tag:theconversation.com,2011:/us/topics/maths-problem-18325/articlesmaths problem – The Conversation2019-05-27T14:02:35Ztag:theconversation.com,2011:article/1171992019-05-27T14:02:35Z2019-05-27T14:02:35ZSouth Africa's voter turnout: a mathematician runs the numbers<figure><img src="https://images.theconversation.com/files/276315/original/file-20190524-187165-ycastt.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">People queue to cast their votes in Johannesburg, South Africa.</span> <span class="attribution"><span class="source">Kim Ludbrook/EPA</span></span></figcaption></figure><p>It has commonly been believed in South Africa that voter turnout in national elections has been fairly high when compared with other countries, among them <a href="https://ewn.co.za/2019/05/09/sa-voter-turnout-compared-to-other-parts-of-the-world">the UK, US and France</a>. </p>
<p>To a mathematician like myself, “voter turnout” is not a very precise term. What exactly does it mean? And how is it calculated? </p>
<p>After South Africa’s recent national and provincial elections, I drew from several definitions and understandings of “voter turnout” to crunch the numbers and work out what country’s real voter turnout was. </p>
<p>The results suggest that, when measured either as a percentage of the voting-eligible population or as a percentage of the voting-age population, South Africa’s voter turnout has been worryingly low by international standards – not just in the 2019 polls, but for the last few elections. </p>
<p>For instance, by one measure – voting-age population, or simply the number of people living within the country who are of voting age – voter turnout in the 2019 elections was 46.7%. This places it among the lowest in the world. </p>
<p>The problem, as I discovered while gathering the numbers, is that very different definitions of “voter turnout” are used in different parts of the world. It is important, for the sake of accuracy, that everybody uses the same denominator. This will allow countries to correctly understand their voter turnouts, and design interventions to improve these where necessary.</p>
<h2>Voting-eligible population turnout</h2>
<p>To begin with, here’s how South Africa’s <a href="https://www.elections.org.za/NPEDashboard/app/dashboard.html">Independent Electoral Commission</a> (IEC) defines “voter turnout”: the number of people who voted divided by the number of registered voters. </p>
<p>But globally, the “gold standard” in measuring voter turnout is to take the number of votes cast and divide by the size of the <em>voting-eligible population</em> (VEP). The VEP is the number of citizens living in the country who are legally allowed to register and vote. </p>
<p>In South Africa, this means all citizens over the age of 18. In other countries it is slightly different; for instance in the US, it means all citizens over the age of 18 who are not felons or mentally incapacitated.</p>
<p>The trouble is that estimating the citizen population living in a country is difficult: counting the population is fairly easy; this is done in a census. But counting the number of <em>citizens</em> is harder and is generally not done in a census.</p>
<p>South Africa’s IEC has sometimes provided such estimates; for instance in the 2019 elections it <a href="https://www.elections.org.za/content/About-Us/News/Over-700-000-new-voters-added-to-the-voters--roll-ahead-of-elections/">estimated</a> that 74.5% of eligible voters were registered. </p>
<p>Since 66% of registered voters cast a vote, we conclude that the VEP turnout was 66% x 74.5% = 49%. </p>
<p>With these calculations, I created a graph comparing the VEP turnout of South African parliamentary elections to the major elections in the UK, US and France, from 1999 to the present day. </p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/276584/original/file-20190527-40038-1sq6vw6.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/276584/original/file-20190527-40038-1sq6vw6.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/276584/original/file-20190527-40038-1sq6vw6.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=291&fit=crop&dpr=1 600w, https://images.theconversation.com/files/276584/original/file-20190527-40038-1sq6vw6.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=291&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/276584/original/file-20190527-40038-1sq6vw6.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=291&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/276584/original/file-20190527-40038-1sq6vw6.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=366&fit=crop&dpr=1 754w, https://images.theconversation.com/files/276584/original/file-20190527-40038-1sq6vw6.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=366&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/276584/original/file-20190527-40038-1sq6vw6.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=366&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption"></span>
