tag:theconversation.com,2011:/fr/topics/mathematical-ability-27502/articlesMathematical ability – The Conversation2022-01-04T19:12:29Ztag:theconversation.com,2011:article/1713902022-01-04T19:12:29Z2022-01-04T19:12:29ZLearn how to make a sonobe unit in origami – and unlock a world of mathematical wonder<figure><img src="https://images.theconversation.com/files/433304/original/file-20211122-13-1uohlvq.jpg?ixlib=rb-1.1.0&rect=0%2C6%2C4031%2C2257&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">
</span> <span class="attribution"><span class="source">Julia Collins</span>, <span class="license">Author provided</span></span></figcaption></figure><p><em>This article is part of a <a href="https://theconversation.com/au/topics/how-to-guides-113946">series</a> explaining how readers can learn the skills to take part in activities that academics love doing as part of their work.</em></p>
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<p>Many of us could happily fold a <a href="https://origami.me/crane/">paper crane</a>, yet few feel confident solving an equation like <em>x</em>³ – 3 <em>x</em>² – <em>x</em> + 3 = 0, to find a value for <em>x</em>.</p>
<p>Both activities, however, share similar skills: precision, the ability to follow an algorithm, an intuition for shape, and a search for pattern and symmetry. </p>
<p>I’m a mathematician whose hobby is origami, and I love introducing people to mathematical ideas through crafts like paper folding. Any piece of origami will contain mathematical ideas and skills, and can take you on a fascinating, creative journey.</p>
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Read more:
<a href="https://theconversation.com/why-bother-calculating-pi-to-62-8-trillion-digits-its-both-useless-and-fascinating-166271">Why bother calculating pi to 62.8 trillion digits? It's both useless and fascinating</a>
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<h2>The ‘building blocks’ of origami models</h2>
<p>As a geometer (mathematician who studies geometry), my favourite technique is modular origami. That’s where you use several pieces of folded paper as “building blocks” to create a larger, often symmetrical structure.</p>
<p>The building blocks, called units, are typically straightforward to fold; the mathematical skill comes in assembling the larger structure and discovering the patterns within them. </p>
<p>Many modular origami patterns, although they may use different units, have a similar method of combining units into a bigger creation. </p>
<p>So, for a little effort you are rewarded with a vast number of models to explore.</p>
<p>My website <a href="https://www.mathscraftaus.org/resources">Maths Craft Australia</a> contains a range of modular origami patterns, as well as patterns for other crafts such as crochet, knitting and stitching. </p>
<p>They require no mathematical background but will take you in some fascinating mathematical directions.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/432338/original/file-20211117-23-ujn9or.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/432338/original/file-20211117-23-ujn9or.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/432338/original/file-20211117-23-ujn9or.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=555&fit=crop&dpr=1 600w, https://images.theconversation.com/files/432338/original/file-20211117-23-ujn9or.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=555&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/432338/original/file-20211117-23-ujn9or.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=555&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/432338/original/file-20211117-23-ujn9or.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=697&fit=crop&dpr=1 754w, https://images.theconversation.com/files/432338/original/file-20211117-23-ujn9or.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=697&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/432338/original/file-20211117-23-ujn9or.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=697&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">This model, folded by the author, uses a design from the book Perfectly Mindful Origami - The Art and Craft of Geometric Origami by Mark Bolitho.</span>
</figcaption>
</figure>
<h2>Building 3D shapes from smaller 2D units</h2>
<p>In mathematics, the shapes with the most symmetry are called the <a href="https://en.wikipedia.org/wiki/Platonic_solid">Platonic solids</a>. They’re named after the ancient Greek philosopher Plato (although they almost certainly predate him and have been discovered in ancient civilisations around the world). </p>
<p>The Platonic solids are 3D shapes made from regular 2D shapes (also known as regular polygons) where every side and angle is identical: equilateral triangles, squares, pentagons.</p>
<p>While there are infinitely many regular polygons, there are, surprisingly, only five Platonic solids: </p>
<ul>
<li><p>the tetrahedron (four triangles)</p></li>
<li><p>the cube (six squares)</p></li>
<li><p>the octahedron (eight triangles)</p></li>
<li><p>the dodecahedron (12 pentagons) and</p></li>
<li><p>the icosahedron (20 triangles). </p></li>
</ul>
<p>To build Platonic solids in origami, the best place to start is to master what’s known as the “<a href="https://momath.org/home/math-monday-introducing-the-sonobe-unit/">sonobe unit</a>”.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/433363/original/file-20211123-25-1r7jrtk.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="Sonobe units, like these ones piled in a stack, can be put together to create 3D shapes." src="https://images.theconversation.com/files/433363/original/file-20211123-25-1r7jrtk.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/433363/original/file-20211123-25-1r7jrtk.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=450&fit=crop&dpr=1 600w, https://images.theconversation.com/files/433363/original/file-20211123-25-1r7jrtk.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=450&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/433363/original/file-20211123-25-1r7jrtk.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=450&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/433363/original/file-20211123-25-1r7jrtk.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=566&fit=crop&dpr=1 754w, https://images.theconversation.com/files/433363/original/file-20211123-25-1r7jrtk.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=566&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/433363/original/file-20211123-25-1r7jrtk.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=566&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Sonobe units, like these ones piled in a stack, can be put together to create 3D shapes.</span>
<span class="attribution"><span class="source">Julia Collins</span>, <span class="license">Author provided</span></span>
</figcaption>
</figure>
<h2>Enter the sonobe unit</h2>
<p>A sonobe unit (sometimes called the sonobe module) looks a bit like a parallelogram with two flaps folded behind.</p>
<p>I’ve got instructions for how to make a sonobe unit <a href="https://static1.squarespace.com/static/59699ab4b8a79b10f84ba4cd/t/59b92716e5dd5b882846ee5b/1505306394805/Sonobe-unit-instructions.pdf">on my website</a> and there are plenty of videos online, like this one:</p>
<figure>
<iframe width="440" height="260" src="https://www.youtube.com/embed/TKGW2W168H0?wmode=transparent&start=0" frameborder="0" allowfullscreen=""></iframe>
<figcaption><span class="caption">How to make a sonobe unit.</span></figcaption>
</figure>
<p>Sonobe units are fast and simple to fold, and can be fitted together to create beautiful, intriguing 3D shapes like these:</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/432142/original/file-20211116-13-1vvh9l2.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/432142/original/file-20211116-13-1vvh9l2.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=338&fit=crop&dpr=1 600w, https://images.theconversation.com/files/432142/original/file-20211116-13-1vvh9l2.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=338&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/432142/original/file-20211116-13-1vvh9l2.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=338&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/432142/original/file-20211116-13-1vvh9l2.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=424&fit=crop&dpr=1 754w, https://images.theconversation.com/files/432142/original/file-20211116-13-1vvh9l2.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=424&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/432142/original/file-20211116-13-1vvh9l2.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=424&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Three sonobe origami models by Julia Collins.</span>
</figcaption>
</figure>
<p>You will need six sonobe units to make a cube like the yellow-blue-green one pictured above, 12 to make an octahedron (the red-pink-purple one), and 30 to make an icosahedron (the golden one). (Interestingly, it’s not possible to build a tetrahedron and dodecahedron from sonobe units).</p>
