tag:theconversation.com,2011:/fr/topics/teaching-maths-24482/articlesTeaching maths – The Conversation2023-08-07T20:02:07Ztag:theconversation.com,2011:article/2070302023-08-07T20:02:07Z2023-08-07T20:02:07Z‘Why would they change maths?’ How your child’s maths education might be very different from yours<figure><img src="https://images.theconversation.com/files/540939/original/file-20230803-27-rfc6b9.jpg?ixlib=rb-1.1.0&rect=0%2C10%2C6709%2C4426&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">
</span> <span class="attribution"><span class="source">Karolina Grabowska/Pexels</span></span></figcaption></figure><p>There is a <a href="https://www.youtube.com/watch?v=3QtRK7Y2pPU">scene</a> in the film Incredibles 2 where young Dash asks his dad Bob for help with his maths homework. Bob obliges and begins to scribble on a notepad. But Dash quickly points to his textbook and says, “that’s not the way you’re supposed to do it, Dad”.</p>
<p>Frustrated, Bob exclaims</p>
<blockquote>
<p>I don’t know that way! Why would they change math? Math is math!</p>
</blockquote>
<p>Many parents trying to help their children with maths may be asking the same sort of question. </p>
<p><a href="https://www.britannica.com/science/Pythagorean-theorem">Pythagoras’ Theorem</a> is as accurate today as it was when it was discovered millennia ago, and it will continue to remain so. But teachers today also teach maths very differently from when parents were at school. </p>
<h2>Mental connections not procedures</h2>
<p>The teaching and learning of maths has undergone a transformation in past 30 years.</p>
<p>In the past there has been a focus on teaching students <a href="https://gargicollege.in/wp-content/uploads/2020/03/Skemp-article.pdf">procedures</a>, such as times tables and how to work out the circumference of a circle or solve an equation.</p>
<p>We now appreciate the importance of forming mental <a href="https://dergipark.org.tr/en/download/article-file/397449">connections between concepts</a>. For example, when students understand the connection between similar triangles and trigonometry they understand the definition of <a href="https://www.cuemath.com/trigonometry/trigonometric-ratios/">trigonometric ratios</a> at a much deeper level. </p>
<p>This is because we know <a href="https://journals.sagepub.com/doi/abs/10.3102/0013189X19890600?journalCode=edra">today’s learners</a> need to be able to transfer their mathematical understanding to complex, unfamiliar situations. </p>
<p>This means they need to be able to do more than apply a formula and get an answer right. They need to be able to <a href="https://www.monash.edu/education/teachspace/articles/seven-reasons-why-maths-is-important-for-21st-century-thinking">solve problems</a> as they arise. </p>
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<iframe width="440" height="260" src="https://www.youtube.com/embed/3QtRK7Y2pPU?wmode=transparent&start=0" frameborder="0" allowfullscreen=""></iframe>
<figcaption><span class="caption">‘Math is math!’ From the Incredibles 2.</span></figcaption>
</figure>
<h2>Problem solving and reasoning</h2>
<p>Today, the shift in mathematics education is reflected in <a href="https://v9.australiancurriculum.edu.au/teacher-resources/understand-this-learning-area/mathematics">key mathematical proficiencies</a> in the Australian school curriculum. These include: </p>
<ul>
<li><p>understanding mathematical concepts and procedures </p></li>
<li><p>being fluent in applying mathematical concepts efficiently and accurately </p></li>
<li><p>drawing on mathematical skills and knowledge to solve challenging questions where solutions are not immediately obvious, and </p></li>
<li><p>developing skills in logical thought, and justifying the use of strategies and conclusions reached.</p></li>
</ul>
<hr>
<p>
<em>
<strong>
Read more:
<a href="https://theconversation.com/will-i-ever-need-math-a-mathematician-explains-how-math-is-everywhere-from-soap-bubbles-to-pixar-movies-204609">Will I ever need math? A mathematician explains how math is everywhere – from soap bubbles to Pixar movies</a>
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<h2>Encouraging inventiveness</h2>
<p>One practical way teachers are developing students’ mathematical proficiency is by encouraging students to be inventive in the way they solve mathematical problems. </p>
<p>They are deliberately showing students different and multiple ways to represent mathematics problems to give students the space to develop understanding. This also give them an opportunity to reason, model and engage in mathematical thinking. </p>
<p>For example, your child might bring home problems to solve using the <a href="https://calculate.org.au/wp-content/uploads/sites/15/2020/05/year-5-multiplication-using-the-area-model.pdf">area model</a> for multiplication, which looks quite different to a traditional method. For example, we teach how 8x27 can be modelled in parts – 8x20 and 8x7. </p>
<p>When adding numbers, we teach students to deeply understand the place value of each number. Doing so makes our calculations efficient and supports the development of mental strategies for computation. </p>
<p>For example, to add 27 and 5, we can consider that 27 is made of 20 and 7.
Now 7 and 5 can be easily added to get to 12, and the final 20 can be added to obtain an answer of 32. </p>
<h2>The world is changing</h2>
<p>Maths teaching has also shifted to keep pace with the development of computer technology. This is influencing maths instruction from early childhood, right through to Year 12. </p>
<p>Apart from anything, we know proficiency in <a href="http://www.sciencedirect.com/science/article/pii/S0747563217301590">information and communications technology</a> is key to educating students for the world they will live and work in. </p>
<p>As an example, the use of dynamic geometry software means teachers can quickly show <a href="https://teacher.desmos.com/activitybuilder/custom/562de119ea7a4dc71545d46e#preview/83062beb-71d5-4269-9fc5-61f4f6a65442">transformations of graphs</a>, giving students a deeper understanding of how variables work, allowing students to apply these concepts to a wide range of graphs. </p>
<p>Animation is also helping in maths classrooms. The video below shows how the formula for solids of revolution – which are formed by taking an enclosed area and rotating that area about an axis to form a solid – is derived more clearly than could be done using pens and a whiteboard. </p>
<figure>
<iframe width="440" height="260" src="https://www.youtube.com/embed/HHCK9iweK6U?wmode=transparent&start=0" frameborder="0" allowfullscreen=""></iframe>
<figcaption><span class="caption">A video explains how to use calculus to find the volume of solids of revolution. Casey Machen.</span></figcaption>
</figure>
<h2>How can parents help?</h2>
<p>For parents, resist the temptation to react like Bob and get angry about how maths has changed. Instead expect the way children are taught today will be different to how you were taught and that is completely OK. </p>
<p>Knowing this, ask questions of your child, such as can you tell me about the thinking behind this method? Or, how do you know that gets the correct answer? </p>
<p>This will allow your child to show their understanding, reasoning, and communication skills. It will also help you find areas that need addressing, which you can investigate together. Who knows, you might learn some new maths and enhance your own skills!</p>
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<strong>
Read more:
<a href="https://theconversation.com/maths-anxiety-is-a-real-thing-here-are-3-ways-to-help-your-child-cope-200822">'Maths anxiety' is a real thing. Here are 3 ways to help your child cope</a>
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<img src="https://counter.theconversation.com/content/207030/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>The authors do not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and have disclosed no relevant affiliations beyond their academic appointment.</span></em></p>In the past, maths teaching has focussed on procedures and right answers. Today, teachers want students to form connections between concepts and solve problems.Ben Zunica, Lecturer in Secondary Maths Education, University of SydneyBronwyn Reid O'Connor, Lecturer in Mathematics Education, University of SydneyEddie Woo, Professor of Practice, Mathematics Education, University of SydneyLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/1943042022-11-29T17:20:59Z2022-11-29T17:20:59ZDyscalculia: how to support your child if they have mathematical learning difficulties<figure><img src="https://images.theconversation.com/files/497621/original/file-20221128-24-ead7sb.jpg?ixlib=rb-1.1.0&rect=0%2C0%2C5607%2C3295&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">
</span> <span class="attribution"><a class="source" href="https://www.shutterstock.com/image-photo/girl-doing-exercise-math-303704459">Lorena Fernandez/Shutterstock</a></span></figcaption></figure><p>A good grasp of maths has been linked to <a href="https://www.nationalnumeracy.org.uk/sites/default/files/2021-04/Counting%20on%20the%20Recovery%20(compressed)%20FINAL.pdf">greater success in employment</a> and better health. But a large proportion of us – <a href="https://www.frontiersin.org/articles/10.3389/fpsyg.2013.00516/full">up to 22%</a> – have mathematical learning difficulties. What’s more, around <a href="https://pure.qub.ac.uk/en/publications/the-prevalence-of-specific-learning-disorder-in-mathematics-and-c">6% of children in primary schools</a> may have dyscalculia, a mathematical learning disability.</p>
<p><a href="https://www.ucl.ac.uk/ioe/departments-and-centres/departments/psychology-and-human-development/child-development-and-learning-difficulties-lab/awareness-developmental-dyscalculia-and-mathematical-difficulties-toolkit-add">Developmental dyscalculia</a> is a persistent difficulty in <a href="https://www.bdadyslexia.org.uk/dyscalculia">understanding numbers</a> which can affect anyone, regardless of age or ability. </p>
<p>If 6% of children have dyscalculia, that would mean one or two children in each primary school class of 30 – about as many children as have been estimated to have <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8183124/">dyslexia</a>. But dyscalculia is less well known, by both the general public and teachers. It is also less well researched in comparison to other learning difficulties.</p>
<p>Children with dyscalculia may struggle to learn foundational mathematical skills and concepts, such as simple counting, adding, subtracting and simple multiplication as well as times tables. Later, they may have difficulty with more advanced mathematical facts and procedures, such as borrowing and carrying over but also understanding fractions and ratios, for instance. Dyscalculia not only affects children during maths lessons: it can have an impact on all areas of the curriculum.</p>
<p>These persistent difficulties cannot be explained by a general below-average ability level, or other developmental disorders. Nevertheless, children with dyscalculia may also experience other learning difficulties, such as <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3514770/">dyslexia and ADHD</a>. </p>
<p>Here are some practical tips to support children with mathematical learning difficulties. </p>
<h2>Use props</h2>
<p>Children with dyscalculia can find additional <a href="https://pubmed.ncbi.nlm.nih.gov/33845673/">practical supports</a> useful when working out even simple sums and maths problems. They may often need to use practical aids, such as their fingers or an abacus. They can benefit from using counters and beads to make sets or groups, as well as using number lines to work out answers to maths problems.</p>
<p><div data-react-class="Tweet" data-react-props="{"tweetId":"1588430330671026176"}"></div></p>
<p>Older children may find it helpful to keep crib sheets handy, which make information such as the times tables or certain formulas easily accessible. Inclusive teaching methods like these are likely to benefit all learners, not just those with dyscalculia.</p>
<h2>Break the problem down</h2>
<p>Research shows that <a href="https://educationendowmentfoundation.org.uk/education-evidence/teaching-learning-toolkit/metacognition-and-self-regulation">metacognition</a> can have a positive effect on maths learning. Metacognition is “thinking about thinking” – for example, thinking about the information you do and don’t know, or self awareness about the strategies you have to work out problems. </p>
<p>Teaching children strategies to identify where to start on a problem and how to break mathematical problems down could be a good starting point. For example, parents and teachers could encourage children to use songs and mnemonics to help them remember strategies to solve particular problems. </p>
<p>For example, the mnemonic DRAW provides students with a strategy for solving addition, subtraction, multiplication, & division problems:</p>
