tag:theconversation.com,2011:/us/topics/algebra-13729/articlesAlgebra – The Conversation2023-02-06T16:59:09Ztag:theconversation.com,2011:article/1956182023-02-06T16:59:09Z2023-02-06T16:59:09Z'Numberless math' gets kids thinking about and visualizing algebra<figure><img src="https://images.theconversation.com/files/506189/original/file-20230124-16-w7tn0k.jpg?ixlib=rb-1.1.0&rect=797%2C0%2C5849%2C3536&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">In elementary school, algebra involves using mathematical models to represent and analyze mathematical situations.</span> <span class="attribution"><span class="source">(Shutterstock)</span></span></figcaption></figure><p>Today, elementary mathematics classrooms look and sound very different than what many parents may have experienced. </p>
<p>Many mathematics education researchers note that getting learners <a href="https://theconversation.com/why-students-need-more-math-talk-104034">talking about mathematics</a> and developing their own explanations is important. </p>
<p>To build students’ comfort and capabilities talking about math, sharing tasks that involve only pictures — what we call numberless or textless tasks — can elicit fascinating mathematical ideas. This puts learners in the driver’s seat,
<a href="https://theconversation.com/mathematics-is-about-wonder-creativity-and-fun-so-lets-teach-it-that-way-120133">noticing, wondering and actually doing</a> mathematics for themselves.</p>
<p>Parents, caregivers and teachers can use these strategies to get learners
talking, explaining and visualizing important mathematical concepts while having quite a bit of fun!</p>
<h2>Making kids notice and wonder</h2>
<p>In our roles as mathematics education teachers at the university level (Marc and Evan) and in elementary school (Matthew), we all ask our students: <a href="https://www.nctm.org/noticeandwonder/">“What do you notice?”</a> and “What do you wonder?” to elicit mathematics conversations. This prompt is recommended by the National Council for Teachers of Mathematics.</p>
<p>Suppose we were asking this question about the picture shown below that features two different lines of images with seahorses, lobster and fish (we’re Maritimers, after all).</p>
<figure class="align-center ">
<img alt="Graphic showing two lines of images which include seahorses, lobster and fish. Each line is split in the middle with a small triangle. The first line shows a line of five seahorses on one side of the triangle and a lobster on the other, and the second line shows a lobster on one side of the triangle, and a seahorse and a fish on the other." src="https://images.theconversation.com/files/505518/original/file-20230120-16-yti1xv.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/505518/original/file-20230120-16-yti1xv.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=487&fit=crop&dpr=1 600w, https://images.theconversation.com/files/505518/original/file-20230120-16-yti1xv.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=487&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/505518/original/file-20230120-16-yti1xv.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=487&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/505518/original/file-20230120-16-yti1xv.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=612&fit=crop&dpr=1 754w, https://images.theconversation.com/files/505518/original/file-20230120-16-yti1xv.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=612&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/505518/original/file-20230120-16-yti1xv.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=612&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">What do you notice?</span>
<span class="attribution"><span class="source">(Marc Husband)</span>, <span class="license">Author provided</span></span>
</figcaption>
</figure>
<p>Almost immediately, we’d hear learners talk about mathematical ideas like these: </p>
<blockquote>
<p>“It looks like a teeter-totter.”</p>
<p>“It’s a balance. The balance is even.”</p>
<p>“The right side of the scale weighs the same as the left side.” </p>
<p>“Five seahorses equals a lobster and a fish, and on the second balance, one lobster equals a seahorse and a fish.”</p>
<p>“These are two equations.”</p>
</blockquote>
<p>These mathematical ideas provide the foundation for algebra. In <a href="https://school.nelson.com/good-questions-great-ways-to-differentiate-mathematics-instruction-3rd-edition">elementary school</a>, algebra involves using mathematical models: balances, in this case, to represent and analyze mathematical situations. </p>
<h2>Invites kids to ask their own questions</h2>
<p>Seeing the picture makes learners wonder and ask questions such as these: “What would happen to the balance if we removed a seahorse?” “How much does a seahorse weigh?” “How much does a fish weigh?” </p>
<p>We think this is significant because rather than the teacher or parent presenting questions for learners to solve, learners pose and investigate their own questions. </p>
<p>For example, we noticed students in our classrooms became super engaged in this task because the pictures prompted them to problem solve without us (their teachers) telling them the problem or how to solve it. </p>
<h2>Now you try</h2>
<p>Before reading further, try solving one of the above questions based on line one (with five seahorses), or line two (with one seahorse) with a friend or family member. </p>
<p>We recommend using paper cut-outs of the visual so you can move them around and record how you solved your problem.</p>
<p>Below are three different approaches for figuring out how much each item weighs.</p>
<h2>Assigning different values</h2>
<p>Learners assign values to the items and check whether their assigned guesses are true for each balance. </p>
<p>For example, if each seahorse is a one, a lobster might be two, and a fish might be three. We often hear, “Let’s check to see if this works on the second balance.” </p>
<p>When learners realize two does not equal three plus one, they reassign values to make the equation true. Interestingly, some learners guess by trying other numbers, such as one seahorse equals five or 10, and so on. </p>
<figure class="align-center ">
<img alt="Graphic showing two lines of images which include seahorses, lobster and fish. Each line is split in the middle with a small triangle. The first line shows a line of five seahorses on one side of the triangle and a lobster on the other, and the second line shows a lobster on one side of the triangle, and a seahorse and a fish on the other. Both groups of sea creatures on the bottom line are circled and an arrow is pointing at the row above." src="https://images.theconversation.com/files/504106/original/file-20230111-24-l6epd3.jpeg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/504106/original/file-20230111-24-l6epd3.jpeg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=414&fit=crop&dpr=1 600w, https://images.theconversation.com/files/504106/original/file-20230111-24-l6epd3.jpeg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=414&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/504106/original/file-20230111-24-l6epd3.jpeg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=414&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/504106/original/file-20230111-24-l6epd3.jpeg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=521&fit=crop&dpr=1 754w, https://images.theconversation.com/files/504106/original/file-20230111-24-l6epd3.jpeg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=521&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/504106/original/file-20230111-24-l6epd3.jpeg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=521&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">If five seahorses balance with one lobster and one fish, how might you start to think about their respective values?</span>
<span class="attribution"><span class="source">(Marc Husband)</span>, <span class="license">Author provided</span></span>
</figcaption>
</figure>
<h2>Combining balances</h2>
<p>Learners combine balances by moving the items on the bottom to the top (or vice-versa) and then pulling items away to maintain the balance. </p>
<p>For example, the lobster in the second balance moves to the side of the five seahorses in the first balance, and the seahorse and fish move up to the fish and lobster side. </p>
<p>We hear learners say, “If we remove the lobsters on both sides and one seahorse from each side, it will stay in balance with four seahorses on one side and two fish on the other.” </p>
<figure class="align-center ">
<img alt="Sea creatures on the bottom are circled with arrows pointing to the top balance." src="https://images.theconversation.com/files/504107/original/file-20230111-34767-gmnxy9.jpeg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/504107/original/file-20230111-34767-gmnxy9.jpeg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=302&fit=crop&dpr=1 600w, https://images.theconversation.com/files/504107/original/file-20230111-34767-gmnxy9.jpeg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=302&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/504107/original/file-20230111-34767-gmnxy9.jpeg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=302&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/504107/original/file-20230111-34767-gmnxy9.jpeg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=380&fit=crop&dpr=1 754w, https://images.theconversation.com/files/504107/original/file-20230111-34767-gmnxy9.jpeg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=380&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/504107/original/file-20230111-34767-gmnxy9.jpeg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=380&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">What about moving creatures on the bottom?</span>
<span class="attribution"><span class="source">(Marc Husband)</span>, <span class="license">Author provided</span></span>
</figcaption>
</figure>
<figure class="align-center ">
<img alt="Graphic showing one line with a triangle in the middle and one side is five lobsters and one seahorse, and on the other side of the triangle is a lobser, fish, seahorse and fish." src="https://images.theconversation.com/files/504108/original/file-20230111-11-50ewoi.jpeg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/504108/original/file-20230111-11-50ewoi.jpeg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=211&fit=crop&dpr=1 600w, https://images.theconversation.com/files/504108/original/file-20230111-11-50ewoi.jpeg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=211&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/504108/original/file-20230111-11-50ewoi.jpeg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=211&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/504108/original/file-20230111-11-50ewoi.jpeg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=265&fit=crop&dpr=1 754w, https://images.theconversation.com/files/504108/original/file-20230111-11-50ewoi.jpeg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=265&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/504108/original/file-20230111-11-50ewoi.jpeg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=265&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">The two balances combined.</span>
<span class="attribution"><span class="source">(Marc Husband)</span>, <span class="license">Author provided</span></span>
</figcaption>
</figure>
<figure class="align-center ">
<img alt="Graphic image showing two lines; the top line shows four seahorses and two fish; the bottom line shows two lobsters atop two sea horses." src="https://images.theconversation.com/files/504109/original/file-20230111-27936-z7fdw6.jpeg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/504109/original/file-20230111-27936-z7fdw6.jpeg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=396&fit=crop&dpr=1 600w, https://images.theconversation.com/files/504109/original/file-20230111-27936-z7fdw6.jpeg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=396&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/504109/original/file-20230111-27936-z7fdw6.jpeg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=396&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/504109/original/file-20230111-27936-z7fdw6.jpeg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=498&fit=crop&dpr=1 754w, https://images.theconversation.com/files/504109/original/file-20230111-27936-z7fdw6.jpeg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=498&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/504109/original/file-20230111-27936-z7fdw6.jpeg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=498&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Could you remove items to maintain balance?</span>
<span class="attribution"><span class="source">(Marc Husband)</span>, <span class="license">Author provided</span></span>
</figcaption>
</figure>
<p>Based on this, learners often conclude that two seahorses equal one fish (or comparatively, for those who may be struggling still: a seahorse equals one, a fish equals two and a lobster equals five. </p>
<p>These approaches remind us that numberless/textless tasks get kids doing elimination in algebra without them knowing. </p>
<h2>Exchanging pictures</h2>
<p>Learners exchange the value of one picture with pictures of equal value. For example, seeing that a lobster is the same as a fish and a seahorse, learners replace the lobster in the first balance with a fish and an seahorse. This leads them to start pulling away, similar to the above strategy, taking one seahorse from each side and so on. </p>
<figure class="align-center ">
<img alt="Two lines of images seen, with each line split by a triangle. The first one shows five seahorses on one side and on the other, one seahorse and one fish. The second shows one lobster on one side of the line and on the other, one seahorse and one fish." src="https://images.theconversation.com/files/504110/original/file-20230111-4958-mqpabm.jpeg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/504110/original/file-20230111-4958-mqpabm.jpeg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=343&fit=crop&dpr=1 600w, https://images.theconversation.com/files/504110/original/file-20230111-4958-mqpabm.jpeg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=343&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/504110/original/file-20230111-4958-mqpabm.jpeg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=343&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/504110/original/file-20230111-4958-mqpabm.jpeg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=431&fit=crop&dpr=1 754w, https://images.theconversation.com/files/504110/original/file-20230111-4958-mqpabm.jpeg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=431&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/504110/original/file-20230111-4958-mqpabm.jpeg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=431&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">How about exchanging one lobster for one seahorse and one fish?</span>
<span class="attribution"><span class="source">(Marc Husband)</span>, <span class="license">Author provided</span></span>
</figcaption>
</figure>
<figure class="align-center ">