<p>I’ve got written instructions for building the cube <a href="https://static1.squarespace.com/static/59699ab4b8a79b10f84ba4cd/t/59b9273ab7411cfd32068b83/1505306433027/Sonobe-unit-instructions-cube.pdf">on my website</a>, and some quick searching online will find you instructions for the larger models.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/433364/original/file-20211123-13-1g66xxm.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="Sonobe units can be put together to build wondrous shapes." src="https://images.theconversation.com/files/433364/original/file-20211123-13-1g66xxm.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/433364/original/file-20211123-13-1g66xxm.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=450&fit=crop&dpr=1 600w, https://images.theconversation.com/files/433364/original/file-20211123-13-1g66xxm.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=450&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/433364/original/file-20211123-13-1g66xxm.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=450&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/433364/original/file-20211123-13-1g66xxm.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=566&fit=crop&dpr=1 754w, https://images.theconversation.com/files/433364/original/file-20211123-13-1g66xxm.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=566&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/433364/original/file-20211123-13-1g66xxm.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=566&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Sonobe units can be put together to build wondrous shapes.</span>
<span class="attribution"><span class="source">Julia Collins</span>, <span class="license">Author provided</span></span>
</figcaption>
</figure>
<h2>Into the mathematical rabbit hole</h2>
<p>Once you’ve mastered the basic structure of each 3D shape, you may find yourself (as <a href="https://www.polypompholyx.com/2017/01/modularorigami/">others have done</a>) pondering deeper mathematical questions.</p>
<p>Can you arrange the sonobe units so two units of the same colour never touch, if you only have three colours? </p>
<p>Are larger symmetric shapes possible? (Answer: yes!) </p>
<p>Are there relationships between the different 3D shapes? (For example, the icosahedron is basically built of triangles, but can you spot the pentagons lurking within? Or the triangles in the dodecahedron?)</p>
<p>One seemingly innocent question can easily lead to a mathematical rabbit hole.</p>
<p>Questions about colouring will lead you to the mathematics of graphs and networks (and big questions that remained <a href="https://en.wikipedia.org/wiki/Four_color_theorem">unsolved for many centuries</a>). </p>
<p>Questions about larger models will lead you to the <a href="https://en.wikipedia.org/wiki/Archimedean_solid">Archimedean solids</a> and the <a href="https://en.wikipedia.org/wiki/Johnson_solid">Johnson solids</a>. These 3D shapes have a lot of symmetry, though not as much as the Platonic solids. </p>
<p>Then, for a truly mind-bending journey, you might land on the concept of <a href="https://en.wikipedia.org/wiki/List_of_regular_polytopes_and_compounds">higher-dimensional symmetric shapes</a>. </p>
<p>Or perhaps your questions will lead you in the opposite direction. </p>
<p>Instead of using origami to explore new ideas in mathematics, some researchers have used mathematical frameworks to explore new ideas in origami.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/433365/original/file-20211123-27-1h94v50.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="Origami can take you into the mathematical rabbit hole." src="https://images.theconversation.com/files/433365/original/file-20211123-27-1h94v50.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/433365/original/file-20211123-27-1h94v50.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=600&fit=crop&dpr=1 600w, https://images.theconversation.com/files/433365/original/file-20211123-27-1h94v50.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=600&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/433365/original/file-20211123-27-1h94v50.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=600&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/433365/original/file-20211123-27-1h94v50.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=754&fit=crop&dpr=1 754w, https://images.theconversation.com/files/433365/original/file-20211123-27-1h94v50.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=754&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/433365/original/file-20211123-27-1h94v50.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=754&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Origami can take you into the mathematical rabbit hole.</span>
<span class="attribution"><span class="source">Julia Collins</span>, <span class="license">Author provided</span></span>
</figcaption>
</figure>
<h2>Solving old problems in new ways</h2>
<p>Perhaps the most famous mathematical origami artist is the US-based former NASA physicist <a href="https://langorigami.com/">Robert Lang</a>, who designs computer programs that generate crease patterns for fantastically complicated models. </p>
<p>His models include segmented tarantulas and ants, stags with twisted antlers and soaring, feathered birds.</p>
<figure>
<iframe width="440" height="260" src="https://www.youtube.com/embed/DJ4hDppP_SQ?wmode=transparent&start=0" frameborder="0" allowfullscreen=""></iframe>
<figcaption><span class="caption">Credit: Great Big Story/YouTube.</span></figcaption>
</figure>
<p>Robert Lang and others have also created crease patterns for use in new engineering contexts such as <a href="https://langorigami.com/article/eyeglass-telescope/">folding telescope lenses</a>, <a href="https://royalsocietypublishing.org/doi/10.1098/rsos.160429">air bags</a> and <a href="https://www.nasa.gov/jpl/news/origami-style-solar-power-20140814">solar panels</a>.</p>
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<p>
<em>
<strong>
Read more:
<a href="https://theconversation.com/curved-origami-offers-a-creative-route-to-making-robots-and-other-mechanical-devices-150253">Curved origami offers a creative route to making robots and other mechanical devices</a>
</strong>
</em>
</p>
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<p>My final example of the power of origami goes back to the cubic equation I mentioned at the outset:</p>
<p><em>x</em>³ – 3 <em>x</em>² – <em>x</em> + 3 = 0</p>
<p>Cubic equations relate to some “impossible” mathematical problems, such as <a href="https://mathworld.wolfram.com/AngleTrisection.html">trisecting an angle</a> (splitting an arbitrary angle into three equal angles). Or <a href="https://en.wikipedia.org/wiki/Doubling_the_cube">doubling a cube</a> (which is finding a cube with double the volume of a given cube). </p>
<figure class="align-right zoomable">
<a href="https://images.theconversation.com/files/433368/original/file-20211123-17-19aazig.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="A blue and purple origami shape sits on a grey background." src="https://images.theconversation.com/files/433368/original/file-20211123-17-19aazig.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/433368/original/file-20211123-17-19aazig.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=554&fit=crop&dpr=1 600w, https://images.theconversation.com/files/433368/original/file-20211123-17-19aazig.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=554&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/433368/original/file-20211123-17-19aazig.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=554&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/433368/original/file-20211123-17-19aazig.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=696&fit=crop&dpr=1 754w, https://images.theconversation.com/files/433368/original/file-20211123-17-19aazig.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=696&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/433368/original/file-20211123-17-19aazig.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=696&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Any piece of origami will contain mathematical ideas and skills.</span>
<span class="attribution"><span class="source">Julia Collins</span>, <span class="license">Author provided</span></span>
</figcaption>
</figure>
<p>Famously, these problems cannot be solved using the classical methods of a straightedge (ruler without the markings) and compass. </p>
<p>In 1980, however, Japanese mathematician Hisashi Abe showed how to <a href="https://plus.maths.org/content/trisecting-angle-origami">solve all these problems using origami</a>.</p>
<p>I am excited to see where mathematics and origami will intersect in future. Grab some paper today, make a few models and start your own journey of mathematical exploration. </p>
<hr>
<p><em>You can read other articles in this series <a href="https://theconversation.com/au/topics/how-to-guides-113946">here</a>.</em></p><img src="https://counter.theconversation.com/content/171390/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Julia Collins 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>I’m a mathematician whose hobby is origami. It has inspired mathematicians to solve problems once thought impossible, and create folding telescope lenses, airbags and solar panels.Julia Collins, Lecturer of Mathematics, Edith Cowan UniversityLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/825522017-09-12T02:18:56Z2017-09-12T02:18:56ZThese four easy steps can make you a math whiz<figure><img src="https://images.theconversation.com/files/184994/original/file-20170906-9830-3mf2kc.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Can you cut it in this math problem?</span> <span class="attribution"><a class="source" href="https://www.shutterstock.com/image-photo/camembert-cheese-brie-575682340">Sergey Lapin/shutterstock.com</a></span></figcaption></figure><p>Many people find mathematics daunting. If true, this piece is for you. If not, this piece is still for you.</p>
<p>What do you think of when you think about mathematics? Perhaps you think about x’s and y’s, intractable fractions, or nonsensical word problems. The cartoonist Gary Larson once depicted hell’s library as containing only giant tomes of word problems. You know, “If a train leaves New York…” </p>
