tag:theconversation.com,2011:/us/topics/mathematics-education-1534/articlesMathematics education – The Conversation2024-08-20T12:19:08Ztag:theconversation.com,2011:article/2338532024-08-20T12:19:08Z2024-08-20T12:19:08ZThe mystic and the mathematician: What the towering 20th-century thinkers Simone and André Weil can teach today’s math educators<figure><img src="https://images.theconversation.com/files/613406/original/file-20240814-17-sjl8lq.png?ixlib=rb-4.1.0&rect=0%2C0%2C720%2C358&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">André Weil (left, in 1956) and Simone Weil (in 1922) were siblings who became prominent in mathematics and philosophy, respectively.</span> <span class="attribution"><a class="source" href="https://commons.wikimedia.org/wiki/File:Simone_Weil_1922.jpg">Konrad Jacobs (right) and Anonymous (left) via Wikimedia Commons</a>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span></figcaption></figure><p><a href="https://scholar.google.com/citations?user=OM2nz9YAAAAJ&hl=en">Like most mathematicians</a>, I hear confessions from complete strangers: the inevitable “I was always bad at math.” I suppress the response, “You are forgiven, my child.” </p>
<p>Why does it feel like a sin to struggle in math? Why are so many traumatized by their mathematics education? Is learning math worthwhile? </p>
<p>Sometimes agreeing and sometimes disagreeing, <a href="https://www.britannica.com/biography/Andre-Weil">André</a> and <a href="https://www.britannica.com/biography/Simone-Weil">Simone Weil</a> were the sort of siblings who would argue about such questions. André achieved renown as a mathematician; Simone was a formidable philosopher and mystic. André focused on applying algebra and geometry to deep questions about the structures of whole numbers, while Simone was concerned with how the world can be soul-crushing. </p>
<p>Both wrestled with the best way to teach math. Their insights and contradictions point to the fundamental role that mathematics and mathematics education play in human life and culture.</p>
<h2>André Weil’s rigorous mathematics</h2>
<figure class="align-right zoomable">
<a href="https://images.theconversation.com/files/611595/original/file-20240805-17-lrfy2k.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="André Weil, aged about 50, stands on a sunny path. A child is standing next to him shielding their face." src="https://images.theconversation.com/files/611595/original/file-20240805-17-lrfy2k.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/611595/original/file-20240805-17-lrfy2k.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=600&h=882&fit=crop&dpr=1 600w, https://images.theconversation.com/files/611595/original/file-20240805-17-lrfy2k.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=600&h=882&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/611595/original/file-20240805-17-lrfy2k.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=600&h=882&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/611595/original/file-20240805-17-lrfy2k.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&h=1109&fit=crop&dpr=1 754w, https://images.theconversation.com/files/611595/original/file-20240805-17-lrfy2k.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=754&h=1109&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/611595/original/file-20240805-17-lrfy2k.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=754&h=1109&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
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<span class="caption">André Weil, pictured here in 1956, was a prominent mathematician in the 20th century.</span>
<span class="attribution"><a class="source" href="https://commons.wikimedia.org/wiki/File:Weil.jpg">Konrad Jacobs via Wikimedia Commons</a>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span>
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<p>Unlike the prominent French mathematicians of previous generations, André, who was born in 1906 and died in 1998, spent little time philosophizing. For him, mathematics was a living subject endowed with a long and substantial history, but as he remarked, he saw “<a href="https://doi.org/10.1007/978-0-8176-4571-7">no need to defend</a> (it).”</p>
<p>In his interactions with people, André was an unsparing critic. <a href="https://www.ams.org/notices/199904/mem-weil.pdf">Although admired</a> by some colleagues, he was <a href="https://nupress.northwestern.edu/9780810142626/at-home-with-andre-and-simone-weil/">feared by</a> and at times <a href="https://doi.org/10.1007/978-3-0348-8634-5">disdainful of</a> his students. He co-founded the <a href="https://theconversation.com/nicolas-bourbaki-the-greatest-mathematician-who-never-was-122845">Bourbaki mathematics collective</a> that used abstraction and logical rigor to <a href="https://doi.org/10.2307/2305937">restructure mathematics</a> from the ground up.</p>
<p>Nicolas Bourbaki’s commitment to proceeding from first principles, however, did not completely encapsulate his conception of what constituted worthwhile mathematics. André <a href="https://link.springer.com/book/9783662452561#bibliographic-information">was attuned</a> to how math should be taught differently to different audiences. </p>
<p>Tempering the Bourbaki spirit, he defined rigor as “(not) proving everything, but … endeavoring to assume as little as possible at every stage.” </p>
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<a href="https://images.theconversation.com/files/611593/original/file-20240805-17-gfnc04.png?ixlib=rb-4.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="A black and white photo of one woman and six men standing in front of a doorway." src="https://images.theconversation.com/files/611593/original/file-20240805-17-gfnc04.png?ixlib=rb-4.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/611593/original/file-20240805-17-gfnc04.png?ixlib=rb-4.1.0&q=45&auto=format&w=600&h=681&fit=crop&dpr=1 600w, https://images.theconversation.com/files/611593/original/file-20240805-17-gfnc04.png?ixlib=rb-4.1.0&q=30&auto=format&w=600&h=681&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/611593/original/file-20240805-17-gfnc04.png?ixlib=rb-4.1.0&q=15&auto=format&w=600&h=681&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/611593/original/file-20240805-17-gfnc04.png?ixlib=rb-4.1.0&q=45&auto=format&w=754&h=855&fit=crop&dpr=1 754w, https://images.theconversation.com/files/611593/original/file-20240805-17-gfnc04.png?ixlib=rb-4.1.0&q=30&auto=format&w=754&h=855&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/611593/original/file-20240805-17-gfnc04.png?ixlib=rb-4.1.0&q=15&auto=format&w=754&h=855&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