<img alt="Two lines of images seen, with each line split by a triangle. The first one shows five seahorses on one side and on the other, two seahorses and two fish. The second shows two lobsters." src="https://images.theconversation.com/files/504111/original/file-20230111-47547-4cfph9.jpeg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/504111/original/file-20230111-47547-4cfph9.jpeg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=371&fit=crop&dpr=1 600w, https://images.theconversation.com/files/504111/original/file-20230111-47547-4cfph9.jpeg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=371&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/504111/original/file-20230111-47547-4cfph9.jpeg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=371&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/504111/original/file-20230111-47547-4cfph9.jpeg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=467&fit=crop&dpr=1 754w, https://images.theconversation.com/files/504111/original/file-20230111-47547-4cfph9.jpeg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=467&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/504111/original/file-20230111-47547-4cfph9.jpeg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=467&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Lobster and seahorse trade places.</span>
<span class="attribution"><span class="source">(Marc Husband)</span>, <span class="license">Author provided</span></span>
</figcaption>
</figure>
<figure class="align-center ">
<img alt="a line separated by a triangle in the middle shows four seahorses one one side and two fish on the other" src="https://images.theconversation.com/files/504112/original/file-20230111-18-im7uuk.jpeg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/504112/original/file-20230111-18-im7uuk.jpeg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=253&fit=crop&dpr=1 600w, https://images.theconversation.com/files/504112/original/file-20230111-18-im7uuk.jpeg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=253&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/504112/original/file-20230111-18-im7uuk.jpeg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=253&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/504112/original/file-20230111-18-im7uuk.jpeg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=318&fit=crop&dpr=1 754w, https://images.theconversation.com/files/504112/original/file-20230111-18-im7uuk.jpeg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=318&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/504112/original/file-20230111-18-im7uuk.jpeg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=318&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Now you could remove items to maintain balance.</span>
<span class="attribution"><span class="source">(Marc Husband)</span>, <span class="license">Author provided</span></span>
</figcaption>
</figure>
<p>This approach reminds us that numberless/textless tasks get kids doing substitution in algebra without them knowing. </p>
<h2>Learners in charge</h2>
<p>We think selecting the right mathematics task for learners to work on with their peers is important because it engages them in fruitful conversations. </p>
<p>As the saying goes, “the person doing the talking is the person doing the learning”. Numberless/textless tasks get learners talking about mathematical ideas — in this case, with concepts like balance, same as and equal. </p>
<p>Learners actually see how to maintain balance by moving items on each side. This visual supports their understanding of an algebraic rule: “Whatever you do to one side of the equation, you need to do to the other side,” which parents might recall from memory. </p>
<p>This task sets the foundation for algebraic thinking without learners even knowing it. Algebra, often perceived as a scary word by many learners, becomes less scary through a numberless/textless task. This could be why many learners find themselves in the driver’s seat — talking, generating questions, visualizing and explaining mathematics!</p><img src="https://counter.theconversation.com/content/195618/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>Working with moveable pictures can help children learn an algebra rule: Whatever you do to one side of the equation, you need to do to the other. Here’s how teachers or caregivers can lead this.Marc Husband, Assistant Professor, School of Education, St. Francis Xavier UniversityEvan Throop Robinson, Associate Professor, School of Education, St. Francis Xavier UniversityMatthew Little, Masters of Education student, Faculty of Education, St. Francis Xavier UniversityLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/1765182022-08-28T12:33:40Z2022-08-28T12:33:40ZThe simple reason a viral math equation stumped the internet<figure><img src="https://images.theconversation.com/files/479476/original/file-20220816-12125-zqcgvx.jpg?ixlib=rb-1.1.0&rect=8%2C16%2C5396%2C3176&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Inappropriate ways of denoting multiplication are everywhere. </span> <span class="attribution"><span class="source">(Shutterstock)</span></span></figcaption></figure><p><a href="https://slate.com/technology/2013/03/facebook-math-problem-why-pemdas-doesnt-always-give-a-clear-answer.html">For about a decade now</a>, mathematicians and mathematics educators have been weighing in on a particular debate rooted in school mathematics that <a href="https://www.suggest.com/viral-simple-math-problem-causes-divide/2653191/">shows no signs of abating</a>. </p>
<p>The debate, covered by <em>Slate</em>, <a href="https://www.popularmechanics.com/science/math/a28569610/viral-math-problem-2019-solved/"><em>Popular Mechanics</em></a>, <a href="https://www.nytimes.com/2019/08/02/science/math-equation-pedmas-bemdas-bedmas.html"><em>The New York Times</em></a> and many other outlets, is focused on an equation that went so “<a href="https://www.lifewire.com/what-does-it-mean-to-go-viral-3486225">viral</a>” that it, eventually, was lumped with other phenomena that have “<a href="https://www.inc.com/dave-kerpen/this-basic-math-problem-is-breaking-internet.html#">broken</a>” or <a href="https://www.cbc.ca/kidsnews/post/a-simple-math-problem-has-divided-the-internet">“divided” the internet</a>. </p>
<p>On the off chance you’ve yet to weigh in, now would be a good time to see where you stand. Please answer the following: </p>
<p><strong>8÷2(2+2)=?</strong> </p>
<p>If you’re like most, your answer was 16 and are flabbergasted someone else can find a different answer. Unless, that is, you’re like most others and your answer was 1 and you’re equally confused about seeing it another way. Fear not, in what follows, we will explain the definitive answer to this question — and why the manner in which the equation is written should be banned. </p>
<p>Our interest was piqued because we have <a href="https://doi.org/10.1007/978-3-319-92390-1_50">conducted research</a> on <a href="https://doi.org/10.1007/s10649-017-9789-9">conventions</a> about following <a href="https://flm-journal.org/Articles/574E410A33668116D7F3326364FA2.pdf">the order of operations</a> — a sequence of steps taken when faced with a math equation — and were a bit befuddled with what all the fuss was about.</p>
<h2>Clearly, the answer is…</h2>
<p>Two viable answers to one math problem? Well, if there’s one thing we all remember from math class: that can’t be right! </p>
<p>Many themes emerged from the plethora of articles explaining how and why this “equation” broke the internet. Entering the expression on calculators, <a href="https://www.insider.com/viral-math-problem-solution-dividing-the-internet-2019-7">some of which are programmed to respect a particular order of operations</a>, was much discussed. </p>
<p>Others, hedging a bit, suggest both <a href="https://www.foxnews.com/tech/viral-math-problem-baffles-many-internet">answers are correct</a> (which is ridiculous).</p>
<p>The most dominant theme simply focused on implementation of the order of operations according to different acronyms. Some commentators said people’s misunderstandings were attributed to <a href="https://www.nytimes.com/2019/08/02/science/math-equation-pedmas-bemdas-bedmas.html">incorrect interpretation of the memorized acronym taught in different countries to remember the order of operations</a> like PEMDAS, sometimes used in the United States: PEMDAS refers to applying parentheses, exponents, multiplication, division, addition and subtraction. </p>
<p>A person following this order would have 8÷2(2+2) become 8÷2(4) thanks to starting with parentheses. Then, 8÷2(4) becomes 8÷8 because there are no exponents, and “M” stands for multiplication, so they multiply 2 by 4. Lastly, according to the “D” for division, they get 8÷8=1. </p>
<figure class="align-center ">
<img alt="Image of the acronym PEMDAS spelled out referring to parentheses, exponents, multiplication, division." src="https://images.theconversation.com/files/478942/original/file-20220812-22-22cm9f.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/478942/original/file-20220812-22-22cm9f.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=300&fit=crop&dpr=1 600w, https://images.theconversation.com/files/478942/original/file-20220812-22-22cm9f.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=300&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/478942/original/file-20220812-22-22cm9f.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=300&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/478942/original/file-20220812-22-22cm9f.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=377&fit=crop&dpr=1 754w, https://images.theconversation.com/files/478942/original/file-20220812-22-22cm9f.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=377&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/478942/original/file-20220812-22-22cm9f.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=377&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Were different ways of teaching the order of operations responsible for confusion?</span>
<span class="attribution"><span class="source">(Shutterstock)</span></span>
</figcaption>
</figure>
<p>By contrast, Canadians may be taught to remember BEDMAS, which stands for applying brackets, exponents, division, multiplication, addition and subtraction. Someone following this order would have 8÷2(2+2) become 8÷2(4) thanks to starting with brackets (the same as parentheses). Then, 8÷2(4) becomes 4(4) because (there are no exponents) and “D” stands for division. Lastly, according to the “M” for multiplication, 4(4)=16. </p>
<h2>Do not omit multiplication symbol</h2>
<p>For us, the expression 8÷2(2+2) is syntactically wrong. </p>
<p>Key to the debate, we contend, is that the multiplication symbol before the parentheses is omitted. </p>
<p>Such an omission is a convention in algebra. For example, in algebra we write 2x or 3a which means 2 × x or 3 × a. When letters are used for variables or constants, the multiplication sign is omitted. Consider the famous equation e=mc<sup>2,</sup> which suggests the computation of energy as e=m×c<sup>2.</sup></p>
<p>The real reason, then, that 8÷2(2+2) broke the internet stems from the practice of omitting the multiplication symbol, which was inappropriately brought to arithmetic from algebra. </p>
<h2>Inappropriate priority</h2>
<p>In other words, had the expression been correctly “spelled out” that is, presented as “8 ÷ 2 × (2 + 2) = ? ”, there would be no going viral, no duality, no broken internet, no heated debates. No fun!</p>
<figure class="align-center ">
<img alt="The equation 8 ÷ 2 × (2 + 2) = ?" src="https://images.theconversation.com/files/479481/original/file-20220816-11082-5eioce.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/479481/original/file-20220816-11082-5eioce.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=169&fit=crop&dpr=1 600w, https://images.theconversation.com/files/479481/original/file-20220816-11082-5eioce.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=169&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/479481/original/file-20220816-11082-5eioce.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=169&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/479481/original/file-20220816-11082-5eioce.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=213&fit=crop&dpr=1 754w, https://images.theconversation.com/files/479481/original/file-20220816-11082-5eioce.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=213&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/479481/original/file-20220816-11082-5eioce.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=213&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Had the problem been correctly presented as 8 ÷ 2 × (2 + 2) = ?, there would be no heated debate.</span>
<span class="attribution"><span class="source">(Egan J. Chernoff)</span>, <span class="license">Author provided</span></span>
</figcaption>
</figure>
<p>Ultimately, omission of the multiplication symbol invites inappropriate priority to multiplication. All commentators agreed that adding the terms in the brackets or parentheses was the appropriate first step. But confusion arose given the proximity of 2 to (4) relative to 8 in 8÷2(4).</p>
<p>We want it known that writing 2(4) to refer to multiplication is inappropriate, but we get that it’s done all the time and everywhere. </p>
<h2>Nice symbol for multiplication</h2>
<p>There is a very nice symbol for multiplication, so let’s use it: 2 × 4. Should you not be a fan, there are other symbols, such as 2•4. Use either, at your pleasure, but do not omit. </p>
<p>As such, for the record, the debate over one versus 16 is now over! The answer is 16. Case closed. Also, there should have never really been a debate in the first place.</p><img src="https://counter.theconversation.com/content/176518/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Egan J Chernoff received and receives funding from SSHRC (Social Sciences and Humanities Research Council of Canada) which is not explicitly related to this article. </span></em></p><p class="fine-print"><em><span>Rina Zazkis received funding from SSHRC (Social Sciences and Humanities Research Council of Canada ) which is not explicitly related to the article</span></em></p>When the equation 8÷2(2+2)=? is written properly and includes a multiplication sign before the first bracket, the answer is clear.Egan J Chernoff, Professor of Mathematics Education, University of SaskatchewanRina Zazkis, Professor, Faculty of Education, Simon Fraser UniversityLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/1651492021-07-29T12:24:04Z2021-07-29T12:24:04ZBob Moses played critical role in civil rights organizing and math literacy for Black students<figure><img src="https://images.theconversation.com/files/413595/original/file-20210728-13-mlpe5w.jpg?ixlib=rb-1.1.0&rect=14%2C9%2C3244%2C2433&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Civil rights activist Bob Moses founded The Algebra Project to help Black students develop strong math skills.</span> <span class="attribution"><a class="source" href="https://www.flickr.com/photos/princetonpubliclibrary/15627635898">Princeton Public Library/Flickr</a>, <a class="license" href="http://creativecommons.org/licenses/by-nd/4.0/">CC BY-ND</a></span></figcaption></figure><p>As an organizer for the <a href="https://snccdigital.org/inside-sncc/the-story-of-sncc/">Student Nonviolent Coordinating Committee</a> during the 1960s, <a href="https://snccdigital.org/people/bob-moses/">Bob Moses</a> traveled to the most dangerous parts of Mississippi to help African Americans end segregation and secure the right to vote. But it would be tutoring students in math 20 years later at his daughter’s racially mixed middle school in Massachusetts that would lead to his life’s work – The Algebra Project.</p>
<p><a href="https://snccdigital.org/events/bob-moses-begins-algebra-project/">The Algebra Project</a> is a nonprofit dedicated to helping students from historically marginalized communities develop math literacy, which is an individual’s ability to formulate, employ and interpret mathematics in a variety of contexts. Moses founded it in 1982.</p>