<p>I was trained as a mathematician, and I will let you in on a trade secret: That is not what mathematics is, nor where it lives. It’s true that learning mathematics often involves solving problems, but it should focus on the joy of solving puzzles, rather than memorizing rules. </p>
<p>I invite you to see yourself as a problem solver and mathematician. And I’d like to introduce you to the man who once invited me to the study of problem solving: George Pólya. </p>
<h1>Math Pólya’s way</h1>
<p>For many reasons, not the least of which is that Pólya <a href="http://articles.latimes.com/1985-09-08/news/mn-2892_1_polya-george-mathematician">died</a> in 1985, you will meet him as I did – through his wildly successful <a href="http://press.princeton.edu/titles/669.html">“How to Solve It</a>.” Penned in 1945, this book went on to sell over one million copies and was translated into 17 languages. </p>
<p>As a mathematician, Pólya worked on a wide range of problems, including the study of heuristics, or how to solve problems. When you read “How to Solve It,” it feels like you’re taking a guided tour of Pólya’s mind. This is because his writing is metacognitive – he writes about how he thinks about thinking. And metacognition is often the heart of problem solving.</p>
<p>Pólya’s problem solving plan breaks down to four simple steps:</p>
<ol>
<li> Make sure you understand the problem.</li>
<li> Make a plan to solve the problem.</li>
<li> Carry out the plan.</li>
<li> Check your work to test your answer.</li>
</ol>
<p>There it is. Problem solving in the palm of your hand – math reduced to four steps.</p>
<p><a href="http://epltt.coe.uga.edu/index.php?title=Situated_Cognition">Here’s a classic problem</a> from research on mathematics education done by Jean Lave. A man, let’s call him John, is making ¾ of a recipe that calls for 2/3 cup of cottage cheese. What do you think John did? What would you do? </p>
<p>If you’re like me, you might immediately dive into calculations, perhaps struggling with what the fractions mean, working to remember the rules for arithmetic. That’s what John seemed to do, at first. But then he had a Eureka! moment.</p>
<p>John measured 2/3 cup of cottage cheese, then dumped it onto a cutting board. He patted the cheese into a circle and drew lines into it, one vertical, one horizontal, dividing the cheese patty into quarters. He then carefully pushed one quarter of the cottage cheese back into its container. Voilá! Three-quarters of 2/3 cup of cottage cheese remained. </p>
<p>John is a mathematician and problem solver. First, he understood the problem: He needed ¾ of what the recipe called for, which was 2/3 cup. Then, he made a plan, most likely visualizing in his head how he would measure and divide the cottage cheese. Finally, he carried out the plan. </p>
<p>Did he check his answer? That remains unclear, but we can check the validity of his work for him. Did he indeed end up with ¾ of 2/3 cup of cottage cheese? Yes, because the full amount was reduced by one-quarter, leaving three-quarters. </p>
<h1>Another approach</h1>
<p>Would this solution work with different foods or serving sizes? So long as a person could divide that serving into quarters, yes, the plan would work. </p>
<p>Could we solve the problem another way with the same result? Sure — there are many ways to solve this problem, and they should all result in the same half-cup answer. Here is one. </p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/185181/original/file-20170907-9603-cugdd3.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/185181/original/file-20170907-9603-cugdd3.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/185181/original/file-20170907-9603-cugdd3.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=265&fit=crop&dpr=1 600w, https://images.theconversation.com/files/185181/original/file-20170907-9603-cugdd3.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=265&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/185181/original/file-20170907-9603-cugdd3.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=265&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/185181/original/file-20170907-9603-cugdd3.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=333&fit=crop&dpr=1 754w, https://images.theconversation.com/files/185181/original/file-20170907-9603-cugdd3.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=333&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/185181/original/file-20170907-9603-cugdd3.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=333&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">How to find ¾ of 2/3.</span>
<span class="attribution"><span class="source">Jennifer Ruef</span>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span>
</figcaption>
</figure>
<p>Notice that this solution uses pictures. <a href="https://www.youcubed.org/downloads/?file_id=1584&file_name=jacmaths-seeing-article&resource_name=visual-math-article-in-journal-of-applied-computational-mathematics">New brain research</a> validates what mathematics educators have been saying for decades: Pictures help us think. Drawing pictures also happens to be another of Pólya’s suggestions.</p>
<p>John probably made use of one of Pólya’s most important suggestions: Can you think of a related problem?</p>
<p>Of course, this is a cheesy problem – sorry, I really didn’t even try to fight that pun – which is a common complaint about story problems. I chose it because it has delighted math researchers for years, and because John is quite clever in his solution. He is also extremely mathematical.</p>
<p>I’ve taught mathematics, and how to teach mathematics, for nearly 30 years. For over a decade, it was my job to convince high school freshmen not only that algebra was meaningful, but that it was meant for them, and they for it. In my work, I’ve met many people who love mathematics and many who find it overwhelming and nonsensical. And so it’s an important part of my work to help people see the beauty and wonder of mathematics, and think of themselves as mathematicians. </p>
<p>These <a href="https://www.youcubed.org/resources/parents-beliefs-math-change-childrens-achievement/">messages</a> are especially important for parents helping children learn mathematics. If you understand the problem you’re trying to solve, you’re well on your way to solving it. And you, yes you, are a problem solver.</p>
<p>We all know it’s not always so simple to solve problems. Pólya did too. That’s the glory of it – the messy, wonderful, powerful adventure.</p><img src="https://counter.theconversation.com/content/82552/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Jennifer Ruef does not work for, consult, own shares in or receive funding from any company or organization that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>Dreading math class as you head back into school? Never fear: Try these tips from famed mathematician George Pólya.Jennifer Ruef, Assistant Professor of Education Studies, University of OregonLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/790592017-06-14T09:33:07Z2017-06-14T09:33:07ZAre left-handed people more gifted than others? Our study suggests it may hold true for maths<figure><img src="https://images.theconversation.com/files/173409/original/file-20170612-9404-1m749h.jpg?ixlib=rb-1.1.0&rect=7%2C33%2C1583%2C845&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Barack Obama signs at his desk</span> <span class="attribution"><a class="source" href="https://commons.wikimedia.org/wiki/File:Barack_Obama_signs_at_his_desk2.jpg">Pete Souza</a></span></figcaption></figure><p>The belief that there is a link between talent and left-handedness has a long history. Leonardo da Vinci was left-handed. So were Mark Twain, Mozart, Marie Curie, Nicola Tesla and Aristotle. It’s no different today – former US president Barack Obama is a left-hander, as is business leader Bill Gates and footballer Lionel Messi. </p>
<p>But is it really true that left-handers are <a href="http://www.independent.co.uk/voices/comment/why-are-left-handed-people-so-brilliant-8919135.html">more likely to be geniuses</a>? Let’s take a look at the latest evidence – including <a href="http://journal.frontiersin.org/article/10.3389/fpsyg.2017.00948/full">our new study</a> on handedness and mathematical ability.</p>
<p>It is <a href="http://www.sciencedirect.com/science/article/pii/002839329290065T">estimated</a> that between 10% and 13.5% of the population are not right-handed. While a few of these people are equally comfortable using either hand, the vast majority are left-handed.</p>
<p>Hand preference is a manifestation of brain function and is therefore related to <a href="http://www.tandfonline.com/doi/abs/10.1080/13803395.2013.778231?journalCode=ncen20">cognition</a>. Left-handers exhibit, on average, <a href="http://www.rightleftrightwrong.com/brain.html">a more developed right brain hemisphere</a>, which is specialised for processes such as <a href="http://psychology.jrank.org/pages/545/Right-Brain-Hemisphere.html">spatial reasoning</a> and the ability to rotate <a href="https://en.wikipedia.org/wiki/Mental_rotation">mental representations</a> of objects. </p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/173420/original/file-20170612-9404-1diev1t.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/173420/original/file-20170612-9404-1diev1t.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/173420/original/file-20170612-9404-1diev1t.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=341&fit=crop&dpr=1 600w, https://images.theconversation.com/files/173420/original/file-20170612-9404-1diev1t.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=341&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/173420/original/file-20170612-9404-1diev1t.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=341&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/173420/original/file-20170612-9404-1diev1t.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=429&fit=crop&dpr=1 754w, https://images.theconversation.com/files/173420/original/file-20170612-9404-1diev1t.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=429&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/173420/original/file-20170612-9404-1diev1t.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=429&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">The corpus callosum.</span>