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<span class="caption">The Bourbaki congress in 1938. Simone, pictured at front left, accompanies André, obscured at back left.</span>
<span class="attribution"><a class="source" href="https://commons.wikimedia.org/wiki/File:Bourbaki_congress1938.png">Unknown author via Wikimedia Commons</a></span>
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<p>In other words, absolute rigor has its place, but teachers must be willing to take their audience into account. He believed that teachers must motivate students by providing them meaningful problems and provocative examples. Excitement for advanced students comes in encountering the unknown; for beginning students, it emerges from solving questions of, as he put it, “theoretical or practical importance.” He insisted that math “must be a source of intellectual excitement.” </p>
<p>André’s own sense of intellectual excitement came from applying insights from one part of mathematics to other parts. In a <a href="https://www.ams.org/notices/200503/fea-weil.pdf">letter to his sister</a>, André described his work as seeking a metaphorical “<a href="https://www.quantamagazine.org/a-rosetta-stone-for-mathematics-20240506/">Rosetta stone</a>” of analogies between advanced versions of three basic mathematical objects: numbers, polynomials and geometric spaces.</p>
<p>André <a href="https://www.ams.org/notices/200503/fea-weil.pdf">described mathematics in romantic terms</a>. Initially, the relationship between the different parts of mathematics is that of passionate lovers, exchanging “furtive caresses” and having “inexplicable quarrels.” But as the analogies eventually give way to a single unified theory, the affair grows cold: “Gone is the analogy: gone are the two theories, their conflicts and their delicious reciprocal reflections … alas, all is just one theory, whose majestic beauty can no longer excite us.”</p>
<p>Despite being passionless, this theory that unifies numbers, polynomials and geometry gets to the heart of mathematics; André pursued it intensely. In the words of a colleague, André sought the “<a href="https://www.ams.org/journals/notices/199904/199904FullIssue.pdf">real meaning</a> of every basic mathematical phenomenon.” For him, unlike his sister, this real meaning was found in the careful definitions, precisely articulated theorems and rigorous proofs of the most advanced mathematics of his time. Romantic language simply described the emotions of the mathematician encountering the mathematics; it did not point to any deeper significance.</p>
<h2>Simone Weil and the philosophy of mathematics</h2>
<p>On the other hand, Simone, who was born three years after André and died 55 years before him, used philosophy and religion to investigate the value of mathematics for nongeniuses, in addition to her work on politics, war, science and suffering. </p>
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<a href="https://images.theconversation.com/files/611596/original/file-20240805-21-iy2ilj.png?ixlib=rb-4.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="A headshot of Simone Weil, around age 16. She has short hair and glasses." src="https://images.theconversation.com/files/611596/original/file-20240805-21-iy2ilj.png?ixlib=rb-4.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/611596/original/file-20240805-21-iy2ilj.png?ixlib=rb-4.1.0&q=45&auto=format&w=600&h=793&fit=crop&dpr=1 600w, https://images.theconversation.com/files/611596/original/file-20240805-21-iy2ilj.png?ixlib=rb-4.1.0&q=30&auto=format&w=600&h=793&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/611596/original/file-20240805-21-iy2ilj.png?ixlib=rb-4.1.0&q=15&auto=format&w=600&h=793&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/611596/original/file-20240805-21-iy2ilj.png?ixlib=rb-4.1.0&q=45&auto=format&w=754&h=996&fit=crop&dpr=1 754w, https://images.theconversation.com/files/611596/original/file-20240805-21-iy2ilj.png?ixlib=rb-4.1.0&q=30&auto=format&w=754&h=996&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/611596/original/file-20240805-21-iy2ilj.png?ixlib=rb-4.1.0&q=15&auto=format&w=754&h=996&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
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<span class="caption">Simone Weil, pictured here in 1925, was a prominent philosopher.</span>
<span class="attribution"><a class="source" href="https://commons.wikimedia.org/wiki/File:Simone_Weil_(1909-1943)_portrait.png">Anonymous via Wikimedia Commons</a></span>
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<p>All of her writing – indeed, her life – has a maddening quality to it. In her polished essays, as well as her private letters and journals, she will often make an extreme assertion or enigmatic comment. Such assertions might concern the motivations of scientists, the psychological state of a sufferer, the nature of labor, an analysis of labor unions or an interpretation of Greek philosophers and mathematicians. She is not a systematic thinker but rather circles around and around clusters of ideas and themes. When I read her writing, I am often taken aback. I start to argue with her, bringing up counterexamples and qualifications, but I eventually end up granting the essence of her point. Simone was known for the single-minded pursuit of her ideals.</p>
<p>Despite the discomfort her viewpoints provoke, they are worth engaging. Although <a href="https://philpapers.org/rec/WEIGTG">her childhood was largely happy</a>, her whole life she <a href="https://openlibrary.org/books/OL4881067M/Simone_Weil">felt stupid in comparison</a> with her brother. She channeled her feelings of inadequacy into an exploration of how to experience a meaningful existence in the face of oppression and affliction. Over her life, she developed an <a href="https://www.harpercollins.com/products/waiting-for-god-simone-weil?variant=32131086975010">interpretation of beauty and suffering</a> intertwined with geometry.</p>
<p>Along with her lifelong mathematical discussions with André, her views were influenced by one of her <a href="https://wipfandstock.com/9781498239202/seventy-letters/">first jobs as a teacher</a>. In a letter to a colleague, she described her pupils as struggling because they “regarded the various sciences as compilations of cut-and-dried knowledge.” Like André, Simone saw the ability to motivate students as the key to good teaching. She taught mathematics as a subject embedded in culture, emphasizing overarching historical themes. Even those students who were “most ignorant in science” followed her lectures with “passionate interest.”</p>