<p>After researching Moses’ role in the civil rights movement for my book – <a href="https://nyupress.org/9780814743317/bloody-lowndes/">“Bloody Lowndes: Civil Rights and Black Power in Alabama’s Black Belt”</a> – and later interviewing him for various projects about SNCC, it became abundantly clear that The Algebra Project sprang directly from his civil rights work in Mississippi. That work helped transform Mississippi from a segregationist stronghold into a focal point of the civil rights revolution.</p>
<p>In his book “<a href="https://www.penguinrandomhouse.com/books/206027/radical-equations-by-robert-p-moses/">Radical Equations</a>,” Moses recalls that in 1982 he was surprised to discover that his daughter, Maisha, who was entering the eighth grade at the Dr. Martin Luther King, Jr. School in Cambridge, Massachusetts, would not be taught algebra because the school did not offer it. Without knowledge of algebra, she could not qualify for honors math and science classes in high school.</p>
<h2>Math endeavors</h2>
<p>As explained in his book, Moses had a background in mathematics. In 1957, before joining the civil rights movement, he earned a master’s degree in philosophy at Harvard University and then taught middle school math for a few years in the Bronx, New York, at Horace Mann School, a prestigious private school just north of where he grew up in Harlem. And from 1969 to 1976, he taught algebra in Tanzania before returning stateside to work on a doctorate in the philosophy of math.</p>
<p>Moses asked Maisha’s teacher if he could provide his daughter with supplemental math lessons in class since Maisha refused to be tutored at home – she opposed doing what she called “two maths.” The teacher consented, but on the condition that Moses instruct some of Maisha’s classmates as well, according to his book.</p>
<p>Moses agreed. Like the teacher, he believed that all children, including those from historically marginalized communities, deserved a chance to take advanced math and science classes in high school.</p>
<p>At the end of the school year, Maisha and the three students who studied with her passed the citywide algebra exam. They were the first from their school to do so, according to his book.</p>
<p>Excited by this success, Maisha’s teacher asked Moses to work his math magic with more students. </p>
<p>But it wasn’t magic.</p>
<p>Moses succeeded in teaching algebra to the students who were frequently <a href="https://files.eric.ed.gov/fulltext/EJ1250375.pdf">tracked</a> into less rigorous classes and courses of study because he believed that Black, brown, working-class and poor children could master algebra - or other advanced classes - even at an early age.</p>
<p>He also knew that these same students would be eager to study math if instruction revolved around their lived experiences. Rote memorization would not work; content had to be relatable.</p>
<p>Moses agreed to teach the incoming eighth graders, even though none of his children were in the class. “I was beginning to think I had found my work,” he wrote in “Radical Equations.” And his work was teaching math literacy in the emerging digital age.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/413604/original/file-20210728-17-1a4o3t7.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="A Black boy works on a math problem in class." src="https://images.theconversation.com/files/413604/original/file-20210728-17-1a4o3t7.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/413604/original/file-20210728-17-1a4o3t7.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/413604/original/file-20210728-17-1a4o3t7.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/413604/original/file-20210728-17-1a4o3t7.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/413604/original/file-20210728-17-1a4o3t7.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/413604/original/file-20210728-17-1a4o3t7.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/413604/original/file-20210728-17-1a4o3t7.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">Math literacy for African American children was an essential aspect of Moses’ philosophy and work.</span>
<span class="attribution"><a class="source" href="https://www.gettyimages.com/detail/photo/african-american-student-studying-at-desk-in-royalty-free-image/135205410?adppopup=true">Ariel Skelley/DigitalVision via Getty Images</a></span>
</figcaption>
</figure>
<h2>Key to a better life</h2>
<p>Moses believed that math proficiency was a gateway to equality in a post-industrial society. <a href="https://www.npr.org/templates/story/story.php?storyId=7495586">He explained in 2007</a>: “In our society, algebra is the place where we ask students to master a quantitative literacy requirement. And so hence, algebra becomes available as an organizing tool now for educational rights and for economic rights.” In other words, math literacy would provide access to the kinds of computer-driven careers that would enable African Americans, and other historically marginalized youth, to permanently improve their life circumstances and the social and economic conditions of their communities.</p>
<p>But Moses wasn’t interested in teaching just a few students, much as he wasn’t interested in registering just a few Black Mississippians. He wanted to instruct as many young people as possible, in the same way he wanted to organize as many Black people in Mississippi as possible. </p>
<p>Reaching more youth, however, required a dramatic shift in the culture of learning at the school. Expectations regarding when young children from marginalized groups should study algebra had to change, which was no small task considering many children weren’t expected to study algebra at all. </p>
<p>Just as he once organized sharecroppers, he began organizing parents.</p>
<h2>Emphasis on independence</h2>
<p>In the civil rights movement, Moses routinely deferred to the wishes and desires of the people he was organizing, so much so that he left the movement in 1965 when he felt people were turning to him too often for solutions to their problems. This was the approach of his mentor, veteran activist and SNCC adviser <a href="https://uncpress.org/book/9780807856161/ella-baker-and-the-black-freedom-movement/">Ella Baker</a>, who led by asking questions, rather than by providing answers. </p>
<p>Moses talked to parents at the school about the lack of opportunities to take algebra, which, he recalled, led them to initiate a survey that showed that – as explained in his book – “All parents thought their child should do algebra, but not all parents thought that every child should do algebra.” </p>
<p>The parents were shocked and somewhat embarrassed by the survey results, leading to a consensus for allowing any seventh or eighth grader to take algebra. </p>
<p>Only two years after Moses’ daughter passed the citywide exam, the King school offered algebra to students in the seventh and eighth grades, and even provided Saturday classes for parents. </p>
<p>Today, The Algebra Project is fighting to ensure students receive the quality math education they deserve by supporting learning cohorts in dozens of schools across the country where students have historically performed poorly in math on eighth grade state tests. The impact of the project at Mansfield Senior High School in Mansfield, Ohio, is <a href="https://www.mansfieldnewsjournal.com/story/news/local/2016/12/20/students-talk-math-algebra-project-founder/95654378/">illustrative</a>. In the eighth grade, the math proficiency of The Algebra Project cohort was 17%. By the 10th grade, that number had <a href="https://iris.siue.edu/math-literacy-archive/files/original/82b7ceef4d303db77a55e7b34e9b6412.pdf">risen to 82%</a>.</p>
<p>Ella Baker was fond of saying, “Give light and people will find the way.” Few did that better than Bob Moses, who died on July 25, 2021. </p>
<p>[<em>Get the best of The Conversation, every weekend.</em> <a href="https://theconversation.com/us/newsletters/weekly-highlights-61?utm_source=TCUS&utm_medium=inline-link&utm_campaign=newsletter-text&utm_content=weeklybest">Sign up for our weekly newsletter</a>.]</p><img src="https://counter.theconversation.com/content/165149/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Hasan Kwame Jeffries tidak bekerja, menjadi konsultan, memiliki saham, atau menerima dana dari perusahaan atau organisasi mana pun yang akan mengambil untung dari artikel ini, dan telah mengungkapkan bahwa ia tidak memiliki afiliasi selain yang telah disebut di atas.</span></em></p>The Algebra Project – a long-standing initiative to teach algebra to Black students who might not otherwise take it – sprang from Bob Moses’ work as a civil rights activist, a historian recounts.Hasan Kwame Jeffries, Associate Professor of History, The Ohio State UniversityLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/1620882021-06-08T20:06:41Z2021-06-08T20:06:41ZThe proposed new maths curriculum doesn't dumb down content. It actually demands more of students<p>In recent days, <a href="https://www.smh.com.au/education/confused-and-confusing-maths-experts-say-curriculum-is-faddish-and-shallow-20210602-p57xj3.html">dozens of maths professors</a> and teachers wrote an open letter airing concerns about a new draft national maths curriculum. Their concerns include <a href="https://www.smh.com.au/education/confused-and-confusing-maths-experts-say-curriculum-is-faddish-and-shallow-20210602-p57xj3.html">dropping “mastery” of times tables</a> and a “faddish” emphasis on student-led learning.</p>
<p>A <a href="https://www.australiancurriculum.edu.au/consultation/">proposed curriculum</a> for foundation to year 10 Australian students was released for consultation at the end of April. For maths, it is actually an improvement on the one we have and demands more of students.</p>
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Read more:
<a href="https://theconversation.com/a-crowded-curriculum-sure-it-may-be-complex-but-so-is-the-world-kids-must-engage-with-157690">A 'crowded curriculum'? Sure, it may be complex, but so is the world kids must engage with</a>
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<p>If applied in the proposed form, the new curriculum would result in a deeper understanding of key concepts. It expects students to be able to explain their maths reasoning rather than present their answer without justification. And talking about <a href="https://www.nctm.org/uploadedFiles/Research_and_Advocacy/research_brief_and_clips/Clip%2017%20Benefits%20of%20Discussion.pdf;https://theconversation.com/why-students-need-more-math-talk-104034">maths is important</a>. Students learn better when they’re able to <a href="https://www.nctm.org/uploadedFiles/Research_and_Advocacy/research_brief_and_clips/Clip%2017%20Benefits%20of%20Discussion.pdf">articulate</a> what they are thinking and explain this to another person.</p>
<h2>It helps students overcome hurdles</h2>
<p>The proposed curriculum is based on <a href="https://www.merga.net.au/Public/Public/Publications/Engaging_the_Australian_curriculum_mathematics_book.aspx">research that demonstrates</a> the importance of building an understanding of concepts, such as multiplication, rather than just teaching kids to memorise times tables. </p>
<p>With every new concept, students experience a hurdle as they need to completely shift the way they previously perceived the concept. For instance, a child will need to move from seeing a triangle as a pointy shape to focusing on the relationship between the length of sides and angles, as well as its properties (such as symmetry). </p>
<p>The proposed curriculum design acknowledges these kinds of learning hurdles. It provides teaching sequences in key areas such as algebra, measurement and numbers, to help teachers make informed decisions about where to target their teaching. </p>
<p>Say a student reaches the hurdle of needing to focus on the relationships among properties of triangles, which is necessary before they can solve geometric proofs. Here, the proposed curriculum prompts teachers to consider a range of real-word examples. It also provides student-centred activities to support the kids in getting over the hurdle. </p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/404999/original/file-20210608-130350-1gn2fl.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="Triangles marked up with algebraic terms." src="https://images.theconversation.com/files/404999/original/file-20210608-130350-1gn2fl.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/404999/original/file-20210608-130350-1gn2fl.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=150&fit=crop&dpr=1 600w, https://images.theconversation.com/files/404999/original/file-20210608-130350-1gn2fl.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=150&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/404999/original/file-20210608-130350-1gn2fl.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=150&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/404999/original/file-20210608-130350-1gn2fl.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=189&fit=crop&dpr=1 754w, https://images.theconversation.com/files/404999/original/file-20210608-130350-1gn2fl.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=189&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/404999/original/file-20210608-130350-1gn2fl.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=189&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Younger students need to completely shift the way they perceive a triangle if they want to learn about the relationship between the length of its sides and angles.</span>
<span class="attribution"><a class="source" href="https://commons.wikimedia.org/wiki/File:Triangle.GeometryArea.svg">Wikimedia Commons</a></span>
</figcaption>
</figure>
<p>Another criticism is that the new curriculum <a href="https://www.smh.com.au/education/sum-of-all-fears-why-australia-s-maths-problem-is-getting-worse-20210604-p57y2n.html">apparently delays linear equations</a> — such as x + 3 = 11 (find the value of x) — from year 7 to year 8. </p>
<p>But the <a href="https://www.australiancurriculum.edu.au/media/7046/mathematics_comparative_information_7-10.pdf">proposed curriculum</a> expects year 7 students to use “algebraic expressions to model situations and represent formulas. Students substitute values into these formulas to determine unknown values and interpret these in the context.” </p>
<p>So, rather than confining students to solving simple linear equations, the new curriculum wants students to consider more complex relationships between numbers. It expects them to understand these, rather than showing them the trivial act of solving simple equations first.</p>
<h2>It helps students build their understanding</h2>
<p>Three content strands in the <a href="https://www.australiancurriculum.edu.au/f-10-curriculum/mathematics/">current curriculum</a> (number and algebra, measurement and geometry, and statistics and probability) have become six (number, algebra, measurement, geometry, statistics, and probability).</p>
<p>Each of these strands appears in all grades, allowing students to build their understanding gradually. </p>
<p>In year 1, students build algebraic understanding by exploring number patters.</p>
<p>The <a href="https://www.australiancurriculum.edu.au/media/7047/mathematics_comparative_information_f-6.pdf">current year 1 curriculum</a> requires students to:</p>
<blockquote>
<p>Investigate and describe number patterns formed by skip-counting and patterns with objects.</p>
</blockquote>
<p>Skip-counting is counting forward by numbers other than one. For example 2, 4, 6, 8, 10 and so on.</p>
<p>The <a href="https://www.australiancurriculum.edu.au/media/7047/mathematics_comparative_information_f-6.pdf">proposed curriculum</a> adds more detail, requiring students to:</p>
<blockquote>
<p>recognise, describe, continue and create growing number patterns formed by skip-counting, initially by twos, fives and tens starting from zero. </p>