<span class="attribution"><a class="source" href="https://commons.wikimedia.org/wiki/File:Corpus_callosum.png">Life Science Databases(LSDB)/wikipedia</a>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span>
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</figure>
<p>Also, the corpus callosum – the bundle of nerve cells connecting the two brain hemispheres – tends to be <a href="http://science.sciencemag.org/content/229/4714/665.long">larger in left-handers</a>. This suggests that some left-handers have an enhanced connectivity between the two hemispheres and hence superior information processing. Why that is, however, is unclear. One theory argues that living in a world designed for right-handers could be forcing left-handers to use both hands – thereby increasing connectivity. This opens up the possibility that we could all achieve enhanced connectivity by training ourselves to use both hands.</p>
<p>These peculiarities may be the reason why left-handers seem to have an edge in several professions and arts. For example, they are over-represented among <a href="http://musicweb.hmtm-hannover.de/sightreading/Kopiez-etal(2006)NP-Laterality.pdf">musicians</a>, <a href="http://journals.lww.com/jonmd/Abstract/2007/10000/Creativity_and_Psychopathology__Higher_Rates_of.6.aspx">creative artists</a>, <a href="http://journals.sagepub.com/doi/abs/10.2466/pms.1977.45.3f.1216">architects</a> and <a href="https://www.researchgate.net/publication/6598587_The_role_of_domain-specific_practice_handedness_and_starting_age_in_chess">chess players</a>. Needless to say, efficient information processing and superior spatial skills are essential in all these activities.</p>
<h2>Handedness and mathematics</h2>
<p>But what about the link between left-handedness and mathematical skill? Unsurprisingly, the role played by handedness in mathematics has long been a matter of interest. More than 30 years ago, a seminal study claimed left-handedness <a href="http://www.sciencedirect.com.liverpool.idm.oclc.org/science/article/pii/0028393286900114">to be a predictor of mathematical precociousness</a>. The study found that the rate of left-handedness among students talented in mathematics was much greater than among the general population.</p>
<p>However, the idea that left-handedness is a predictor of superior intellectual ability has been challenged recently. Several scholars have claimed that left-handedness is not related to any advantage in cognitive skills, and may even exert detrimental effects on general cognitive function and, hence, academic achievement.</p>
<p>For example, one study discovered that left-handed children <a href="https://link.springer.com/article/10.1353/dem.0.0053">slightly under-performed</a> in a series of developmental measures. Also, a recent review <a href="http://www.sciencedirect.com/science/article/pii/S0149763415001712">reported</a> that left-handers appear to be slightly over-represented among people with intellectual disabilities. Another large study found that left-handers <a href="http://onlinelibrary.wiley.com/doi/10.1111/j.1467-985X.2012.01074.x/abstract">performed more poorly</a> in mathematical ability in a sample of children aged five to 14.</p>
<h2>Carefully designed experiment</h2>
<p>Interestingly, these past studies, just like many others, differed from each other in how handedness was measured and how participants were categorised – some of them simply asked people what their hand preference was in general. And, most importantly, they had different approaches to measuring mathematical ability – ranging from simple arithmetic to complex problem solving. These discrepancies in the experimental design may be the cause of the mixed observed results.</p>
<p>To get more reliable results, we decided to carry out a whole <a href="http://journal.frontiersin.org/article/10.3389/fpsyg.2017.00948/full">series of experiments</a> including more than 2,300 students (in primary school and high school). These experiments varied in terms of type and difficulty of mathematical tasks.</p>
<p>To assure comparability, we used the same questionnaire – <a href="http://www.sciencedirect.com/science/article/pii/0028393271900674">the Edinburgh Inventory</a> – to assess handedness in all the experiments. This questionnaire asks people which hand they prefer for writing, drawing, throwing, brushing and other things. It assesses to what extent someone prefers their right or left – it’s a scale rather than a categorical left versus right assessment. This specific feature allowed us to build more reliable and powerful statistical models. </p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/173426/original/file-20170612-10249-1nmc0gj.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/173426/original/file-20170612-10249-1nmc0gj.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=402&fit=crop&dpr=1 600w, https://images.theconversation.com/files/173426/original/file-20170612-10249-1nmc0gj.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=402&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/173426/original/file-20170612-10249-1nmc0gj.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=402&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/173426/original/file-20170612-10249-1nmc0gj.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=505&fit=crop&dpr=1 754w, https://images.theconversation.com/files/173426/original/file-20170612-10249-1nmc0gj.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=505&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/173426/original/file-20170612-10249-1nmc0gj.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=505&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Could training to use both hands boost mathematical ability?</span>
<span class="attribution"><a class="source" href="https://commons.wikimedia.org/wiki/File:Math_class_in_Da_Ji_Junior_High_School_2006-12-1.jpg">enixii/flickr</a></span>
</figcaption>
</figure>
<p>The results, published in Frontiers, show that left-handers outperformed the rest of the sample when the tasks involved difficult problem-solving, such as associating mathematical functions to a given set of data. This pattern of results was particularly clear in male adolescents. By contrast, when the task was not so demanding, such as when doing simple arithmetic, there was no difference between left- and right-handers. We also discovered that extreme right-handers – individuals who said they prefer to use their right hand for all items on the handedness test – under-performed in all the experiments compared to moderate right-handers and left-handers.</p>
<p>Left-handers seem to have, on average, an edge when solving demanding mathematical tasks – at least during primary school and high school. Also, being strongly right-handed may represent a disadvantage for mathematics. Taken together, these findings show that handedness, as an indicator of connectivity between brain hemispheres, does influence cognition to some extent.</p>
<p>That said, handedness is just an indirect expression of brain function. For example, only <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3258574/">a third</a> of the people with a more developed right hemisphere are left-handed. So plenty of right-handed people will have a similar brain structure as left-handers. Consequently, we need to be cautious in interpreting people’s hand preference – whether we see it as a sign of genius or a marker for cognitive impairment.</p><img src="https://counter.theconversation.com/content/79059/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Giovanni Sala receives funding from the University of Liverpool.</span></em></p><p class="fine-print"><em><span>Fernand Gobet receives funding from the University of Liverpool and is Research Associate at the Centre for Philosophy of Natural & Social Science, London School of Economics. He gets royalties from a number of books on expertise and talent.</span></em></p>People who have an extreme preference for using their right hand may be worse at maths, according to new research.Giovanni Sala, PhD Candidate - Cognitive Psychology, University of LiverpoolFernand Gobet, Professor of Decision Making and Expertise, University of LiverpoolLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/702892016-12-13T09:47:31Z2016-12-13T09:47:31ZPressured South African schools had no choice but to relax maths pass mark<figure><img src="https://images.theconversation.com/files/149835/original/image-20161213-1615-vu7id5.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">By the time pupils who struggle with Maths reach Grade 9, there are huge bottlenecks in the system.</span> <span class="attribution"><span class="source">REUTERS/Ryan Gray</span></span></figcaption></figure><p>Starting now, South Africa’s pupils will be able to obtain as <a href="http://www.education.gov.za/Portals/0/Documents/Publications/Circular%20A3%20of%202016.pdf?ver=2016-12-07-154005-723">little as 20%</a> in mathematics in Grades 7, 8 and 9 and still progress to the next year of learning. This has been touted by many as evidence of an alleged inexorable decline in educational standards.</p>
