training</a> certificate required for admission to a technical college.</p>
<p>In the past, officials and schools have often suspended the “max four years” criterion to give pupils another opportunity to try and attain a recognisable school leaving qualification, requiring a maths score of higher than 40%. For pupils who have been failing maths for years, this is almost <a href="http://www.iol.co.za/dailynews/news/dismal-10-average-for-grade-9-maths-1791182">impossible</a>.</p>
<p>The pressure to move learners through the system is immense. Each year, principals and senior teachers suffer validation meetings, an event where schools justify their decisions to the provincial education department about whether students who failed should repeat or progress.</p>
<p>As a former mathematics Head of Department who has attended such meetings, I came to appreciate the lottery involved about who was “progressed” and who was not, as officials clandestinely tweak results until the number of students moved through was politically acceptable. Often those with 20% or more would have their marks “adjusted” to 30% for what is referred to as a “condoned pass”. As teachers, we are told to “find marks” in assessments to justify passing or condoning borderline students.</p>
<p>But sometimes there are just not enough marks to find.</p>
<h2>Huge learning backlogs</h2>
<p>The second policy that adds to the conundrum is the Curriculum and Assessment Policy Statement (CAPS). This demands strict adherence to pacing and content. Mathematics in CAPS moves at breakneck speed: ten jam-packed weeks of content per term, even though there are often only eight weeks of actual lessons.</p>
<p>Curriculum advisers regularly correct teachers who deviate from the stated content and pacing of curriculum documents. That means a teacher who has the confidence and ability to address learning backlogs by professionally interpreting the curriculum to meet a pupil’s needs is often criticised for doing so. Teachers without this confidence or skill will not even attempt the task.</p>
<p>Such rigidity is in stark contradiction to the National Policy Pertaining to the Promotion Requirements, which is peppered with phrases regarding tailoring learning to address backlogs and learning barriers.</p>
<p>Primary schools pragmatically push over-age (16 years old) Grade 7 pupils through to Grade 8 in senior schools. Senior schools then receive under-prepared pupils who are too old to refer to schools of skills or special needs schools – the maximum referral age is 14. There is nothing to be done but to try and teach struggling learners, knowing they will be pushed up into Grade 9 where they will get stuck or <a href="https://africacheck.org/spot_check/south-africas-matric-pass-rate-obscures-dropout-rate/">drop out</a>. After Grade 9, the pupil enrolment dwindles rapidly as students lose the protection of being pushed through by the conveyor belt.</p>
<p>Together, these policies effectively put pupils on a one way track into Grade 9 irrespective of their performance in mathematics at lower grades. Then it has kept them in Grade 9 by insisting they meet the pass criteria… until now.</p>
<h2>Struggling mathematics teachers</h2>
<p>Two urgent issues, most concentrated in schools that serve the country’s poorest learners, further exacerbate what is already an obviously disastrous situation.</p>
<p>Firstly, the mathematics abilities of primary school teachers is a problem experienced in many countries, including the <a href="http://washingtonmonthly.com/2016/06/15/elementary-school-teachers-struggle-with-common-core-math-standards/">US</a> and the <a href="https://www.theguardian.com/education/2010/feb/14/primary-teachers-fail-maths-tests">UK</a>, but particularly in <a href="http://www.cde.org.za/wp-content/uploads/2013/10/MATHEMATICS%20OUTCOMES%20IN%20SOUTH%20AFRICAN%20SCHOOLS.pdf">South Africa</a>. Mathematics specialists are appointed in high schools. Primary school teachers are trained as generalists. Yet it is in primary school where the learning backlog begins.</p>
<p>Secondly, teachers’ working conditions in poorer schools are abysmal. Those teachers who can leave often do, and mathematics teachers in particular often possess transferable skills. They <a href="http://www.education.gov.za/Portals/0/Documents/Reports/Teachers%20for%20the%20future%2016%20NOV%202005.pdf?ver=2008-03-05-111025-000">relocate</a> to other schools or other industries for better working conditions.</p>
<p>Primary schools thus struggle to provide the crucial foundations for maths, and secondary schools struggle to retain the specialists who might be able to address the problem later.</p>
<h2>Relieving the self-applied pressure</h2>
<p>It’s no wonder then that <a href="http://www.education.gov.za/Portals/0/Documents/Publications/Education%20Statistic%202013.pdf?ver=2015-03-30-144732-767">Grade 9 is the largest cohort in South Africa’s senior schools</a>. Nor should it come as a surprise that large percentages of these classes are extremely weak at mathematics. Many pupils have barriers to learning that have been unaddressed for so long that there is little to be done at this late stage.</p>
<p>The Department of Basic Education has snookered itself by applying tight Grade 9 promotion criteria based on mathematics, without providing the means to meet them. This latest move is simply a welcome, realistic – and long overdue – acknowledgement that the ability to factorise quadratic functions is not a prerequisite for an educated child.</p><img src="https://counter.theconversation.com/content/70289/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Sara Muller works at the University of Cape Town as a researcher and PhD candidate.
She receives funding from the Canon Collins Educational and Legal Assistance Trust in support of her PhD research, and is an active member of the Education Fishtank group, an open forum for engaging in education discussions in Cape Town.
All opinions expressed in her articles are her own.</span></em></p>The truth behind South Africa's decision to allow 20% as a maths pass mark in some grades is a little more complex than many have suggested.Sara Black, Researcher: Teacher Development and Sociology of Education, University of Cape TownLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/659632016-10-13T18:06:42Z2016-10-13T18:06:42ZYes, mathematics can be decolonised. Here's how to begin<figure><img src="https://images.theconversation.com/files/139426/original/image-20160927-14593-1rf92dt.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">
</span> <span class="attribution"><span class="source">Shutterstock</span></span></figcaption></figure><p>At a time when decolonisation, part of which involves changing the content of what’s taught, is dominating debate at many universities, the discipline of mathematics presents an interesting case. </p>
<p>But it’s not obvious how mathematics can be decolonised at the level of content. This means that those within the discipline must consider other aspects: curriculum processes, such as critical thinking and problem solving; pedagogy – how the subject is taught and, as a number of people have <a href="https://www.routledge.com/Mathematical-Relationships-in-Education-Identities-and-Participation/Black-Mendick-Solomon/p/book/9780415996846">argued</a>, addressing the issue of identity. </p>
<p>Students’ mathematical identities – how they see themselves as learners of mathematics and the extent to which mathematics is meaningful to them – are important when thinking about teaching and learning in mathematics.</p>
<p>In his book <em><a href="https://www.amazon.com/Leading-Change-leadership-university-Educational/dp/113889026X">Leading for change</a></em>, South African educationist Jonathan Jansen suggests that transforming university campuses into deracialised spaces requires attention to both the academic and the human project. I take the human project to mean how students see themselves. What might this mean for mathematics?</p>
<h2>So what is mathematics?</h2>
<p>For starters, it’s important to explore what mathematics actually is.</p>
<p>Mathematician and academic Jo Boaler <a href="http://eu.wiley.com/WileyCDA/WileyTitle/productCd-0470894520.html">points out</a> that mathematics is the only subject where students and mathematicians give very different answers to this question.</p>
<p>Mathematicians view the subject as an exciting, creative endeavour in which problem solving, curiosity, excitement, intuition and perseverance play important roles – albeit in relation to abstract objects of study. </p>
<p>For school and even undergraduate mathematics students, these aspects of mathematics are often not experienced and remain opaque. Students tend to believe that mathematics is a set of procedures to be followed. They think only particularly gifted people can do and understand these procedures. This suggests that the way mathematics is usually taught doesn’t provide opportunities for accessing mathematical knowledge. It doesn’t allow students to identify with mathematics, nor make them aspire to become mathematicians.</p>
<p>As a result, mathematics has a problem with diversity. All over the world, black and women mathematicians remain rare. They <a href="https://www.youcubed.org/think-it-up/ideas-of-giftedness-hurt-students/">simply don’t</a> take mathematics at higher academic levels as much as their white and male peers.</p>
<p>One reason for this is given by <a href="https://www.youcubed.org/think-it-up/ideas-of-giftedness-hurt-students/">a study</a> in the US, which showed that the more a field attributes success to giftedness rather than effort, the fewer female and black academics are in that field. This is because the field perpetuates stereotypes about who belongs in the field. The same study found that mathematics professors hold the most fixed ideas about giftedness. </p>
<p>But this view of giftedness versus effort is not borne out by research. A number of scholars <a href="http://www.ams.org/notices/200102/rev-devlin.pdf">have argued</a> that all people are capable of learning mathematics, to high levels.</p>