<span class="attribution"><span class="source">Source: International IDEA, Eurostat, United States Elections Project, IEC, Schulz-Herzenberg, C & Southall, R. (eds), Election 2014 South Africa: The campaigns, results and future prospects, Jacana Media & Konrad Adenauer Stiftung</span></span>
</figcaption>
</figure>
<p>From the graph it’s clear that, since the 2004 elections, the VEP turnout in South Africa has consistently been much lower than in the UK and France, and a little lower than in the US. Also, VEP voter turnout plunged dramatically in South Africa from 57% in 2014 to 49% in 2019.</p>
<h2>Voting-age population turnout worldwide</h2>
<p>Next I turned to a more readily available statistic, <a href="http://www.electproject.org/home/voter-turnout/faq/denominator">the voting-age population</a> (VAP). Using this in my calculations allowed me to compare South Africa’s voter turnout to a broader spectrum of countries.</p>
<p>The VAP is simply the number of people living within the country of voting age; in South Africa, as in most countries, that means older than 18. This number is usually fairly accurately known from census data.</p>
<p>VAP turnout is not as good a measurement of voter turnout as VEP turnout, since it does not account for non-citizens and other residents of a population not eligible to vote. However, it is a more accurate reflection of voter engagement than only dividing by the number of <em>registered</em> voters, as it takes into account those who do not register.</p>
<p>The <a href="https://www.idea.int/data-tools/data/voter-turnout">International Institute for Democracy and Electoral Assistance</a> keeps tabs on VAP turnout data. The VAP turnout for the 2019 South African elections is not yet in the IDEA database, but we can readily compute it ourselves. The official Stats SA figures allow one to estimate that there were 37.8 million people aged 18 years and older living in South Africa in mid-year 2018. Furthermore, 17.6 million votes were cast. This means the VAP turnout can be estimated at about 46.7%. </p>
<p>Here is a graph of VAP turnout in the most recent major national election for all <a href="https://freedomhouse.org/report/freedom-world-2018-table-country-scores">“free” or “partly free”</a> countries in the world with greater than 1 million inhabitants. Some countries have been highlighted for the sake of comparison. </p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/276578/original/file-20190527-40012-b521ui.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/276578/original/file-20190527-40012-b521ui.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/276578/original/file-20190527-40012-b521ui.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=287&fit=crop&dpr=1 600w, https://images.theconversation.com/files/276578/original/file-20190527-40012-b521ui.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=287&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/276578/original/file-20190527-40012-b521ui.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=287&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/276578/original/file-20190527-40012-b521ui.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=361&fit=crop&dpr=1 754w, https://images.theconversation.com/files/276578/original/file-20190527-40012-b521ui.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=361&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/276578/original/file-20190527-40012-b521ui.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=361&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
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<span class="caption"></span>
<span class="attribution"><span class="source">Source: International IDEA</span></span>
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<p>We see from the graph that the VAP voter turnout of 46.7% in the South African 2019 elections was the 5th lowest out of the 59 countries in the dataset, higher only than Switzerland, Cyprus, Mali and Jamaica. In recent weeks, the media has <a href="https://www.businesslive.co.za/bd/opinion/columnists/2019-05-17-genevieve-quintal-frustration-rather-than-apathy-behind-low-turnout/">started to</a> <a href="https://www.dailymaverick.co.za/opinionista/2019-05-14-2019-elections-proof-that-the-people-have-withdrawn-their-consent/">realise</a> <a href="https://www.dailymaverick.co.za/opinionista/2019-05-17-south-africans-must-overcome-the-elections-apathy-and-hold-the-government-to-account/">the problem of voter apathy</a> and various experts’ concerns are being heard.</p>