<p>The country is already known for its <a href="http://www.telegraph.co.uk/education/2016/11/29/revealed-world-pupil-rankings-science-maths-timss-results/">poor performance</a> in international standardised assessments in mathematics. This latest move may be misconstrued as condoning such poor achievement.</p>
<p>But the truth is a little more complex.</p>
<p>For Grades 7 and 8 – when pupils should be between 14 and 15 years of age – this strategy of “pushing through” to avoid repeated student retention is not new. It has been part of standard policy. This means that by the time pupils reach Grade 9, there’s a bottleneck in the system. It was inevitable that this pressure would need to be relieved.</p>
<p>To understand why, one must consider the confluence of a number of factors, including: the over-inflated importance of mathematics; a curriculum packed too full to allow for any slip-ups or slower learning rates, and the country’s struggling maths teachers. <a href="http://mg.co.za/article/2016-12-09-00-home-is-where-the-learning-is">Maths performance correlates directly with poverty factors</a>, meaning these challenges affect more than 75% of South Africa’s schools. </p>
<h2>Inflated value of maths</h2>
<p>In the past 20 years there’s been a major shift internationally towards thinking of education in purely economic terms (as opposed to critical citizenry, creativity or self-actualization). This reduction of education to purely economic ends, coupled with the conflation between mathematical prowess and problem-solving skills for the “knowledge economy”, has resulted in mathematics being isolated as “essential knowledge”. Its proponents insist that maths is required for an education of value.</p>
<p>To fully appreciate this shift in thinking, South Africans need to suspend their collective amnesia: passing mathematics was not a requirement to move into Grade 10 a generation ago. And yet adults from this era are often economically productive, creative and academically accomplished. Many would publicly acknowledge their own struggles with numbers.</p>
<p>The vast majority of jobs of many flavours and incomes do not require the type of maths taught even in Grade 9. This is forgotten when mathematics is positioned as supremely important for the job market, or for students’ personal development.</p>
<h2>Moving targets</h2>
<p>Against the backdrop of this increased emphasis on mathematics, it’s useful to consider key features of the <a href="http://www.education.gov.za/Portals/0/Documents/Policies/PolicyProgPromReqNCS.pdf?ver=2015-02-03-154857-397">National Policy Pertaining to the Promotion Requirements of the National Curriculum Statement</a>.</p>
<p>An excessive emphasis on mathematics permeates this policy. Passing mathematics with “moderate” performance (that is, 40% or more) is now a criterion for passing in every grade. It’s a criterion many students <a href="http://www.education.gov.za/Portals/0/Documents/Reports/REPORT%20ON%20THE%20ANA%20OF%202014.pdf?ver=2014-12-04-104938-000">do not meet</a>.</p>
<p>The second issue is the “maximum four years in phase” policy. According to this, a pupil may not repeat more than one year in each three year phase of compulsory schooling. If a pupil has already repeated a year in a phase, they are “progressed” through into the next grade – whether they meet the promotion/pass criteria or not.</p>
<p>This “maximum four years in phase” policy bites at the end of Grade 9. Pushing pupils through without passing maths was a viable option in lower grades, as there was a “next grade” to progress to. But leaving Grade 9 without passing means leaving school without the <a href="http://www.saqa.org.za/docs/pol/2003/getc.pdf">General Education and Training</a> certificate required for admission to a technical college.</p>
<p>In the past, officials and schools have often suspended the “max four years” criterion to give pupils another opportunity to try and attain a recognisable school leaving qualification, requiring a maths score of higher than 40%. For pupils who have been failing maths for years, this is almost <a href="http://www.iol.co.za/dailynews/news/dismal-10-average-for-grade-9-maths-1791182">impossible</a>.</p>
<p>The pressure to move learners through the system is immense. Each year, principals and senior teachers suffer validation meetings, an event where schools justify their decisions to the provincial education department about whether students who failed should repeat or progress.</p>
<p>As a former mathematics Head of Department who has attended such meetings, I came to appreciate the lottery involved about who was “progressed” and who was not, as officials clandestinely tweak results until the number of students moved through was politically acceptable. Often those with 20% or more would have their marks “adjusted” to 30% for what is referred to as a “condoned pass”. As teachers, we are told to “find marks” in assessments to justify passing or condoning borderline students.</p>
<p>But sometimes there are just not enough marks to find.</p>
<h2>Huge learning backlogs</h2>
<p>The second policy that adds to the conundrum is the Curriculum and Assessment Policy Statement (CAPS). This demands strict adherence to pacing and content. Mathematics in CAPS moves at breakneck speed: ten jam-packed weeks of content per term, even though there are often only eight weeks of actual lessons.</p>
<p>Curriculum advisers regularly correct teachers who deviate from the stated content and pacing of curriculum documents. That means a teacher who has the confidence and ability to address learning backlogs by professionally interpreting the curriculum to meet a pupil’s needs is often criticised for doing so. Teachers without this confidence or skill will not even attempt the task.</p>
<p>Such rigidity is in stark contradiction to the National Policy Pertaining to the Promotion Requirements, which is peppered with phrases regarding tailoring learning to address backlogs and learning barriers.</p>
<p>Primary schools pragmatically push over-age (16 years old) Grade 7 pupils through to Grade 8 in senior schools. Senior schools then receive under-prepared pupils who are too old to refer to schools of skills or special needs schools – the maximum referral age is 14. There is nothing to be done but to try and teach struggling learners, knowing they will be pushed up into Grade 9 where they will get stuck or <a href="https://africacheck.org/spot_check/south-africas-matric-pass-rate-obscures-dropout-rate/">drop out</a>. After Grade 9, the pupil enrolment dwindles rapidly as students lose the protection of being pushed through by the conveyor belt.</p>
<p>Together, these policies effectively put pupils on a one way track into Grade 9 irrespective of their performance in mathematics at lower grades. Then it has kept them in Grade 9 by insisting they meet the pass criteria… until now.</p>
<h2>Struggling mathematics teachers</h2>
<p>Two urgent issues, most concentrated in schools that serve the country’s poorest learners, further exacerbate what is already an obviously disastrous situation.</p>
<p>Firstly, the mathematics abilities of primary school teachers is a problem experienced in many countries, including the <a href="http://washingtonmonthly.com/2016/06/15/elementary-school-teachers-struggle-with-common-core-math-standards/">US</a> and the <a href="https://www.theguardian.com/education/2010/feb/14/primary-teachers-fail-maths-tests">UK</a>, but particularly in <a href="http://www.cde.org.za/wp-content/uploads/2013/10/MATHEMATICS%20OUTCOMES%20IN%20SOUTH%20AFRICAN%20SCHOOLS.pdf">South Africa</a>. Mathematics specialists are appointed in high schools. Primary school teachers are trained as generalists. Yet it is in primary school where the learning backlog begins.</p>
<p>Secondly, teachers’ working conditions in poorer schools are abysmal. Those teachers who can leave often do, and mathematics teachers in particular often possess transferable skills. They <a href="http://www.education.gov.za/Portals/0/Documents/Reports/Teachers%20for%20the%20future%2016%20NOV%202005.pdf?ver=2008-03-05-111025-000">relocate</a> to other schools or other industries for better working conditions.</p>
<p>Primary schools thus struggle to provide the crucial foundations for maths, and secondary schools struggle to retain the specialists who might be able to address the problem later.</p>
<h2>Relieving the self-applied pressure</h2>
<p>It’s no wonder then that <a href="http://www.education.gov.za/Portals/0/Documents/Publications/Education%20Statistic%202013.pdf?ver=2015-03-30-144732-767">Grade 9 is the largest cohort in South Africa’s senior schools</a>. Nor should it come as a surprise that large percentages of these classes are extremely weak at mathematics. Many pupils have barriers to learning that have been unaddressed for so long that there is little to be done at this late stage.</p>
<p>The Department of Basic Education has snookered itself by applying tight Grade 9 promotion criteria based on mathematics, without providing the means to meet them. This latest move is simply a welcome, realistic – and long overdue – acknowledgement that the ability to factorise quadratic functions is not a prerequisite for an educated child.</p><img src="https://counter.theconversation.com/content/70289/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Sara Muller works at the University of Cape Town as a researcher and PhD candidate.