<p>This suggests that a lot of the “bad press” around mathematics as a subject and discipline lies with how it is taught and learned.</p>
<h2>What is learning?</h2>
<p>When scholars theorise learning, the thinking always happens in two directions: to the past, and to the future. </p>
<p>Some see learning as building on current knowledge in a <a href="http://rer.sagepub.com/content/57/2/175.full.pdf">step-wise linear way</a>. Some see it as working <a href="http://infed.org/mobi/jerome-bruner-and-the-process-of-education/">in a spiral</a> –- coming back to old ideas in new ways. Still others view learning as <a href="https://books.google.co.za/books?id=x2XRiIm-3vAC&pg=RA4-PA1946&lpg=RA4-PA1946&dq=a+conception+of+knowledge+acquisition+and+its+implications+for+education&source=bl&ots=tO3N1Cedy1&sig=UYT9tzDyfKDubi0MxKIJK0eOS1A&hl=en&sa=X&redir_esc=y#v=onepage&q=a%20conception%20of%20knowledge%20acquisition%20and%20its%20implications%20for%20education&f=false">disrupting or transforming</a> current knowledge.</p>
<p>For teachers, working with current knowledge means finding ways to ascertain, predict, anticipate and think about students’ ideas – and finding ways to engage with these. An important part of students’ ideas about mathematics is how they see themselves in relation to mathematics. Research in schools <a href="http://eu.wiley.com/WileyCDA/WileyTitle/productCd-0470894520.html">has shown</a> that one of the key factors in students’ mathematics achievement is a teacher who believes that they can do mathematics.</p>
<p>The future is important because universities must produce future thinkers, leaders, professionals and citizens. These institutions are the bridge between the past and the future. </p>
<p>Educational theorist Etienne Wenger argues that learning is fundamentally about becoming <a href="https://www.amazon.com/Communities-Practice-Cognitive-Computational-Perspectives/dp/0521663636">a certain kind of person</a>. At universities, students are inducted into disciplines, fields and professions that require them to be certain kinds of people with certain orientations to the world, to knowledge, to other people and to practice. </p>
<p>Traditionally universities have focused on knowledge and hoped that identity will follow. This hasn’t been entirely unsuccessful. But to genuinely transform the academic project, universities must do explicit identity work with their students. Academics must engage in the human project, thinking about who their students are and what their previous experiences of mathematics and of learning mathematics have been.</p>
<h2>Towards genuine change</h2>
<p>There have been attempts to transform the content of school mathematics curricula. These include <a href="http://www.maa.org/publications/periodicals/maa-focus/ethnomathematics-shows-students-their-connections-math">ethnomathematics</a>, which excavates the mathematics in cultural objects, artefacts and practices; and critical mathematics, where mathematics is used to critique aspects of society and where students critique mathematics, for example, how algorithms structure our lives in ways which <a href="http://nymag.com/thecut/2016/09/cathy-oneils-weapons-of-math-destruction-math-is-biased.html">reproduce inequality</a>.</p>
<p>However, not all of mathematics can be accessed in these ways. For true epistemological access to mathematics, students need to study it systematically, as a body of knowledge in and of itself. This can be both empowering or disempowering.</p>
<p>Much, though certainly not all, of mathematics was created by <a href="https://theconversation.com/its-time-to-take-the-curriculum-back-from-dead-white-men-40268">dead white men</a>. But maths should and does belong to everybody. Everybody deserves access to its beauty and its power – and everybody should be able to push back when the discipline is used to destroy and oppress.</p>
<p>To transform mathematics teaching and learning in ways that empower students, universities need to give students the theoretical grounding they need to access the subject and support them to identify with it –- to want to learn it, to become the mathematicians of the future, to enjoy and critique mathematics and its applications. </p>
<p>This means that as teachers, my colleagues and I need to believe – to know – that all students can do mathematics. This knowledge must be transmitted to them. They must be shown that mathematics is a human enterprise: it belongs to all, and it can be taken forward to transform society.</p><img src="https://counter.theconversation.com/content/65963/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Karin Brodie receives funding from the NRF for a research project on mathematical identities of high school learners. </span></em></p>Some have suggested that deracialising the academy requires all researchers, teachers and students to link knowledge and identity. What might this mean for mathematics?Karin Brodie, Professor of Education and Mathematics Education, University of the WitwatersrandLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/564232016-03-18T02:09:16Z2016-03-18T02:09:16ZUniversities should require science, engineering and commerce students to know their maths<figure><img src="https://images.theconversation.com/files/115536/original/image-20160317-3199-1abn0jp.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Many university degrees require a high level of maths skill.</span> <span class="attribution"><span class="source">Shutterstock</span></span></figcaption></figure><p>In 2013, a meeting of academics specialising in teaching first year undergraduate mathematics (known as the <a href="http://fyimaths.org.au/">FYiMaths</a> network) identified that the broad removal of mathematics prerequisites for many undergraduate degrees had created the biggest challenge they faced in teaching. </p>
<p>Many individuals had made attempts to pass this message up the management line at their universities. But at that time, staff believed that reintroducing prerequisites would never happen. </p>
<p>However, earlier this year The University of Sydney announced it <a href="http://sydney.edu.au/news-opinion/news/2016/02/01/mathematics-to-become-a-prerequisite-for-university-of-sydney-ad.html">would do exactly that</a>, by requiring students studying science, engineering, commerce and IT to have completed at least intermediate level mathematics in high school.</p>
<p>The Australian Academy of Science’s <a href="https://www.science.org.au/support/analysis/decadal-plans-science/decadal-plan-mathematical-sciences-australia-2016-2025">Decadal Plan for the Mathematical Sciences</a>, launched in Canberra yesterday, continues this push. One of its <a href="https://www.science.org.au/files/userfiles/support/reports-and-plans/2016/mathematics-decade-plan-2016-vision-for-2025.pdf">key recommendations</a> is the reinstatement of mathematics prerequisites for science, engineering and commerce degrees. </p>
<p>But will it improve the level of maths education? Will it bolster mathematics skills in those studying science, engineering and commerce?</p>
<h2>Opting out</h2>
<p>A prerequisite study for entry to a degree is considered to be essential background knowledge that students need in order to be successful in that degree. A student cannot be selected into the degree if they do not have the stated prerequisite or an equivalent to it. </p>
<p>Over the past two decades, most universities have moved away from mathematics prerequisites, replacing them with assumed knowledge statements. This means that students can be selected without verifying that they have in fact completed this background study.</p>
<p>So what’s wrong with that?</p>
<p>In most cases, the assumed knowledge statements are unclear and often difficult to find, so students may not be aware of the assumed requirements. The removal of mathematics prerequisites also grossly underplays the level of mathematical facility required for these courses and trivialises the learning and skill development required to acquire it. </p>
<p>It places the burden on students to decide what should or should not be known in order to succeed in a course, and to assume the risk of those decisions, even though they are in no position to know what the risks are.</p>
<p>As a consequence, large numbers of students have been enrolling in mathematics-dependent courses without the assumed knowledge. </p>
<p>Over the last decade or more, numbers of students studying intermediate and advanced level mathematics in school has been in steady decline. Students have been free to make subject choices based on maximising their ATAR score rather than choosing the subjects that will best prepare them for their chosen career. </p>
<p>Since intermediate and advanced mathematics subjects are seen as hard and deemed not necessary for entry, students have been allowed – in some cases even encouraged – to opt out.</p>
<p>On the other side of the enrolment gate, consequences for students include being required to undertake bridging courses (some at extra cost) and having limited pathways through their degrees. Students do not generally know this at the end of Year 10 when they decide on which subjects they will choose for their Year 12.</p>
<p>Neither do they know that these choices may impact on their ability to succeed in their tertiary studies. Failure and attrition rates are generally high in first-year STEM subjects. And lack of the requisite background in mathematics plays a significant part in this. </p>
<p>Students who enter university without the assumed knowledge in mathematics also generally have lower success rates than students who have the assumed knowledge from school, even after they have completed bridging courses. In consumer terms, this buyer beware approach is not working.</p>
<p>So, where does that leave us?</p>
<h2>One piece of the puzzle</h2>
<p>Universities have a responsibility to determine what minimum background knowledge students require to be successful in a course. Once that determination is made, they should be required to ensure that the students they accept have that required knowledge. </p>