<p><em>Author’s note: A longer form of this article containing more detailed and interactive graphs, further details about the sources for the data and the computations behind them, can be found on my <a href="https://math.sun.ac.za/bbartlett/assets/voter-turnout-long.html">webpage</a>.</em></p><img src="https://counter.theconversation.com/content/117199/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Bruce Bartlett 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>To a mathematician the idea of, "voter turnout" is not a very precise term. What exactly does it mean? And how is it calculated?Bruce Bartlett, Senior Lecturer in Mathematics, Stellenbosch UniversityLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/1156462019-05-16T10:31:15Z2019-05-16T10:31:15ZHow to overcome a fear of maths<figure><img src="https://images.theconversation.com/files/274417/original/file-20190514-60529-ktb5t3.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>It’s fair to say maths is not everyone’s favourite subject. In fact, for many people, the feelings of tension and anxiety that arise when trying to solve a mathematical problem can be all consuming. This is known as maths anxiety – and this feeling of being a failure at maths can affect people’s <a href="https://www.theguardian.com/commentisfree/2019/mar/14/panic-maths-anxiety-studying-pupils-schools">self-worth for years to come</a>.</p>
<p>For those who suffer with maths anxiety, it can be difficult to shift from a mindset of failure to a more positive outlook when it comes to dealing with numbers. This is why, for many people, maths anxiety can become a lifelong issue.</p>
<p>But <a href="https://www.amazon.co.uk/Elephant-Classroom-Helping-Children-Learn/dp/0285643185/ref=tmm_pap_swatch_0_encoding=UTF8&qid=1557478920&sr=8-1-fkmrnull">research</a> shows that if teachers tackle maths anxiety in the classroom and encourage children to try to approach a problem in a different way – by shifting their mindset – this can be an empowering experience. This is especially the case for pupils from a disadvantaged background. </p>
<h2>Mindset theory</h2>
<p>US psychology professor, Carol Dweck, came up with the idea of “<a href="https://www.ted.com/talks/carol_dweck_the_power_of_believing_that_you_can_improve/discussion">mindset theory</a>”. Dweck realised that people can often be categorised into two groups, those who believe they are bad at something and cannot change, and those who believe their abilities can grow and improve. </p>
<p>This formed the basis of her mindset theory, which states that some people have a “fixed mindset”, meaning they believe their ability to be set in stone and unable to be improved. Other people have a “growth mindset” meaning they believe their ability can change and improve over time with effort and practice. </p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/274418/original/file-20190514-60554-phm6ki.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/274418/original/file-20190514-60554-phm6ki.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/274418/original/file-20190514-60554-phm6ki.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/274418/original/file-20190514-60554-phm6ki.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/274418/original/file-20190514-60554-phm6ki.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/274418/original/file-20190514-60554-phm6ki.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/274418/original/file-20190514-60554-phm6ki.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">Maths can be fun – if only it’s taught properly.</span>
<span class="attribution"><span class="source">Shutterstock</span></span>
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</figure>
<p>Jo Boaler, the British education author and professor of mathematics education, applied mindset theory to mathematics, subsequently naming her recommendations “mathematical mindsets”. </p>
<p>She has used this theory to encourage learners to develop a growth mindset <a href="https://www.youcubed.org/mathematical-mindset-teaching-guide-teaching-video-and-additional-resources">in the context of mathematics</a>. The idea is that the problems themselves can help to promote a growth mindset in pupils – without them <a href="https://www.amazon.co.uk/tale-Lucy-wanted-learn-spell/dp/1533445001">having to think about their mindset</a> intentionally. </p>
<h2>New ways of thinking</h2>
<p>But while this all sounds well and good, one of the issues with mindset theory is that it is often presented in terms of <a href="https://theconversation.com/what-is-brain-plasticity-and-why-is-it-so-important-55967">brain plasticity</a> or the <a href="https://www.conted.ox.ac.uk/about/brain-resources">brain’s ability to grow</a>. This has lead to <a href="http://www.danielwillingham.com/daniel-willingham-science-and-education-blog/march-13th-2019">complaints</a> about a shortage of neurological evidence supporting mindset theory. <a href="https://authors.elsevier.com/c/1YsHs7sy6LOEAw">Our latest research</a> aimed to address this lack of neurological research. </p>