She receives funding from the Canon Collins Educational and Legal Assistance Trust in support of her PhD research, and is an active member of the Education Fishtank group, an open forum for engaging in education discussions in Cape Town.
All opinions expressed in her articles are her own.</span></em></p>The truth behind South Africa’s decision to allow 20% as a maths pass mark in some grades is a little more complex than many have suggested.Sara Black, Researcher: Teacher Development and Sociology of Education, University of Cape TownLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/659632016-10-13T18:06:42Z2016-10-13T18:06:42ZYes, mathematics can be decolonised. Here’s how to begin<figure><img src="https://images.theconversation.com/files/139426/original/image-20160927-14593-1rf92dt.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">
</span> <span class="attribution"><span class="source">Shutterstock</span></span></figcaption></figure><p>At a time when decolonisation, part of which involves changing the content of what’s taught, is dominating debate at many universities, the discipline of mathematics presents an interesting case. </p>
<p>But it’s not obvious how mathematics can be decolonised at the level of content. This means that those within the discipline must consider other aspects: curriculum processes, such as critical thinking and problem solving; pedagogy – how the subject is taught and, as a number of people have <a href="https://www.routledge.com/Mathematical-Relationships-in-Education-Identities-and-Participation/Black-Mendick-Solomon/p/book/9780415996846">argued</a>, addressing the issue of identity. </p>
<p>Students’ mathematical identities – how they see themselves as learners of mathematics and the extent to which mathematics is meaningful to them – are important when thinking about teaching and learning in mathematics.</p>
<p>In his book <em><a href="https://www.amazon.com/Leading-Change-leadership-university-Educational/dp/113889026X">Leading for change</a></em>, South African educationist Jonathan Jansen suggests that transforming university campuses into deracialised spaces requires attention to both the academic and the human project. I take the human project to mean how students see themselves. What might this mean for mathematics?</p>
<h2>So what is mathematics?</h2>
<p>For starters, it’s important to explore what mathematics actually is.</p>
<p>Mathematician and academic Jo Boaler <a href="http://eu.wiley.com/WileyCDA/WileyTitle/productCd-0470894520.html">points out</a> that mathematics is the only subject where students and mathematicians give very different answers to this question.</p>
<p>Mathematicians view the subject as an exciting, creative endeavour in which problem solving, curiosity, excitement, intuition and perseverance play important roles – albeit in relation to abstract objects of study. </p>
<p>For school and even undergraduate mathematics students, these aspects of mathematics are often not experienced and remain opaque. Students tend to believe that mathematics is a set of procedures to be followed. They think only particularly gifted people can do and understand these procedures. This suggests that the way mathematics is usually taught doesn’t provide opportunities for accessing mathematical knowledge. It doesn’t allow students to identify with mathematics, nor make them aspire to become mathematicians.</p>
<p>As a result, mathematics has a problem with diversity. All over the world, black and women mathematicians remain rare. They <a href="https://www.youcubed.org/think-it-up/ideas-of-giftedness-hurt-students/">simply don’t</a> take mathematics at higher academic levels as much as their white and male peers.</p>
<p>One reason for this is given by <a href="https://www.youcubed.org/think-it-up/ideas-of-giftedness-hurt-students/">a study</a> in the US, which showed that the more a field attributes success to giftedness rather than effort, the fewer female and black academics are in that field. This is because the field perpetuates stereotypes about who belongs in the field. The same study found that mathematics professors hold the most fixed ideas about giftedness. </p>
<p>But this view of giftedness versus effort is not borne out by research. A number of scholars <a href="http://www.ams.org/notices/200102/rev-devlin.pdf">have argued</a> that all people are capable of learning mathematics, to high levels.</p>
<p>This suggests that a lot of the “bad press” around mathematics as a subject and discipline lies with how it is taught and learned.</p>
<h2>What is learning?</h2>
<p>When scholars theorise learning, the thinking always happens in two directions: to the past, and to the future. </p>
<p>Some see learning as building on current knowledge in a <a href="http://rer.sagepub.com/content/57/2/175.full.pdf">step-wise linear way</a>. Some see it as working <a href="http://infed.org/mobi/jerome-bruner-and-the-process-of-education/">in a spiral</a> –- coming back to old ideas in new ways. Still others view learning as <a href="https://books.google.co.za/books?id=x2XRiIm-3vAC&pg=RA4-PA1946&lpg=RA4-PA1946&dq=a+conception+of+knowledge+acquisition+and+its+implications+for+education&source=bl&ots=tO3N1Cedy1&sig=UYT9tzDyfKDubi0MxKIJK0eOS1A&hl=en&sa=X&redir_esc=y#v=onepage&q=a%20conception%20of%20knowledge%20acquisition%20and%20its%20implications%20for%20education&f=false">disrupting or transforming</a> current knowledge.</p>
<p>For teachers, working with current knowledge means finding ways to ascertain, predict, anticipate and think about students’ ideas – and finding ways to engage with these. An important part of students’ ideas about mathematics is how they see themselves in relation to mathematics. Research in schools <a href="http://eu.wiley.com/WileyCDA/WileyTitle/productCd-0470894520.html">has shown</a> that one of the key factors in students’ mathematics achievement is a teacher who believes that they can do mathematics.</p>
<p>The future is important because universities must produce future thinkers, leaders, professionals and citizens. These institutions are the bridge between the past and the future. </p>
<p>Educational theorist Etienne Wenger argues that learning is fundamentally about becoming <a href="https://www.amazon.com/Communities-Practice-Cognitive-Computational-Perspectives/dp/0521663636">a certain kind of person</a>. At universities, students are inducted into disciplines, fields and professions that require them to be certain kinds of people with certain orientations to the world, to knowledge, to other people and to practice. </p>
<p>Traditionally universities have focused on knowledge and hoped that identity will follow. This hasn’t been entirely unsuccessful. But to genuinely transform the academic project, universities must do explicit identity work with their students. Academics must engage in the human project, thinking about who their students are and what their previous experiences of mathematics and of learning mathematics have been.</p>
<h2>Towards genuine change</h2>
<p>There have been attempts to transform the content of school mathematics curricula. These include <a href="http://www.maa.org/publications/periodicals/maa-focus/ethnomathematics-shows-students-their-connections-math">ethnomathematics</a>, which excavates the mathematics in cultural objects, artefacts and practices; and critical mathematics, where mathematics is used to critique aspects of society and where students critique mathematics, for example, how algorithms structure our lives in ways which <a href="http://nymag.com/thecut/2016/09/cathy-oneils-weapons-of-math-destruction-math-is-biased.html">reproduce inequality</a>.</p>
<p>However, not all of mathematics can be accessed in these ways. For true epistemological access to mathematics, students need to study it systematically, as a body of knowledge in and of itself. This can be both empowering or disempowering.</p>
<p>Much, though certainly not all, of mathematics was created by <a href="https://theconversation.com/its-time-to-take-the-curriculum-back-from-dead-white-men-40268">dead white men</a>. But maths should and does belong to everybody. Everybody deserves access to its beauty and its power – and everybody should be able to push back when the discipline is used to destroy and oppress.</p>
<p>To transform mathematics teaching and learning in ways that empower students, universities need to give students the theoretical grounding they need to access the subject and support them to identify with it –- to want to learn it, to become the mathematicians of the future, to enjoy and critique mathematics and its applications. </p>
<p>This means that as teachers, my colleagues and I need to believe – to know – that all students can do mathematics. This knowledge must be transmitted to them. They must be shown that mathematics is a human enterprise: it belongs to all, and it can be taken forward to transform society.</p><img src="https://counter.theconversation.com/content/65963/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Karin Brodie receives funding from the NRF for a research project on mathematical identities of high school learners. </span></em></p>Some have suggested that deracialising the academy requires all researchers, teachers and students to link knowledge and identity. What might this mean for mathematics?Karin Brodie, Professor of Education and Mathematics Education, University of the WitwatersrandLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/636492016-08-24T20:26:14Z2016-08-24T20:26:14ZWhy STEM subjects and fashion design go hand in hand<p>The fashion industry evokes images of impossibly beautiful people jet setting around the world in extravagant finery. Like a moth to the flames, it draws many of our most creative young minds. Often, the first instinct of high school students who want to work in creative industries is to drop all their math and science subjects to take up textiles and art. </p>