<p>Reintroducing appropriate mathematics prerequisites should increase participation in intermediate and advanced level mathematics at school. It has to. </p>
<p>We want students to take full advantage of the excellent education that is available to them through our secondary school system rather than trying to play catchup for years later.</p>
<p>Engaging students in the study of mathematics at school needs to be addressed on many levels. Certainly, making strong statements about prerequisites is one piece of the puzzle, but not the only one.</p>
<p>The Decadal Plan also calls for an urgent increase in the provision of professional development for teachers, especially those teaching mathematics out-of-field. It is essential that we support our teachers at all levels of education, so that we can give students the best possible education in mathematics that we must.</p><img src="https://counter.theconversation.com/content/56423/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>John Rice is the Executive Director of the Australian Council of Deans of Science (ACDS). The opinions expressed in this article, however, are his own, and do not necessarily reflect those of the ACDS.</span></em></p><p class="fine-print"><em><span>Deborah King ne travaille pas, ne conseille pas, ne possède pas de parts, ne reçoit pas de fonds d'une organisation qui pourrait tirer profit de cet article, et n'a déclaré aucune autre affiliation que son poste universitaire.</span></em></p>Lowering maths prerequisites to study science, engineering and commerce at university has led to students playing catch up for years. This should be fixed.Deborah King, Associate Professor in Mathematics, University of MelbourneJohn Rice, Honorary Professor, University of SydneyLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/440162015-07-09T20:07:01Z2015-07-09T20:07:01ZKids prefer maths when you let them figure out the answer for themselves<figure><img src="https://images.theconversation.com/files/86784/original/image-20150630-9056-j7udzd.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">New research for primary and junior secondary schools shows kids prefer to nut out maths problems without the teacher's help. </span> <span class="attribution"><span class="source">from www.shutterstock.com</span></span></figcaption></figure><p>A common view is that students learn maths best when teachers give clear explanations of mathematical concepts, usually in isolation from other concepts, and students are then given opportunities to practise what they have been shown. </p>
<p>I’ve recently undertaken research at primary and junior secondary levels exploring a different approach. This approach involves posing questions like the following and expecting (in this case, primary level) students to work out their own approaches to the task for themselves prior to any instruction from the teacher:</p>
<blockquote>
<p>The minute hand of a clock is on two, and the hands make an acute angle. What might be the time?</p>
</blockquote>
<p>There are three ways that this question is different from conventional questions. First, it focuses on two aspect of mathematics together, time and angles. Contrasting two concepts helps students see connections and move beyond approaching mathematics as a collection of isolated facts. </p>
<figure class="align-left ">
<img alt="" src="https://images.theconversation.com/files/86791/original/image-20150630-9090-1qt7nzc.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/86791/original/image-20150630-9090-1qt7nzc.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/86791/original/image-20150630-9090-1qt7nzc.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/86791/original/image-20150630-9090-1qt7nzc.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/86791/original/image-20150630-9090-1qt7nzc.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/86791/original/image-20150630-9090-1qt7nzc.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/86791/original/image-20150630-9090-1qt7nzc.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">Questions posed to students as part of the research are different to conventional math problems.</span>
<span class="attribution"><span class="source">from www.shutterstock.com</span></span>
</figcaption>
</figure>
<p>Second, the question has more than one correct answer. Having more than one correct answer means students have opportunities to make decisions about their own answer and then have something unique to contribute to discussions with other students. </p>
<p>Third, students can respond at different levels of sophistication: some students might find just one answer, while other students might find all of the possibilities and formulate generalisations.</p>
<p>The task is what is described as appropriately challenging. The solutions and solution pathways are not immediately obvious for middle primary students but the task draws on ideas with which they are familiar. An explicit advantage of posing such challenging tasks is that the need for students to apply themselves and persist is obvious to the students, even if the task seems daunting at first.</p>
<p>After the students have worked on the task for a time, the teacher manages a discussion in which students share their insights and solutions. This is an important opportunity for students to see what other students have found, and especially to realise that in many cases there are multiple ways of solving mathematics problems. </p>
<p>It is suggested to teachers that they use a data projector or similar technology to project students’ actual work. This saves time rewriting the work, presents the students’ work authentically and illustrates to students the benefits of writing clearly and explaining thinking fully.</p>
<p>Subsequently, the teacher poses a further task in which some aspects are kept the same and some aspects changed, such as:</p>
<blockquote>
<p>The minute hand of a clock is on eight, and the hands make an obtuse angle. What might be the time?</p>
</blockquote>
<figure class="align-right ">
<img alt="" src="https://images.theconversation.com/files/86793/original/image-20150630-9062-1n5cj6w.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/86793/original/image-20150630-9062-1n5cj6w.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/86793/original/image-20150630-9062-1n5cj6w.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/86793/original/image-20150630-9062-1n5cj6w.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/86793/original/image-20150630-9062-1n5cj6w.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/86793/original/image-20150630-9062-1n5cj6w.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/86793/original/image-20150630-9062-1n5cj6w.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">The tasks given to students are appropriately challenging.</span>
<span class="attribution"><span class="source">from www.shutterstock.com</span></span>
</figcaption>
</figure>
<p>The intention is that students learn from the thinking activated by working on the first task and from the class discussion, then apply that learning to the second task.</p>
<p>The research aims to identify tasks that not only are appropriately challenging but can be adapted to suit the needs of particular students. For example, there may be some students for whom the first task is too difficult. Those students might be asked to work on a question like:</p>
<blockquote>
<p>What is a time at which the hands of a clock make an acute angle?</p>
</blockquote>
<p>The intention is that those students then have more chance of engaging with the original task. Of course, there are also students who can find answers quickly and are then ready for further challenges. Those students might be posed questions like:</p>
<blockquote>
<p>With the minute hand on two, why are there six times for which the hands make an acute angle? Is there a number to which the minute hand might point for which there are not six possibilities?</p>
</blockquote>
<p>There might even be advanced students who could be asked:</p>
<blockquote>
<p>What are some times for which the hands on a clock make a right angle?</p>
</blockquote>
<p>The combination of the students’ own engagement with the problem and the different levels of prompts means the students’ work contains rich and useful information about what the students know. Teachers can use this not only to give the students feedback but also to plan subsequent teaching.</p>
<h2>Students welcomed the challenge</h2>
<figure class="align-left ">
<img alt="" src="https://images.theconversation.com/files/86786/original/image-20150630-9102-1rd26jj.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/86786/original/image-20150630-9102-1rd26jj.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/86786/original/image-20150630-9102-1rd26jj.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/86786/original/image-20150630-9102-1rd26jj.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/86786/original/image-20150630-9102-1rd26jj.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/86786/original/image-20150630-9102-1rd26jj.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/86786/original/image-20150630-9102-1rd26jj.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">The project found that students prefer to work out solutions and representations by themselves or with other students.</span>
<span class="attribution"><span class="source">from www.shutterstock.com</span></span>
</figcaption>
</figure>
<p>The project found that, contrary to the preconceptions of some teachers, many students do not fear challenges in mathematics but welcome them. Rather than preferring teachers to instruct them on solution methods, many students prefer to work out solutions by themselves or by working with other students. </p>
<p>The project also established that students learn substantive mathematics content from working on challenging tasks and are willing and able to develop ways of articulating their reasoning.</p><img src="https://counter.theconversation.com/content/44016/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Peter Sullivan receives funding from The Australian Research Council.