<p>Generally speaking, for every problem in mathematics there is more than one way to solve it. If someone asks you what three multiplied by four is, you can calculate the answer either as 4+4+4 or as 3+3+3+3, depending on your preference. But if you have not developed sufficient mathematical maturity or have maths anxiety, it can prevent you from <a href="https://www.researchgate.net/publication/326424642_Non-adaptive_strategy_selection_in_adults_with_high_mathematical_anxiety">seeing multiple ways of solving problems</a>. But <a href="https://authors.elsevier.com/c/1YsHs7sy6LOEAw">our new study</a> shows that a “growth mindset” can make maths anxiety a thing of the past.</p>
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Read more:
<a href="http://theconversation.com/maths-six-ways-to-help-your-child-love-it-96441">Maths: six ways to help your child love it</a>
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<p>We measured participants’ motivation to solve mathematical problems by asking about motivation both before and after each problem was presented. We also measured participants’ brain activity, specifically looking at areas associated with motivation, while they solved each problem. This was done using an electroencephalogram (EEG) which records patterns of activation across the brain.</p>
<p>In our research, we phrased questions in different ways to assess how question structure may affect both our participants’ ability to answer the questions and their motivation while tackling maths problems.</p>
<p>Each question appeared in two formats: one of typical mathematical teaching and another adhering to the recommendations of mathematical mindset theory. Both questions asked essentially the same question and had the same answer, like in the following simplified example:</p>
<p>“Find the number which is the sum of 20,000 and 30,000 divided by two” (a typical mathematical problem) and “Find the midpoint number between 20,000 and 30,000” (an example of a mathematical mindset problem). </p>
<h2>Growth mindsets</h2>
<p>Our study provides two important findings.</p>
<p>The first is that participants’ motivation was greater when solving mathematical mindset versions of problems compared to the standard versions – as measured by their brain response when solving the problems. It is assumed this is because the mathematical mindset wording encourages students to treat numbers as points in the space and manipulate spatial constructions.</p>
<p>The second is that participants’ subjective reports of motivation were significantly decreased after attempting the more standard maths questions. </p>
<p>Our research is immediately actionable in that it shows how opening up problems so that there are multiple methods to solving them, or adding a visual component, allows learning to become an empowering experience for all students.</p>
<p>So for people with maths anxiety, you will be relieved to know that you are not innately “bad” at maths and your ability is not fixed. It is actually just a bad habit you have developed due to bad teaching. And the good news is, it can be reverted.</p><img src="https://counter.theconversation.com/content/115646/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>Maths anxiety can be made a thing of the past, as new research shows.Alexei Vernitski, Senior Lecturer in Mathematics, University of EssexIan Daly, Lecturer in Brain-Computer Interfaces, University of EssexJake Bourgaize, PhD Candidate, University of EssexLicensed as Creative Commons – attribution, no derivatives.tag: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>
<figure class="align-center ">
<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">
<figcaption>
<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>
</figcaption>
</figure>
<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/601682016-06-01T20:16:40Z2016-06-01T20:16:40ZWill computers replace humans in mathematics?<figure><img src="https://images.theconversation.com/files/124546/original/image-20160531-13810-16flhes.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Computers are coming up with proofs in mathematics that are almost impossible for a human to check.</span> <span class="attribution"><span class="source">Shutterstock/Fernando Batista</span></span></figcaption></figure><p>Computers can be valuable tools for helping mathematicians solve problems but they can also play their own part in the discovery and proof of mathematical theorems.</p>