<p>As a fashion and textile designer myself, I would like to explain how this is a bad strategy and how the future of fashion requires <a href="https://www.academia.edu/27951162/SO_YOU_WANT_TO_BECOME_A_FASHION_DESIGNER_..%E2%80%8B">science, technology, engineering and mathematics</a> (STEM skills) more than ever.</p>
<p>Beneath the glamorous façade, the fashion industry is undergoing disruptive changes due to rapid advances in technology. We take it for granted that you can use your Iphone to watch a fashion runway show on YouTube, Google the garment to find an online retailer like Net-A-Porter, pay for it using PayPal and then upload a selfie onto Snapchat. None of these services even existed 20 years ago.</p>
<p>Materials that were theoretical thirty years ago have become pervasive. So when you buy yoga clothing from Lululemon that are “anti-bacterial” you are actually wearing fabrics that are coated in silver <a href="http://eng.thesaurus.rusnano.com/wiki/article1257">nano-whiskers</a>. Sportswear companies such as <a href="http://www.materialise.com/cases/software-solutions-help-nike-in-supporting-great-art">Nike</a> and <a href="http://www.materialise.com/cases/adidas-futurecraft-the-ultimate-3d-printed-personalized-shoe">Adidas</a> engage in a technological arms race of materials and technology. The reason why their latest shoes look like something out of science fiction is because the technology is truly cutting edge science.</p>
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<a href="https://images.theconversation.com/files/135242/original/image-20160824-30249-1wzaln.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/135242/original/image-20160824-30249-1wzaln.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/135242/original/image-20160824-30249-1wzaln.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/135242/original/image-20160824-30249-1wzaln.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/135242/original/image-20160824-30249-1wzaln.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/135242/original/image-20160824-30249-1wzaln.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/135242/original/image-20160824-30249-1wzaln.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/135242/original/image-20160824-30249-1wzaln.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=503&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Actor Gwendoline Christie models a creation by Iris van Herpen.</span>
<span class="attribution"><span class="source">Benoit Tessier/Reuters</span></span>
</figcaption>
</figure>
<p>In 2011, Parisian High Fashion forever changed when designer Iris van Herpen was <a href="http://www.materialise.com/cases/iris-van-herpen-s-escapism">invited as a guest member</a> of La Chambre Syndicale de La Haute Couture. Van Herpen, who makes liberal use of hi tech materials such as magnetic fabric, laser cutters and custom developed thermoplastics which are 3D printed, was embraced by the oldest establishment as “Haute Couture”.</p>
<p>Even the supermodel Karlie Kloss advocates the importance of STEM skills for future careers in the tech industry and has a scholarship program <a href="http://kodewithklossy.com/">Kode with Klossy</a> that teaches young girls computer coding.</p>
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<a href="https://images.theconversation.com/files/135055/original/image-20160823-18708-1v506s6.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/135055/original/image-20160823-18708-1v506s6.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/135055/original/image-20160823-18708-1v506s6.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=879&fit=crop&dpr=1 600w, https://images.theconversation.com/files/135055/original/image-20160823-18708-1v506s6.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=879&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/135055/original/image-20160823-18708-1v506s6.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=879&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/135055/original/image-20160823-18708-1v506s6.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=1105&fit=crop&dpr=1 754w, https://images.theconversation.com/files/135055/original/image-20160823-18708-1v506s6.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=1105&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/135055/original/image-20160823-18708-1v506s6.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=1105&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
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<span class="caption">Karlie Kloss: a fan of coding.</span>
<span class="attribution"><span class="source">Danny Moloshok/Reuters</span></span>
</figcaption>
</figure>
<p>Fashion is a unique blend of business, science, art and technology. It requires a polymath, a person who can understand all of these skills. The most compelling reasons to learn STEM skills is because technology and rapidly changing business models have made surviving in the business more competitive than ever. </p>
<p>If you are running a fashion label you will probably need a business loan or have to justify what you are spending your money on. No matter how brilliant your ideas, the people who control money are only swayed by arguments based on sound financial reasoning. Rates of return, accounting and interest rates are all ideas that can only be well understood using mathematics.</p>
<p>Mathematics is mandatory for financial literacy. It introduces ideas such as optimisation, understanding statistics and problem solving and forms a language that allows designers to talk to scientists, engineers and business people.</p>
<p>If you are going to study fashion in college, you will need to learn about fabrics, which are material science. No matter how advanced the school syllabus in textiles, by the time you get to college there will be new materials and technology that did not exist before you got there. If you learn chemistry and physics you will understand the underlying scientific principles on a deeper level, making new material science really easy in the future.</p>
<p>Learning chemistry in school introduces you to lab protocols, taking measurements and accurately recording experiments. These are the exact skills you will need when working with dyes and pigments in textiles. </p>
<p>Using dyes to change the colour of textiles is essentially carbon chemistry. To do this a designer must change the acidity or alkalinity of the fabric - known as the PH level. This allows the “chromophores,” which are the parts of the dye molecule that create colour, to embed into the fabric. The PH scale in chemistry is a logarithmic scale and this is one place where abstract mathematical ideas are actually used in practice.</p>
<h2>Maths and creativity</h2>
<p>Mathematics can also push the boundaries of creativity in fashion. Designer Dai Fujiwara collaborated with legendary 1982 <a href="https://www.britannica.com/topic/Fields-Medal">Fields Medal</a> winning mathematician William Thurston to create radically different garments inspired by geometry and topology. </p>
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<span class="caption">A 2011 creation by Dai Fujiwara.</span>
<span class="attribution"><span class="source">Benoit Tessier/Reuters</span></span>
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<p>In his 1 32 5 collection Fujiwara collaborated with computer scientist Jun Mitani to create mathematical folding algorithms generating innovative clothing. My own PhD research explores <a href="http://newsroom.uts.edu.au/news/2016/08/disruptive-fashion?utm_source=disruptive_gk6&G3utm_medium=gk&utm_campaign=disruptive_aug16">the underlying geometry of how clothing is made</a> and has even been used to teach abstract mathematical concepts through making fashion garments. </p>
<p>For a socially minded designer, STEM skills are essential to understanding environmental sustainability. Fashion used to have seasons, but now with fast fashion companies such as Zara and H&M, new clothing is coming into stores in each week. Fast fashion companies are often criticised for being unsustainable and exploiting workers. </p>
<p>Sustainability in the fashion industry is an extremely complex issue. It requires an understanding of the underlying science, economic behaviour and business practises of the fashion industry and their environmental impact. </p>
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<p>The fashion industry is full of “Greenwash,” fake sustainable marketing which has no scientific basis. STEM skills allow you to navigate these complex issues and try to address them for yourself.</p>