<p>For Simone, however, the primary purpose of mathematics education was to develop the virtue of attention. Mathematics confronts us with our mistakes, and the <a href="https://www.harpercollins.com/products/waiting-for-god-simone-weil?variant=32131086975010">contemplation of these inadequacies</a> brings the ability to concentrate on one thing, at the exclusion of all else, to the fore. As a math teacher, I frequently see students grit their teeth and furrow their brow, developing only a headache and resentment. According to Simone, however, true attention arises from joy and desire. We hold our knowledge lightly and wait with detached thought for light to arrive.</p>
<p>For Simone, the “first duty” of teachers is to help students develop, through their studies, the ability to apprehend God, which she conceptualized as a blending of Plato’s description of the <a href="https://plato.stanford.edu/entries/plato/#PlaCenDoc">ultimate Good</a> with Christian conceptions of the self-abnegating God. A true understanding of God results in love for the afflicted. </p>
<p>Simone might even locate the lingering anxiety and frustration of many former math students in the absence of attention paid to them by their teachers.</p>
<h2>Authors grapple with the Weil legacy</h2>
<p>Recently, others have wrestled with the Weil legacy.</p>
<p>Sylvie Weil, André’s daughter, was born shortly before Simone’s death. Her <a href="https://nupress.northwestern.edu/9780810142626/at-home-with-andre-and-simone-weil/">family experience</a> was that of being mistaken for her aunt, ignored or demeaned by her father and not being acknowledged and appreciated by those in her orbit. </p>
<p>Similarly, author <a href="https://www.bloomsbury.com/uk/weil-conjectures-9781526607546/">Karen Olsson</a> uses Simone and André to explore her own conflicted relationship with mathematics. Her forlorn quest to understand André’s mathematics eerily reflects Simone’s desire to understand André’s work and Sylvie’s desire to be seen as her own person, to not be in Simone’s shadow. Olsson studied with exceptional math teachers and students, all the while feeling out of place, overwhelmed and intimidated by her fellow students. Most painfully, in the process of writing her book on the Weil siblings, Olsson asks a mathematician, who had been a student with her, for help in understanding some aspect of André’s mathematics. She was ignored. Both Sylvie Weil and Karen Olsson are living witnesses to Simone’s observation that each of us cries out to be seen.</p>
<p>Christopher Jackson, on the other hand, gives testimony to how mathematics can live up to Simone’s vision. Jackson is incarcerated in a federal prison but found a new life through mathematics. His correspondence with mathematician Francis Su is the backbone of Su’s 2020 book “<a href="https://yalebooks.yale.edu/book/9780300258516/mathematics-for-human-flourishing/">Mathematics for Human Flourishing</a>,” which uses Simone’s observation that “every being cries out silently to be read differently” as a leitmotif. Su identifies aspects of mathematics that promote human flourishing, such as beauty, truth, freedom and love. In their own ways, both Simone and André would likely agree.</p><img src="https://counter.theconversation.com/content/233853/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Scott Taylor receives funding from the National Science Foundation and the John and Mary Neff Foundation. </span></em></p>André Weil was a mathematician. His sister Simone Weil was a philosopher. They both thought deeply about the nature and value of mathematics and mathematics education.Scott Taylor, Professor of Mathematics, Colby CollegeLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/2172472023-12-26T17:15:34Z2023-12-26T17:15:34ZHow counting by 10 helps children learn about the meaning of numbers<figure><img src="https://images.theconversation.com/files/564574/original/file-20231208-15-3eojg4.jpg?ixlib=rb-4.1.0&rect=0%2C242%2C4383%2C3017&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Using concrete tools or objects matters for fostering mathematical development – but how can children best learn to count by 10?
</span> <span class="attribution"><span class="source">(Shutterstock)</span></span></figcaption></figure><iframe style="width: 100%; height: 100px; border: none; position: relative; z-index: 1;" allowtransparency="" allow="clipboard-read; clipboard-write" src="https://narrations.ad-auris.com/widget/the-conversation-canada/how-counting-by-10-helps-children-learn-about-the-meaning-of-numbers" width="100%" height="400"></iframe>
<p>When children start school, they learn how to recite their numbers (“one, two, three…”) and how to write them (1, 2, 3…). Learning about what those numbers mean is even more challenging, and this becomes trickier yet when numbers have more than one digit — such as 42 and 608. </p>
<p>It turns out that the meaning of such “multidigit” numbers cannot be gleaned from simply looking at them or by performing calculations with them. Our number system has many hidden meanings that are not transparent, making it <a href="https://doi.org/10.1037/dev0001145">difficult for children</a> to comprehend it. </p>
<p>In collaboration with elementary teachers, the Mathematics Teaching and Learning Lab at <a href="https://www.concordia.ca/">Concordia University</a> explores tools that can support young children’s understanding of multidigit numbers.</p>
<p>We investigate the impact of using concrete objects (like bundling straws into groups of 10). We also investigate the use of visual tools, such as number lines and charts, or words to represent numbers (the word for 40 is “forty”) and written notation (for example, 42). </p>
<p>Our recent research examined whether the “hundreds chart” — 10 by 10 grids containing numbers from one to 100, with each row in the chart containing numbers in groups of 10 — could be useful for teaching children about counting by 10, something foundational for understanding how numbers work. </p>
<h2>What’s in a number?</h2>
<p>Most adults know that the placement of the “4” and “2” in 42 means four tens and two ones, respectively. </p>
<p>But when young children start learning about numbers, they do not naturally see 10s and ones in a number like 42. They think the number represents 42 things counted from one to 42 without distinguishing between the meaning of the digits “4” and “2.” Over time, through counting and other activities, children see the four as a <a href="https://doi.org/10.1177/1053451221994827">collection of 40 ones</a>. </p>
<p>This realization is not sufficient, however, for <a href="https://doi.org/10.1111/ssm.12258">learning more advanced topics</a> in math. </p>
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Read more:
<a href="https://theconversation.com/mathematical-thinking-begins-in-the-early-years-with-dialogue-and-real-world-exploration-128282">Mathematical thinking begins in the early years with dialogue and real-world exploration</a>