<p>D: discover the sign – the student finds, circles, and says the name of the operator (+,-, x or /).</p>
<p>R: read the problem – the student reads the equation.</p>
<p>A: answer – the student draws tallies or circles to find the answer, and checks it over.</p>
<p>W: write the answer – the student writes out the answer to the problem.</p>
<h2>Find out where help is needed</h2>
<p>Children with mathematical learning difficulties often get stuck with maths problems and may quickly give up. Teachers and parents should ask children what they find difficult – <a href="https://www.repository.cam.ac.uk/bitstream/handle/1810/290514/Szucs%2041179%20-%20Main%20Public%20Output%208%20March%202019.pdf?sequence=1&isAllowed=y">even young children can explain this</a> – and provide explicit instruction to support them with what they find difficult. </p>
<h2>Focus on one thing at a time</h2>
<p>As mathematical problems can be confusing for young people with mathematical difficulties, make sure to only work on one problem at the time. This could mean covering other maths questions on the page, and removing irrelevant pictures. Provide immediate feedback on both correct and incorrect answers. This will help children learn from their practice and understand the difference between correct and incorrect problem-solving strategies.</p>
<figure class="align-center ">
<img alt="Mother and daughter doing maths and counting on fingers" src="https://images.theconversation.com/files/497698/original/file-20221128-4861-j9nre0.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/497698/original/file-20221128-4861-j9nre0.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/497698/original/file-20221128-4861-j9nre0.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/497698/original/file-20221128-4861-j9nre0.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/497698/original/file-20221128-4861-j9nre0.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/497698/original/file-20221128-4861-j9nre0.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/497698/original/file-20221128-4861-j9nre0.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">Focus on one topic or problem at a time.</span>
<span class="attribution"><a class="source" href="https://www.shutterstock.com/image-photo/patient-mom-teaching-daughter-schoolwork-home-489146788">Ground Picture/Shutterstock</a></span>
</figcaption>
</figure>
<p>It may also help to provide plenty of repetition and revisiting, teach short and frequent sessions, and make sure learners know what they should do if they get stuck, such as ask an adult for help. </p>
<h2>Use the right vocabulary</h2>
<p>Mathematical language and symbols can also be confusing. For example, a negative number carries a minus sign, but a minus sign can also be used to define an operation such as subtraction. We often use the word “minus” for both – for instance, saying “14 minus minus 9” (14 – –9). This can be difficult to interpret. Various different words, such as subtract, minus and take away, can describe the same concept. </p>
<p>It is important to use clear language (for instance, “14 take away negative 9”). Helping children expand their maths vocabulary, as well as checking their understanding, will also be useful. </p>
<h2>Play games</h2>
<p>Mathematics is everywhere around us in the environment and what is learned in the classroom also applies to our daily lives. <a href="https://www.nuffieldfoundation.org/project/improving-preschoolers-number-foundations">Our own research</a> has shown that young children benefit from playing short mathematical games using the tools and materials around them. </p>
<p>Counting and collecting sets of items can be done in any place: at the dining table, in the bath, or when out and about. Practice-based <a href="https://www.nuffieldfoundation.org/wp-content/uploads/2022/05/Can-Maths-Apps-Add-Value-to-Young-Childrens-Learning-A-Systematic-Review-and-Content-Analysis_Web_final_v2.pdf">educational apps</a> can also help children master foundational maths skills.</p>
<h2>Be positive</h2>
<p>Finally, it is crucial to promote positive feelings towards mathematics. This might include not voicing your own concerns and negative feelings about maths. Rather, foster an interest in maths that will help children persevere and overcome their difficulties.</p><img src="https://counter.theconversation.com/content/194304/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Jo Van Herwegen receives funding from Higher Education Innovation Funding - UKRI. </span></em></p><p class="fine-print"><em><span>Elisabeth Herbert and Laura Outhwaite do not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and have disclosed no relevant affiliations beyond their academic appointment.</span></em></p>Use props, break the problem down and stay positive.Jo Van Herwegen, Associate Professor in the department of Psychology and Human Development, UCLElisabeth Herbert, Associate Professor, Department of Psychology and Human Development, IOE UCLUCL IOE. Programme Director for MA SpLD dyslexia and Programme route leader for the MA in Special and Inclusive Education Specific Learning Difficulties route, UCLLaura Outhwaite, Senior Research Fellow in the Centre for Education Policy and Equalising Opportunities, UCLLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/1574072021-06-07T20:06:37Z2021-06-07T20:06:37ZWhy too many recorded lecture videos may be bad for maths students’ learning<figure><img src="https://images.theconversation.com/files/397499/original/file-20210428-23-1e1lnuo.jpg?ixlib=rb-1.1.0&rect=0%2C0%2C6000%2C3988&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">
</span> <span class="attribution"><a class="source" href="https://www.shutterstock.com/image-photo/asian-woman-teacher-teaching-online-lesson-1783859852">Shutterstock</a></span></figcaption></figure><p>Screen-based devices have <a href="https://www.nielsen.com/au/en/insights/article/2018/screen-time-skyrocketing/">increasingly</a> become part of our human experience – <a href="https://www.washingtonpost.com/technology/2020/03/24/screen-time-iphone-coronavirus-quarantine-covid/">even more so</a> since the pandemic began. This trend includes watching more and more videos. For example, before COVID-19, the average American watched <a href="http://www.nielsen.com/us/en/insights/reports/2018/q1-2018-total-audience-report.html">about six hours of videos a day</a> on devices ranging from televisions to desktop computers and mobile phones. By <a href="https://www.wsj.com/articles/how-covid-19-has-transformed-the-amount-of-time-we-spend-online-01596818846">one estimate</a>, this figure has “surged” more than 40% during the pandemic.</p>
<p>In higher education, the online use of recorded lecture videos has also increased greatly. How is this affecting learning? For undergraduate mathematics, a <a href="https://link.springer.com/article/10.1007/s13394-021-00369-8">recently published review</a> confirmed the findings of a <a href="https://www.tandfonline.com/doi/full/10.1080/0020739X.2011.646325?src=recsys">2012 study</a> that, overall, the more often students watched such videos the poorer their performance in their course. </p>
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<em>
<strong>
Read more:
<a href="https://theconversation.com/covid-killed-the-on-campus-lecture-but-will-unis-raise-it-from-the-dead-152971">COVID killed the on-campus lecture, but will unis raise it from the dead?</a>
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<p>Recent research has identified a possible reason for this. It might help explain why the findings of these two reviews differ from those of studies of learning from videos in other disciplines. </p>
<h2>How might videos depress learning?</h2>
<p>Of course, correlation is not causation. It’s possible, for example, that weaker mathematics students tend to rely on videos more than stronger students. </p>
<p>However, an equally plausible explanation is that regular use of these videos is somehow depressing students’ learning. A two-part study was designed to investigate this possibility. </p>
<p>The <a href="https://www.tandfonline.com/doi/abs/10.1080/0020739X.2018.1458339">first study</a> involved two groups of students studying engineering mathematics courses in Australia and the UK. At the beginning and end of each course, students completed a questionnaire to assess how they approached their studying. </p>
<p>In both settings, regular video users were found to become more surface learners over the course of the semester. Those accessing few or no videos were unchanged in their study approaches. This was despite regular video users, as compared to low users, being older in Australia and initially better at mathematics in the UK.</p>
<p>This gave rise to a <a href="https://doi.org/10.1080/0020739X.2021.1930221">second study</a> that used interviews with Australian participants to explore how they were using the videos to advance their understanding of mathematics. First, to provide some insight into underlying processes and thus the design of the second study, a review of the cognitive research on the use of television was conducted. <a href="https://books.google.com.au/books?hl=en&lr=&id=Sicxx9FBZWMC&oi=fnd&pg=PR11&dq=Kubey,+Csikszentmihalyi&ots=EXvADrPNHw&sig=pimBbUjZ9JTdx7GMnjVmXWSsOMs#v=onepage&q=Kubey%2C%20Csikszentmihalyi&f=false">Kubey and Csikszentmihalyi</a> sum up this research:</p>
<blockquote>
<p>“[…] in every sample we have studied, with different demographic groups and with subjects ranging in age from 10 to 82, and with groups from more than one country, it has been found that people consistently report their experiences with television as being passive, relaxing, and involving relatively little concentration.” </p>
</blockquote>
<figure class="align-center ">
<img alt="couple's feet in socks in front of a TC screen" src="https://images.theconversation.com/files/397500/original/file-20210428-13-yiqrvv.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/397500/original/file-20210428-13-yiqrvv.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/397500/original/file-20210428-13-yiqrvv.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/397500/original/file-20210428-13-yiqrvv.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/397500/original/file-20210428-13-yiqrvv.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/397500/original/file-20210428-13-yiqrvv.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/397500/original/file-20210428-13-yiqrvv.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">For many people, the TV screen is the cue for a passive and relaxing experience, involving relatively little concentration.</span>
<span class="attribution"><a class="source" href="https://www.shutterstock.com/image-photo/couple-socks-woolen-stockings-watching-movies-1275793150">Shutterstock</a></span>
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<p>
<em>
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Read more:
<a href="https://theconversation.com/who-learns-in-maths-classes-depends-on-how-maths-is-taught-21013">Who learns in maths classes depends on how maths is taught</a>
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<p>With this understanding, cognitive processes associated with the use of lecture videos were considered as a dual-process system, meaning people tend to think using two channels:</p>
<ol>
<li><p>“type 1” thinking: fast and intuitive with little to no working memory used.</p></li>
<li><p>“type 2” thinking: slow and analytical with working memory used. </p></li>
</ol>
<p>Working memory has been <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4207727/">defined as</a> “the small amount of information that can be held in mind and used in the execution of cognitive tasks”. </p>
<p>The mathematics videos were viewed outside of typical lecture or classroom settings. Students actively controlled their use. Therefore, the second study interview questions focused on the critical point at which students judge their own learning to determine, for example, whether they move on to new learning or not. </p>
<p>All Australian participants were interviewed at the end of the course. The analysis of their responses showed regular users were more prone to type 1 thinking when judging their learning. They relied mostly on “feelings of rightness” rather than, for example, checking that correct procedures were followed. In mathematics, the former may lead to wrong (“<a href="https://link.springer.com/article/10.1023/A:1002998529016">pseudo-analytical</a>”) thinking, while the latter typically results in the correct solution. </p>
<h2>Findings differ in other disciplines. Why?</h2>
<p>At first glance, this discovery contrasts sharply with findings from a <a href="https://journals.sagepub.com/doi/10.3102/0034654321990713">recent systematic review</a> that <a href="https://theconversation.com/videos-wont-kill-the-uni-lecture-but-they-will-improve-student-learning-and-their-marks-142282">concluded</a> the use of videos was “consistently good for learning”. However, a closer look at the review reveals almost all the included studies (96%) related to instruction in applied undergraduate disciplines, such as health sciences, which represented over 80% of the included studies. Studies on the use of video in mathematics or other abstract disciplines that demand high-level conceptual thinking were not part of the review. </p>