</blockquote>
<p>Growing patterns this way is a building block for times tables. In this case, the two-times table.</p>
<p>By year 4, students have progressed to dealing with more complex patterns and numbers, including those in multiplication tables. They are developing increasingly efficient mental strategies such as doubling and halving. Research shows <a href="http://amsi.org.au/wp-content/uploads/sites/15/2014/03/Module-1.pdf">these are effective approaches</a> for everyday computations. </p>
<hr>
<p>
<em>
<strong>
Read more:
<a href="https://theconversation.com/jump-split-or-make-to-the-next-10-strategies-to-teach-maths-have-changed-since-you-were-at-school-150262">Jump, split or make to the next 10: strategies to teach maths have changed since you were at school</a>
</strong>
</em>
</p>
<hr>
<p>Research also shows it is important to <a href="https://www.utas.edu.au/__data/assets/pdf_file/0003/1094475/BPME-Report.pdf">build a solid foundation</a> from the early school years, while building students’ confidence and success from grade to grade.</p>
<h2>The new curriculum sets higher standards</h2>
<p>The new maths curriculum actually sets higher standards for students. For example, compare the achievement standards for year 2 geometry:</p>
<p>The current achievement standards are that:</p>
<ul>
<li><p>students compare and order different shapes and objects using informal units (for example, measuring the length of a table using handspans or paperclips) </p></li>
<li><p>they use calendars to identify dates and seasons</p></li>
<li><p>they draw two-dimensional shapes and describe one-step transformations (such as rotating a shape or image or flipping it along a line). </p></li>
</ul>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/405000/original/file-20210608-17-ncy6x8.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="Wooden blocks." src="https://images.theconversation.com/files/405000/original/file-20210608-17-ncy6x8.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/405000/original/file-20210608-17-ncy6x8.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/405000/original/file-20210608-17-ncy6x8.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/405000/original/file-20210608-17-ncy6x8.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/405000/original/file-20210608-17-ncy6x8.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/405000/original/file-20210608-17-ncy6x8.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/405000/original/file-20210608-17-ncy6x8.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">Patterns in number sequences are the building blocks of learning times tables.</span>
<span class="attribution"><a class="source" href="https://www.shutterstock.com/image-photo/block-338482094">Shutterstock</a></span>
</figcaption>
</figure>
<p>Proposed achievement standards are that:</p>
<ul>
<li><p>students use consistent informal units repeatedly to compare different measurements of shapes and objects (that is, students need to use a single repeated unit, such as a paper clip, rather than mixing units, which children commonly do when learning to measure)</p></li>
<li><p>explain the effects of one-step transformations and compare shapes and objects describing features and properties using spatial terms (such as sides, angles, symmetry, location and direction) </p></li>
<li><p>identify relative positions, locate things on two-dimensional representations (flat shapes) and move within a space by giving and following directions and pathways (using slides, turns and flips).</p></li>
</ul>
<p>Opportunities for students to explain their reasoning using more complex language helps them to build connections between maths ideas and lays the foundations for deeper understanding. Listening to students provides guidance for teachers in planning their lessons.</p>
<h2>More training for teachers</h2>
<p>Neither the current nor proposed curriculum prescribes a particular approach to teaching. Teaching is not a one-size-fits-all activity. Teachers remain free to approach teaching maths in ways that suit their students, using a wide range of activities and resources. </p>
<p>In Australia, many schools struggle to attract qualified secondary maths teachers. Many teachers for whom maths is not a specialisation may fall back on the way they were taught. For instance, they could show students a few worked examples, followed by asking them to complete every second exercise. </p>
<p>Research shows <a href="https://www.utas.edu.au/__data/assets/pdf_file/0003/1094475/BPME-Report.pdf">schools successful in maths</a> focus on helping students develop a deep understanding of concepts, using a variety of teaching approaches. </p>
<p>Despite the adoption of a well-structured curriculum, Australia needs to develop a targeted strategy for increasing the number of qualified secondary mathematics teachers in our schools.</p>
<hr>
<p>
<em>
<strong>
Read more:
<a href="https://theconversation.com/1-in-4-australian-year-8s-have-teachers-unqualified-in-maths-this-hits-disadvantaged-schools-even-harder-161100">1 in 4 Australian year 8s have teachers unqualified in maths — this hits disadvantaged schools even harder</a>
</strong>
</em>
</p>
<hr>
<p>The implementation of the new curriculum will also require professional learning for teachers to understand the teaching implications of how students develop maths concepts. Research shows <a href="https://www.utas.edu.au/__data/assets/pdf_file/0003/1094475/BPME-Report.pdf">professional learning that is relevant</a> to teachers, and requires teachers to develop their teaching, results in improved maths outcomes. </p>
<p>The proposed maths curriculum has the potential to provide a bridge between teaching, learning and assessment that should, in time, lead to improved maths outcomes.</p><img src="https://counter.theconversation.com/content/162088/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Rosemary Callingham has received research funding from the Australian Research Council and the Australian Federal Government and the Tasmanian State Government in the past. She is not currently in receipt of any external funding. </span></em></p><p class="fine-print"><em><span>Penelope Baker does not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>The proposed maths curriculum would result in a deeper understanding of key concepts. It expects students to explain their maths reasoning rather than present their answer without justification.Penelope Baker, Professor, Mathematics Education, University of New EnglandRosemary Callingham, Adjunct Associate Professor, University of TasmaniaLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/1452442020-08-31T05:41:07Z2020-08-31T05:41:07ZIs mathematics real? A viral TikTok video raises a legitimate question with exciting answers<figure><img src="https://images.theconversation.com/files/355476/original/file-20200831-14-174eny.jpg?ixlib=rb-1.1.0&rect=10%2C10%2C3449%2C2456&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>While filming herself getting ready for work recently, TikTok user <a href="https://www.tiktok.com/@gracie.ham/video/6864198263063448837">@gracie.ham</a> reached deep into the ancient foundations of mathematics and found an absolute gem of a question: </p>
<blockquote>
<p>How could someone come up with a concept like algebra? </p>
</blockquote>
<p>She also asked what the ancient Greek philosopher Pythagoras might have used mathematics for, and other questions that revolve around the age-old conundrum of whether mathematics is “real” or something humans just made up.</p>
<p>Many responded negatively to the post, but others — including mathematicians like me — found the questions quite insightful.</p>
<h2>Is mathematics real?</h2>
<p>Philosophers and mathematicians have been <a href="https://en.wikipedia.org/wiki/Philosophy_of_mathematics#Mathematical_realism">arguing over this</a> for centuries. Some believe mathematics is universal; others consider it only as real as anything else humans have invented. </p>
<p>Thanks to @gracie.ham, Twitter users have now vigorously joined the debate. </p>
<p><div data-react-class="Tweet" data-react-props="{"tweetId":"1298372968838508546"}"></div></p>
<p><div data-react-class="Tweet" data-react-props="{"tweetId":"1299273758256115713"}"></div></p>
<p>For me, part of the answer lies in history.</p>
<p>From one perspective, mathematics is a universal language used to describe the world around us. For instance, two apples plus three apples is always five apples, regardless of your point of view. </p>
<p>But mathematics is also a language used by humans, so it is not independent of culture. History shows us that different cultures had their own understanding of mathematics.</p>
<p>Unfortunately, most of this ancient understanding is now lost. In just about every ancient culture, a few scattered texts are all that remain of their scientific knowledge.</p>
<p>However, there is one ancient culture that left behind an absolute abundance of texts.</p>
<h2>Babylonian algebra</h2>
<p>Buried in the deserts of modern Iraq, clay tablets from ancient Babylon have survived intact for about 4,000 years. </p>
<p>These tablets are slowly being translated and what we have learned so far is that the Babylonians were practical people who were highly numerate and knew how to solve sophisticated problems with numbers. </p>
<p>Their arithmetic was different from ours, though. They didn’t use zero or negative numbers. They even mapped out the motion of the planets without using calculus as we do. </p>
<hr>
<p>
<em>
<strong>
Read more:
<a href="https://theconversation.com/written-in-stone-the-worlds-first-trigonometry-revealed-in-an-ancient-babylonian-tablet-81472">Written in stone: the world's first trigonometry revealed in an ancient Babylonian tablet</a>
</strong>
</em>
</p>
<hr>
<p>Of particular importance for @gracie.ham’s question about the origins of algebra is that they knew that the numbers 3, 4 and 5 correspond to the lengths of the sides and diagonal of a rectangle. They also knew these numbers satisfied the fundamental relation 3² + 4² = 5² that ensures the sides are perpendicular.</p>
<figure class="align-right zoomable">
<a href="https://images.theconversation.com/files/355413/original/file-20200830-14-fbzxlx.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/355413/original/file-20200830-14-fbzxlx.png?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/355413/original/file-20200830-14-fbzxlx.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=450&fit=crop&dpr=1 600w, https://images.theconversation.com/files/355413/original/file-20200830-14-fbzxlx.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=450&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/355413/original/file-20200830-14-fbzxlx.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=450&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/355413/original/file-20200830-14-fbzxlx.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=566&fit=crop&dpr=1 754w, https://images.theconversation.com/files/355413/original/file-20200830-14-fbzxlx.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=566&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/355413/original/file-20200830-14-fbzxlx.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=566&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">No theorems were harmed (or used) in the construction of this rectangle.</span>
</figcaption>
</figure>
<p>The Babylonians did all this without modern algebraic concepts. We would express a more general version of the same idea using Pythagoras’ theorem: any right-angled triangle with sides of length <em>a</em> and <em>b</em> and hypotenuse <em>c</em> satisfies <em>a</em>² + <em>b</em>² = <em>c</em>². </p>
<p>The Babylonian perspective omits algebraic variables, theorems, axioms and proofs not because they were ignorant but because these ideas had not yet developed. In short, these social constructs began more than 1,000 years later, in ancient Greece. The Babylonians happily and productively did mathematics and solved problems without any of these relatively modern notions.</p>
<h2>What was it all for?</h2>
<p>@gracie.ham also asks how Pythagoras came up with his theorem. The short answer is: he didn’t.</p>
<p>Pythagoras of Samos (c. 570-495 BC) probably heard about the idea we now associate with his name while he was in Egypt. He may have been the person to introduce it to Greece, but we don’t really know.</p>
<p>Pythagoras didn’t use his theorem for anything practical. He was primarily interested in numerology and the mysticism of numbers, rather than the applications of mathematics.</p>
<hr>
<p>
<em>
<strong>
Read more:
<a href="https://theconversation.com/curious-kids-how-was-maths-discovered-who-made-up-the-numbers-and-rules-121509">Curious Kids: how was maths discovered? Who made up the numbers and rules?</a>
</strong>
</em>
</p>
<hr>
<p>The Babylonians, on the other hand, may well have used their knowledge of right triangles for more concrete purposes, although we don’t really know. We do have evidence from ancient India and Rome showing the dimensions 3-4-5 were used as a simple but effective way to create right angles in the construction of religious altars and surveying.</p>
<p>Without modern tools, how do you make right angles <em>just right</em>? Ancient Hindu religious texts give instructions for making a rectangular fire altar using the 3-4-5 configuration with sides of length 3 and 4, and diagonal length 5. These measurements ensure that the altar has right angles in each corner.</p>
<figure class="align-center ">
<img alt="A man sits at a fire altar" src="https://images.theconversation.com/files/355415/original/file-20200830-20-1fbfao.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/355415/original/file-20200830-20-1fbfao.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=450&fit=crop&dpr=1 600w, https://images.theconversation.com/files/355415/original/file-20200830-20-1fbfao.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=450&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/355415/original/file-20200830-20-1fbfao.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=450&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/355415/original/file-20200830-20-1fbfao.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=566&fit=crop&dpr=1 754w, https://images.theconversation.com/files/355415/original/file-20200830-20-1fbfao.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=566&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/355415/original/file-20200830-20-1fbfao.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">
<figcaption>
<span class="caption">A rectangular fire altar.</span>
<span class="attribution"><a class="source" href="https://en.wikipedia.org/wiki/Vedi_(altar)#/media/File:Homa_during_Sri_Thimmaraya_swamy_Pratishthapana..jpg">Madhu K / Wikipedia</a>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span>
</figcaption>
</figure>
<h2>Big questions</h2>
<p>In the 19th century, the German mathematician <a href="https://en.wikipedia.org/wiki/Leopold_Kronecker">Leopold Kronecker</a> said “God made the integers, all else is the work of man”. I agree with that sentiment, at least for the positive integers — the whole numbers we count with — because the Babylonians didn’t believe in zero or negative numbers.</p>
<p>Mathematics has been happening for a very, very long time. Long before ancient Greece and Pythagoras. </p>