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<p>An important next step is to see that 42 is made up of four distinct groups of 10 and two ones, and that the four 10s can be counted as if they were ones (for example, 42 is one, two, three, four 10s and one, two, “ones”). </p>
<p>Ultimately, one of the most challenging aspects of understanding numbers is that groups of ten and ones are <a href="https://doi.org/10.17226/12519">different kinds of units</a>. </p>
<h2>Numbers can be arranged in different ways</h2>
<p>The numbers in hundreds charts can be arranged in different ways. A top-down hundreds chart has the digit “1” in the top-left corner and 100 in the bottom-right corner. </p>
<p>The numbers increase by 10 moving downward one row at a time, like going from 24 to 34 using one hop down, for instance. A second type of chart is the “bottom-up” chart, which has the numbers increasing in the opposite direction. </p>
<h2>Counting by 10s</h2>
<p>Children can move from one number to another in the chart to <a href="https://doi.org/10.5951/teacchilmath.24.3.00e1">solve problems</a>. Considering 24 + 20, for example, children could start on 24 and move 20 spaces to land on 44. </p>
<p>Another way would be to move up (or down, depending on the chart) two rows (for example, counting “one,” “two”) until they land on 44. This second method shows a developing understanding of multidigit numbers being composed of distinct groups of 10, which is critical for an advanced knowledge of the number system. </p>
<p>For her master’s research at Concordia University, Vera Wagner, one of the authors of this story, thought children might find it more intuitive to solve problems with the bottom-up chart, where the numbers get larger with upward movement. </p>
<p>After all, plants grow taller and liquid rises in a glass as it is filled. Because of such <a href="https://doi.org/10.1111/tops.12278">familiar experiences</a>, she thought children would move by tens more frequently in the bottom-up chart than in the top-down chart. </p>
<h2>Study with kindergarteners, Grade 1 students</h2>
<p>To examine this hypothesis, we worked with 47 kindergarten and first grade students in Canada and the United States. All the children but one spoke English at home. In addition to English, 14 also spoke French, four spoke Spanish, one spoke Russian, one spoke Arabic, one spoke Mandarin and one communicated to some extent in ASL at home. </p>
<p>We assigned all child participants in the study an online version of <a href="http://mathchart.ca/chart.html#nt">either a top-down</a> or <a href="https://mathchart.ca/chart.html#reversednt">bottom-up</a> hundreds chart, programmed by research assistant André Loiselle, to solve arithmetic word problems. </p>
<p><a href="https://doi.org/10.1111/ssm.12593">What we found surprised us</a>: children counted by tens more often with the top-down chart than the bottom-up one. This was the exact opposite of what we thought they might do!</p>
<p>This finding suggests that the top-down chart fosters children’s counting by tens as if they were ones (that is, up or down one row at a time), an important step in their mathematical development. Children using the bottom-up chart were more likely to confuse the digits and move in the wrong direction. </p>
<h2>Tools can impact learning</h2>
<p>Our research suggests that the types of tools used in the math classroom can impact children’s learning in <a href="https://doi.org/10.1016/j.learninstruc.2008.03.005">different ways</a>. </p>
<p>One advantage of the top-down chart could be the corresponding left-to-right and downward movement that matches the direction in which children learn to read in English and French, the official languages of instruction in the schools in our study. Children who learn to read in a different direction (for example, <a href="https://escholarship.org/uc/item/4tt0k00j">from right to left, as in Arabic</a>) may interact with some math tools differently from children whose first language is English or French. </p>
<p>The role of cultural experiences in math learning opens up questions about the design of teaching tools for the classroom, and the relevance <a href="https://theconversation.com/culturally-responsive-teaching-in-a-globalized-world-109881">of culturally responsive</a> mathematics teaching. Future research could seek to directly examine the relation between reading direction and the use of the hundreds chart.</p><img src="https://counter.theconversation.com/content/217247/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Helena Osana received funding from the Social Sciences and Humanities Research Council of Canada for this research. </span></em></p><p class="fine-print"><em><span>Jairo A. Navarrete-Ulloa receives funding from the National Agency for Research and Development (ANID) in Chile. </span></em></p><p class="fine-print"><em><span>Vera Wagner 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>Findings of a study suggest using a ‘hundreds chart’ showing numbers one through 100, beginning with one in the top-left corner, fosters children’s counting by 10s.Helena Osana, Professor, Principal Investigator of the Mathematics Teaching and Learning Lab, Concordia UniversityJairo A. Navarrete-Ulloa, Adjunct assistant professor, Institute of Education Sciences, Universidad de O’Higgins (Chile)Vera Wagner, Research Assistant, Mathematics Teaching and Learning Lab, Concordia UniversityLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/2093092023-08-10T12:45:59Z2023-08-10T12:45:59ZHeritage algorithms combine the rigors of science with the infinite possibilities of art and design<figure><img src="https://images.theconversation.com/files/541961/original/file-20230809-29902-o57gog.png?ixlib=rb-4.1.0&rect=53%2C0%2C7168%2C4088&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Artist AbdulAlim U-K (Aikin Karr) combines the fractal structure of traditional African architecture with emerging technologies in computer graphics.</span> <span class="attribution"><a class="source" href="https://www.instagram.com/p/Cge-WOAsrkz/?img_index=2">AbdulAlim U-K</a>, <a class="license" href="http://creativecommons.org/licenses/by-nd/4.0/">CC BY-ND</a></span></figcaption></figure><p>The model of democracy in the 1920s is sometimes called “<a href="https://www.populismstudies.org/Vocabulary/melting-pot/">the melting pot</a>” – the dissolution of different cultures into an American soup. An update for the 2020s might be “open source,” where cultural mixing, sharing and collaborating can build bridges between people rather than create divides.</p>