<p>Perhaps the first major result by a computer came 40 years ago, with proof for the <a>four-color theorem</a> – the assertion that any map (with certain reasonable conditions) can be coloured with just four distinct colours.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/124539/original/image-20160531-13773-1iv50nc.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/124539/original/image-20160531-13773-1iv50nc.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/124539/original/image-20160531-13773-1iv50nc.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=450&fit=crop&dpr=1 600w, https://images.theconversation.com/files/124539/original/image-20160531-13773-1iv50nc.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=450&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/124539/original/image-20160531-13773-1iv50nc.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=450&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/124539/original/image-20160531-13773-1iv50nc.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=566&fit=crop&dpr=1 754w, https://images.theconversation.com/files/124539/original/image-20160531-13773-1iv50nc.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=566&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/124539/original/image-20160531-13773-1iv50nc.png?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">No more that four colours are needed in this picture to make sure that no two touching shapes share the same colour.</span>
<span class="attribution"><a class="source" href="https://commons.wikimedia.org/wiki/File:Four_Colour_Map_Example.svg">Wikimedia/Inductiveload</a>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span>
</figcaption>
</figure>
<p>This was first proved by computer in 1976, although flaws were later found, and a <a href="http://www.ams.org/notices/200811/tx081101382p.pdf">corrected proof</a> was not completed until 1995.</p>
<p>In 2003, Thomas Hales, of the University of Pittsburgh, published a computer-based proof of <a href="http://experimentalmath.info/blog/2014/08/formal-proof-completed-for-keplers-conjecture-on-sphere-packing/">Kepler’s conjecture</a> that the familiar method of stacking oranges in the supermarket is the most space-efficient way of arranging equal-diameter spheres.</p>
<p>Although Hales published a proof in 2003, many mathematicians were not satisfied because the proof was accompanied by two gigabytes of computer output (a large amount at the time), and some of the computations could not be certified.</p>
<p>In response, Hales produced a <a href="http://experimentalmath.info/blog/2014/08/formal-proof-completed-for-keplers-conjecture-on-sphere-packing/">computer-verified formal proof</a> in 2014.</p>
<h2>The new kid on the block</h2>
<p>The latest development along this line is the <a href="http://www.nature.com/news/two-hundred-terabyte-maths-proof-is-largest-ever-1.19990">announcement this month in Nature</a> of a computer proof for what is known as the Boolean Pythagorean triples problem. </p>
<p>The assertion here is that the integers from one to 7,824 can be coloured either red or blue with the property that no set of three integers a, b and c that satisfy a<sup>2</sup> + b<sup>2</sup> = c<sup>2</sup> (Pythagoras’s Theorem where a, b and c form the sides of a right triangle) are all the same colour. For the integers from one to 7,825, this cannot be done.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/124463/original/image-20160530-7678-sa7jwl.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/124463/original/image-20160530-7678-sa7jwl.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/124463/original/image-20160530-7678-sa7jwl.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=360&fit=crop&dpr=1 600w, https://images.theconversation.com/files/124463/original/image-20160530-7678-sa7jwl.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=360&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/124463/original/image-20160530-7678-sa7jwl.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=360&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/124463/original/image-20160530-7678-sa7jwl.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=452&fit=crop&dpr=1 754w, https://images.theconversation.com/files/124463/original/image-20160530-7678-sa7jwl.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=452&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/124463/original/image-20160530-7678-sa7jwl.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=452&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Pythagoras’s theorem for a right-angled triangle.</span>
<span class="attribution"><span class="source">The Conversation</span>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span>
</figcaption>
</figure>
<p>Even for small integers, it is hard to find a non-monochrome colouring. For instance, if five is red then one of 12 or 13 must be blue, since 5<sup>2</sup> + 12<sup>2</sup> = 13<sup>2</sup>; and one of three or four must also be blue, since 3<sup>2</sup> + 4<sup>2</sup> = 5<sup>2</sup>. Each choice has many constraints.</p>