The Australian Research Council funding the research from which the article is drawn. There is no conflict of interest between the ARC and this article.</span></em></p>Rather than having teachers instruct students on solution methods, many students prefer to work out solutions by themselves or by working with other students.Peter Sullivan, Professor of Science, Mathematics and Technology Education, Monash UniversityLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/430532015-06-25T10:36:38Z2015-06-25T10:36:38ZDon't freak if you can't solve a math problem that's gone viral<figure><img src="https://images.theconversation.com/files/86322/original/image-20150624-31526-1jbqvaz.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Are you smarter than a third grader in Vietnam?</span> <span class="attribution"><a class="source" href="http://www.shutterstock.com/pic.mhtml?id=252676549&src=lb-29877982">Woman image via www.shutterstock.com</a></span></figcaption></figure><p>It’s been quite a year for mathematics problems on the internet. In the last few months, three questions have been online everywhere, causing consternation and head-scratching and blowing the minds of adults worldwide as examples of what kids are expected to know these days.</p>
<p>As a mathematician, I suppose I should subscribe to the “no such thing as bad publicity” theory, except that problems of this ilk a) usually aren’t that difficult once you get the trick, b) sometimes aren’t even math problems and c) fuel the defeatist “I’m not good at math” fire that pervades American culture. The inability to solve such a problem quickly is certainly not indicative of a person’s overall math skill, nor should it prompt a crisis of confidence about the state of American math aptitude.</p>
<h2>When is Cheryl’s birthday?</h2>
<p>In April, the internet erupted with shock that 10-year-olds in Singapore were asked to answer the following question on an exam.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/86311/original/image-20150624-31504-1omznvi.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/86311/original/image-20150624-31504-1omznvi.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/86311/original/image-20150624-31504-1omznvi.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=436&fit=crop&dpr=1 600w, https://images.theconversation.com/files/86311/original/image-20150624-31504-1omznvi.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=436&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/86311/original/image-20150624-31504-1omznvi.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=436&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/86311/original/image-20150624-31504-1omznvi.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=548&fit=crop&dpr=1 754w, https://images.theconversation.com/files/86311/original/image-20150624-31504-1omznvi.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=548&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/86311/original/image-20150624-31504-1omznvi.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=548&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 logic puzzle from the Singapore and Asian Math Olympiads.</span>
</figcaption>
</figure>
<p>Except that it wasn’t for elementary school students at all; rather it appeared on an Asian Olympiad exam designed for mathematically talented high school students. What’s more, this isn’t even a math problem, but a logic problem. It’s true that students tend to learn formal logic via mathematics (plane geometry in particular), so it is common to see problems of this type in mathematics competitions. When I was in junior high, we spent a good deal of time on these puzzles in my language arts class, and I met them again when taking the GRE prior to entering graduate school (the test contains a whole section of them). </p>
<p>If you’re stumped, check out a <a href="http://www.independent.co.uk/news/world/asia/singapore-maths-problem-can-you-solve-the-viral-maths-question-that-was-set-to-children-in-singapore-10173090.html">solution to the problem</a>.</p>
<h2>Vietnamese eight-year-olds do arithmetic</h2>
<p>A month later, we heard about a third grade teacher in Vietnam who set the following puzzle for his students. Place the digits from 1 to 9 in this grid, using each only once (the : represents division).</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/85280/original/image-20150616-5829-129m39.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/85280/original/image-20150616-5829-129m39.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=433&fit=crop&dpr=1 600w, https://images.theconversation.com/files/85280/original/image-20150616-5829-129m39.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=433&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/85280/original/image-20150616-5829-129m39.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=433&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/85280/original/image-20150616-5829-129m39.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=545&fit=crop&dpr=1 754w, https://images.theconversation.com/files/85280/original/image-20150616-5829-129m39.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=545&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/85280/original/image-20150616-5829-129m39.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=545&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">A puzzle for Vietnamese children.</span>
<span class="attribution"><span class="source">VN Express</span></span>
</figcaption>
</figure>
<p>This reminds me of the (probably apocraphyl) story of one of the greatest mathematicians in history, <a href="https://en.wikipedia.org/wiki/Carl_Friedrich_Gauss">Carl Friedrich Gauss</a>. Legend has it that when Gauss was seven or eight, his teacher, wanting to occupy his students for a while, told the class to add up the numbers from 1 to 100. Gauss thought about it for 30 seconds or so and wrote the correct answer, 5,050, on his slate and turned it in.</p>
<p>The puzzle above has a similar feel. It’s really a question about knowing the order of arithmetic operations (multiplication/division, addition/subtraction, in that order). Beyond that, it just takes trial and error; that is, it’s kind of just busy work. Someone who knows some algebra might be able to generate some equations to gain insight into how you might find a <a href="http://www.theguardian.com/science/alexs-adventures-in-numberland/2015/may/20/can-you-do-the-maths-puzzle-for-vietnamese-eight-year-olds-that-has-stumped-parents-and-teachers">solution</a>.</p>
<p>Another approach would be to open up a spreadsheet program and just try all the possibilities. Since there are nine choices for the first box, then eight choices for the second, and so on, there are only (9)(8)(7)(6)(5)(4)(3)(2)(1) = 362,880 possible configurations, of which only a few will give a valid equation. This can be programmed with very little effort.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/86327/original/image-20150624-31495-14gxmoi.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/86327/original/image-20150624-31495-14gxmoi.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/86327/original/image-20150624-31495-14gxmoi.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=480&fit=crop&dpr=1 600w, https://images.theconversation.com/files/86327/original/image-20150624-31495-14gxmoi.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=480&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/86327/original/image-20150624-31495-14gxmoi.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=480&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/86327/original/image-20150624-31495-14gxmoi.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=603&fit=crop&dpr=1 754w, https://images.theconversation.com/files/86327/original/image-20150624-31495-14gxmoi.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=603&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/86327/original/image-20150624-31495-14gxmoi.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=603&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Yellow or orange, students didn’t find the problem sweet.</span>
<span class="attribution"><a class="source" href="http://www.shutterstock.com/pic-275334341/stock-photo-orange-and-yellow-jelly-candies-closeup-sweet-background.html">Candy image via www.shutterstock.com</a></span>
</figcaption>
</figure>
<h2>Hannah’s sweets</h2>
<p>Just a couple of weeks ago, students in the UK vented their frustration via social media about a problem on the Edexcel GCSE (General Certificates of Secondary Education) mathematics exam. It is a probability question: Hannah has a bag containing <em>n</em> candies, six of which are orange and the rest of which are yellow. She takes two candies out of the bag and eats them. The probability that she ate two orange candies is 1/3. Given this, show that <em>n² - n - 90 = 0</em>. The students’ complaint? It’s too difficult.</p>
