tag:theconversation.com,2011:/au/topics/algebra-13729/articlesAlgebra – The Conversation2018-07-16T10:39:19Ztag: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>
<figcaption>
<span class="caption"></span>
<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>
</figcaption>
</figure>
<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.