<p>As it turns out, the number of possible ways to colour the integers from one to 7,825 is gigantic – more than 10<sup>2,300</sup> (a one followed by 2,300 zeroes). This number is far, far greater than the number of fundamental particles in the visible universe, which is a mere <a>10<sup>85</sup></a>. </p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/124693/original/image-20160601-1964-12ufmt4.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/124693/original/image-20160601-1964-12ufmt4.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/124693/original/image-20160601-1964-12ufmt4.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=524&fit=crop&dpr=1 600w, https://images.theconversation.com/files/124693/original/image-20160601-1964-12ufmt4.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=524&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/124693/original/image-20160601-1964-12ufmt4.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=524&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/124693/original/image-20160601-1964-12ufmt4.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=658&fit=crop&dpr=1 754w, https://images.theconversation.com/files/124693/original/image-20160601-1964-12ufmt4.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=658&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/124693/original/image-20160601-1964-12ufmt4.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=658&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 numbers one to 7,824 can be coloured either red or blue so that no trio a, b and c that satisfies Pythagoras’s theorem are all the same colour. A white square can be either red or blue.</span>
<span class="attribution"><span class="source">Marijn Heule</span></span>
</figcaption>
</figure>
<p>But the researchers were able to sharply reduce this number by taking advantage of various symmetries and number theory properties, to “only” one trillion. The computer run to examine each of these one trillion cases required two days on 800 processors of the University of Texas’ <a>Stampede supercomputer</a>.</p>
<p>While direct applications of this result are unlikely, the ability to solve such difficult colouring problems is bound to have implications for coding and for security.</p>
<p>The Texas computation, which we estimate performed roughly 10<sup>19</sup> arithmetic operations, is still not the largest mathematical computation. A 2013 <a href="http://www.ams.org/notices/201307/rnoti-p844.pdf">computation</a> of digits of pi<sup>2</sup> by us and two IBM researchers did twice this many operations. </p>
<p>The Great Internet Mersenne Prime Search (<a href="http://www.mersenne.org">GIMPS</a>), a global network of computers search for the largest known prime numbers, routinely performs a total of <a href="https://www.sciencedaily.com/releases/2016/01/160120084917.htm">450 trillion calculations per second</a>, which every six hours exceeds the number of operations performed by the Texas calculation. </p>
<p>In computer output, though, the Texas calculation takes the cake for a mathematical computation – a staggering 200 terabytes, namely 2✕10<sup>14</sup> bytes, or 30,000 bytes for every human being on Earth.</p>
<p>How can one check such a sizeable output? Fortunately, the Boolean Pythagorean triple program produced a solution (shown in the image, above) that can be checked by a much smaller program.</p>
<p>This is akin to factoring a very large number c into two smaller factors a and b by computer, so that c = a ✕ b. It is often quite difficult to find the two factors a and b, but once found, it is a trivial task to multiply them together and verify that they work.</p>
<h2>Are mathematicians obsolete?</h2>
<p>So what do these developments mean? Are research mathematicians soon to join the ranks of <a href="http://www.nytimes.com/1997/05/12/nyregion/swift-and-slashing-computer-topples-kasparov.html">chess grandmasters</a>, <a href="http://www.nytimes.com/2011/02/17/science/17jeopardy-watson.html">Jeopardy champions</a>, <a href="http://www.geekwire.com/2016/more-layoffs-at-nordstrom/">retail clerks</a>, <a href="https://www.theguardian.com/technology/2016/feb/10/black-cab-drivers-uber-protest-london-traffic-standstill">taxi drivers</a>, <a href="http://www.cnet.com/news/driverless-truck-convoy-platoons-across-europe/">truck drivers</a>, <a href="http://www.huffingtonpost.com/entry/ibm-watson-radiology_us_55cbccf9e4b0898c48867c56">radiologists</a> and other professions threatened with obsolescence due to rapidly advancing technology?</p>
<p>Not quite. Mathematicians, like many other professionals, have for the large part embraced computation as a new mode of mathematical research, a development known as experimental mathematics, which has far-reaching implications.</p>
<p>So what exactly is experimental mathematics? It is best defined as a mode of research that employs computers as a “laboratory,” in the same sense that a physicist, chemist, biologist or engineer performs an experiment to, for example, gain insight and intuition, test and falsify conjecture, and confirm results proved by conventional means.</p>