<p>Is it real? Most cultures agree about some basics, like the positive integers and the 3-4-5 right triangle. Just about everything else in mathematics is determined by the society in which you live.</p><img src="https://counter.theconversation.com/content/145244/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Daniel Mansfield does not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>What did Pythagoras do with all those triangles, anyway?Daniel Mansfield, Lecturer in Mathematics, UNSW SydneyLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/958962018-07-16T10:39:19Z2018-07-16T10:39:19ZWhy I teach math through knitting<figure><img src="https://images.theconversation.com/files/225197/original/file-20180627-112628-1tr48e8.jpg?ixlib=rb-1.1.0&rect=0%2C145%2C752%2C598&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Math in yarn.</span> <span class="attribution"><span class="source">Carthage College</span>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span></figcaption></figure><p>One snowy January day, I asked a classroom of college students to tell me the first word that came to mind when they thought about mathematics. The top two words were “calculation” and “equation.” </p>
<p>When I asked a room of professional mathematicians the same question, neither of those words were mentioned; instead, they offered phrases like “critical thinking” and “problem-solving.”</p>
<p>This is unfortunately common. What professional mathematicians think of as mathematics is entirely different from what the general population thinks of as mathematics. When so many describe mathematics as synonymous with calculation, it’s no wonder we hear “I hate math” so often. </p>
<p>So I set out to solve this problem in a somewhat unconventional way. I decided to offer a class called “The Mathematics of Knitting” at my institution, Carthage College. In it, I chose to eliminate pencil, paper, calculator (gasp) and textbook from the classroom completely. Instead, we talked, used our hands, drew pictures and played with everything from beach balls to measuring tapes. For homework, we reflected by blogging. And of course, we knit.</p>
<h2>Same but different</h2>
<p>One crux of mathematical content is the equation, and crucial to this is the equal sign. An equation like x = 5 tells us that the dreaded x, which represents some quantity, has the same value as 5. The number 5 and the value of x must be exactly the same. </p>
<p>A typical equal sign is very strict. Any small deviation from “exactly” means that two things are not equal. However, there are many times in life where two quantities are not exactly the same, but are essentially the same by some meaningful criteria.</p>
<p>Imagine, for example, that you have two square pillows. The first is red on top, yellow on the right, green on bottom and blue on the left. The second is yellow on the top, green on the right, blue on bottom, and red on the left.</p>
<p>The pillows aren’t exactly the same. One has a red top, while one has a yellow top. But they’re certainly similar. In fact, they would be exactly the same if you turned the pillow with the red top once counterclockwise.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/224747/original/file-20180625-19396-1ychj4o.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/224747/original/file-20180625-19396-1ychj4o.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/224747/original/file-20180625-19396-1ychj4o.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=281&fit=crop&dpr=1 600w, https://images.theconversation.com/files/224747/original/file-20180625-19396-1ychj4o.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=281&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/224747/original/file-20180625-19396-1ychj4o.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=281&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/224747/original/file-20180625-19396-1ychj4o.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=353&fit=crop&dpr=1 754w, https://images.theconversation.com/files/224747/original/file-20180625-19396-1ychj4o.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=353&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/224747/original/file-20180625-19396-1ychj4o.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=353&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Rotating two square pillows.</span>
<span class="attribution"><span class="source">Sara Jensen</span></span>
</figcaption>
</figure>
<p>How many different ways could I put the same pillow down on a bed, but make it look like a different one? A little homework shows there are 24 possible colored throw pillow configurations, though only eight of them can be obtained from moving a given pillow. </p>
<p>Students demonstrated this by knitting throw pillows, consisting of two colors, from knitting charts.</p>
<figure class="align-right zoomable">
<a href="https://images.theconversation.com/files/224748/original/file-20180625-19390-jal506.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/224748/original/file-20180625-19390-jal506.png?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/224748/original/file-20180625-19390-jal506.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=271&fit=crop&dpr=1 600w, https://images.theconversation.com/files/224748/original/file-20180625-19390-jal506.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=271&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/224748/original/file-20180625-19390-jal506.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=271&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/224748/original/file-20180625-19390-jal506.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=341&fit=crop&dpr=1 754w, https://images.theconversation.com/files/224748/original/file-20180625-19390-jal506.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=341&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/224748/original/file-20180625-19390-jal506.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=341&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">A knitting chart for a throw pillow.</span>
<span class="attribution"><span class="source">Sara Jensen</span></span>
</figcaption>
</figure>
<p>The students created square knitting charts where all eight motions of the chart resulted in a different-looking picture. These were then knit into a throw pillow where the equivalence of the pictures could be demonstrated by actually moving the pillow.</p>
<h2>Rubber sheet geometry</h2>
<p>Another topic we covered is a subject sometimes referred to as “rubber sheet geometry.” The idea is to imagine the whole world is made of rubber, then reimagine what shapes would look like. </p>
<p>Let’s try to understand the concept with knitting. One way of knitting objects that are round – like hats or gloves – is with special knitting needles called double pointed needles. While being made, the hat is shaped by three needles, making it look triangular. Then, once it comes off the needles, the stretchy yarn relaxes into a circle, making a much more typical hat. </p>
<figure class="align-right ">
<img alt="" src="https://images.theconversation.com/files/225196/original/file-20180627-112601-d7vuv0.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/225196/original/file-20180627-112601-d7vuv0.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=899&fit=crop&dpr=1 600w, https://images.theconversation.com/files/225196/original/file-20180627-112601-d7vuv0.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=899&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/225196/original/file-20180627-112601-d7vuv0.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=899&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/225196/original/file-20180627-112601-d7vuv0.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=1129&fit=crop&dpr=1 754w, https://images.theconversation.com/files/225196/original/file-20180627-112601-d7vuv0.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=1129&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/225196/original/file-20180627-112601-d7vuv0.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=1129&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Knitting to learn.</span>
<span class="attribution"><span class="source">Carthage College</span>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span>
</figcaption>
</figure>
<p>This is the concept that “rubber sheet geometry” is trying to capture. Somehow, a triangle and a circle can be the same if they’re made out of a flexible material. In fact, all polygons become circles in this field of study. </p>
<p>If all polygons are circles, then what shapes are left? There are a few traits that are distinguishable even when objects are flexible – for example, if a shape has edges or no edges, holes or no holes, twists or no twists. </p>
<p>One example from knitting of something that is not equivalent to a circle is an infinity scarf. If you want to make a paper infinity scarf at home, take a long strip of paper and glue the short edges together by attaching the top left corner to the bottom right corner, and the bottom left corner to the top right corner. Then draw arrows pointing up the whole way around the object. Something cool should happen. </p>
<p>Students in the course spent some time knitting objects, like infinity scarves and headbands, that were different even when made out of flexible material. Adding markings like arrows helped visualize exactly how the objects were different. </p>
<h2>Different flavors</h2>
<figure class="align-left zoomable">
<a href="https://images.theconversation.com/files/224776/original/file-20180625-19399-y5ti0b.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/224776/original/file-20180625-19399-y5ti0b.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/224776/original/file-20180625-19399-y5ti0b.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=900&fit=crop&dpr=1 600w, https://images.theconversation.com/files/224776/original/file-20180625-19399-y5ti0b.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=900&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/224776/original/file-20180625-19399-y5ti0b.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=900&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/224776/original/file-20180625-19399-y5ti0b.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=1131&fit=crop&dpr=1 754w, https://images.theconversation.com/files/224776/original/file-20180625-19399-y5ti0b.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=1131&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/224776/original/file-20180625-19399-y5ti0b.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>
<figcaption>
<span class="caption">An infinity scarf.</span>
<span class="attribution"><span class="source">Carthage College</span></span>
</figcaption>
</figure>
<p>If the things described in this article don’t sound like math to you, I want to reinforce that they very much are. The subjects discussed here – abstract algebra and topology – are typically reserved for math majors in their junior and senior years of college. Yet the philosophies of these subjects are very accessible, given the right mediums. </p>
<p>In my view, there’s no reason these different flavors of math should be hidden from the public or emphasized less than conventional mathematics. Further, <a href="https://files.eric.ed.gov/fulltext/ED321967.pdf">studies have shown</a> that using materials that can be physically manipulated can improve mathematical learning at all levels of study. </p>
<p>If more mathematicians were able to set aside classical techniques, it seems possible the world could overcome the prevailing misconception that computation is the same as mathematics. And just maybe, a few more people out there could embrace mathematical thought; if not figuratively, then literally, with a throw pillow.</p><img src="https://counter.theconversation.com/content/95896/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Sara Jensen works for Carthage College. She is a member of the Mathematical Association of America, and is a Project NExT red dot ('15).</span></em></p>In this professor’s class, there are no calculators. Instead, students learn advanced math by talking, drawing pictures, playing with beach balls – and knitting.Sara Jensen, Assistant Professor of Mathematics, Carthage CollegeLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/843322017-09-21T11:02:15Z2017-09-21T11:02:15ZFive ways ancient India changed the world – with maths<figure><img src="https://images.theconversation.com/files/186896/original/file-20170920-16437-hxdak9.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Bakhshali manuscript.</span> <span class="attribution"><span class="source">Bodleian Libraries, University of Oxford</span></span></figcaption></figure><p>It should come as no surprise that the first recorded use of the number zero, <a href="https://www.theguardian.com/science/2017/sep/14/much-ado-about-nothing-ancient-indian-text-contains-earliest-zero-symbol">recently discovered</a> to be made as early as the 3rd or 4th century, happened in India. Mathematics on the Indian subcontinent has a rich history <a href="http://www.tifr.res.in/%7Earchaeo/FOP/FoP%20papers/ancmathsources_Dani.pdf">going back over 3,000 years</a> and thrived for centuries before similar advances were made in Europe, with its influence meanwhile spreading to China and the Middle East.</p>
<p>As well as giving us the concept of zero, Indian mathematicians made seminal contributions to the study of <a href="http://www-history.mcs.st-andrews.ac.uk/HistTopics/Indian_mathematics.html">trigonometry, algebra, arithmetic and negative numbers among other areas</a>. Perhaps most significantly, the decimal system that we still employ worldwide today was first seen in India.</p>
<h2>The number system</h2>
<p>As far back as 1200 BC, mathematical knowledge was being written down as part of a large body of knowledge known as <a href="https://www.ancient.eu/The_Vedas/">the Vedas</a>. In these texts, numbers were commonly expressed as <a href="http://www.thehindu.com/sci-tech/science/understanding-ancient-indian-mathematics/article2747006.ece">combinations of powers of ten</a>. For example, 365 might be expressed as three hundreds (3x10²), six tens (6x10¹) and five units (5x10⁰), though each power of ten was represented with a name rather than a set of symbols. It is <a href="http://www.tifr.res.in/%7Earchaeo/FOP/FoP%20papers/ancmathsources_Dani.pdf">reasonable to believe</a> that this representation using powers of ten played a crucial role in the development of the decimal-place value system in India.</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/186842/original/file-20170920-16414-6jy46n.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/186842/original/file-20170920-16414-6jy46n.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=126&fit=crop&dpr=1 600w, https://images.theconversation.com/files/186842/original/file-20170920-16414-6jy46n.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=126&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/186842/original/file-20170920-16414-6jy46n.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=126&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/186842/original/file-20170920-16414-6jy46n.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=158&fit=crop&dpr=1 754w, https://images.theconversation.com/files/186842/original/file-20170920-16414-6jy46n.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=158&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/186842/original/file-20170920-16414-6jy46n.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=158&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Brahmi numerals.</span>
<span class="attribution"><a class="source" href="https://en.wikipedia.org/wiki/Brahmi_numerals#/media/File:Indian_numerals_100AD.svg">Wikimedia</a></span>
</figcaption>
</figure>
<p>From the <a href="http://www-history.mcs.st-andrews.ac.uk/HistTopics/Indian_mathematics.html">third century BC</a>, we also have written evidence of the <a href="http://www-history.mcs.st-andrews.ac.uk/HistTopics/Indian_numerals.html">Brahmi numerals</a>, the precursors to the modern, Indian or Hindu-Arabic numeral system that most of the world uses today. Once zero was introduced, almost all of the mathematical mechanics would be in place to enable ancient Indians to study higher mathematics. </p>