<div data-react-class="Tweet" data-react-props='{"tweetId":"606837485239463937"}'></div>
<p>I’ve taught math long enough to recognize the pitfalls of setting this problem. The students actually have the knowledge to do it, if they know basic probability, but it is unlike problems they would have practiced. A typical question would indicate the total number of candies in the bag and ask students to compute the probability of a certain outcome. This question gives the probability and asks for a condition on the number of candies. It’s just algebra. You may read the solution (and some humorous memes about the question) <a href="http://www.telegraph.co.uk/education/11652918/Students-vent-their-frustration-at-Edexcel-GCSE-maths-exam.html">here</a>.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/86326/original/image-20150624-31495-1wy27l9.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/86326/original/image-20150624-31495-1wy27l9.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/86326/original/image-20150624-31495-1wy27l9.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=450&fit=crop&dpr=1 600w, https://images.theconversation.com/files/86326/original/image-20150624-31495-1wy27l9.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=450&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/86326/original/image-20150624-31495-1wy27l9.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=450&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/86326/original/image-20150624-31495-1wy27l9.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=566&fit=crop&dpr=1 754w, https://images.theconversation.com/files/86326/original/image-20150624-31495-1wy27l9.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=566&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/86326/original/image-20150624-31495-1wy27l9.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">What does his lifelong future with math look like?</span>
<span class="attribution"><a class="source" href="https://www.flickr.com/photos/grahams__flickr/360774920">Prisoner 5413</a>, <a class="license" href="http://creativecommons.org/licenses/by-nc/4.0/">CC BY-NC</a></span>
</figcaption>
</figure>
<h2>A nation at risk?</h2>
<p>Mathematicians dread cocktail parties because we inevitably have to endure the response we receive when asked what we do: “Oh, I hated (or am terrible at) math.” No other subject in school receives such scorn, nor would we find it acceptable for an adult to admit they are terrible at reading or writing. So when these “unsolvable” problems pop up, they simply reinforce our culture’s math anxiety. </p>
<p>And that’s a real shame, because everyone likes math when they’re young. We all like to count. We like playing with blocks and shapes. We all use math daily whether we realize it or not – reading maps, planning routes, calculating tips. I once had a flooring installer tell me he was bad at math <em>while I watched him lay tile</em>. <a href="http://www.theatlantic.com/education/archive/2013/10/the-myth-of-im-bad-at-math/280914/">It’s a myth</a> that all these people can’t do math. When people say they are “bad at math,” they usually mean that they had trouble with algebra, although if you corner them and ask the right questions you can usually make them realize that they use algebra all the time without noticing it. This leads to <a href="https://grantwiggins.wordpress.com/2013/04/10/my-100th-post-so-why-not-bash-algebra/">valid criticisms</a> of how we teach math, but it doesn’t mean we’re a nation of math idiots.</p>
<p>So, the next time one of these outrageous problems comes along, instead of giving in to anxiety, why not think about it for a few minutes and try to find a solution? You might be surprised how satisfying it can be.</p><img src="https://counter.theconversation.com/content/43053/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Kevin Knudson 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>People shouldn't let these tricky puzzlers reinforce their misguided notion that they stink at math.Kevin Knudson, Professor of Mathematics, University of FloridaLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/429672015-06-09T20:01:58Z2015-06-09T20:01:58ZBeliefs about innate talent may dissuade students from STEM<figure><img src="https://images.theconversation.com/files/84312/original/image-20150609-27420-2vh3ds.jpg?ixlib=rb-1.1.0&rect=0%2C0%2C1099%2C799&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Believing "math isn't for everyone" may steer kids away from tackling the challenge.</span> <span class="attribution"><a class="source" href="https://www.flickr.com/photos/susanrm/6012030753/">susanrm8</a>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span></figcaption></figure><p>“It’s OK – not everyone can do difficult math.”</p>
<p>Believing such messages may deter both boys and girls from later majoring in physical science, technology, engineering and mathematics (STEM) fields, according to <a href="http://journal.frontiersin.org/article/10.3389/fpsyg.2015.00530/full">a new national, longitudinal study</a> in the US published in Frontiers in Psychology. </p>
<p>“Students may need to hear that encountering difficulty during classwork is expected and normal,” argued <a href="http://perezfelkner.wordpress.com/research/">Lara Perez-Felkner</a>, a coauthor of the study and assistant professor of higher education and sociology at Florida State University. </p>
<p>The study used data from 4,450 US adolescents who later entered college to probe why some students shun math-intensive fields. Believing that solving tough math problems requires innate abilities might discourage students, the researchers reasoned.</p>
<p>“Most people believe they can do some mathematics, such as splitting a dinner bill with friends,” said Samantha Nix, lead author and doctoral student at Florida State University. “But fewer believe they can do mathematics they perceive as ‘difficult.’”</p>
<p>It’s almost silly if you think about it: you don’t take classes to study topics you’ve already mastered. Yet saying “I can’t do math” is often accepted with head nods from others. Saying “I can’t do reading” might instead be met with looks of disbelief or encouragement to work harder on learning language skills.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/84412/original/image-20150609-10672-u3g00k.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/84412/original/image-20150609-10672-u3g00k.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/84412/original/image-20150609-10672-u3g00k.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=399&fit=crop&dpr=1 600w, https://images.theconversation.com/files/84412/original/image-20150609-10672-u3g00k.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=399&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/84412/original/image-20150609-10672-u3g00k.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=399&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/84412/original/image-20150609-10672-u3g00k.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=501&fit=crop&dpr=1 754w, https://images.theconversation.com/files/84412/original/image-20150609-10672-u3g00k.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=501&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/84412/original/image-20150609-10672-u3g00k.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=501&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Naturally ‘good at math’ or working to master the subject?</span>
<span class="attribution"><a class="source" href="https://www.flickr.com/photos/departmentofed/13130915094">US Department of Education</a>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span>
</figcaption>
</figure>
<h2>What the national study found</h2>
<p>High school students who believed they could master the toughest math concepts were more likely to major in math-intensive fields two years after graduation. Similar results were found for students who believed “most people can learn to be good at math” – a belief <a href="http://dx.doi.org/10.1080/00461520.2012.722805">psychologists call</a> a “growth mindset.”</p>
<p>Beliefs still mattered even after statistically correcting for some other factors such as demographics and science coursework. However, these controls were somewhat limited. Math grades were omitted, for instance.</p>
<p>Performance on a difficult math test was used as a control. But students had <a href="http://journal.frontiersin.org/article/10.3389/fpsyg.2015.00530/full">“almost no probability”</a> of correctly answering the test’s problems. This fact limits how well the test can measure individual differences in math performance, since everyone was bound to bomb it. </p>
<p>Nevertheless, the encouraging results echo <a href="http://www.aauw.org/research/why-so-few/">experiments</a> in <a href="http://dx.doi.org/10.1177/0956797615571017">actual classrooms</a> that better control for prior mathematics background.</p>
<figure>