<hr>
<p>
<em>
<strong>
Read more:
<a href="https://theconversation.com/videos-wont-kill-the-uni-lecture-but-they-will-improve-student-learning-and-their-marks-142282">Videos won't kill the uni lecture, but they will improve student learning and their marks</a>
</strong>
</em>
</p>
<hr>
<p>This might suggest the use of video will help learning if the level of thinking required is relatively low, such as learning medical procedures, but not necessarily where it is high, such as gaining conceptual understanding in mathematics.</p>
<p>More research is certainly needed. We still know very little about thought processes when viewing lecture videos. </p>
<p>One question arising from research in undergraduate mathematics is: have we somehow become conditioned by almost a century of television use so that when presented with a simple video recording of a lecture, the medium subconsciously signals its viewers to tone down any mental effort? This is enough to achieve better learning outcomes where low-level cognitive processing is sufficient, but could be detrimental where high-level processing is required.</p>
<p>Put another way, and more broadly, under what circumstances and with which people can screens act as cognitive cues signalling us to relax mentally, in much the same way <a href="https://pubmed.ncbi.nlm.nih.gov/27820842/">viewing food can make us salivate</a>?</p><img src="https://counter.theconversation.com/content/157407/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Sven Trenholm 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>Why might maths students’ performance suffer from relying on videos? A new study suggests we might be conditioned to watch video in a way that hinders the sort of thinking needed in maths.Sven Trenholm, Adjunct Lecturer in Mathematics Education, University of South AustraliaLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/1502622020-12-16T03:32:42Z2020-12-16T03:32:42ZJump, split or make to the next 10: strategies to teach maths have changed since you were at school<figure><img src="https://images.theconversation.com/files/375248/original/file-20201215-15-1lh0rut.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">
</span> <span class="attribution"><a class="source" href="https://www.shutterstock.com/image-photo/portrait-african-girl-writing-solution-sums-1078335890">Shutterstock</a></span></figcaption></figure><p>I’m sure most people can remember trying to master a certain maths rule or procedure in primary or secondary school.</p>
<p>My elderly mother has a story about a time her father was helping her with arithmetic homework. She remembers getting upset because her father did not do it “the school way”. I suspect her father was able to do the calculation mentally rather than the school way, which was to use the vertical algorithm.</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/375289/original/file-20201216-19-1saklzv.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/375289/original/file-20201216-19-1saklzv.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=166&fit=crop&dpr=1 600w, https://images.theconversation.com/files/375289/original/file-20201216-19-1saklzv.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=166&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/375289/original/file-20201216-19-1saklzv.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=166&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/375289/original/file-20201216-19-1saklzv.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=208&fit=crop&dpr=1 754w, https://images.theconversation.com/files/375289/original/file-20201216-19-1saklzv.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=208&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/375289/original/file-20201216-19-1saklzv.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=208&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
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<span class="attribution"><a class="license" href="http://creativecommons.org/licenses/by-nd/4.0/">CC BY-ND</a></span>
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<p>Students are expected to add the numbers in the ones (right) column first, before adding the numbers in the tens (left) column. The task becomes more difficult when the total of the ones column is more than 10 — as you then have to “trade” ten ones for one ten.</p>
<p>Students who give the answer as 713 rather than the correct answer of 83 may well have started with the tens column first. Or they may have written 13 in the ones column rather than trading ten ones for one ten.</p>
<p>The formal school algorithms are still used for larger numbers and decimals but we encourage students to use whichever strategy they prefer for two-digit addition. </p>
<p>The trouble with teaching rules is many students then struggle to remember when to apply the rule because they don’t understand how or why the rule works. </p>
<hr>
<p>
<em>
<strong>
Read more:
<a href="https://theconversation.com/weapons-of-maths-destruction-are-calculators-killing-our-ability-to-work-it-out-in-our-head-44900">Weapons of maths destruction: are calculators killing our ability to work it out in our head?</a>
</strong>
</em>
</p>
<hr>
<p>The <a href="https://australiancurriculum.edu.au/f-10-curriculum/mathematics/">Australian Curriculum: Mathematics</a> states that by the end of year 2, students will “perform simple addition and subtraction calculations using a range of strategies”. By the end of year 4, they will “identify and explain strategies for finding unknown quantities in number sentences”. </p>
<p><div data-react-class="Tweet" data-react-props="{"tweetId":"1143666051793788928"}"></div></p>
<p>We want children to remember how to do these equations in their head, rather than relying on writing down the process. Here are three strategies schools use to teach children how to add and subtract two-digit numbers.</p>
<h2>1. Split strategy</h2>
<p>This is sometimes called the decomposition, partitioning or partial-sums strategy. </p>
<p>You can add or subtract the tens separately to the ones (or units). For example, using the split strategy to add 46 + 23, you would:</p>
<ul>
<li><p>split each number (decompose) into tens and ones: 46 + 23 = 40 + 6 + 20 + 3</p></li>
<li><p>rearrange the tens and ones: 40 + 20 + 6 + 3</p></li>
<li><p>add the tens and then the ones 60 + 9 = 69</p></li>
</ul>
<p>Using the split strategy for addition such as 37 + 65 would be similar, but there would be an extra step:</p>
<ul>
<li><p>split or decompose the numbers into tens and ones: 30 + 7 + 60 + 5</p></li>
<li><p>rearrange the tens and ones: 30 + 60 + 7 + 5</p></li>
<li><p>add the tens and then the ones: 90 + 12 </p></li>
<li><p>split 12 (10 + 2) to give: 90 + 10 + 2 = 100 + 2 = 102</p></li>
</ul>
<p>Many students find the split strategy more difficult for subtraction than addition. This is because there are more steps if performing this strategy mentally. </p>
<p>For a subtraction such as 69 – 46, you would:</p>
<ul>
<li><p>split or decompose each number into tens and ones: 60 + 9 – (40 + 6)</p></li>
<li><p>remove bracket: 60 + 9 – 40 – 6</p></li>
<li><p>rearrange tens and ones: (60 – 40) + (9 – 6)</p></li>
<li><p>subtract the tens, then the ones: 20 + 3 = 23</p></li>
</ul>
<p>Students often make mistakes in the third step. Successful students may say: “I take 40 from 60, then 6 from 9”. Unsuccessful students will say “I take 40 from 60 then add 6 and 9”. </p>
<p>Students who use this strategy successfully are showing they understand place value (the value of each digit in a number) and their knowledge of maths rules needed for algebra. </p>
<h2>2. Jump strategy</h2>
<p>This is sometimes called the sequencing or cumulative sums strategy. The actual steps taken depend on the confidence and ability of the students. </p>
<p>Some students add increments of tens or ones, while others add or subtract multiples of tens then ones. </p>
<p>For example, adding 46 + 23 using the jump strategy might look like this:</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/375290/original/file-20201216-13-1odomar.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/375290/original/file-20201216-13-1odomar.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=158&fit=crop&dpr=1 600w, https://images.theconversation.com/files/375290/original/file-20201216-13-1odomar.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=158&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/375290/original/file-20201216-13-1odomar.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=158&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/375290/original/file-20201216-13-1odomar.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=199&fit=crop&dpr=1 754w, https://images.theconversation.com/files/375290/original/file-20201216-13-1odomar.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=199&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/375290/original/file-20201216-13-1odomar.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=199&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
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</figure>
<ul>
<li><p>add two lots of ten to 46: 46 + 10 = 56, then 56 + 10 = 66</p></li>
<li><p>add the remaining 3: 66 + 3 = 69 </p></li>
</ul>
<p>or </p>
<ul>
<li><p>add 20 to 46 which becomes 66</p></li>
<li><p>add the remaining 3: 66 + 3 = 69</p></li>
</ul>
<p>The two versions of this strategy can be shown using an empty number line. Using a blank or empty number line allows student to record their thinking and for teachers to analyse their thinking and determine the strategy they have attempted to use.</p>
<p>Subtracting 69 – 46 with the jump strategy could be done by:</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/375291/original/file-20201216-19-n4muyw.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/375291/original/file-20201216-19-n4muyw.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=149&fit=crop&dpr=1 600w, https://images.theconversation.com/files/375291/original/file-20201216-19-n4muyw.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=149&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/375291/original/file-20201216-19-n4muyw.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=149&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/375291/original/file-20201216-19-n4muyw.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=187&fit=crop&dpr=1 754w, https://images.theconversation.com/files/375291/original/file-20201216-19-n4muyw.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=187&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/375291/original/file-20201216-19-n4muyw.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=187&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
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<ul>
<li><p>subtracting four lots of ten (40) from 69: 69 – 10 = 59; 59 – 10 = 49; 49 – 10 = 39; 39 – 10 = 29 </p></li>
<li><p>then finally subtracting the remaining 6: 29 – 6 = 23 </p></li>
</ul>
<p>or</p>
<ul>
<li><p>subtract 40: 69 – 40 = 29 </p></li>
<li><p>then subtract 6: 29 – 6 = 23</p></li>
</ul>
<h2>3. ‘Make to the next ten’ strategy</h2>
<p>This is sometimes called the compensation or shortcut strategy. It involves adjusting one number to make the task easier to solve. </p>
<p>The “make to the next ten” strategy builds on the “friends of ten” strategy. </p>
<p>Many students in the first years of primary school create all the combinations of two single digit numbers that give a total of ten. </p>
<blockquote>
<p>9 + 1, 8 + 2, 7 + 3, 6 + 4, 5 + 5 … </p>
</blockquote>
<p>These are sometimes called the rainbow facts as the children create rainbows as they connect two numbers together. For instance, 9 may be on one end of a rainbow colour and 1 on the other. </p>
<p>By combining the numbers in this way teachers hope students will realise the answer for 9 + 1 is the same as 1 + 9.</p>
<p>In the “make to the next ten” strategy, you add or subtract a number larger than the number given (such as the next multiple of ten) and then readjust the number by subtracting what was added or adding what was subtracted.</p>