<h2>The concept of zero</h2>
<p>Zero itself has a much longer history. The <a href="http://www.bodleian.ox.ac.uk/news/2017/sep-14">recently dated first recorded zeros</a>, in what is known as the Bakhshali manuscript, were simple placeholders – a tool to distinguish 100 from 10. Similar marks had already been seen in the <a href="https://www.scientificamerican.com/article/history-of-zero/">Babylonian and Mayan cultures in the early centuries AD</a> and arguably in <a href="https://www.scientificamerican.com/article/history-of-zero/">Sumerian mathematics as early as 3000-2000 BC</a>.</p>
<p>But only in India did the placeholder symbol for nothing progress to become a <a href="https://theconversation.com/nothing-matters-how-the-invention-of-zero-helped-create-modern-mathematics-84232">number in its own right</a>. The advent of the concept of zero allowed numbers to be written efficiently and reliably. In turn, this allowed for effective record-keeping that meant important financial calculations could be checked retroactively, ensuring the honest actions of all involved. Zero was a significant step on the route to the <a href="https://blog.sciencemuseum.org.uk/illuminating-india-starring-oldest-recorded-origins-zero-bakhshali-manuscript/">democratisation of mathematics</a>.</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/187001/original/file-20170921-8179-14x8av.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/187001/original/file-20170921-8179-14x8av.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/187001/original/file-20170921-8179-14x8av.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/187001/original/file-20170921-8179-14x8av.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/187001/original/file-20170921-8179-14x8av.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/187001/original/file-20170921-8179-14x8av.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/187001/original/file-20170921-8179-14x8av.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">No abacus needed.</span>
<span class="attribution"><span class="source">Shutterstock</span></span>
</figcaption>
</figure>
<p>These accessible mechanical tools for working with mathematical concepts, in combination with a strong and open scholastic and scientific culture, meant that, by around 600AD, all the ingredients were in place for an explosion of mathematical discoveries in India. In comparison, these sorts of tools were not popularised in the West until the early 13th century, though <a href="http://www.springer.com/gb/book/9780387407371">Fibonnacci’s book liber abaci</a>. </p>
<h2>Solutions of quadratic equations</h2>
<p>In the seventh century, the first written evidence of the rules for working with zero were formalised in the <a href="https://archive.org/details/Brahmasphutasiddhanta_Vol_1">Brahmasputha Siddhanta</a>. In his seminal text, the astronomer <a href="http://www.storyofmathematics.com/indian_brahmagupta.html">Brahmagupta</a> introduced rules for solving quadratic equations (so beloved of secondary school mathematics students) and for computing square roots.</p>
<h2>Rules for negative numbers</h2>
<p>Brahmagupta also demonstrated rules for working with negative numbers. He referred to <a href="https://nrich.maths.org/5961">positive numbers as fortunes and negative numbers as debts</a>. He wrote down rules that have been interpreted by translators as: “A fortune subtracted from zero is a debt,” and “a debt subtracted from zero is a fortune”.</p>
<p>This latter statement is the same as the rule we learn in school, that if you subtract a negative number, it is the same as adding a positive number. Brahmagupta also knew that “The product of a debt and a fortune is a debt” – a positive number multiplied by a negative is a negative. </p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/187008/original/file-20170921-8211-139ku7u.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/187008/original/file-20170921-8211-139ku7u.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/187008/original/file-20170921-8211-139ku7u.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/187008/original/file-20170921-8211-139ku7u.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/187008/original/file-20170921-8211-139ku7u.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/187008/original/file-20170921-8211-139ku7u.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/187008/original/file-20170921-8211-139ku7u.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">Negative cows.</span>
<span class="attribution"><span class="source">Shutterstock</span></span>
</figcaption>
</figure>
<p>For the large part, European mathematicians were reluctant to accept negative numbers as meaningful. Many took the view that <a href="https://books.google.co.uk/books?id=STKX4qadFTkC&pg=PA56&redir_esc=y#v=onepage&q&f=false">negative numbers were absurd</a>. They reasoned that numbers were developed for counting and questioned what you could count with negative numbers. Indian and Chinese mathematicians recognised early on that one answer to this question was debts.</p>
<p>For example, in a primitive farming context, if one farmer owes another farmer 7 cows, then effectively the first farmer has -7 cows. If the first farmer goes out to buy some animals to repay his debt, he has to buy 7 cows and give them to the second farmer in order to bring his cow tally back to 0. From then on, every cow he buys goes to his positive total. </p>
<h2>Basis for calculus</h2>
<p>This reluctance to adopt negative numbers, and indeed zero, held European mathematics back for many years. Gottfried Wilhelm Leibniz was one of the first Europeans to use zero and the negatives in a systematic way in his <a href="https://books.google.co.uk/books?id=CXG6CgAAQBAJ&pg=PA165&lpg=PA165&dq=Leibniz+zero+negatives+calculus&source=bl&ots=NsKOzdZL7Y&sig=dE2KJvCXPFovF4uyFdgHMJOAQr8&hl=en&sa=X&ved=0ahUKEwjdxKv8_LPWAhXhAcAKHR0XBcUQ6AEIMjAC#v=onepage&q=Leibniz%20zero%20negatives%20calculus&f=false">development of calculus</a> in the late 17th century. Calculus is used to measure rates of changes and is important in almost every branch of science, notably underpinning many key discoveries in modern physics.</p>
<figure class="align-right ">
<img alt="" src="https://images.theconversation.com/files/187017/original/file-20170921-8194-1ypsmdq.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/187017/original/file-20170921-8194-1ypsmdq.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=759&fit=crop&dpr=1 600w, https://images.theconversation.com/files/187017/original/file-20170921-8194-1ypsmdq.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=759&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/187017/original/file-20170921-8194-1ypsmdq.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=759&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/187017/original/file-20170921-8194-1ypsmdq.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=954&fit=crop&dpr=1 754w, https://images.theconversation.com/files/187017/original/file-20170921-8194-1ypsmdq.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=954&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/187017/original/file-20170921-8194-1ypsmdq.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=954&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Leibniz: Beaten to it by 500 years.</span>
</figcaption>
</figure>
<p>But <a href="http://www-groups.dcs.st-and.ac.uk/history/Biographies/Bhaskara_II.html">Indian mathematician Bhāskara</a> had already discovered many of Leibniz’s ideas <a href="https://ijrier.com/published-papers/volume-1/issue-8/origin-of-concept-of-calculus-in-india.pdf">over 500 years earlier</a>. Bhāskara, also made major contributions to algebra, arithmetic, geometry and trigonometry. He provided many results, for example on the solutions of certain “Doiphantine” equations, that <a href="https://www.amazon.co.uk/Mathematical-Achievements-Pre-modern-Mathematicians-Elsevier/dp/0123979137#reader_0123979137">would not be rediscovered in Europe for centuries</a>.</p>
<p><a href="https://link.springer.com/referenceworkentry/10.1007%2F978-1-4020-4425-0_8683">The Kerala school of astronomy and mathematics</a>, founded by <a href="https://en.wikipedia.org/wiki/Madhava_of_Sangamagrama">Madhava of Sangamagrama</a> in the 1300s, was responsible for many firsts in mathematics, including the use of mathematical induction and some early calculus-related results. Although no systematic rules for calculus were developed by the Kerala school, its proponents first conceived of many of the results that would <a href="http://www.jstor.org/stable/1558972?origin=crossref&seq=1#page_scan_tab_contents">later be repeated in Europe</a> including Taylor series expansions, infinitessimals and differentiation. </p>
<p>The leap, made in India, that transformed zero from a simple placeholder to a number in its own right indicates the mathematically enlightened culture that was flourishing on the subcontinent at a time when Europe was stuck in the dark ages. Although its reputation <a href="http://www.cbc.ca/news/technology/calculus-created-in-india-250-years-before-newton-study-1.632433">suffers from the Eurocentric bias</a>, the subcontinent has a strong mathematical heritage, which it continues into the 21st century by <a href="http://m.ranker.com/list/famous-mathematicians-from-india/reference?page=1">providing key players at the forefront of every branch of mathematics</a>.</p><img src="https://counter.theconversation.com/content/84332/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Christian Yates 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>High school students can blame ancient India for quadratic equations and calculus.Christian Yates, Senior Lecturer in Mathematical Biology, University of BathLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/842322017-09-20T14:14:28Z2017-09-20T14:14:28ZNothing matters: how the invention of zero helped create modern mathematics<figure><img src="https://images.theconversation.com/files/186837/original/file-20170920-16403-yazsqf.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>A small dot on an old piece of birch bark marks one of the biggest events in the history of mathematics. The bark is actually part of an ancient Indian mathematical document known as the Bakhshali manuscript. And the dot is the first known recorded use of the number zero. What’s more, researchers from the University of Oxford <a href="https://www.theguardian.com/science/2017/sep/14/much-ado-about-nothing-ancient-indian-text-contains-earliest-zero-symbol">recently discovered</a> the document is 500 years older than was previously estimated, dating to the third or fourth century – a breakthrough discovery.</p>
<p>Today, it’s difficult to imagine how you could have mathematics without zero. In a positional number system, such as the decimal system we use now, the location of a digit is really important. Indeed, the real difference between 100 and 1,000,000 is where the digit 1 is located, with the symbol 0 serving as a punctuation mark.</p>
<p>Yet for thousands of years we did without it. The Sumerians of 5,000BC employed a <a href="http://www.storyofmathematics.com/sumerian.html">positional system</a> but without a 0. In some <a href="https://www.scientificamerican.com/article/what-is-the-origin-of-zer/">rudimentary form</a>, a symbol or a space was used to distinguish between, for example, 204 and 20000004. But that symbol was never used at the end of a number, so the difference between 5 and 500 had to be determined by context.</p>
<p>What’s more, 0 at the end of a number makes multiplying and dividing by 10 easy, as it does with adding numbers like 9 and 1 together. The invention of zero immensely simplified computations, freeing mathematicians to develop vital mathematical disciplines such as algebra and calculus, and eventually the basis for computers. </p>
<figure>
<iframe width="440" height="260" src="https://www.youtube.com/embed/pV_gXGTuWxY?wmode=transparent&start=0" frameborder="0" allowfullscreen=""></iframe>
</figure>
<p>Zero’s late arrival was partly a reflection of the negative views some cultures held for the concept of nothing. Western philosophy is plagued with grave misconceptions about nothingness and the mystical powers of language. The fifth century BC <a href="https://plato.stanford.edu/entries/parmenides/">Greek thinker Parmenides</a> proclaimed that nothing cannot exist, since to speak of something is to speak of something that exists. This Parmenidean approach kept prominent <a href="http://www.nothing.com/Heath.html">historical figures</a> busy for a long while. </p>
<p>After the advent of Christianity, religious leaders in Europe argued that since God is in everything that exists, anything that represents nothing must be satanic. In an attempt to save humanity from the devil, they <a href="http://yaleglobal.yale.edu/history-zero">promptly banished</a> zero from existence, though merchants continued secretly to use it.</p>
<p>By contrast, in Buddhism the concept of nothingness is not only devoid of any demonic possessions but is actually a central idea worthy of much study en route to nirvana. <a href="http://www.huffingtonpost.com/lewis-richmond/emptiness-most-misunderstood-word-in-buddhism_b_2769189.html">With such a mindset</a>, having a mathematical representation for nothing was, well, nothing to fret over. In fact, the English word “zero” is <a href="http://www.etymonline.com/index.php?term=zero">originally derived</a> from the Hindi “sunyata”, which means nothingness and is a central concept in Buddhism.</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/186836/original/file-20170920-16445-1yaf95v.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/186836/original/file-20170920-16445-1yaf95v.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=299&fit=crop&dpr=1 600w, https://images.theconversation.com/files/186836/original/file-20170920-16445-1yaf95v.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=299&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/186836/original/file-20170920-16445-1yaf95v.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=299&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/186836/original/file-20170920-16445-1yaf95v.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=376&fit=crop&dpr=1 754w, https://images.theconversation.com/files/186836/original/file-20170920-16445-1yaf95v.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=376&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/186836/original/file-20170920-16445-1yaf95v.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=376&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">The Bakhshali manuscript.</span>
<span class="attribution"><span class="source">Bodleian Libraries</span></span>
</figcaption>
</figure>
<p>So after zero finally emerged in ancient India, it took almost 1,000 years to set root in Europe, much longer than in China or the Middle East. In 1200 AD, the Italian mathematician Fibonacci, who brought the decimal system to Europe, <a href="http://www.springer.com/gb/book/9780387407371">wrote that</a>: </p>
<blockquote>
<p>The method of the Indians surpasses any known method to compute. It’s a marvellous method. They do their computations using nine figures and the symbol zero.</p>
</blockquote>
<p>This superior method of computation, clearly reminiscent of our modern one, freed mathematicians from tediously simple calculations, and enabled them to tackle more complicated problems and study the general properties of numbers. For example, it led to the work of the seventh century Indian <a href="http://www.storyofmathematics.com/indian_brahmagupta.html">mathematician and astronomer Brahmagupta</a>, considered to be the beginning of modern algebra.</p>