<iframe width="440" height="260" src="https://www.youtube.com/embed/NhY6ilZLMNQ?wmode=transparent&start=0" frameborder="0" allowfullscreen></iframe>
<figcaption><span class="caption">Why children need to learn to fail. Jessica Lahey, educator and writer of the upcoming book “The Gift of Failure,” discusses the broader research on growth mindsets.</span></figcaption>
</figure>
<h2>Equal benefits for boys and girls</h2>
<p>Endorsing a “growth mindset” seemed to equally benefit boys and girls, the study found. Regardless of gender, these beliefs predicted later majoring in math-intensive fields. <a href="http://www.baruch.cuny.edu/wsas/academics/psychology/cgood.htm">Catherine Good</a>, associate professor of psychology at Baruch College, told me this finding did not necessarily surprise her. <a href="http://dx.doi.org/10.1037/a0026659">She’s also found</a> benefits of growth mindset messages for both genders. </p>
<p>Gender gaps in beliefs were also modest. In 12th grade, boys rated their math abilities higher than girls did by 0.2 points on a 4-point scale, for instance. Accounting for gaps in self-perceived abilities did not explain the much larger gaps in majors. Men outnumber women 3-to-1 among <a href="http://dx.doi.org/10.3389/fpsyg.2015.00037">college graduates in math-intensive STEM majors</a>, for instance. </p>
<p>Other studies suggest nuance about whether beliefs concerning hard work contribute to the low numbers of women in STEM. Professor Good, for instance, told me that messages attaching great value to hard work may particularly benefit females facing “identity threat.” Such threats include taking a math test <a href="http://dx.doi.org/10.1016/j.appdev.2003.09.002">described as diagnostic of math intelligence</a>, for example.</p>
<p>Professors prizing innate “genius” may also discourage women more than men, cautioned <a href="http://www.psychology.illinois.edu/people/acimpian">Andrei Cimpian</a>, associate professor of psychology at University of Illinois at Urbana-Champaign. “Women’s personal growth mindsets – although undoubtedly beneficial – may not be sufficient to buffer them against an environment that cherishes innate talent,” he told me.</p>
<p>Across 30 academic fields, philosophy and math professors were the most likely to say that success in their fields depends on innate talent, according to <a href="http://dx.doi.org/10.1126/science.1261375">a recent study</a> Cimpian helped lead. Fewer women were found in fields that idolized “brilliance” over hard work. This remained true even after statistically correcting for other factors such as the <a href="http://internal.psychology.illinois.edu/%7Eacimpian/GRE_results.pdf">math performance of graduate school applicants</a>. </p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/84443/original/image-20150609-10689-w3s2p1.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/84443/original/image-20150609-10689-w3s2p1.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/84443/original/image-20150609-10689-w3s2p1.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=387&fit=crop&dpr=1 600w, https://images.theconversation.com/files/84443/original/image-20150609-10689-w3s2p1.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=387&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/84443/original/image-20150609-10689-w3s2p1.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=387&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/84443/original/image-20150609-10689-w3s2p1.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=487&fit=crop&dpr=1 754w, https://images.theconversation.com/files/84443/original/image-20150609-10689-w3s2p1.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=487&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/84443/original/image-20150609-10689-w3s2p1.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=487&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Fewer women are found in “genius” fields in which professors believe that success depends on innate talent, not hard work.</span>
<span class="attribution"><span class="source">Adapted from Figure 1 in Leslie, Cimpian, Meyer, and Freeland (2015).</span>, <a class="license" href="http://creativecommons.org/licenses/by-nc-nd/4.0/">CC BY-NC-ND</a></span>
</figcaption>
</figure>
<p>“It is crucial to look not just at what’s in people’s heads but also at the ability beliefs that are ‘in the air,’” Cimpian concluded. Teachers who believe that math intelligence is fixed can <a href="http://dx.doi.org/10.1016/j.jesp.2011.12.012">both comfort and demotivate students</a> with messages such as “It’s ok – not everyone can be good at math,” for instance.</p>
<h2>How interventions can help</h2>
<figure class="align-right ">
<img alt="" src="https://images.theconversation.com/files/84323/original/image-20150609-5896-rg0oim.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/84323/original/image-20150609-5896-rg0oim.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=899&fit=crop&dpr=1 600w, https://images.theconversation.com/files/84323/original/image-20150609-5896-rg0oim.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=899&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/84323/original/image-20150609-5896-rg0oim.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=899&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/84323/original/image-20150609-5896-rg0oim.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=1130&fit=crop&dpr=1 754w, https://images.theconversation.com/files/84323/original/image-20150609-5896-rg0oim.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=1130&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/84323/original/image-20150609-5896-rg0oim.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=1130&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Brains are not ‘hardwired’ and static, but rather grow even in adulthood.</span>
<span class="attribution"><span class="source">deadstar</span></span>
</figcaption>
</figure>
<p>Our brains develop <a href="http://dx.doi.org/10.1016/j.tics.2011.08.002">even in adulthood</a>. Experiences such as <a href="http://dx.doi.org/10.1002/hipo.20233">learning to navigate in London</a> and even <a href="http://dx.doi.org/10.1186/1756-0500-2-174">playing Tetris</a> can make important neural regions grow. Basic cognitive abilities <a href="http://dx.doi.org/10.1177/0963721413484756">can improve too</a>, even among students with <a href="http://dx.doi.org/10.1016/j.lindif.2012.03.012">exceptionally strong math backgrounds</a>.</p>
<p>Teaching this brain science to students may even help them learn. Struggling students’ grades improve when they hear that intelligence can grow with hard work, according to <a href="http://dx.doi.org/10.1177/0956797615571017">a new study</a> on “mindset interventions” involving 1,594 students in 13 high schools. In the study’s growth mindset intervention, students spent roughly 45 minutes to read – and do two writing exercises related to – an article about the brain’s ability to grow.</p>
<p>Improvement in grades was roughly one-tenth of a letter grade – a modest, but still impressive, improvement considering the intervention lasted less than an hour.</p>
<p>That study’s authors argue the effects were <a href="http://dx.doi.org/10.3102/0034654311405999">“not magic”</a> because they depend on opportunities such as having supportive teachers and peers. Mindset interventions “encourage students to take advantage of such opportunities and may be ineffective if these opportunities are absent,” the authors write.</p>
<p><a href="http://d-miller.github.io/pubs/">My research</a> has looked at how opportunities such as sketching engineering designs shape basic spatial skills such as mentally rotating objects. These skills <a href="http://dx.doi.org/10.1037/a0016127">are important</a> to success in math-intensive careers, yet <a href="http://www.nap.edu/catalog/11019/learning-to-think-spatially-gis-as-a-support-system-in">often neglected</a> in education. </p>
<p>“Oh, but you can’t teach those skills,” teachers often say when I’ve discussed my research with them. Contrary to such beliefs, <a href="http://dx.doi.org/10.1016/j.lindif.2012.03.012">I found</a> that 12 hours of spatial instruction improved students’ spatial skills and grades in a challenging calculus-based physics course. In fact, <a href="http://dx.doi.org/10.1037/a0028446">a quantitative review</a> of 217 related studies found training spatial skills was “effective, durable, and transferable.”</p>
<figure>
<iframe width="440" height="260" src="https://www.youtube.com/embed/YPBI8-isxCM?wmode=transparent&start=0" frameborder="0" allowfullscreen></iframe>
<figcaption><span class="caption">Parents can help children learn spatial skills too. Susan Levine, professor of psychology at University of Chicago, talks about her research. Parents who use spatial words like ‘circle’ and ‘tall’ engage their children in spatial learning.</span></figcaption>
</figure>