<p>In the diagrams the relationships are indicated by the use of arrows. </p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/375301/original/file-20201216-21-1lvy5ai.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/375301/original/file-20201216-21-1lvy5ai.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=156&fit=crop&dpr=1 600w, https://images.theconversation.com/files/375301/original/file-20201216-21-1lvy5ai.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=156&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/375301/original/file-20201216-21-1lvy5ai.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=156&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/375301/original/file-20201216-21-1lvy5ai.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=196&fit=crop&dpr=1 754w, https://images.theconversation.com/files/375301/original/file-20201216-21-1lvy5ai.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=196&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/375301/original/file-20201216-21-1lvy5ai.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=196&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
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<span class="attribution"><a class="license" href="http://creativecommons.org/licenses/by-nd/4.0/">CC BY-ND</a></span>
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</figure>
<p>So, to add 37 + 65, you would</p>
<ul>
<li><p>add 3 to 37 to give 40. </p></li>
<li><p>subtract 3 from 65 to get 62</p></li>
<li><p>this becomes: 40 + 62 = 102.</p></li>
</ul>
<p>If subtracting 102 – 65, you would:</p>
<ul>
<li><p>subtract 2 from 102 to make 100</p></li>
<li><p>subtract 2 from 65 to maintain the balance</p></li>
<li><p>this becomes 100 – 63 = 37.</p></li>
</ul>
<p>Many students using this strategy incorrectly add 2 to 65 instead of subtracting 2.</p>
<h2>Why these strategies?</h2>
<p>Students would have been using all these strategies, or some forms of them, in their head for generations. But for many years, the expectation was that students use the formal written algorithm rather than their own mental strategies. </p>
<p>The introduction of the empty or blank number line allowed students to record their mental strategies, which allowed teachers and parents to see them. Naming these strategies has allowed teachers and students to discuss possible strategies using a common vocabulary.</p>
<hr>
<p>
<em>
<strong>
Read more:
<a href="https://theconversation.com/kids-prefer-maths-when-you-let-them-figure-out-the-answer-for-themselves-44016">Kids prefer maths when you let them figure out the answer for themselves</a>
</strong>
</em>
</p>
<hr>
<p>Rather than teach rules and procedures, we now need to encourage students to explain their strategies using both concrete materials and diagrams to demonstrate their knowledge of addition and subtraction.</p><img src="https://counter.theconversation.com/content/150262/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Cath Pearn is affiliated with both the Australian Council for Educational Research and the Melbourne Graduate School of Education, The University of Melbourne. </span></em></p>For years you may have been adding and subtracting numbers in your head in a certain way, but these strategies were never formally taught at school. Now they are, and they all have names.Cath Pearn, Senior Research Fellow, Australian Council for Educational ResearchLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/1295632020-01-09T17:08:57Z2020-01-09T17:08:57ZWhy South Africa’s declining maths performance is a worry<figure><img src="https://images.theconversation.com/files/309280/original/file-20200109-80107-18meh2w.jpg?ixlib=rb-1.1.0&rect=1015%2C134%2C2726%2C2345&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Basic Education Minister Angie Motshekga announces South Africa's 2019 matric results and congratulates top achievers.</span> <span class="attribution"><a class="source" href="https://flickr.com/photos/governmentza/49349311572/in/dateposted/">Flickr/GCIS</a></span></figcaption></figure><p>South Africa’s Department of Basic Education recently released the country’s National Senior Certificate results for the <a href="https://www.education.gov.za/Portals/0/Documents/Reports/2019%20NSC%20Examination%20Report.pdf?ver=2020-01-07-155811-230">class of 2019</a>. These are commonly known as the “matric results” and they determine school-leavers’ admission and placement into tertiary level study. About 81.3% of those who wrote the matriculation exams passed. There has been much well-deserved celebration of this achievement of the highest post-apartheid national matric pass rate. </p>
<p>What the country is not hearing about from the Minister of Basic Education, Angie Motshekga, is the drop in performance in mathematics. It is one of the <a href="https://www.iol.co.za/mercury/news/schools-warned-against-scrapping-hard-subjects-to-achieve-100-pass-marks-30974367">“gateway” subjects</a>, subjects which are considered critical for the country’s economic growth and development.</p>
<p>This decline can be measured in two ways. There is a reduction in the number of students writing mathematics from 270,516 in 2018 to 222,034 <a href="https://www.education.gov.za/Resources/Reports.aspx">in 2019</a>. The second measure is the performance: only 54% of the pupils who wrote the exam passed it. This pass rate is down from 58% in 2018. The minimum score for a pass is 30%. This means only 54% of mathematics exam candidates achieved a mark of at least 30%. Of all the maths candidates only 2% (4,415) <a href="https://www.education.gov.za/Resources/Reports.aspx">achieved distinctions</a>. A distinction is a score of 80%-100%. This is down from 2.5% in 2018.</p>
<h2>Why does this matter?</h2>
<p>The drop in numbers of pupils writing the grade 12 mathematics exam should be of great concern. Performance in mathematics matters for university entrance. Without it, school leavers are not eligible for programmes at university in science or engineering or some in commerce. A decline signals the closing of the doors of opportunity in these fields to a growing number of students. This will only increase inequality. Economics researcher Nic Spaull’s <a href="https://link.springer.com/chapter/10.1007%2F978-3-030-18811-5_1">research</a> has shown that the top 200 high schools in the country produce 97% of the mathematics distinctions. The majority of these schools charge significant fees. </p>
<p>The deterioration in performance is also of great concern. Getting a pass (30%) may secure a diploma or university entrance but these low pass marks will not prepare students to succeed at mathematics at university level. </p>
<p>This development runs contrary to the needs of the <a href="https://www.britannica.com/topic/The-Fourth-Industrial-Revolution-2119734">fourth industrial revolution</a>, which requires highly competent graduates in the science, technology, engineering and maths areas. Strong performance in mathematics is essential for careers in computing, programming, finance and machine learning. </p>
<h2>Universities need to shoulder the blame</h2>
<p>Universities cannot absolve themselves of this national challenge. At the University of Cape Town data from the <a href="https://www.uct.ac.za/main/teaching-and-learning/courses-impeding-graduation">Courses Impeding Graduation</a> project is being analysed to better understand incoming students’ challenges, specifically in courses like Mathematics 1. </p>
<p>In this course a worrying pattern of performance emerged. A minimum mark of 70% for maths in matric is needed to get into Mathematics 1 at the university. Based on several years of data, an average of 33% of students fail this course. Those students who enter with a 90% mark for maths in matric score a pass in Mathematics 1 with an average mean of 64%. Those students who achieved between 80% and 89% in matric fail the course with an average mean of 47%. Those who achieved between 70% and 79% in matric fail with an average mean of 43%. </p>
<p>Unless a student achieved a distinction for mathematics at school level they are at risk of failing it at university level. Students who fail Mathematics 1 will inevitably take longer to complete their degree and are at higher risk of being excluded from the university.</p>
<h2>Dealing with the problem</h2>
<p>The University of Cape Town is taking responsibility for its share in these dismal results. A number of interventions have been put in place over recent years to provide additional support to students. These include “maths labs”, Saturday workshops, and even providing multilingual resources to support students who are not yet fluent in the medium of instruction.</p>
<p>Expert maths teachers have been appointed to lecture this challenging course. But the overall failure rates of approximately one third of the class have remained stubbornly in place. A decision was taken in 2019 to revise the Mathematics 1 curriculum to ensure a greater alignment between schooling and university curriculum. </p>
<p>This kind of curriculum review raises a number of complex issues: what is the appropriate content to ensure a relatively seamless transition from school maths to university maths? Do different disciplinary areas like actuarial science, chemistry and engineering need different kinds of mathematics courses? How can the pacing of the curriculum accommodate different learning needs? How can educational technology support innovative forms of teaching and learning mathematics? These are global issues, not unique to South Africa.</p>
<p>The national euphoria around the national pass rate means nothing if it hides problems such as declining mathematics performance.</p><img src="https://counter.theconversation.com/content/129563/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Suellen Shay 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>Performance in mathematics matters for university entrance. Without it, school leavers are not eligible for many programmes.Suellen Shay, Professor, University of Cape TownLicensed 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>
<figure class="align-center ">
<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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</figure>
<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">
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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>
</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 organisation 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/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="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Harry Zandberg Wiggins 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>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/853272017-10-25T19:06:31Z2017-10-25T19:06:31ZTeaching kids about maths using money can set them up for financial security<figure><img src="https://images.theconversation.com/files/191138/original/file-20171020-1082-atxtty.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">There are plenty of opportunities when you are out shopping to include your child in discussions about financial decisions.</span> <span class="attribution"><span class="source">Shutterstock</span></span></figcaption></figure><p><em>As the world of finance becomes more complex, most of us aren’t keeping up. In this series we’re exploring <a href="https://theconversation.com/au/topics/what-is-financial-literacy-45200">what it means to be financially literate</a>.</em></p>
<hr>
<p>One of the most common complaints children have about learning maths is its lack of relevance to their lives outside school. When they fail to see the importance of maths to their current and future lives, they often <a href="https://link.springer.com/article/10.1007/s13394-011-0020-5">lose interest</a>. </p>
<p>This results in opting out of mathematics study as soon as they can, and proclaiming they are <a href="https://theconversation.com/saying-im-not-good-at-maths-is-not-cool-negative-attitudes-are-affecting-business-53298">“not good at maths”</a>.</p>
<p>Financial literacy – learning about budgeting, saving, investing and basic financial decision making – taught by both parents and teachers can help keep them engaged.</p>
<h2>Three strategies for teachers</h2>
<p><a href="http://www.aamt.edu.au/">The Australian Association of Mathematics Teachers</a> promote the teaching of financial literacy through maths with the help of contemporary teaching and learning resources that reflect students’ interests. These include lesson plans, units of work, children’s literature, and interactive digital resources such as games.</p>
<p>A wide range of resources are available from websites such as <a href="https://www.moneysmart.gov.au/">MoneySmart</a> and <a href="http://finlit.org.au/resources/programs-and-information/">Financial Literacy Australia</a>. These are an excellent way to begin teaching financial literacy concepts, with some units of work specifically designed with a mathematics focus. However, these units can and should be adjusted to suit the specific needs of the students in your classroom.</p>
<p>Additionally, teachers should consider using resources that are familiar to students’ everyday lives. These could include items that are in the news media, shopping catalogues, television commercials etc. Keep watch for interesting photographs or misleading advertisements. They are great for starting discussions about maths. Questions such as “is this really a good deal?”, “what is the best deal?” or even “what mathematics do we need to know and understand to work out if this advertisement is offering a bargain?” could begin discussions. </p>