<h2>Algorithms and calculus</h2>
<p>The Indian method is so powerful because it means you can draw up simple rules for doing calculations. Just imagine trying to explain long addition without a symbol for zero. There would be too many exceptions to any rule. The ninth century Persian mathematician Al-Khwarizmi was the first to meticulously note and exploit these arithmetic instructions, which <a href="https://books.google.co.uk/books?id=zTQrDwAAQBAJ&pg=PA47&lpg=PA47&dq=al+khwarizmi+abacus&source=bl&ots=PakFxbCVwk&sig=FWjwHlnppHAU9i_zgAficOcw4ug&hl=en&sa=X&ved=0ahUKEwii-46257PWAhUhBcAKHaWtCRcQ6AEIajAP#v=onepage&q=al%20khwarizmi%20abacus&f=false">would eventually</a> make the abacus obsolete.</p>
<p>Such mechanical sets of instructions illustrated that portions of mathematics could be automated. And this would eventually lead to the development of modern computers. In fact, the word “algorithm” to describe a set of simple instructions <a href="https://en.oxforddictionaries.com/definition/algorithm">is derived</a> from the name “Al-Khwarizmi”.</p>
<p>The invention of zero also created a new, more accurate way to describe fractions. Adding zeros at the end of a number increases its magnitude, with the help of a decimal point, adding zeros at the beginning decreases its magnitude. Placing infinitely many digits to the right of the decimal point corresponds to <a href="https://www.youtube.com/watch?v=JmyLeESQWGw&list=PLYoCqdGsxmn9HOU84Ln2PhPKpxRfaEO9h&index=17">infinite precision</a>. That kind of precision was exactly what 17th century thinkers Isaac Newton and Gottfried Leibniz needed to develop calculus, the study of continuous change.</p>
<p>And so algebra, algorithms, and calculus, three pillars of modern mathematics, are all the result of a notation for nothing. Mathematics is a science of invisible entities that we can only understand by writing them down. India, by adding zero to the positional number system, unleashed the true power of numbers, advancing mathematics from infancy to adolescence, and from rudimentary toward its current sophistication.</p><img src="https://counter.theconversation.com/content/84232/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Ittay Weiss 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>Turning zero from a punctuation mark into a number paved the way for everything from algebra to algorithms.Ittay Weiss, Teaching Fellow, Department of Mathematics, University of PortsmouthLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/771292017-06-02T02:52:29Z2017-06-02T02:52:29ZHow math education can catch up to the 21st century<figure><img src="https://images.theconversation.com/files/171036/original/file-20170525-23232-hokg03.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">A student in Cape Coast solves a math problem.</span> <span class="attribution"><a class="source" href="https://www.flickr.com/photos/worldbank/5321246556/in/photolist-97dMdh-8rMuqU-8rMtNd-7tub8y-97aGaD-8rJogv-fAuDrE-fAfmgK-eokjMs-8rJoSr-fAuC9W-fAfkp2-8rJo7R-2HGirM-fAuBG5-fAuDPW-8rJnn8-fAfjkT-8rJp4i-fAfkv4-fAfkAp-fAfmUr-fAfmPB-fAuDJY-8rJp7V-5ua7bz-fAuCwJ-fAuBM1-5dsFUh-fAuDfj-8rMufd-8cwJHq-fAfjZi-5domkt-8rJoNM-8q1J3r-fAfmzM-5dom5D-5cWBvd-5domaD-8rJoZi-8rMu6s-fAfm4t-8rMuc3-aYTnYM-aRAYUk-8rMuGq-fAfkMr-fAuCZ7-7tq9jP">World Bank/flickr</a>, <a class="license" href="http://creativecommons.org/licenses/by-nc-nd/4.0/">CC BY-NC-ND</a></span></figcaption></figure><p>In 1939, the fictional professor J. Abner Pediwell published a curious book called “<a href="https://www.amazon.com/Saber-Tooth-Curriculum-Classic-Abner-Peddiwell-ebook/dp/B00G6DSY7Q">The Saber-Tooth Curriculum</a>.”</p>
<p>Through a series of satirical lectures, Pediwell (or the actual author, education professor <a href="http://education.stateuniversity.com/pages/1783/Benjamin-H-R-W-1893-1969.html">Harold R. W. Benjamin</a>) describes a Paleolithic curriculum that includes lessons in grabbing fish with your bare hands and scaring saber-toothed tigers with fire. Even after the invention of fishnets proves to be a far superior method of catching fish, teachers continued teaching the bare-hands method, claiming that it helps students develop “generalized agility.” </p>
<p>Pedwill showed how curricula can become entrenched and ritualistic, failing to respond to changes in the world around it. In math education, the problem is not quite so dire – but it’s time to start breaking a few of our own traditions. There’s a growing interest in emphasizing problem-solving and understanding concepts over skills and procedures. While memorized skills and procedures are useful, knowing the underlying meanings and understanding how they work builds problem-solving skills so that students may go beyond solving the standard book chapter problem. </p>
<p>As education researchers, we see two different ways that educators can build alternative mathematics courses. These updated courses work better for all students by changing what they teach and how they teach it.</p>
<h1>New paths in math</h1>
<p>In math, the usual curricular pathway – or sequence of courses – starts with algebra in eighth or ninth grade. This is followed by geometry, second-year algebra and trigonometry, all the way up to calculus and differential equations in college. </p>
<p>This pathway still serves science, technology, engineering and mathematics (STEM) majors reasonably well. However, some educators are now concerned about students who may have other career goals or interests. These students are stuck on largely the same path, but many end up terminating their mathematics studies at an earlier point along the way. </p>
<p>In fact, students who struggle early with the traditional singular STEM pathway are more likely to fall out of the higher education pipeline entirely. Many institutions have identified <a href="http://www.npr.org/sections/ed/2014/10/09/354645977/who-needs-algebra">college algebra courses</a> as a key roadblock leading to students dropping out of college altogether. </p>
<p>Another issue is that there is a <a href="http://www.cnbc.com/2015/06/15/math-science-skills-add-up-to-more-job-opportunities-survey.html">growing need</a> for new quantitative skills and reasoning in a wide variety of careers – not just STEM careers. In the 21st century, workers across many fields need to know how to deal effectively with data (statistical reasoning), detect trends and patterns in huge amounts of information (“big data”), use computers to solve problems (computational thinking) and make predictions about the relationships between different components of a system (mathematical modeling). </p>
<figure class="align-left ">
<img alt="" src="https://images.theconversation.com/files/171035/original/file-20170525-23230-15qjovd.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/171035/original/file-20170525-23230-15qjovd.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=450&fit=crop&dpr=1 600w, https://images.theconversation.com/files/171035/original/file-20170525-23230-15qjovd.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=450&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/171035/original/file-20170525-23230-15qjovd.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=450&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/171035/original/file-20170525-23230-15qjovd.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=566&fit=crop&dpr=1 754w, https://images.theconversation.com/files/171035/original/file-20170525-23230-15qjovd.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=566&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/171035/original/file-20170525-23230-15qjovd.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">
<figcaption>
<span class="caption">New technology offers unprecedented mathematical capabilities.</span>
<span class="attribution"><a class="source" href="https://www.flickr.com/photos/kosheahan/4009828196/in/photolist-77kqdU-4HZ3fV-4J4cuJ-4TC8zV-21EH7M-fh2dMQ-8Ahkos-5LJbpH-QMsaX-aPr5mn-6MH67-CpmDp-aPqYNZ-aPr3AP-CiKXa-aPr1sV-Cpmvg-zQ3Q3-6HVf6B-4v6ue-QMbRz-CpmoE-4J4dCo-4PENW-CiKQr-5CmU4d-bRwBMZ-QLpSA-QLpAj-zQ3fK-bsKvA-4sobRc-zQ37h-4ssaJm-FvWeu-h5PJV-4so3kv-h5PbY-4ssfLQ-4ss6eb-h5Pjw-4so6QX-4ss4xE-4so5Ec-4sseyU-njnKGG-4sseHA-4so3sa-4ss4tJ-4ssae7">kosheahan/flickr</a>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span>
</figcaption>
</figure>
<p>What’s more, sophisticated computational tools provide us with mathematical capabilities far beyond arithmetic calculations. For example, large numerical data sets can be visually examined for patterns using computer graphing software. Other tools can derive predictive equations that would be impractical for anyone to compute with paper and pencil. What’s really needed are people who can make use of those tools productively, by posing the right questions and then interpreting the results sensibly.</p>
<p>The quest to improve student retention has led schools to consider other pathways that would provide students with the quantitative skills they need. For example, <a href="http://www.educationworld.com/a_curr/mathchat/mathchat025.shtml">courses that use spreadsheets</a> extensively for mathematical modeling and powerful statistical software packages have been developed as part of <a href="http://epublications.bond.edu.au/cgi/viewcontent.cgi?article=1178&context=ejsie">an alternative pathway</a> designed for students with interests in business and economics. </p>
<p>The Carnegie Foundation for the Advancement of Teaching has created alternative math curricula called <a href="https://www.carnegiefoundation.org/in-action/carnegie-math-pathways/">Quantway and Statway</a> as examples of alternative pathways – used primarily in community colleges – that focus on quantitative reasoning and statistics/data analysis, respectively. </p>
<h1>Lectures aren’t enough</h1>
<p>These alternative pathways involve activities that go beyond students writing examples down in their notebooks. Students might use software, build mathematical models or exercise other skills – all of which require flexible instruction.</p>
<p>Both new and old pathways can benefit from new and more flexible methods. In 2012, the President’s Council of Advisors on Science and Technology <a href="http://files.eric.ed.gov/fulltext/ED541511.pdf">called for a 34 percent increase</a> in the number of STEM graduates by 2020. Their report suggested current STEM teaching practices could improve through evidence-based approaches like active learning.</p>
<figure class="align-right ">
<img alt="" src="https://images.theconversation.com/files/168417/original/file-20170508-20738-1hz978a.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/168417/original/file-20170508-20738-1hz978a.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/168417/original/file-20170508-20738-1hz978a.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/168417/original/file-20170508-20738-1hz978a.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/168417/original/file-20170508-20738-1hz978a.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/168417/original/file-20170508-20738-1hz978a.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/168417/original/file-20170508-20738-1hz978a.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">Calculating the best way to learn math.</span>
<span class="attribution"><a class="source" href="https://www.shutterstock.com/image-photo/schoolgirl-glasses-solving-math-problem-on-167441195?src=msBh2Y81MF_nzOMV89qUTw-1-39">ESB Professional/Shutterstock</a></span>
</figcaption>
</figure>
<p>In a traditional classroom, students act as passive observers, watching an expert correctly work out problems. This approach doesn’t foster an environment where mistakes can be made and answers can be questioned. Without mistakes, students lack the opportunity to more deeply explore how and why things don’t work. They then tend to view mathematics as a series of isolated problems for which the solution is merely a prescribed formula. </p>
<p>Mathematician <a href="http://launchings.blogspot.com/2011/07/the-worst-way-to-teach.html">David Bressoud</a> summarized this well:</p>
<blockquote>
<p>“No matter how engaging the lecturer may be, sitting still, listening to someone talk, and attempting to transcribe what they have said into a notebook is a very poor substitute for actively engaging with the material at hand, for doing mathematics.”</p>
</blockquote>
<p>Conversely, classrooms that incorporate active learning allow students to ask questions and explore. Active learning is not a specifically defined teaching technique. Rather, it’s a spectrum of instructional approaches, all of which involve students actively participating in lessons. For example, teachers could pose questions during class time for students to answer with an electronic clicker. Or, the class could skip the lecture entirely, leaving students to work on problems in groups. </p>
<p>While the idea of active learning has existed for decades, there has been a greater push for widespread adoption in recent years, as more scientific research has emerged. <a href="http://www.pnas.org/content/111/23/8410.full">A 2014 analysis</a> looked at 225 studies comparing active learning with traditional lecture in STEM courses. Their findings unequivocally support using active learning and question whether or not lecture should even continue in STEM classrooms. If this were a medical study in which active learning was the experimental drug, the authors write, trials would be “stopped for benefit” – because active learning is so clearly beneficial for students. </p>
<p>The studies in this analysis varied greatly in the level of active learning that took place. In other words, active learning, no matter how minimal, leads to greater student achievement than a traditional lecture classroom. </p>
<p>Regardless of pathway, all students can benefit from active engagement in the classroom. As mathematician <a href="http://www.jstor.org/stable/2319737?seq=1#page_scan_tab_contents">Paul Halmos</a> put it: “The best way to learn is to do; the worst way to teach is to talk.”</p><img src="https://counter.theconversation.com/content/77129/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Mary E. Pilgrim receives funding from National Science Foundation. </span></em></p><p class="fine-print"><em><span>Thomas Dick receives funding from National Science Foundation.</span></em></p>By embracing a style beyond the typical classroom lecture, math education can serve all of our students better.Mary E. Pilgrim, Assistant Professor of Mathematics Education, Colorado State UniversityThomas Dick, Professor of Mathematics, Oregon State UniversityLicensed 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>
<figure class="align-right zoomable">
<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>
<figcaption>
<span class="caption"></span>
<span class="attribution"><a class="source" href="http://thenewpress.com/books/math-myth">The New Press</a></span>
</figcaption>
</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">
<figcaption>
<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/513592015-11-27T13:42:24Z2015-11-27T13:42:24ZHow to solve a Rubik's cube in five seconds<figure><img src="https://images.theconversation.com/files/103380/original/image-20151126-28303-18sdw23.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.flickr.com/photos/theilr/345056969">theilr</a>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span></figcaption></figure><p>This week, 14-year-old Lucas Etter set a new world record for solving the classic Rubik’s cube in Clarksville, Maryland, in the US, solving the scrambled cube in <a href="http://www.guinnessworldrecords.com/news/2015/11/confirmed-teenager-lucas-etter-sets-new-fastest-time-to-solve-a-rubiks-cube-wor">an astonishing 4.904 seconds</a>.</p>