<p>Teachers who continue to believe that “your basic intelligence can’t change” – despite evidence to the contrary – may rob students of opportunities to learn and grow. Computer science and math instructors who endorse such beliefs, for instance, <a href="http://dx.doi.org/10.1016/j.jesp.2011.12.012">report being more likely to advise</a> struggling undergraduates to drop their classes. </p>
<p>We need to abandon dangerous ideas that some people just can’t do math. Neuroscience and educational research flatly contradict such beliefs. As the new study suggests, valuing hard work over innate “genius” might even spur students to tackle new challenges.</p><img src="https://counter.theconversation.com/content/42967/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>David Miller receives funding from the National Science Foundation.</span></em></p>Kids who think being good at mathematics is just a matter of God-given talent are less likely to pursue math-related fields. But research says this kind of belief is misguided.David Miller, Doctoral Student in Psychology, Northwestern UniversityLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/403272015-06-07T23:16:35Z2015-06-07T23:16:35ZA little number theory makes the times table a thing of beauty<figure><img src="https://images.theconversation.com/files/83790/original/image-20150603-10701-1j85hpt.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Doing the 9 times table.</span> <span class="attribution"><a class="source" href="https://www.flickr.com/photos/jimmiehomeschoolmom/4007733107">Flickr/Jimmie</a>, <a class="license" href="http://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA</a></span></figcaption></figure><p>Most people will probably remember the times tables from primary school quizzes. There might be patterns in some of them (the simple doubling of the 2 times table) but others you just learnt by rote. And it was never quite clear just why it was necessary to know what 7 x 9 is off the top of your head.</p>
<p>Well, have no fear, there will be no number quizzes here.</p>
<p>Instead, I want to show you a way to build numbers that gives them some structure, and how multiplication uses that structure.</p>
<h2>Understanding multiplication</h2>
<p>Multiplication simply gives you the area of a rectangle, if you know the lengths of the sides. Pick any square in the grid, (for example, let’s pick the 7th entry in the 5th row) and colour a rectangle from that square to the top left corner.</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/78485/original/image-20150418-3220-1km7pgm.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/78485/original/image-20150418-3220-1km7pgm.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=600&fit=crop&dpr=1 600w, https://images.theconversation.com/files/78485/original/image-20150418-3220-1km7pgm.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=600&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/78485/original/image-20150418-3220-1km7pgm.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=600&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/78485/original/image-20150418-3220-1km7pgm.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=754&fit=crop&dpr=1 754w, https://images.theconversation.com/files/78485/original/image-20150418-3220-1km7pgm.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=754&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/78485/original/image-20150418-3220-1km7pgm.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">A rectangle of size 5 × 7 in the multiplication table.</span>
</figcaption>
</figure>
<p>This rectangle has length 7 and height 5, and the area (the number of green squares) is found in the blue circle in the bottom right corner! This is true no matter which pair of numbers in the grid you pick.</p>
<p>Now let’s take this rectangle and flip it around the main diagonal (the red dotted line).</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/83911/original/image-20150604-1003-1ijb31i.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/83911/original/image-20150604-1003-1ijb31i.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=600&fit=crop&dpr=1 600w, https://images.theconversation.com/files/83911/original/image-20150604-1003-1ijb31i.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=600&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/83911/original/image-20150604-1003-1ijb31i.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=600&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/83911/original/image-20150604-1003-1ijb31i.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=754&fit=crop&dpr=1 754w, https://images.theconversation.com/files/83911/original/image-20150604-1003-1ijb31i.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=754&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/83911/original/image-20150604-1003-1ijb31i.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">The same rectangle, flipped.</span>
</figcaption>
</figure>
<p>The length and height of the rectangle have swapped, but the area hasn’t changed. So from this we can see that 5 × 7 is the same as 7 × 5. This holds true for any pair of numbers — in mathematics we say that multiplication is commutative.</p>
<p>But this fact means that there is a symmetry in the multiplication table. The numbers above the diagonal line are like a mirror image of the numbers below the line.</p>
<p>So if your aim is to memorise the table, you really only need to memorise about half of it.</p>
<h2>The building blocks of numbers</h2>
<p>To go further with multiplication we first need to do some dividing. Remember that dividing a number just means breaking it into pieces of equal size.</p>
<blockquote>
<p>12 ÷ 3 = 4</p>
</blockquote>
<p>This means 12 can be broken into 3 pieces, each of size 4.</p>
<p>Since 3 and 4 are both whole numbers, they are called factors of 12, and 12 is said to be divisible by 3 and by 4. If a number is only divisible by itself and 1, it is called a prime number.</p>
<p>But there’s more than one way to write 12 as a product of two numbers:</p>
<blockquote>
<p>12 × 1</p>
<p>6 × 2</p>
<p>4 × 3</p>
<p>3 × 4</p>
<p>2 × 6</p>
<p>1 × 12</p>
</blockquote>
<p>In fact, we can see this if we look at the multiplication table.</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/79013/original/image-20150423-29743-1tzio4a.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/79013/original/image-20150423-29743-1tzio4a.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=600&fit=crop&dpr=1 600w, https://images.theconversation.com/files/79013/original/image-20150423-29743-1tzio4a.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=600&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/79013/original/image-20150423-29743-1tzio4a.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=600&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/79013/original/image-20150423-29743-1tzio4a.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=754&fit=crop&dpr=1 754w, https://images.theconversation.com/files/79013/original/image-20150423-29743-1tzio4a.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=754&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/79013/original/image-20150423-29743-1tzio4a.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">The occurrences of 12 in the multiplication table.</span>
</figcaption>
</figure>
<p>The number of coloured squares in this picture tells you there are six ways you can make a rectangle of area 12 with whole number side lengths. So it’s also the number of ways you can write 12 as a product of two numbers.</p>
<p>Incidentally, you might have noticed that the coloured squares seem to form a smooth curve — they do! The curve joining the squares is known as a hyperbola, given by the equation a × b = 12, where ‘a’ and ‘b’ are not necessarily whole numbers.</p>
<p>Let’s look again at the list of products above that are equal to 12. Every number listed there is a factor of 12. What if we look at factors of factors? Any factor that is not prime (except for 1) can be split into further factors, for example</p>
<blockquote>
<p>12 = 6 × 2 = (2 × 3) × 2</p>
<p>12 = 4 × 3 = (2 × 2) × 3</p>
</blockquote>
<p>No matter how we do it, when we split the factors until we’re left only with primes, we always end up with two 2’s and one 3.</p>
<p>This product</p>
<blockquote>
<p>2 × 2 × 3</p>
</blockquote>
<p>is called the prime decomposition of 12 and is unique to that number. There is only one way to write a number as a product of primes, and each product of primes gives a different number. In mathematics this is known as the <a href="https://www.mathsisfun.com/numbers/fundamental-theorem-arithmetic.html">Fundamental Theorem of Arithmetic</a>.</p>
<p>The prime decomposition tells us important things about a number, in a very condensed way.</p>
<p>For example, from the prime decomposition 12 = 2 × 2 × 3, we can see immediately that 12 is divisible by 2 and 3, and not by any other prime (such as 5 or 7). We can also see that it’s divisible by the product of any choice of two 2’s and one 3 that you want to pick.</p>