<p>There are also a range of apps that could be used alongside maths and financial literacy explorations, including budgeting apps and supermarket apps such as <em>TrackMySpend</em>, <em>Smart Budget</em>, or <em>My Student Budget Planner</em> . If you like using picture books to introduce and teach concepts, the <a href="http://moneyandstuff.info/books/">Money & Stuff website</a> has an extensive list of books relating to financial literacy.</p>
<h2>The money connection</h2>
<p>One way to improve engagement with mathematics is for schools to teach it in ways that children are familiar with. Most children are familiar with money, and many are already consumers of financial services from a young age. <a href="https://www.acer.org/files/PISA_2012_Financial_Literacy.pdf">Research</a> has found that it’s not uncommon for children to have accounts with access to online payment facilities or to use mobile phones during the primary school years. It’s clear that financial literacy and mathematics skills would be beneficial when using such products.</p>
<p><a href="http://www.bea.asn.au/cms/files/cms_files/content/Financial_literacy/NationalConsumerFinancialLiteracyFramework_2011.pdf">Financial education</a> programs for young people can be essential in nurturing sound financial knowledge and behaviour in students from a young age. <a href="https://eric.ed.gov/?id=ED572577">Using real-life contexts</a> involving financial literacy can help children learn a range of mathematical concepts and numeracy skills like lending and borrowing, budgeting, and interest rates. They are more likely to remember and understand what they have learned because they applied mathematics to something they’re interested in and something that they can use in their lives. </p>
<p><a href="https://www.moneysmart.gov.au/media/560516/moneymathematicsengagement_final_report_14_september_2016-assoc-prof-catherine-attard.pdf">Research</a> into the teaching of financial literacy combined with mathematics in primary schools shows how important it is for all children to understand the importance and value of money and recognise the maths that underpins consumer and financial literacy. </p>
<p>They also need to engage in real world projects and investigations relating to consumer and financial literacy to understand how mathematics is applied in everyday decisions that could influence life opportunities.</p>
<h2>Shopping is a teaching opportunity for parents</h2>
<p>Many young children don’t understand where money comes from. It’s important that they begin to develop some understanding of how our economy works, even from a young age. <a href="https://www.moneysmart.gov.au/media/560516/moneymathematicsengagement_final_report_14_september_2016-assoc-prof-catherine-attard.pdf">Research</a> has found a pattern emerging where children whose parents talk to them about money develop an earlier understanding of its importance. They are also provided with more opportunities to deal with making decisions about money. </p>
<p>If you have young children in primary school, it’s a great time to start their financial literacy and mathematics education. There are plenty of opportunities when you are out shopping to include your child in discussions and decisions where appropriate, or explain the financial decisions you make on their behalf. Talk about the mathematics involved in financial decision-making. Where possible, encourage children to make their own financial decisions with things like pocket money or savings. If you feel you need to improve your own financial literacy first, there are many <a href="https://www.moneysmart.gov.au/">resources</a> available for adults. </p>
<p>Teaching children about money through mathematics helps children learn. It helps them use mathematics in real-life scenarios and, more importantly, can help set them up for future financial security.</p><img src="https://counter.theconversation.com/content/85327/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Catherine Attard received a grant from Financial Literacy Australia in 2014 to investigate the use of financial literacy education to improve student engagement with mathematics in primary schools from low socio-economic areas. </span></em></p>Learning about real-life money decisions from a young age helps kids learn maths and improves their financial literacy.Catherine Attard, Associate Professor, Mathematics Education, Western Sydney 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/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 organisation 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/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/702822016-12-12T11:52:27Z2016-12-12T11:52:27ZWhy it doesn’t help – and may harm – to fail pupils with poor maths marks<figure><img src="https://images.theconversation.com/files/149628/original/image-20161212-31402-1xeb6x.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Learning deficits in Maths compound over time.</span> <span class="attribution"><span class="source">Shutterstock</span></span></figcaption></figure><p>Many South Africans were <a href="https://businesstech.co.za/news/finance/146351/sa-government-lowers-pass-mark-for-maths-to-20/">outraged</a> by the <a href="http://www.education.gov.za/Portals/0/Documents/Publications/Circular%20A3%20of%202016.pdf?ver=2016-12-07-154005-723">recent announcement</a> that for 2016, pupils in Grades 7 to 9 could progress to the next grade with only 20% in Mathematics. </p>
<p>The usual minimum has been 40%, provided that all other requirements for promotion are met. Pupils with less than 30% in Mathematics in grade 9 must take <a href="http://www.education.gov.za/Portals/0/CD/National%20Curriculum%20Statements%20and%20Vocational/CAPS%20FET%20_%20MATHEMATICAL%20LITERACY%20_%20GR%2010-12%20_%20Web_DDA9.pdf?ver=2015-01-27-154330-293">Mathematical Literacy</a> (this involves what the Department of Basic Education calls “the use of elementary mathematical content” and is not the same as Mathematics) as a matric subject.</p>
<p>Public concern is understandable. South Africans should be deeply worried about the state of mathematics teaching and learning. The country was placed <a href="http://www.telegraph.co.uk/education/2016/11/29/revealed-world-pupil-rankings-science-maths-timss-results/">second from last</a> for mathematics achievement in the latest Trends in International Maths and Science Study. </p>
<p>Research closer to home has <a href="http://www.ekon.sun.ac.za/wpapers/2014/wp272014">shown</a> that pupils, particularly from poorer and less well resourced schools, are under performing in mathematics relative to the curriculum outcomes. These learning deficits compound over time, which makes it increasingly difficult to address learning difficulties in mathematics in the higher grades. </p>
<p>All of this means that children and young people may be in Mathematics classes but are not learning. But the answer to this problem does not lie with making pupils repeat an entire grade because of poor mathematical performance. There’s extensive research evidence to suggest that grade repetition does more harm than good.</p>
<h2>Repetition is not effective</h2>
<p>Grade repetition is practised worldwide – despite there being very little evidence for its effectiveness. In fact, it can be argued that its consequences are mainly negative for repeating pupils. Grade repetition is a predictor of <a href="http://www.create-rpc.org/pdf_documents/PTA16.pdf">early school leaving</a>, sometimes called “drop out”. </p>
<p>Pupils who repeat grades and move out of their age cohort become <a href="https://www.oecd.org/economy/grade-repetition-a-comparative-study-of-academic-and-non-academic-consequences.pdf">disaffected with school</a>. They disengage from learning. </p>
<p>Repeating a grade <a href="http://www.unesco.org/iiep/PDF/Edpol6.pdf">lowers motivation</a> towards learning and is <a href="https://books.google.co.za/books?id=yxDawksXxn0C&printsec=frontcover&dq=International+guide+to+student+achievement&hl=en&sa=X&redir_esc=y#v=onepage&q=grade%20retention%20is%20not%20associated%20with%20academic%20growth&f=false">seldom associated with improved learning outcomes</a>. </p>
<p>South Africa’s rates of grade repetition are high. Research by the Department of Basic Education <a href="http://www.education.gov.za/Portals/0/Documents/Publications/General%20Household%20Survey%202013.pdf?ver=2015-07-07-111309-287">shows</a> that on average, 12% of all pupils from grades one to 12 repeat a year. The grades with the highest repetition rates are grade 9 (16.3%), grade 10 (24.2%) and grade 11 (21.0%).</p>
<p>And grade repetition is an equity issue. <a href="http://www.socialsurveys.co.za/factsheets/AcessToEducation-TechnicalReport/29ff21.pdf">The Social Survey-CALS (2010)</a> report found that black children are more likely to repeat grades than their white or Indian peers. This reflects the fracture lines that signal socioeconomic disadvantage in South Africa.</p>
<p>Repetition rates decrease as the education level of the household head increases. Poor access to infrastructural resources, like piped water and flush toilets, are associated with higher rates of grade repetition. Boys are more likely to repeat than girls. There’s also an uncertain link between pupil achievement and grade repetition, particularly for black learners in high schools. </p>
<p>So why does grade repetition persist?</p>
<h1>Beliefs about the benefits of repetition</h1>
<p>Schools and societies <a href="http://www.journals.uchicago.edu/doi/abs/10.1086/667655?journalCode=cer">still believe</a> in the value of making children repeat grades, despite evidence to the contrary.</p>
<p>A recent survey of 95 teachers in Johannesburg – which is currently under review for publication in a journal – showed how teachers believe the additional time spent in a repeated year allows pupils to “catch up” and be better prepared for the subsequent grade. This view is reflected in recent <a href="http://www.groundup.org.za/article/teachers-oppose-20-pass-mark-maths/">reports</a> that teachers are against the new 20% concession which has stirred so much controversy. Their opposition is echoed by countless callers to talk shows, who all seem to assume that repeating subject content results in improved understanding.</p>
<p>But unless the reasons for a pupil’s misunderstanding of concepts are identified and addressed, any improvement is unlikely. Given that the deficits in mathematical understanding may stretch back to the foundation phase (Grades 1 - 3), it’s doubtful that merely repeating a grade in the senior phase is going to be sufficient for remediation. </p>
<p>And teachers may struggle to provide support to pupils repeating a grade. Research conducted in South Africa <a href="http://www.tandfonline.com/doi/abs/10.1080/13603116.2015.1095250?journalCode=tied20">reveals</a> that teachers lack confidence in their ability to teach pupils who experience learning difficulties. They would prefer to refer such pupils to learning support specialists and psychologists who are seen to have more expertise. </p>
<p>Many of the teachers we surveyed believe that grade repetition solves problems intrinsic to pupils. Immaturity is seen as one reason for learning difficulties and teachers expect that the repeated year compensates for this. Other teachers regard the threat of retention as a means to motivate pupils who are not sufficiently diligent or who are “slow” or “weak”. When learning difficulties are seen as being intrinsic to pupils, it is less likely that factors within the education system will be considered as the cause of barriers to learning.</p>
<h2>Failing pupils is not the solution</h2>
<p>Poor achievement in mathematics is not going to be solved by making pupils repeat their grade. Repetition effectively makes pupils and their families pay an additional – financial and emotional – cost for the system’s failure.</p>
<p>Repetition because of poor mathematics achievement during the senior phase compounds the bleak outlook for these pupils. They already have a minimal grasp of mathematics, which denies them access to Science, Technology, Engineering and mathematics (STEM) subjects and careers. Then they’re also at risk of leaving school early and joining the ranks of <a href="https://theconversation.com/how-two-crucial-trends-are-affecting-unemployment-in-south-africa-56296">the unemployed</a>.</p>