<p>Our research on <a href="https://doi.org/10.5209/rev_TEKN.2016.v13.n2.52843">heritage algorithms</a> aims to build such a bridge. We develop <a href="https://csdt.org">digital tools</a> to teach students about the complex mathematical sequences and patterns present in different cultures’ artistic, architectural and design practices.</p>
<p>By combining computational thinking and cultural creative practices, our work provides an entry point for students who are disproportionately left out of STEM careers, whether by race, class or gender. Even those who feel at home with equations and abstraction can benefit from narrowing the gap between the arts and sciences.</p>
<h2>What are heritage algorithms?</h2>
<p>Traditional STEM curricula often present science as a ladder you climb. For example, you might be told that math starts with counting, then goes to algebra, then calculus and so on. </p>
<p>But our research has found that the global history of science is more like a bush: Each culture has its own branching set of discoveries. Some of these discoveries offer a perspective that’s different from the theorem-proof approach for math or hypothesis-experiment approach for biology. Understanding the rules and techniques that create cultural patterns from the maker’s point of view can help bridge the gap between knowledge branches. We refer to these hybrids of computation and culture as <a href="https://doi.org/10.5209/rev_TEKN.2016.v13.n2.52843">heritage algorithms</a>, and there are examples everywhere. </p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/537365/original/file-20230713-17-2yr2er.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="Two photos. On the left, one man in a hat is sitting holding a book, and another person crouches next to him pointing at the page. On the right, two people stand above a table and the person on the right is stamping a blank page." src="https://images.theconversation.com/files/537365/original/file-20230713-17-2yr2er.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/537365/original/file-20230713-17-2yr2er.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=600&h=225&fit=crop&dpr=1 600w, https://images.theconversation.com/files/537365/original/file-20230713-17-2yr2er.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=600&h=225&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/537365/original/file-20230713-17-2yr2er.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=600&h=225&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/537365/original/file-20230713-17-2yr2er.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&h=283&fit=crop&dpr=1 754w, https://images.theconversation.com/files/537365/original/file-20230713-17-2yr2er.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=754&h=283&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/537365/original/file-20230713-17-2yr2er.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=754&h=283&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">The authors learn from artisans. Left: Ron Eglash discusses fractal patterns with an Ethiopian crafter. Right: Audrey Bennett tries her hand at Adinkra stamping in Ghana.</span>
<span class="attribution"><span class="source">Ron Eglash</span>, <a class="license" href="http://creativecommons.org/licenses/by-nd/4.0/">CC BY-ND</a></span>
</figcaption>
</figure>
<p>Flying over an African village, you can see the recursive geometry of <a href="https://www.rutgersuniversitypress.org/african-fractals/9780813526140">African fractals</a> in their architecture: circles of circles, rectangles within rectangles, and other “self-similar” structures. These fractal patterns also appear in their textiles, carvings, paintings, ironwork and more.</p>
<p>Other kinds of <a href="https://doi.org/10.1007/s11423-019-09728-6">algorithms underlie</a> the repeating sequences of bent wood arcs that make up Native American wigwams, canoes and cradles. Even <a href="https://csdt.org/culture/henna/index.html">henna tattoos</a> demonstrate the interactions among computation, nature and culture.</p>
<p>These heritage algorithms challenge the <a href="https://www.routledge.com/The-Reinvention-of-Primitive-Society-Transformations-of-a-Myth/Kuper/p/book/9781138282650">myth of “primitive cultures”</a> – the idea that early Africans had no math past counting on fingers or that Native American agriculture lacked sophistication.</p>
<p>The computational thinking that is embedded in Indigenous artifacts and other creative practices, such as weaving, beadwork and quilting, is not merely decorative. It also reflects different ways of <a href="https://doi.org/10.1007/978-3-031-31293-9_18">thinking about the world</a>. Our interviews with artisans revealed how they visualize <a href="https://doi.org/10.1525/aa.1997.99.1.112">spiritual concepts</a> in formal techniques and numerical sequences. </p>
<h2>Bringing heritage algorithms to the classroom</h2>
<p>Heritage algorithms give students a way to blend the abstract rigors of math, the grounded legacies of culture and the infinite possibilities of art. To bring these algorithms to the classroom, <a href="https://csdt.org">we have created</a> interactive computer programs and simulations that we call <a href="https://www.jstor.org/stable/3804796">culturally situated design tools</a>, or CSDTs.</p>
<p>Each CSDT was created in collaboration with Indigenous elders, street artists, traditional crafters and others. With the creators’ permission, we transfer their knowledge of pattern creation into digital tools that students enjoy using and teachers enjoy implementing in their lesson plans.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/540603/original/file-20230801-29684-6okmwr.png?ixlib=rb-4.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="A close up of a brown and white woven fabric" src="https://images.theconversation.com/files/540603/original/file-20230801-29684-6okmwr.png?ixlib=rb-4.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/540603/original/file-20230801-29684-6okmwr.png?ixlib=rb-4.1.0&q=45&auto=format&w=600&h=290&fit=crop&dpr=1 600w, https://images.theconversation.com/files/540603/original/file-20230801-29684-6okmwr.png?ixlib=rb-4.1.0&q=30&auto=format&w=600&h=290&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/540603/original/file-20230801-29684-6okmwr.png?ixlib=rb-4.1.0&q=15&auto=format&w=600&h=290&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/540603/original/file-20230801-29684-6okmwr.png?ixlib=rb-4.1.0&q=45&auto=format&w=754&h=364&fit=crop&dpr=1 754w, https://images.theconversation.com/files/540603/original/file-20230801-29684-6okmwr.png?ixlib=rb-4.1.0&q=30&auto=format&w=754&h=364&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/540603/original/file-20230801-29684-6okmwr.png?ixlib=rb-4.1.0&q=15&auto=format&w=754&h=364&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">In a woven Navajo blanket, the line y=x forms a 30-degree angle with the horizontal axis.</span>