<p>We have written on this topic at some length elsewhere – see our <a href="http://www.experimentalmath.info/books/">books</a> and <a href="https://www.carma.newcastle.edu.au/jon/papers.html#PAPERS">papers</a> for full technical details.</p>
<p>In one sense, there there is nothing fundamentally new in the experimental methodology of mathematical research. In the third century BCE, the great Greek mathematician Archimedes <a href="https://books.google.com/books?id=Vvj_AwAAQBAJ&pg=PA314#v=onepage">wrote</a>:</p>
<blockquote>
<p>For it is easier to supply the proof when we have previously acquired, by the [experimental] method, some knowledge of the questions than it is to find it without any previous knowledge.</p>
</blockquote>
<p>Galileo once reputedly wrote:</p>
<blockquote>
<p>All truths are easy to understand once they are discovered; the point is to discover them.</p>
</blockquote>
<p>Carl Friederich Gauss, 19th century mathematician and physicist, frequently employed computations to motivate his remarkable discoveries. He once wrote:</p>
<blockquote>
<p>I have the result, but I do not yet know how to get [prove] it.</p>
</blockquote>
<p>Computer-based experimental mathematics certainly has technology on its side. With every passing year, computer hardware advances with <a href="http://www.intel.com/content/www/us/en/silicon-innovations/moores-law-technology.html">Moore’s Law</a>, and mathematical computing software packages such as Maple, Mathematica, Sage and others become ever more powerful.</p>
<p>Already these systems are powerful enough to solve virtually any equation, derivative, integral or other task in undergraduate mathematics.</p>
<p>So while ordinary human-based proofs are still essential, the computer leads the way in assisting mathematicians to identify new theorems and chart a route to formal proof.</p>
<p>What’s more, one can argue that in many cases computations are more compelling than human-based proofs. Human proofs, after all, are subject to mistakes, oversights, and reliance on earlier results by others that may be unsound. </p>
<p><a href="http://www.intel.com/content/www/us/en/silicon-innovations/moores-law-technology.html">Andrew Wiles’</a> initial proof of <a href="http://simonsingh.net/books/fermats-last-theorem/the-whole-story/">Fermat’s Last Theorem</a> was later found to be flawed. This was fixed later.</p>
<p>Along this line, recently Alexander Yee and Shigeru Kondo computed <a href="http://www.numberworld.org/misc_runs/pi-12t/">12.1 trillion digits of pi</a>. To do this, they first computed somewhat more than 10 trillion base-16 digits, then they checked their computation by computing a section of base-16 digits near the end by a completely different algorithm, and compared the results. They matched perfectly.</p>
<p>So which is more reliable, a human-proved theorem hundreds of pages long, which only a handful of other mathematicians have read and verified in detail, or the Yee-Kondo result? Let’s face it, computation is arguably more reliable than proof in many cases.</p>
<h2>What does the future hold?</h2>
<p>There is every indication that research mathematicians will continue to work in respectful symbiosis with computers for the foreseeable future. Indeed, as this relationship and computer technology mature, mathematicians will become more comfortable leaving certain parts of a proof to computers. </p>
<p>This very question was discussed in a June 2014 <a href="http://experimentalmath.info/blog/2014/11/breakthrough-prize-recipients-give-math-seminar-talks/">panel discussion</a> by the five inaugural <a href="https://breakthroughprize.org/?controller=Page&action=news&news_id=18">Breakthrough Prize in Mathematics</a> recipients for mathematics. The Australian-American mathematician Terence Tao expressed their consensus in these terms:</p>
<blockquote>
<p>Computers will certainly increase in power, but I expect that much of mathematics will continue to be done with humans working with computers.</p>
</blockquote>
<p>So don’t toss your algebra textbook quite yet. You will need it!</p><img src="https://counter.theconversation.com/content/60168/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Jonathan Borwein (Jon) receives funding from the Australian Research Council.</span></em></p><p class="fine-print"><em><span>David H. Bailey 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>Computers are increasingly used to prove mathematical theorems. So does that mean human mathematicians will become obselete?Jonathan Borwein (Jon), Laureate Professor of Mathematics, University of NewcastleDavid H. Bailey, PhD; Lawrence Berkeley Laboratory (retired) and Research Fellow, University of California, DavisLicensed 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.