<p>The Department of Basic Education’s 20% concession indicates that it knows grade repetition won’t achieve much. The public outcry should not be that these learners are being given a “free pass” and don’t deserve to be promoted. Instead, civil society needs to hold the government accountable for addressing the crisis in mathematics teaching and learning across all grades – and particularly in the crucial primary school years.</p><img src="https://counter.theconversation.com/content/70282/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Elizabeth Walton 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 extensive research evidence to suggest that grade repetition does more harm than good.Elizabeth Walton, Associate professor, University of the WitwatersrandLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/649762016-09-08T20:02:42Z2016-09-08T20:02:42ZTeaching maths – what does the evidence say actually works?<figure><img src="https://images.theconversation.com/files/136854/original/image-20160907-25260-1mqs01j.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Teachers can help parents support their child's maths learning at home.</span> <span class="attribution"><span class="source">from www.shutterstock.com</span></span></figcaption></figure><p><em>In our series, <a href="https://theconversation.com/au/topics/better-teachers-30749">Better Teachers</a>, we’ll explore how to improve teacher education in Australia. We’ll look at what the evidence says on a range of themes including how to raise the status of the profession and measure and improve teacher quality.</em></p>
<hr>
<p>“<a href="https://www.theguardian.com/education/2016/mar/26/reckon-you-were-born-without-a-brain-for-maths-highly-unlikely?CMP=share_btn_link">I’m just so bad at maths!</a>” Too often we hear that claim uttered in fear and frustration. </p>
<p>It’s not just students who say this, but also their parents and, in some cases, their teachers, particularly in primary schools.</p>
<p>Research on the topic of maths anxiety is inconclusive and scarce. But there are a few things we know. </p>
<p>A <a href="http://www.ncbi.nlm.nih.gov/pubmed/26779093">recent study</a> found that “maths anxiety and maths performance can influence one another in a vicious cycle”. Research is unclear as to whether poor maths performance triggers maths anxiety, or whether maths anxiety reduces maths performance. It seems likely to be a combination of both negatively reinforcing each other.</p>
<p>Psychologists have found there can be a very real physical response to maths in both adults and children. This includes the <a href="http://www.ncbi.nlm.nih.gov/pubmed/21707166">release of stress hormones like cortisol</a>, which are characteristic of the fight or flight response. </p>
<p>One <a href="http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3485285/">study even found that anticipating</a> a maths test activates the brain’s <a href="http://www.bbc.com/future/story/20150619-do-you-have-maths-anxiety">“pain matrix</a>” – the regions that might light up if you had injured yourself.</p>
<p>It is an affliction that appears to affect females more than males. Cultural expectations may be to blame – <a href="http://www.ncbi.nlm.nih.gov/pubmed/23985576">girls are more likely to catch maths anxiety</a> (particularly from female teachers), perhaps because of <a href="http://www.ncbi.nlm.nih.gov/pubmed/23087633">stereotypes that girls are “not very good” at maths</a>.</p>
<h2>What works in maths teaching?</h2>
<p>When maths teachers see students struggling, they often give more of the same work, rather than going back and plugging the hole or gap in understanding. </p>
<p>No matter what age the student, before moving on, the teacher should always go back to the preceding skill, or take the concept back to the hands-on concrete phase, until the student has confidently mastered that skill or concept.</p>
<p><strong>Learning the language of maths</strong></p>
<p>It is important to teach children that maths is a language of its own. If students can’t speak the language of maths fluently, they don’t really understand the fundamental concepts. </p>
<p>Sentence frames, co-operative learning tasks and frequent problem-solving linked to real-world examples ensure students <a href="https://%20www.ascd.org/publications/books/.../Mathematics-as-Language.aspx">“talk” maths fluently</a> and accurately to gain mastery over the language of maths. </p>
<p>Using a sentence frame such as “A ______ has four sides and four corners” or “one metre is equal to ___ centimetres” during warm-ups or plenary sessions allows students to develop fluency in maths vocabulary and a deeper understanding of the targeted mathematical concepts.</p>
<p><strong>More emphasis on formative assessment</strong></p>
<p>To reduce maths test anxiety, teachers should put more emphasis on <a href="http://edglossary.org/formative-assessment/">formative assessment</a> – on-the-spot monitoring and feedback – rather than relying mainly on summative assessment where a student is assessed at the end of a period of learning. </p>
<p>Giving specific, purposeful and timely feedback to maximise learning opportunities is more effective than missing key learning opportunities waiting for children to fail an end-of-unit assignment.</p>
<p><strong>Mastering maths concepts</strong></p>
<p>Being able to give students many opportunities to <a href="https://research.acer.edu.au/cgi/viewcontent.cgi?article=1022&context=aer">practise and master</a> the concrete, pictorial and abstract phases of development for each maths concept is essential for them to learn successfully. </p>
<p><strong>Understand which level each student is at</strong></p>
<p><a href="http://www.apa.org/monitor/2010/07-08/gender-gap.aspx">Some research</a> suggests that girls rely more on manipulatives – using concrete materials to support their problem-solving – and that boys move on more quickly to mental cognitive strategies, such as mental strategies like doubles or bridge to ten. </p>
<p>Regular, informative assessment ensures the teacher has evidence of what each student knows and what they need to learn next when they are <a href="http://grattan.edu.au/wp-content/uploads/2015/07/827-Targeted-Teaching.pdf">ready to move on</a>.</p>
<p><strong>Adapt teaching and resources</strong></p>
<p>Effective maths teachers are able to <a href="http://www.edu.gov.on.ca/eng/literacynumeracy/inspire/research/different_math.pdf">differentiate their teaching practices</a>, curriculum and resources to ensure all students are accessing the maths curriculum, feeling sufficiently challenged, but not overly anxious, and working to grow their potential in learning maths.</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/137022/original/image-20160908-25253-1fzs6l0.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/137022/original/image-20160908-25253-1fzs6l0.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/137022/original/image-20160908-25253-1fzs6l0.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/137022/original/image-20160908-25253-1fzs6l0.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/137022/original/image-20160908-25253-1fzs6l0.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/137022/original/image-20160908-25253-1fzs6l0.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/137022/original/image-20160908-25253-1fzs6l0.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">Some states are moving towards appointing specialist maths teachers in primary schools.</span>
<span class="attribution"><span class="source">from www.shutterstock.com</span></span>
</figcaption>
</figure>
<h2>Do we need specialist maths teachers in primary school?</h2>
<p>In some states, such as <a href="https://www.teach.nsw.edu.au/exploreteaching/types-of-teachers/specialist-teachers">New South Wales</a>, there is a move to appoint specialist maths teachers in primary schools. </p>
<p>On the surface, this appears to be a tempting solution to raising student performance in maths, but what is the price to be paid? </p>
<p>The primary school system of the generalist classroom teacher intentionally differs from secondary schooling in order to meet the social and emotional needs of young children, as well as their academic needs. </p>
<p>Generalist teachers take responsibility for developing the whole child. A focus on coaching and feedback for primary teachers to improve their mathematics instruction would ensure the learning of maths remains the responsibility of everyone in a school. It would also combat the myth that some people simply can’t do maths. </p>
<p>The federal government has committed A$54 million over four years to the <a href="http://archive.industry.gov.au/ministerarchive2013/chrisevans/mediareleases/pages/investinginscienceandmathsforasmarterfuture.aspx.htm">Investing in Science and Maths for a Smarter Future</a> initiative. In 2012, the Office for Learning and Teaching funded five research projects on the teaching of maths and science. These projects were funded in response to recommendations from Ian Chubb, then Australia’s chief scientist, in his <a href="http://www.chiefscientist.gov.au/wp-content/uploads/Office-of-the-Chief-Scientist-MES-Report-8-May-2012.pdf">2012 report</a> Mathematics, Engineering and Science: in the national interest.</p>
<p><a href="http://remstep.org.au/about-the-project/">One of these projects</a> focuses on the training of specialist maths and science teachers in primary and secondary schools. Findings from these projects are due to be delivered in 2017.</p>
<h2>Confident maths teachers, confident maths students</h2>
<p>Not all students will enjoy learning maths. But this is more likely to happen if maths is well taught from early childhood. And the key to confident maths students is confident teachers.</p>
<p>An instructional practices survey at a <a href="http://www.det.wa.edu.au/schoolsonline/ind_rvw_rpt.do?schoolID=5824&pageID=AD28">primary school in Western Australia</a> found that 40% of teachers believed their confidence in teaching maths would benefit from explicit coaching and feedback. </p>
<p>In response to this, the school maths leadership team designed a series of professional learning sessions to build staff confidence; created mentors for other staff members to help reduce their levels of anxiety about teaching maths; and ran workshops for parents every few weeks to inform them of the latest research in maths, the common areas that children are having difficulty with, and provide practical ways that parents can support their children’s maths learning at home. </p>
<p>By informing parents about key maths skills children are learning at school, teachers showed parents how everyday home activities, such as cooking and shopping, can help reinforce mathematical thinking. </p>
<p>Parents were made aware of the subliminal messages they might send to their children when they say things like, “It’s ok you’re no good at maths, because I wasn’t either.” Parents were then given alternative language to use with their children when they talked about maths.</p>
<p>As a result, teachers and parents at the school have gained confidence in their ability to teach maths and this is reflected in the students’ learning and attitude to maths. </p>
<p>The school is analysing a range of data to see how improving students’ confidence in learning maths can translate to improvements in maths performance. </p>
<p><a href="https://grattan.edu.au/report/targeted-teaching-how-better-use-of-data-can-improve-student-learning/">Research from the Grattan Institute</a> reinforces the point that teachers and schools need to collect and use evidence of students’ learning achievements and progress over time to know what works to improve student learning and to change what doesn’t. </p>
<h2>Top tips for improving teaching in maths</h2>
<p>To improve maths teaching, teachers should do the following:</p>
<ul>
<li><p>If they suffer from maths anxiety, don’t suffer alone. Identify the sources of anxiety and seek help from mentors and coaches to improve knowledge of maths concepts. Then use this expertise to build a broader repertoire of effective maths teaching strategies.</p></li>
<li><p>Demonstrate a <a href="https://www.ted.com/talks/carol_dweck_the_power_of_believing_that_you_can_improve?language=en">growth mindset</a> to students in their attitude towards teaching and learning maths.</p></li>
<li><p>Over time, regularly collect and <a href="https://grattan.edu.au/report/targeted-teaching-how-better-use-of-data-can-improve-student-learning/">analyse a range of evidence</a> for each student’s individual achievement and progress to understand what each student knows and what they are ready to learn next.</p></li>
<li><p>Join a <a href="http://www.aamt.edu.au/">professional maths teaching association</a> and/or professional learning community to engage in regular discussion with other teachers about what teaching strategies are working for particular students and what are not.</p></li>
</ul>
<p><em>• This piece was co-authored by Jacki McMahon, teacher at Makybe Rise Primary School and recent winner of a national ChooseMaths 16 teaching award, and Steph McDonald, principal of Makybe Rise Primary School in Western Australia.</em></p>