<span class="attribution"><span class="source">Ron Eglash</span>, <a class="license" href="http://creativecommons.org/licenses/by-nd/4.0/">CC BY-ND</a></span>
</figcaption>
</figure>
<p>It’s important to craft each CSDT to reflect the way those artisans think about the cultural practice. For instance, the slope of the line y=x, mathematically calculated as “rise over run,” is 1 – for every unit you move up the line, you move a unit to the right. This line forms a 45-degree angle with the x-axis. But when Navajo weavers use this “up one, over one” pattern, the slope is closer to a 30-degree angle. This is because they weave yarn horizontally through vertical cords that are thicker than the yarn. So we made sure to preserve this feature in the weaving simulation we built.</p>
<p>A crucial aspect of CSDTs is that students may use them to follow their interests. This freedom and independence lets students encounter new cultures, delve deeper into their own identity or mix designs from different cultures to create something completely new. </p>
<p>We have seen Black students choose an <a href="https://csdt.org/culture/quilting/appalachian.html">Appalachian quilting simulation</a>, Native American students choose <a href="https://csdt.org/culture/cornrowcurves/index.html">cornrow simulations</a> and white students create <a href="https://csdt.org/culture/beadloom/index.html">beadwork simulations</a>. Students’ creative designs often mix many cultures together – cornrows become “<a href="https://csdt.org/news/powwow/">powwow braids</a>,” and African fractal simulations turn into plants, lungs and river deltas.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/537364/original/file-20230713-27-a3kuf7.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="A collage of several images, some depicting students holding up a quilt, another of a student working on the quilt, and another of a computer program featuring the quilt design" src="https://images.theconversation.com/files/537364/original/file-20230713-27-a3kuf7.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/537364/original/file-20230713-27-a3kuf7.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=600&h=414&fit=crop&dpr=1 600w, https://images.theconversation.com/files/537364/original/file-20230713-27-a3kuf7.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=600&h=414&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/537364/original/file-20230713-27-a3kuf7.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=600&h=414&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/537364/original/file-20230713-27-a3kuf7.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&h=521&fit=crop&dpr=1 754w, https://images.theconversation.com/files/537364/original/file-20230713-27-a3kuf7.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=754&h=521&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/537364/original/file-20230713-27-a3kuf7.jpg?ixlib=rb-4.1.0&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"></a>
<figcaption>
<span class="caption">Students from Harlem Academy create designs using the Appalachian and Lakota quilt CSDTs. Many Appalachian quilts contained the ‘radical rose,’ symbolizing support for abolition.</span>
<span class="attribution"><span class="source">Ron Eglash</span>, <a class="license" href="http://creativecommons.org/licenses/by-nd/4.0/">CC BY-ND</a></span>
</figcaption>
</figure>
<p>Heritage algorithms and CSDTs provide a powerful starting place for students to improve their <a href="https://doi.org/10.1353/cye.2009.0024">computing skills and confidence</a>. These tools even provide a foundation for a variety of careers, from <a href="https://blog.ted.com/architecture-infused-with-fractals-ron-eglash-and-xavier-vilalta/">architecture</a> to <a href="https://csdt.org/culture/anishinaabearcs/2017overview.html">environmental engineering</a>.</p>
<h2>When computation and culture collide</h2>
<p>The reach of heritage algorithms has recently extended beyond learning environments to contemporary art spaces. Artists are generating a bold new creative style using “ethnocomputing” – an understanding of computer science from a cultural perspective.</p>
<p>You can see fresh interpretations of heritage algorithms in the African fractals embedded in the work of visual artist <a href="https://www.artforum.com/print/reviews/202007/tendai-mupita-83726">Tendai Mupita</a>, the cornrow simulations integrated in the work of <a href="https://www.nytimes.com/2022/02/24/arts/rashaad-newsome-assembly-exhibit.html">Rashaad Newsome</a>, the blending of the African diaspora and technology by <a href="https://nettricegaskins.medium.com/afrofuturist-software-from-conception-to-manifestation-d05389d0874">Nettrice Gaskins</a> and the creative duo <a href="https://iconeye.com/?p=44925">Tosin Oshinowo and Chrissy Amuah</a>.</p>
<p><a href="https://www.hauserwirth.com/hauser-wirth-exhibitions/35571-the-new-bend/#about">An exhibition</a> on display <a href="https://static1.squarespace.com/static/5dc84bade8c8347aab560645/t/647f625bf3739e5c9f84d163/1686069851882/Press-Release_TheNewBend_HWNY22-1-1.pdf">in New York City</a>, <a href="https://vip-hauserwirth.com/the-new-bend-somerset/">the U.K.</a> <a href="https://vip-hauserwirth.com/the-new-bend-los-angeles/">and Los Angeles</a> explores the textile techniques of artists inspired by the African American <a href="https://www.arts.gov/stories/blog/2015/quilts-gees-bend-slideshow">quilting tradition of Gee’s Bend, Alabama</a>. </p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/537363/original/file-20230713-21522-dovvzg.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="A dark-skinned girl wearing glasses sits in front of a computer screen. Conrow patterns are visible on the screen behind her, and imposed on the right side of the image." src="https://images.theconversation.com/files/537363/original/file-20230713-21522-dovvzg.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/537363/original/file-20230713-21522-dovvzg.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=600&h=331&fit=crop&dpr=1 600w, https://images.theconversation.com/files/537363/original/file-20230713-21522-dovvzg.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=600&h=331&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/537363/original/file-20230713-21522-dovvzg.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=600&h=331&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/537363/original/file-20230713-21522-dovvzg.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&h=416&fit=crop&dpr=1 754w, https://images.theconversation.com/files/537363/original/file-20230713-21522-dovvzg.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=754&h=416&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/537363/original/file-20230713-21522-dovvzg.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=754&h=416&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 high school student uses a CSDT to simulate cornrow hairstyle patterns.</span>