<p>The maximum number of face turns needed to solve the classic Rubik’s cube, one that is segmented into squares laid out 3x3 on each face, is 20, and the maximum number of quarter turns is 26. It took 30 years to discover these numbers, which were <a href="http://cube20.org/">finally proved</a> by Tomas Rokicki and Morley Davidson using a mixture of mathematics and computer calculation. The puzzle does have 43,252,003,274,489,856,000 (43 times 10<sup>18,</sup> or 43 quintillion) possible configurations after all. </p>
<p>So how do the likes of Lucas Etter work out how to solve Rubik’s cube so quickly? They could read instructions, but that rather spoils the fun. If you want to work out how to do it yourself, you need to develop cube-solving tools. In this sense, a tool is a short sequence of turns which results in only a few of the individual squares on the cube’s faces changing position. When you have discovered and memorised enough tools, you can execute them one after the other in order as required to return the cube to its pristine, solved condition.</p>
<p>These tools require experimentation to discover. Here’s how I did it myself: go on holiday with a Rubik’s cube and a screwdriver. Do experiments to find tools. The trouble is that most experiments just scramble the cube horribly and you forget what you did so you cannot undo your moves. </p>
<p>Now you have a choice, either buy another Rubik’s cube, or take out your trusty screwdriver. Turn one face through 45 degrees, and place the screwdriver under a central piece of the rotated face. Using the screwdriver as a lever to gently prise it out, it’s then easy to take the cube apart completely and reassemble it in pristine form. </p>
<p>The final move of reassembly will be the reverse of the screwdriver trick: rotate one face 45 degrees and apply gentle pressure to put the final piece back in place.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/103381/original/image-20151126-28287-pcbdy2.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/103381/original/image-20151126-28287-pcbdy2.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/103381/original/image-20151126-28287-pcbdy2.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/103381/original/image-20151126-28287-pcbdy2.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/103381/original/image-20151126-28287-pcbdy2.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/103381/original/image-20151126-28287-pcbdy2.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/103381/original/image-20151126-28287-pcbdy2.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/103381/original/image-20151126-28287-pcbdy2.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">It’s a common problem.</span>
<span class="attribution"><a class="source" href="https://www.flickr.com/photos/tangi_bertin/2445931396">tangi_bertin</a>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span>
</figcaption>
</figure>
<p>Sequences of moves of a cube form something that mathematicians call a
group. If <em>A</em> is a sequence of moves, then let <em>A<sup>-1</sup></em> (that’s “A inverse”) be the same sequence of moves performed in reverse. So if you perform <em>A</em> and then <em>A<sup>-1</sup></em>, the cube will be in the same state as was it when you began. The same is true if you first perform <em>A<sup>-1</sup></em> followed by <em>A</em>. </p>
<p>Now suppose that <em>B</em> is another sequence of moves. Many tools have the form of what mathematicians call a commutator: do <em>A</em>, then <em>B</em>, then <em>A<sup>-1</sup></em> and finally <em>B<sup>-1</sup></em>. If <em>A</em> and <em>B</em> commute, so that performing <em>A</em> then <em>B</em> is the same as doing <em>B</em> then <em>A</em>, then the commutator does nothing. From a mathematical point of view, a commutator measures failure to commute, and is a key notion in group theory. When I had a Rubik’s cube in one hand, and a screwdriver in the other, it was natural to look at how commutators behave.</p>
<p>Think of the overall structure of the different configurations of a Rubik’s cube as a labyrinth, which has that many chambers, each of which contains a Rubik’s cube in the state which corresponds to that chamber. From each chamber there are 12 doors leading to other chambers, each door corresponding to a quarter turn of one of the six faces of a cube. The type of turn needed to pass through each door is written above it, so you know which door is which. Your job is to navigate your way from a particular chamber to the one where the cube on the table is in perfect condition.</p>
<p>The tools that you have discovered are ways of getting nearer to the goal. So you don’t need to plan your route in advance, you just execute the rotations of each tool so that you get steadily closer to and finally reach the winning chamber. The mathematical result in Rokicki and Davidson’s paper shows that, no matter where you are in the labyrinth, it’s possible to reach the winning chamber by passing through at most 26 doors – although the route you find using your tools is not likely to be that efficient.</p>
<p>How to put this to use to solve the cube in five seconds? Someone like young Lucas Etta who is interested in speed solutions will not only have memorised a large number of tools, they’ll also have practised them until they can perform it very quickly. This is mostly a matter of dexterity and practice, but it’s also important to have a high-quality cube that can be manipulated smoothly and with great precision.</p>
<figure>
<iframe width="440" height="260" src="https://www.youtube.com/embed/0RfJbcydNJ0?wmode=transparent&start=0" frameborder="0" allowfullscreen=""></iframe>
</figure>
<p>Others, rather than going for speed, develop the skill of solving Rubik’s cube while blindfolded or with the cube held behind their back. In the competitive version of this variation, the solver is given a limited amount of time to study the scrambled cube and plan their solution, before they have to carry out their solution from memory without looking at the cube again. </p>
<p>In terms of our metaphor of a labyrinth, this corresponds to all the Rubik’s cubes in all the chambers being removed, except for the one on which you start. You can’t take that cube with you, but you can study it carefully and plan your whole route to the winning chamber in advance. Quite a feat of memory, and not for those with just a passing interest in the cube.</p><img src="https://counter.theconversation.com/content/51359/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Geoff Smith 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>Don’t worry – there are only 43,252,003,274,489,856,000 configurations.Geoff Smith, Senior Lecturer in Mathematics, University of BathLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/341942014-11-28T02:41:18Z2014-11-28T02:41:18ZDomino's square pizza is value for money – with the right toppings<figure><img src="https://images.theconversation.com/files/65580/original/image-20141126-4240-1b5sygt.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Getting to grips with Domino's square pizzas is easy with a bit of algebra.</span> <span class="attribution"><a class="source" href="https://www.flickr.com/photos/roboppy/6187692163/in/photostream/">Robyn Lee/Flickr</a>, <a class="license" href="http://creativecommons.org/licenses/by-nc-nd/4.0/">CC BY-NC-ND</a></span></figcaption></figure><p>Consider a standard pizza box containing a standard circular pizza. How much more would you be willing to pay for a square pizza that filled the box?</p>
<p>Clearly the square pizza contains more pizza: but is it worth the extra A$2 that Domino’s Pizza is asking? Domino’s has, for a limited time only (presumably before too many people examine the mathematics), resumed the offer of a surcharge of A$2 on any pizza to obtain a box full <a href="http://www.dominos.com.au/menu/pizzas">Square Puff</a> option – a square rather than the traditional circle.</p>
<figure>
<iframe width="440" height="260" src="https://www.youtube.com/embed/5dxn63EU1Xo?wmode=transparent&start=0" frameborder="0" allowfullscreen=""></iframe>
<figcaption><span class="caption">An ad from 2010 for Domino’s square pizza.</span></figcaption>
</figure>
<p>This concept is common enough: larger servings of food cost more to produce so need to be sold for more to cover costs. But given that there are always initial costs of production and hence economies of scale, a 200g packet of mozzarella will not cost twice the price of a 100g packet.</p>
<p>To compare the pizzas with similar products: most fast food outlets offering chips give increasing value to customers when selling them as larger portions. </p>
<p>A regular carton of chips from KFC costs A$2.95 and gives the consumer 912kJ of energy; a large carton costs A$4.75 for 2070kJ. To the nearest 10kJ, the energy per dollar of the small chips is 310 whereas for the large chips it is 440.</p>
<p>But what about the pizza deal? A little mathematics will help. </p>
<h2>Squaring the circle</h2>
<p>Take a square with side length <strong>s</strong>. The area of this square is simply** s<sup>2</sup> <strong>. The largest circle that can fit inside has radius</strong> s/2 <strong>. It therefore has area</strong>(¼)π s<sup>2**.</sup> </p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/65575/original/image-20141126-4237-1v6dxip.png?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/65575/original/image-20141126-4237-1v6dxip.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/65575/original/image-20141126-4237-1v6dxip.png?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=356&fit=crop&dpr=1 600w, https://images.theconversation.com/files/65575/original/image-20141126-4237-1v6dxip.png?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=356&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/65575/original/image-20141126-4237-1v6dxip.png?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=356&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/65575/original/image-20141126-4237-1v6dxip.png?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=448&fit=crop&dpr=1 754w, https://images.theconversation.com/files/65575/original/image-20141126-4237-1v6dxip.png?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=448&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/65575/original/image-20141126-4237-1v6dxip.png?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=448&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption"></span>
<span class="attribution"><span class="license">Author provided</span></span>
</figcaption>
</figure>
<p>Then suppose the price of a circular (standard) pizza is <strong>x</strong>. We can compare the pizza’s value (with “value” meaning pizza area per dollar: the cheaper or bigger the pizza, the higher the value): </p>
<ul>
<li>the value of a square pizza is <strong>s<sup>2/(x+2)</sup></strong></li>
<li>the value of a circular pizza is <strong>π s<sup>2/(4x)</sup></strong>. </li>
</ul>
<p>A little algebra shows that the standard pizza is better value provided its price is less than <strong>2π/(4–π) = 7.3195</strong>. Therefore, to the nearest 5 cents, you are better off buying the standard pizza whenever it costs A$7.30 or less. </p>
<p>At first glance this appears counterintuitive: why would there be less value when the pizza cost, say A$5, than if the pizza cost A$9? The answer is that the extra money being paid, namely A$2, is a greater percentage of A$5 than it is of A$9.</p>
<p>Note that we did not need to know the size of the pizza box: when comparing the value the side length <strong>s</strong> vanished. This shows that the problem is independent of the length of the box. Incidentally, this shows (for free!) that we needn’t concern ourselves with crust. </p>
<figure class="align-right zoomable">
<a href="https://images.theconversation.com/files/65526/original/image-20141125-4217-g2njlu.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/65526/original/image-20141125-4217-g2njlu.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/65526/original/image-20141125-4217-g2njlu.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=600&h=600&fit=crop&dpr=1 600w, https://images.theconversation.com/files/65526/original/image-20141125-4217-g2njlu.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=600&h=600&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/65526/original/image-20141125-4217-g2njlu.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=600&h=600&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/65526/original/image-20141125-4217-g2njlu.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&h=754&fit=crop&dpr=1 754w, https://images.theconversation.com/files/65526/original/image-20141125-4217-g2njlu.jpg?ixlib=rb-1.1.0&q=30&auto=format&w=754&h=754&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/65526/original/image-20141125-4217-g2njlu.jpg?ixlib=rb-1.1.0&q=15&auto=format&w=754&h=754&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
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<span class="attribution"><a class="source" href="https://www.flickr.com/photos/kga245/15074511896">Kelly Abbott/Flickr</a>, <a class="license" href="http://creativecommons.org/licenses/by-nc/4.0/">CC BY-NC</a></span>
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<p>Suppose that you hated crust so much that you discard it at once. You are therefore not interested in the total pizza, but in the total area of toppings on the pizza. By throwing out the strip of crust around the pizza, you are, in effect, making the “actual toppings” into a smaller pizza. But, since the figure of A$7.30 is independent of the size of the box, this critical figure remains.</p>
<p>The contrary assumption – that you love the puff pastry crust that comes with the square pizza – complicates matters. In the time-honoured fashion I leave it as an exercise to the reader to develop a model of value based on the increased desire for the crust. </p>
<p>The question now is: what does a standard pizza cost? To the nearest dollar, the “Value Range” of pizzas seems to kick off at A$5; this extends to A$9 for the more exotic “Traditional Range” of pizzas. </p>
<p>The A$5 pizza with the A$2 square surcharge is not a good deal. The Value pizza becomes an oxymoron.</p>
<p>One may then ask: if not a A$2 surcharge, then what? What could Domino’s offer as the surcharge such that the Value pizza, in square form, was actually value for money? Again, a little algebra, based on a Value pizza of A$5, shows that the square pizza would be better value if the surcharge was A$1.35 or less.</p>
<h2>So does Domino’s make a profit?</h2>
<p>Assume, that this advertising campaign does not induce anyone to buy more pizzas: it is simply a choice on whether to “upgrade”. Domino’s clearly makes money on the Value pizzas (A$5) and loses money on Traditional pizzas (A$9). </p>
<p>One would have to add each of the pizzas sold, noting the loss or gain on each one to determine if this is a money-making exercise. This is possible, and I have no doubt that some pizza boffins have noted that, with the concept of A$2 representing a single piece of low-value currency, they are onto a winner.</p><img src="https://counter.theconversation.com/content/34194/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Tim Trudgian receives funding from the ARC.</span></em></p>Consider a standard pizza box containing a standard circular pizza. How much more would you be willing to pay for a square pizza that filled the box? Clearly the square pizza contains more pizza: but is…Tim Trudgian, Research Fellow in Mathematics, Australian National UniversityLicensed as Creative Commons – attribution, no derivatives.