<p><em>• <a href="https://theconversation.com/au/topics/better-teachers-30749">Read more</a> articles in the series</em></p><img src="https://counter.theconversation.com/content/64976/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Claire Brown received funding from the Higher Education, Participation and Partnerships Program (HEPPP) to implement and research the Advancement via Individual Determination (AVID) program in Australia. That funding has now ended.</span></em></p>Here are some strategies that can help boost both teachers’ and their students’ confidence in maths.Dr Claire Brown, Associate Director, The Victoria Institute; National Director, AVID Australia, Victoria UniversityLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/625082016-07-20T07:42:53Z2016-07-20T07:42:53ZMastery over mindset: the cost of rolling out a Chinese way of teaching maths<figure><img src="https://images.theconversation.com/files/131058/original/image-20160719-13854-1p5ih80.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">Oleksandr Berezko/www.shutterstock.com</span></span></figcaption></figure><p>Half of primary schools in England <a href="https://www.gov.uk/government/news/south-asian-method-of-teaching-maths-to-be-rolled-out-in-schools">will receive £41m over four years</a> to teach mathematics using a method called the “mastery approach” that is used in Chinese schools. Yet during the last ten years many primary schools in England have started embracing another method, called the “<a href="https://www.eescpdportal.org/_images/96/flyers/EES_for_Schools_Mindset_Flyer_DP.pdf">mindset approach</a>”, which is supported by 30 years of psychological research in the US and <a href="https://v1.educationendowmentfoundation.org.uk/uploads/pdf/Changing_Mindsets.pdf">more recently in the UK</a>. </p>
<p>While the <a href="https://theconversation.com/explainer-what-is-the-mastery-model-of-teaching-maths-25636">mastery model</a> breaks down learning into small goals which have to be achieved before moving on, the mindset maths model aims to get pupils to develop an intuitive understanding of mathematical concepts before learning formal procedures such as addition or multiplication. Can the two work together in schools to help young children learn maths?</p>
<h2>Mindset is about motivation</h2>
<p>Mindset theory has established that each person has ingrained beliefs regarding each subject, which pre-determine whether learning the subject will be successful. Carol Dweck, the most prominent researcher in this field, has demonstrated spectacularly that <a href="https://psychology.stanford.edu/sites/all/files/Implicit%20Theories,%20Attributions%20and%20Coping_0.pdf">a wrong kind of environment can destroy a learner’s motivation</a>, possibly with long-lasting effects. </p>
<p>Vice versa, it is possible to build learning environments which nurture a learner’s motivation and so <a href="https://psychology.stanford.edu/sites/all/files/cdwecklearning%20success_0.pdf">make learning possible</a>. Mathematics, a subject which is often portrayed as difficult in the media, has immediately <a href="http://www.youcubed.org/wp-content/uploads/14_Boaler_FORUM_55_1_web.pdf">attracted the attention of researchers</a>, and a <a href="http://www.growthmindsetmaths.com/">number of principles</a> have evolved for developing classroom environments which are conducive to learning mathematics.</p>
<p>One thing that proponents of both the mastery and mindset theory agree with is avoiding the idea that a child can have a <a href="https://psychology.stanford.edu/sites/all/files/cdweckmathgift_0.pdf">gift for maths</a> so that high expectations are set for all pupils. According to mindset theory, a learner’s belief that they are or are not talented is a dangerous illusion, which damages their progress.</p>
<p>Some of the suggestions for organising in-class activities are <a href="https://www.ncetm.org.uk/resources/45776">similar in the mastery and mindset approaches</a>. For example, all pupils spend the same period of time working on the same problem, rather than some pupils racing forward to the next problem. This may involve some pupils starting work, perhaps unsuccessfully, on some deeper or more general versions of the problem. </p>
<h2>A shortcut to maths concepts</h2>
<p>While on the surface the mastery approach talks of a curriculum designed to enable deep conceptual, as well as procedural, knowledge, a look at its principles and sample lessons reveals undue concentration on numerous rigid objectives, achieved in a strict linear order. </p>
<p>In one <a href="https://theconversation.com/explainer-what-is-the-mastery-model-of-teaching-maths-25636">implementation of the mastery approach</a>, the process of adding two numbers is split into 23 consecutive learning objectives, which all need to be mastered separately, and may be tested separately by the teacher. </p>
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<img alt="" src="https://images.theconversation.com/files/131144/original/image-20160719-8008-fxbx4p.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/131144/original/image-20160719-8008-fxbx4p.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=401&fit=crop&dpr=1 600w, https://images.theconversation.com/files/131144/original/image-20160719-8008-fxbx4p.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=401&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/131144/original/image-20160719-8008-fxbx4p.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=401&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/131144/original/image-20160719-8008-fxbx4p.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=504&fit=crop&dpr=1 754w, https://images.theconversation.com/files/131144/original/image-20160719-8008-fxbx4p.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=504&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/131144/original/image-20160719-8008-fxbx4p.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=504&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">The goal: avoid maths anxiety.</span>
<span class="attribution"><span class="source">J2R/www.shutterstock.com</span></span>
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</figure>
<p>Mindset approach is completely different. It aims to be more exploratory, and tends to start from the “complicated” concepts, rather than reaching them by a long path. This stems from the belief that most ideas of mathematics are not abstract, but directly correspond to what can be demonstrated by real-world models. For example, a mindset-approach teacher can introduce addition via joining two heaps of cardboard counters (or other props) together, <a href="https://www.youcubed.org/math-topic/addition/">explore properties of addition via activities</a>, and only then break the process of adding numbers into procedural steps.</p>
<h2>Developing number sense</h2>
<p>Mindset approach advocates that pupils develop what is called “number sense”: an ability to choose (or invent) those methods of working with numbers which are more convenient and efficient in a particular situation. </p>
<p>As a simple example, a person possessing number sense will never calculate 5x3 as 3+3+3+3+3, because the process of adding threes is slower, less convenient and more prone to error than adding fives, that is, 5+5+5. </p>
<p>A person with a developed number sense may have gaps in the knowledge they store in their memory (for example, they will not necessarily know the whole times table by heart), but they will be effective at using the principles they have internalised. If a certain mathematical fact needs to be rediscovered because they do not remember it, they will be able to do it effortlessly. Some argue that <a href="https://www.youcubed.org/fluency-without-fear/">a person possessing number sense will not have maths anxiety</a> in their later life and, therefore, will be better prepared for learning more mathematics.</p>
<h2>Too fast, too soon?</h2>
<p>What is concerning at this stage is that <a href="https://www.gov.uk/government/uploads/system/uploads/attachment_data/file/495939/Resilience_posters_FINAL.pdf">unlike the mindset approach</a> there is, as yet, no indication that the mastery approach is being systematically and empirically tested. Only one preliminary study in the UK has been <a href="https://www.gov.uk/government/publications/evaluation-of-the-maths-teacher-exchange-china-and-england">published</a>, and it expresses only cautious optimism regarding the mastery model. </p>
<p>There are striking differences between mindset-approach and mastery-approach classroom activities, and we are not sure that in the long term, the mastery approach is better. The new government focus on the mastery method may also be confusing for teachers, after they have been encouraged to teach with the mindset approach. </p>
<p>We are left with some worrying unanswered questions – particularly as half of UK schools are now being encouraged to move to a mastery approach before any empirical evaluation of its impact on learning has taken place.</p><img src="https://counter.theconversation.com/content/62508/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Sherria Hoskins has received funding from the Education Edowment Foundation (two grants) to carry out Randomised Controlled Trials into Mindset.</span></em></p><p class="fine-print"><em><span>Alexei Vernitski 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>£41m will be spent on ‘mastery learning’ – will it improve learning in primary schools?Alexei Vernitski, Senior Lecturer in Mathematics, University of EssexSherria Hoskins, Head of Psychology, University of PortsmouthLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/557402016-03-04T19:18:30Z2016-03-04T19:18:30Z‘The Math Myth’ fuels the algebra wars, but what’s the fight really about?<figure><img src="https://images.theconversation.com/files/113910/original/image-20160304-17753-1dd1vyc.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">A confused student might not be leaving a math classroom....</span> <span class="attribution"><a class="source" href="http://www.shutterstock.com/pic.mhtml?id=295860566">Student image via www.shutterstock.com.</a></span></figcaption></figure><p>I discovered recently that my calculus students do not know the meaning of the word “quorum.” Since a course in American government is a high school graduation requirement in most states (including here in Florida), I was taken aback.</p>
<p>How should I react? Should I take to the editorial pages of <em>The New York Times</em>, bemoaning the state of civics education? Should I call out political scientists and high school history teachers for their failures?</p>
<p>Surely you’d admonish me to calm down a bit and perhaps not venture into disciplines where I’m not an expert.</p>
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<a href="https://images.theconversation.com/files/113885/original/image-20160304-17753-1oywxos.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/113885/original/image-20160304-17753-1oywxos.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/113885/original/image-20160304-17753-1oywxos.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=900&fit=crop&dpr=1 600w, https://images.theconversation.com/files/113885/original/image-20160304-17753-1oywxos.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=900&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/113885/original/image-20160304-17753-1oywxos.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=900&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/113885/original/image-20160304-17753-1oywxos.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=1131&fit=crop&dpr=1 754w, https://images.theconversation.com/files/113885/original/image-20160304-17753-1oywxos.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=1131&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/113885/original/image-20160304-17753-1oywxos.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=1131&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
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<span class="attribution"><a class="source" href="http://thenewpress.com/books/math-myth">The New Press</a></span>
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</figure>
<p>Yet Andrew Hacker, professor emeritus of political science at the City University of New York, recently <a href="http://www.nytimes.com/2016/02/28/opinion/sunday/the-wrong-way-to-teach-math.html">took this exact approach</a> to attack the teaching of algebra in American schools. He also <a href="http://thenewpress.com/books/math-myth">wrote a book</a>. And he’s <a href="http://www.nytimes.com/2012/07/29/opinion/sunday/is-algebra-necessary.html">done it before</a>.</p>
<p>Nor is he alone. Novelist Nicholson Baker <a href="https://harpers.org/archive/2013/09/wrong-answer/">wrote a piece</a> for <em>Harper’s</em> in 2013 that got the math community talking. The real target of Baker’s piece was the accountability movement and the associated standardized testing, but he chose mathematics as his straw man because it (a) is easy, and (b) will sell magazines. He manages to boil the modern course in Algebra II down to this:</p>
<blockquote>