<span class="attribution"><span class="source">Ron Eglash</span>, <a class="license" href="http://creativecommons.org/licenses/by-nd/4.0/">CC BY-ND</a></span>
</figcaption>
</figure>
<p>Our research on heritage algorithms is partially driven by a philosophical desire to reframe STEM as a source of <a href="https://nmaahc.si.edu/explore/stories/black-joy-resistance-resilience-and-reclamation">radical joy</a> for every ethnicity and identity. Inspired by the radical feminist phrase “sex-positive feminism,” we sometimes call our perspective “<a href="https://www.researchgate.net/publication/340418728_Race-positive_Design_A_Generative_Approach_to_Decolonizing_Computing">race-positive design</a>” – thinking of race not in purely negative terms of oppression but instead as a rich source of creativity, liberation and a <a href="https://doi.org/10.1007/s11528-022-00815-9">free-thinking mindset</a> for curiosity and scientific inquiry.</p>
<p>This philosophical stance also has <a href="https://csdt.org/publications/">a practical side</a>: <a href="https://www.researchgate.net/publication/314263728_From_Sports_to_Science_Using_Basketball_Analytics_to_Broaden_the_Appeal_of_Math_and_Science_Among_Youth">statistically significant</a> <a href="https://doi.org/10.17583/remie.2015.1399">improvement</a> <a href="https://doi.org/10.1145/2037276.2037281">in STEM scores</a> <a href="https://doi.org/10.1525/aa.2006.108.2.347">for underrepresented students</a>. Many teachers have recognized the potential of heritage algorithms for getting students invested in STEM. One teacher using the <a href="https://csdt.org/culture/graffiti/index.html">graffiti tool</a> told us this was the first time students asked if they could stay in her math class after school. Another said she would never teach negative numbers again without the <a href="https://csdt.org/culture/beadloom/index.html">bead loom CSDT</a>.</p>
<p>Heritage algorithms, both in the classroom and beyond, open up a two-way bridge between humanistic and technical knowledge. They offer a space where everyone – teacher and student, young and old, geek and artist – can learn, share and collaborate.</p><img src="https://counter.theconversation.com/content/209309/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Audrey G. Bennett receives funding from the NEH and NSF. </span></em></p><p class="fine-print"><em><span>Ron Eglash receives funding from the NSF.</span></em></p>By bridging culture and computation, heritage algorithms challenge the myth of ‘primitive cultures’ and forge a new understanding of science and art.Audrey G. Bennett, University Diversity and Social Transformation Professor, Stamps School of Art & Design, University of MichiganRon Eglash, Professor of Information, University of MichiganLicensed 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-4.1.0&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-4.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/478942/original/file-20220812-22-22cm9f.jpg?ixlib=rb-4.1.0&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-4.1.0&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-4.1.0&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-4.1.0&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-4.1.0&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-4.1.0&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-4.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/479481/original/file-20220816-11082-5eioce.jpg?ixlib=rb-4.1.0&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-4.1.0&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-4.1.0&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-4.1.0&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-4.1.0&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-4.1.0&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/1351142020-05-05T13:43:05Z2020-05-05T13:43:05Z4 things we’ve learned about math success that might surprise parents<figure><img src="https://images.theconversation.com/files/332066/original/file-20200501-42942-go7fsv.jpg?ixlib=rb-4.1.0&rect=47%2C228%2C5272%2C3133&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">The good news: your child can use their fingers and you can too. </span> <span class="attribution"><span class="source">(Shutterstock)</span></span></figcaption></figure><p>School <a href="https://www.theguardian.com/world/2020/apr/30/coronavirus-scientists-caution-against-reopening-schools">closures due to conronavirus</a> have put parents in the challenging position of home-schooling their children.</p>
<p>In mathematics education programs for future math teachers, we often discuss the <a href="https://link.springer.com/article/10.1007/s10857-012-9208-1">traditional classroom</a> that those studying to become teachers are familiar with. We’re interested in how their own experiences as students can influence their teaching.</p>
<p>Traditional modes of instruction have emphasized that math is best learned through <a href="https://www.jstor.org/stable/40247978">studying and memorizing alone, with the teacher demonstrating procedures and then checking students’ answers</a>.</p>
<p>If parents grew up with this style of instruction, their ideal home-math classroom might look like strict scheduling, workbooks, a child working alone in silence and parents telling children how to solve problems. But if parents enforce this approach, there could be conflicts and maybe even some crying. </p>
<p>But parents, like future educators, can also learn from newer approaches. Here are some practical tips for a different form of home learning. </p>
<h2>1. Talking about math</h2>
<p>Gone are the days of students sitting quietly while the math teacher does all the talking at the chalkboard. <a href="https://www.jstor.org/stable/749877">Discussion</a> is important in the mathematics classroom. </p>
<p>Parents should be explicit. Tell your child “we learn by sharing ideas and listening to each other.”</p>
<p>Model active listening skills. Show your child that you are listening by asking questions about what they said to clarify your understanding of their idea. Try saying “tell me more …” or asking “how do you know that?”</p>