<p>The future of fashion is uncharted territory, but STEM skills make a budding fashion designer smart and adaptable.</p><img src="https://counter.theconversation.com/content/63649/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Mark Liu receives funding for an Australian Postgraduate Award from the Australian Government Department of Education and Training. </span></em></p>The fashion industry attracts creative young minds. But to succeed as a designer in a time of rapid technological change, knowledge of maths and science is invaluable.Mark Liu, PhD Philosophy, Fashion and Textiles Designer, University of Technology SydneyLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/591432016-05-16T10:48:53Z2016-05-16T10:48:53ZWhat makes a mathematical genius?<figure><img src="https://images.theconversation.com/files/121865/original/image-20160510-20731-150nn8p.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">An early understanding of numbers may be a sign of mathematical ability.</span> <span class="attribution"><span class="source">Oksana Kuzmina</span></span></figcaption></figure><p>The film <a href="http://www.imdb.com/title/tt0787524/">The Man Who Knew Infinity</a> tells the gripping <a href="https://theconversation.com/the-man-who-knew-infinity-a-mathematicians-life-comes-to-the-movies-50777">story of Srinivasa Ramanujan</a>, an exceptionally talented, self-taught Indian mathematician. While in India, he was able to develop his own ideas on summing geometric and arithmetic series without any formal training. Eventually, his raw talent was recognised and he got a post at the University of Cambridge. There, he worked with Professor G.H. Hardy until his untimely death at the age of 32 in 1920.</p>
<figure class="align-right ">
<img alt="" src="https://images.theconversation.com/files/121862/original/image-20160510-20721-1jgsbcn.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/121862/original/image-20160510-20721-1jgsbcn.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=822&fit=crop&dpr=1 600w, https://images.theconversation.com/files/121862/original/image-20160510-20721-1jgsbcn.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=822&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/121862/original/image-20160510-20721-1jgsbcn.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=822&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/121862/original/image-20160510-20721-1jgsbcn.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=1033&fit=crop&dpr=1 754w, https://images.theconversation.com/files/121862/original/image-20160510-20721-1jgsbcn.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=1033&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/121862/original/image-20160510-20721-1jgsbcn.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=1033&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Srinivasa Ramanujan.</span>
<span class="attribution"><span class="source">wikimedia</span></span>
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<p>Despite his short life, Ramanujan made substantial contributions to number theory, elliptic functions, infinite series and continued fractions. The story seems to suggest that mathematical ability is something at least partly innate. But what does the evidence say?</p>
<h2>From language to spatial thinking</h2>
<p>There are many different theories about what mathematical ability is. One is that it is closely tied to the capacity for understanding and building language. Just over a decade ago, a study <a href="http://www.ncbi.nlm.nih.gov/pubmed/15319490">examined members of an Amazonian tribe</a> whose counting system comprised words only for “one”, “two” and “many”. The researchers found that the tribe were exceptionally poor at performing numerical thinking with quantities greater than three. They argued this suggests language is a prerequisite for mathematical ability. </p>
<p>But does that mean that a mathematical genius should be better at language than the average person? There is some evidence for this. In 2007, researchers scanned the brains of 25 adult students while they were solving multiplication problems. The study found that individuals with higher mathematical competence <a href="http://www.ncbi.nlm.nih.gov/pubmed/17851092">appeared to rely more strongly on language-mediated processes</a>, associated with brain circuits in the <a href="http://brainmadesimple.com/parietal-lobe.html">parietal lobe</a>. </p>
<p>However, recent findings have challenged this. One <a href="http://www.pnas.org/content/113/18/4909.abstract">study</a> looked at the brain scans of participants, including professional mathematicians, while they evaluated mathematical and non-mathematical statements. They found that instead of the left hemisphere regions of the brain typically involved during language processing and verbal semantics, high level mathematical reasoning was linked with activation of a bilateral network of brain circuits associated with processing numbers and space. </p>
<p>In fact, the brain activation in professional mathematicians in particular showed minimal use of language areas. The researchers argue their results support previous studies that have found that knowledge of numbers and space during early childhood can predict mathematical achievement.</p>
<p>For example, a <a href="http://www.sciencedirect.com/science/article/pii/S0022096515003057">recent study of 77 eight- to 10-year-old children</a> demonstrates that visuo-spatial skills (the capacity to identify visual and spatial relationships among objects) have an important role in mathematical achievement. As part of the study, they took part in a “<a href="http://www.tandfonline.com/doi/abs/10.1080/87565640801982361">number line estimation task</a>”, in which they had to position a series of numbers at appropriate places on a line where only the start and end numbers of a scale (such as 0 and 10) were given.</p>
<p>The study also looked at the children’s overall mathematical ability, visuospatial skills and visuomotor integration (for example, copying increasingly complex images using pencil and paper). It found that children’s scores on visuospatial skill and visuomotor integration strongly predicted how well they would do on number line estimation and mathematics. </p>
<h2>Hidden structures and genes</h2>
<p>An alternative definition of mathematical ability is that it represents the capacity to recognise and exploit hidden structures in data. This may account for an <a href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.581.27&rep=rep1&type=pdf">observed overlap</a> between mathematical and musical ability. Similarly, it could also explain why training in chess can benefit <a href="http://www.sciencedirect.com/science/article/pii/S1747938X16300112">children’s ability to solve mathematical problems</a>. Albert Einstein famously claimed that images, feelings and musical structures formed the basis of his reasoning rather than logical symbols or mathematical equations.</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/122689/original/image-20160516-15920-1624g0m.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/122689/original/image-20160516-15920-1624g0m.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=834&fit=crop&dpr=1 600w, https://images.theconversation.com/files/122689/original/image-20160516-15920-1624g0m.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=834&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/122689/original/image-20160516-15920-1624g0m.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=834&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/122689/original/image-20160516-15920-1624g0m.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=1048&fit=crop&dpr=1 754w, https://images.theconversation.com/files/122689/original/image-20160516-15920-1624g0m.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=1048&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/122689/original/image-20160516-15920-1624g0m.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=1048&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Albert Einstein playing the violin.</span>
<span class="attribution"><span class="source">E. O. Hoppe</span></span>
</figcaption>
</figure>
<p>However, the extent to which mathematical ability relies on innate or environmental factors remains controversial. A <a href="http://www.nature.com/ncomms/2014/140708/ncomms5204/full/ncomms5204.html">recent large scale twin and genome-wide analysis</a> of 12-year-old children found that genetics could explain around half of the observed correlation between mathematical and reading ability. Although this is quite substantial, it still means that the learning environment has an important role to play. </p>
<p>So what does all this tell us about geniuses like Ramanujan? If mathematical ability does stem from a core non-linguistic capacity to reason with spatial and numerical representation, this can help explain how a prodigious talent could blossom in the absence of training. While language might still play a role, the nature of the numerical representations being manipulated could be crucial. </p>
<p>The fact that genetics seems to be involved also helps shed light on the case – Ramanujan could have simply inherited the ability. Nevertheless, we should not forget the important contribution of environment and education. While Ramanujan’s raw talent was sufficient to attract attention to his remarkable ability, it was the <a href="https://theconversation.com/the-man-who-taught-infinity-how-gh-hardy-tamed-srinivasa-ramanujans-genius-57585">later provision</a> of more formal mathematical training in India and England that allowed him to reach his full potential.</p><img src="https://counter.theconversation.com/content/59143/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>David Pearson 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>You may have got what it takes to be a mathematical genius without even being aware of it.David Pearson, Reader of Cognitive Psychology, Anglia Ruskin UniversityLicensed as Creative Commons – attribution, no derivatives.