<p>It’s a highly efficient engine for the creation of math rage: a dead scrap heap of repellent terminology, a collection of spiky, decontextualized, multistep mathematical black-box techniques that you must practice over and over and get by heart in order to be ready to do something interesting later on, when the time comes.</p>
</blockquote>
<p>At least Baker is an entertaining writer.</p>
<p>Hacker makes many of the same points in his <em>Times</em> articles, decrying algebra as a high school graduation requirement that holds back far too many students from having a productive life. He argues instead for “numeracy” and suggests what such a course should contain. It’s mostly statistics and financial mathematics, and lessons in visualizing and analyzing data.</p>
<p>To fight off the counterassertion that it’s possible to learn this material in a high school advanced placement statistics course, Hacker comes up with lists of obscure terminology: “The A.P. [Statistics] syllabus is practically a research seminar for dissertation candidates. Some typical assignments: binomial random variables, least-square regression lines, pooled sample standard errors.”</p>
<h2>It’s not just happening in math</h2>
<p>Every subject in school has been broken down into a string of often unrelated facts or tasks, not just mathematics.</p>
<p>I recall an episode from my own son’s experience in ninth grade while taking “Honors Pre-AP English I” (yes, that’s the real name of the course, not some Orwellian nightmare). His teacher led the class through the “CD/CM method” of essay writing, which goes like this. Fill out a worksheet with the “funnel” (4-7 sentence introduction), the thesis statement, and then for each of three paragraphs create 11 (!) sentences – the topic sentence (fine) and then CD#1, CM#1, CD#2,CM#2,…,CD#5,CM#5. What is a CD, you ask? Concrete Detail. A CM? Comment, of course.</p>
<p>Now, this is really just a superextended outline for an essay, but my son was extremely frustrated by this, eventually exclaiming, “I just want to write the damn paper!”</p>
<p>Is this example from the humanities really any different from what Hacker and Baker complain about?</p>
<p>Hacker is not completely wrong, however. School mathematics <em>has</em> largely been drained of context and beauty. University mathematicians complain about this, too.</p>
<p>For example, my son has also brought home worksheets full of dozens of polynomials with the simple instruction: Factor. But why?</p>
<figure class="align-left ">
<img alt="" src="https://images.theconversation.com/files/113797/original/image-20160303-13754-1e63a0r.png?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/113797/original/image-20160303-13754-1e63a0r.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=295&fit=crop&dpr=1 600w, https://images.theconversation.com/files/113797/original/image-20160303-13754-1e63a0r.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=295&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/113797/original/image-20160303-13754-1e63a0r.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=295&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/113797/original/image-20160303-13754-1e63a0r.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=370&fit=crop&dpr=1 754w, https://images.theconversation.com/files/113797/original/image-20160303-13754-1e63a0r.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=370&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/113797/original/image-20160303-13754-1e63a0r.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=370&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">Light rays striking a parabolic mirror reflect to a common point called the focus (point F above).</span>
<span class="attribution"><span class="source">created in Geogebra by the author</span></span>
</figcaption>
</figure>
<p>There is no context given for why we care about polynomial equations, no discussion of why parabolas (graphs of quadratic equations) are useful things. Maybe we should explain that without parabolas, we wouldn’t have good headlights on our cars or all those pretty pictures of deep space from the Hubble telescope. But just as mathematicians would not argue for the elimination of English or civics from the high school curriculum, Hacker shouldn’t be arguing for the elimination of algebra.</p>
<p>Let’s be honest. Mostly because of the accountability movement and high-stakes testing, K-12 education suffers from these types of problems in every subject. Picking on math alone because it’s particularly vexing for some people is unsporting.</p>
<h2>Credibility gap</h2>
<p>Of course, Hacker and Baker have proposals for how to fix this mess. The problem is that the major prerequisite for much of what Hacker proposes is, ironically, algebra. Not so much the grinding, symbol-driven form of algebra taught in school today, but algebra nonetheless. Reading bar graphs in the newspaper is a skill that we should expect high school graduates to be able to do, but nontrivial calculations with data require at least some facility with algebra. Hacker surely knows this, but it would undermine his argument to admit it.</p>
<p>He’s certainly not wrong that some students fall by the wayside, and the way we teach algebra and geometry in the middle grades is largely to blame. Stanford mathematician Keith Devlin wrote a <a href="http://www.huffingtonpost.com/dr-keith-devlin/andrew-hacker-and-the-cas_b_9339554.html">wonderful response</a> to Hacker’s recent piece, pointing out how his ideas may actually be correct but misguided:</p>
<blockquote>
<p>Not only did that suggestion [the elimination of algebra from the high school curriculum] alienate accomplished scientists and engineers and a great many teachers – groups you’d want on your side if your goal is to change math education – it distracted attention from what was a very powerful argument for introducing the teaching of algebra into our schools, something I and many other mathematicians would enthusiastically support.</p>
</blockquote>
<p>Unfortunately, Hacker undermines his credibility by stating falsehoods. For example, he claims “Coding is not based on mathematics … Most people who do coding, programming, software design, don’t do any mathematics at all.” It may be true that these individuals are not crunching numbers all day (that’s what software is for, of course), but the algorithmic processes underlying coding are the very essence of mathematics. To say otherwise is just delusional.</p>
<p>Hacker also asks, “Would you go to a mathematician to tell us what to do in Syria? It just defies comprehension.” Actually, it shouldn’t. The Central Intelligence Agency and other national security groups <a href="https://www.cia.gov/careers/opportunities/analytical">employ thousands of mathematicians to analyze data</a> associated with foreign affairs, looking for patterns amid the chaos. So, Hacker is just plain wrong about some things, even if his overall idea has merit. </p>
<h2>We’re all on the same team</h2>
<p>You see, college math professors <em>know</em> there is a problem with K-12 mathematics. We see the results in our classrooms on campus. As much as Hacker would like to believe his <em>ad hominem</em> assertions about math faculties at high schools and colleges, we really just want our students well-prepared for the beautiful, fascinating and, yes, useful material we have to offer.</p>
<p>Algebra is a beautiful baby; it would be a shame to throw it out with some dirty bathwater.</p><img src="https://counter.theconversation.com/content/55740/count.gif" alt="The Conversation" width="1" height="1" />
A new book criticizes how and what American math classes are teaching. Singling out math instruction in this age of high-stakes testing and accountability is unsporting.Kevin Knudson, Professor of Mathematics, University of FloridaLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/540342016-02-04T19:05:03Z2016-02-04T19:05:03ZFinger tracing can help students solve maths problems<figure><img src="https://images.theconversation.com/files/110068/original/image-20160203-6944-4v60rq.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">The index finger plays a vital role in early learning.</span> <span class="attribution"><span class="source">from www.shutterstock.com</span></span></figcaption></figure><p>Using the index finger to trace over advanced and multi-step maths problems can help students with problem solving, <a href="http://onlinelibrary.wiley.com/doi/10.1002/acp.3171/abstract">new research</a> shows.</p>
<p>Tracing can assist learning not only for spatial topics such as triangles and angle relationships, but also for non-spatial tasks such as learning the order of tasks in arithmetic problems. </p>
<p>For instance, students who traced over the addition, subtraction, multiplication, division and brackets symbols in problems such as 7 x (31 – 20) + 56 ÷ (5 – 3) = ? solved more problems correctly on a subsequent test. </p>
<p>We also found that students who traced over key elements of maths problems (eg, the arithmetic symbols +, -, ÷, x, and brackets used in order of operations problems) were able to solve other questions that extended the initial maths problem further. Superior performance on such “transfer” problems indicates students who traced weren’t simply memorising solutions to problems. Instead, tracing was helping them develop a deeper, more flexible understanding of the problem-solving methods. </p>
<h2>Why tracing against a surface may enhance learning</h2>
<p>The index finger plays a vital role in early learning. The specific gesture of pointing with the index finger is common across all cultures as a means of guiding attention. </p>
<p>As young as nine months of age, babies learn to manage their conversations with caregivers by <a href="http://onlinelibrary.wiley.com/doi/10.1111/j.1551-6709.2011.01228.x/abstract">pointing to things</a> in the environment. When the caregiver names the object, this helps build the child’s vocabulary.</p>
<p>Hand movements (including tracing and pointing gestures) may also help us form and organise spatial images in our conscious mind. </p>
<p>We have evolved to pay close attention to things that our eyes can easily see. This means that objects near our hands are more quickly recognised and receive prolonged scrutiny. So, when using an index finger to physically touch while tracing visual stimuli, the stimuli receive processing priority. </p>
<p>Gestures, including tracing, may play an important role in helping learners combine or “chunk” different sources of information (eg, text and diagrams) into an integrated, coherent understanding of a problem <a href="http://onlinelibrary.wiley.com/doi/10.1111/j.1551-6709.2010.01102.x/abstract">(Ping & Goldin-Meadow, 2010)</a>. Chunking acts to reduce the load on working memory, and can support more effective learning <a href="http://www.sciencedirect.com/science/article/pii/0959475294900035">(Sweller, 1994)</a>.</p>
<h2>Montessori education</h2>
<p>Finger-tracing has been used by teachers for more than a century. </p>
<p>In the early 1900s, Italian educator, <a href="http://infed.org/mobi/maria-montessori-and-education/">Maria Montessori</a> – who developed an educational method that builds on the way children naturally learn – got young children to trace over letters of the alphabet made from sandpaper with their index fingers.</p>
<p>This technique was based on the intuition that a multi-sensory approach (i.e., visual, auditory, tactile, and kinaesthetic) would benefit young children. </p>
<p>Subsequent studies over the past 40 years have confirmed Montessori’s intuitions for topics relevant to early childhood education, including letter recognition <a href="http://www.sciencedirect.com/science/article/pii/S0885201404000425">(Bara et al., 2004)</a> and geometrical shape recognition <a href="http://jbd.sagepub.com/content/35/1/18.refs">(Kalenine et al., 2011)</a>.</p>
<p>Our research shows the benefits of tracing extend well beyond early childhood, to complex topics in primary and secondary maths.</p>
<p>This simple, no-cost teaching strategy can enhance the effectiveness of maths instruction. </p>
<p>At the classroom level, teachers can assist students to learn new mathematical content by incorporating instructions to “trace over” the important elements of maths problems that already appear in mathematics textbooks or worksheets. </p>
<p>Tracing may also be useful in computer-based maths tutorials where students can then trace out animated maths lessons.</p><img src="https://counter.theconversation.com/content/54034/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>The authors do not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and have disclosed no relevant affiliations beyond their academic appointment.</span></em></p>Tracing over maths problems can enhance students’ learning of more advanced mathematical content and multi-step problems.Paul Ginns, Senior Lecturer in Educational Psychology, University of SydneyJanette Bobis, Professor of Mathematics Education, University of SydneyLicensed as Creative Commons – attribution, no derivatives.