<p>Furthermore, any multiple of 12 will also be divisible by the same numbers. Consider 11 x 12 = 132. This result is also divisible by 1, 2, 3, 4, 6 and 12, just like 12. Multiplying each of these with the factor of 11, we find that 132 is also divisible by 11, 22, 33, 44, 66 and 132.</p>
<p>It’s also easy to see if a number is the square of another number: In that case there must be an even number of each prime factor. For example, 36 = 2 × 2 × 3 × 3, so it’s the square of 2 × 3 = 6.</p>
<p>The prime decomposition can also make multiplication easier. If you don’t know the answer to 11 × 12, then knowing the prime decomposition of 12 means you can work through the multiplication step by step.</p>
<blockquote>
<p>11 x 12</p>
<p>= 11 x 2 × 2 × 3</p>
<p>= ((11 x 2) × 2) × 3</p>
<p>= (22 × 2) × 3</p>
<p>= 44 × 3</p>
<p>= 132</p>
</blockquote>
<p>If the primes of the decomposition are small enough (say 2, 3 or 5), multiplication is nice and easy, if a bit paper-consuming. Thus multiplying by 4 (= 2 x 2), 6 (= 2 x 3), 8 (= 2 x 2 x 2), or 9 (= 3 x 3) doesn’t need to be a daunting task!</p>
<p>For example, if you can’t remember the 9 times table, it doesn’t matter as long as you can multiply by 3 twice. (However this method doesn’t help with multiplying by larger primes, here new methods are required – if you haven’t seen the trick for the 11 times tables <a href="https://www.youtube.com/watch?v=rRzOWmyulS4">watch this video</a>).</p>
<p>So the ability to break numbers into their prime factors can make complicated multiplications much simpler, and it’s even more useful for bigger numbers.</p>
<p>For example, the prime decomposition of 756 is 2 x 2 x 3 x 3 x 3 x 7, so multiplying by 756 simply means multiplying by each of these relatively small primes. (Of course, finding the prime decomposition of a large number is usually very difficult, so it’s only useful if you already know what the decomposition is.)</p>
<p>But more than this, prime decompositions give fundamental information about numbers. This information is widely useful in mathematics and other fields such as cryptography and internet security. It also leads to some surprising patterns – to see this, try colouring all multiples of 12 in the times table and see what happens. I’ll leave that for homework.</p><img src="https://counter.theconversation.com/content/40327/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Anita Ponsaing receives funding from the ARC Centre of Excellence for Mathematical & Statistical Frontiers (ACEMS).</span></em></p>If you know a bit about how numbers are made then you don't need to work out all 144 calculations in a 12 by 12 times table.Anita Ponsaing, Research Associate in Mathematics, University of MelbourneLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/404712015-04-30T20:48:09Z2015-04-30T20:48:09ZHere's how to get kids to remember times tables<figure><img src="https://images.theconversation.com/files/79051/original/image-20150423-3136-uu2cl6.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">It's hard for kids to remember a string of arbitrary numbers</span> <span class="attribution"><span class="source">from www.shutterstock.com.au</span></span></figcaption></figure><p>Lots of kids have trouble remembering their times tables. Learning them by rote can mean a child can accurately recite the times tables, but has no idea what the numbers actually mean or how to apply this knowledge in a maths problem. </p>
<p>Practice is essential to effective learning, but it is important to keep a balance between practice and application.</p>
<h2>Children need to know why they need to learn times tables</h2>
<p>The number system underlying the times tables can often seem fairly arbitrary. A child can be forgiven for thinking it’s just a complex system of numbers that they have to learn because the teacher says so. </p>
<p>If you have no idea why you are required to learn something, it is very difficult to develop sufficient motivation to persist with the practice that is often necessary to master the material.</p>
<p>One way to demonstrate the usefulness of the times tables is to engage a child in a counting task. The task could be timed, such that determining the number of elements quickly will mean something for the ultimate result (for example, beating a time limit results in a positive outcome, failing to beat the limit means the task starts again). </p>
<p>The multiplication facts in the times tables can then be demonstrated as short cuts in the counting process (if you can arrange the elements into four groups of eight, then knowing the answer to “4 x 8 = ?” will result in faster performance than having to rely on counting all of the 32 elements). </p>
<p>Many computer games and apps (like <a href="http://www.ixl.com/">IXL Maths</a>) possess this feature. They also involve many other features designed to maintain the interest of a child, which can help keep them motivated enough to persist with the task. Ultimately, the more practice, the better the knowledge.</p>
<h2>Get to know the sums individually, not as a song lyric</h2>
<p>Memory can often be a good reflection of what we do. If we regularly sing along to a favourite song, each line tends to remind us of the next line. However, if we then try to sing the song by ourselves, without the aid of an accompanying recording, we often find that forgetting one line means subsequent lines also can’t be recalled. </p>
<p>A similar thing can happen if we engage in rote recitation of the times tables. This method is only useful if we want to have a method to fall back upon when all other methods fail. Basically this method can only produce the equivalent of a song lyric where, remembering what “4 8s” are is only possible if you can remember “4 6s are 24” and “4 7s are 28”.</p>
<p>A better form of knowledge is one where a child knows the answer to each multiplication problem as soon as they see it, much like being able to read a word as soon as you see it. </p>
<p>Knowing the answer to each problem is then independent of knowing the answer to other times table problems. This type of knowledge can be gained only by practice at producing the answer. </p>
<p>One method for undertaking this type of practice is something like the old flash-card method. Write a problem on one side of a card (4 x 8 = ?), and the answer on the other side. With a shuffled deck of cards representing all of the problems in the times tables, a child can practise producing the answer to each problem, and then check their response by turning over the card. </p>
<p>Occasionally an adult can ask the child to do this out loud to ensure they are doing the task correctly. Initially the child may have to guess the correct answer, or work it out with their fingers or some other method. But they always have the benefit of immediate feedback by turning over the card. </p>
<p>Eventually, with enough practice, the constant association of the problem with the correct answer will begin to stick in their memory. A similar method can be easily programmed on to a computer or tablet. Plenty of <a href="http://www.bigbrainz.com">commercial apps</a> are available that will mimic this procedure.</p>
<h2>Apply the times tables knowledge</h2>
<p>Knowledge of the times tables is not useful by itself. A child must learn to apply the knowledge in a mathematical context. </p>
<p>It is important, though, that a child’s knowledge of the times tables is not allowed to remain as a list of independent facts. A child needs to engage in activities that demonstrate the connections between the multiplication facts in the times tables. It is important to see how 4 x 8 and 8 x 4 are connected. </p>
<p>Ultimately they will also need to see how 4 x 8 = ? and 32 ÷ 8 = ? are connected. To achieve this the child should be provided with activities that require the application of their arithmetic knowledge in a way that can demonstrate and lead the child to uncover these connections. </p>
<p>Practice with this sort of material can help kids develop a knowledge base that results in reliable retrieval of facts and the sort of flexible application of this knowledge that is required in higher-order problems, such as solving for x in 2x + 3 = 11. If you struggle to come up with an answer to this problem, I would not suggest relying on a times tables song to help you out.</p><img src="https://counter.theconversation.com/content/40471/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Craig Speelman receives funding from the Australian Research Council, Edith Cowan University, the WA Department of Education, the Association of Independent Schools WA, and the Collier Charitable Fund.</span></em></p>Lots of kids have trouble remembering their times tables. Learning them by rote can mean a child knows the numbers but not what they mean.Craig Speelman, Professor of Psychology, Edith Cowan UniversityLicensed as Creative Commons – attribution, no derivatives.