<p>Try setting aside your own idea(s) so you can listen and build on their ideas. Instead of saying “yes, but …,” use “<a href="https://link.springer.com/chapter/10.1007/978-3-319-78928-6_3">yes, and …</a>” to help children feel that they’re not being judged and their ideas are important.</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/332071/original/file-20200501-42942-wr4tlc.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/332071/original/file-20200501-42942-wr4tlc.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/332071/original/file-20200501-42942-wr4tlc.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/332071/original/file-20200501-42942-wr4tlc.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/332071/original/file-20200501-42942-wr4tlc.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/332071/original/file-20200501-42942-wr4tlc.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/332071/original/file-20200501-42942-wr4tlc.jpg?ixlib=rb-4.1.0&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">In today’s mathematics classrooms, discussion is important.</span>
<span class="attribution"><span class="source">(Shutterstock)</span></span>
</figcaption>
</figure>
<h2>2. Attitude</h2>
<p>Researchers have identified <a href="https://link.springer.com/article/10.1007/s10857-009-9134-z?shared-article-renderer">three underlying interconnected aspects of childrens’ relationships</a> with math that impact how they engage with math: emotional disposition (“I like math”), perceived competence (“I am good at math”) and their vision of math: whether math is about problem solving and understanding or math is about memorization and regurgitation.</p>
<hr>
<p>
<em>
<strong>
Read more:
<a href="https://theconversation.com/mathematics-is-about-wonder-creativity-and-fun-so-lets-teach-it-that-way-120133">Mathematics is about wonder, creativity and fun, so let's teach it that way</a>
</strong>
</em>
</p>
<hr>
<p>Parents can set a positive attitude for children by being mindful not to say things like “I don’t like math” or “I’m not a math person.” Your child might think they don’t have a chance because <a href="https://www.researchgate.net/publication/249054114_A_Quantitative_and_Qualitative_Study_of_Math_Anxiety_Among_Preservice_Teachers">you didn’t pass on a math mind</a>. </p>
<p>Academics have debunked common beliefs about the “<a href="https://www.ams.org/journals/notices/200102/rev-devlin.pdf">math gene</a>” and explain that there’s <a href="https://books.google.ca/books?hl=en&lr=&id=K1Ld7FgOdtoC&oi=fnd&pg=PT17&dq=%22math+gene%22+parent&ots=Bxk5UApbwY&sig=dMLYhCKH%20K7mHhHOvfy8SOEc_es&redir_esc=y#v=onepage&q=%22math%20gene%22%20parent&f=false">lots involved in being good at math</a>. Celebrate the process and not just the final answer. Give high fives for sharing solution strategies, developing a plan to tackle the problem and for not giving up.</p>
<p>Make it clear that <a href="https://books.google.ca/books?hl=en&lr=&id=bOGHDQAAQBAJ&oi=fnd&pg=PR9&dq=dweck+mindset+mistakes&ots=YMX--knDci&sig=y07leb0VLednZ4ZhScAAYsKCkyE#v=onepage&q=dweck%20mindset%20mistakes&f=false">making mistakes</a> is OK and can even be a good thing. Many highly successful people see mistakes as learning opportunities and an indication that learning is happening.</p>
<h2>3. Working in partnership</h2>
<p><a href="https://www.advance-he.ac.uk/knowledge-hub/engagement-through-partnership-students-partners-learning-and-teaching-higher">A partnership</a> is about working together and can include seeing the <a href="https://www.semanticscholar.org/paper/We-are-the-Process%3A-Reflections-on-the-of-Power-in-Kehler-Verwoord/aeecc3e2e8e352474a24ce4ccd407f62629d6f56">teacher as a learner and the student as a teacher</a>. It isn’t about the teacher being “all-knowing” and making all the decisions. </p>
<p>Traditional math teaching, where the teacher assumes an authoritative role, is a major cause of <a href="https://doi.org/10.1177/1365480214521457">math anxiety</a>. Researchers have found that not all <a href="https://doi.org/10.1016/j.cedpsych.2019.101784">math homework help</a> is beneficial. There is a difference between parents being controlling and being supportive.</p>
<p>With this in mind, wait for your child to ask for help. Try not to control everything. Focus on asking questions about their decisions that will help them figure out possible limitations and benefits of their decisions. </p>
<p>Let children fail. Failure can <a href="https://books.google.ca/books?id=q0VZwEZoniUC&lpg=PP1&dq=The%20Optimistic%20Child%3A%20A%20Proven%20Program%20to%20Safeguard%20Children%20Against%20Depression%20and%20Build%20Lifelong%20Resilience&lr&pg=PP1#v=onepage&q&f=false">build confidence</a>. Confidence can come from mastery; mastery can come from <a href="https://www.jstor.org/stable/40248303">practice</a>. Good practice includes analyzing what went wrong and what went right.</p>
<p>Don’t worry about being the expert. Be honest and say “I’m not sure. Let’s figure it out together.” </p>
<p>Start with <a href="https://books.google.ca/books?id=Irq913lEZ1QC&lpg=PR13&lr&pg=PP1#v=onepage&q&f=false">what children already know</a>. When your child is stuck, ask them to talk through what they are doing.</p>
<p>Take turns doing questions and talking about solution strategies.</p>
<p>Follow your child’s interests <a href="https://theconversation.com/eight-ways-to-keep-your-kids-smart-over-the-summer-break-100132">and ideas</a>. Let them take the lead, even if you think your approach is better.</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/332073/original/file-20200501-42951-pxpcn5.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/332073/original/file-20200501-42951-pxpcn5.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/332073/original/file-20200501-42951-pxpcn5.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/332073/original/file-20200501-42951-pxpcn5.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/332073/original/file-20200501-42951-pxpcn5.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/332073/original/file-20200501-42951-pxpcn5.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/332073/original/file-20200501-42951-pxpcn5.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=754&h=503&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Focus on asking your child questions that will help them figure out possible limitations and benefits of their decisions.</span>
<span class="attribution"><span class="source">(Shutterstock)</span></span>
</figcaption>
</figure>
<h2>4. Basic math skills</h2>
<p>If you grew up with traditional math instruction and haven’t thought about math since your school days, it might surprise you to learn that there are multiple ways to solve problems.</p>
<p>You could ask your child to share their way of solving the problem and also share your way. </p>
<p>For instance: What is 24 x 6? </p>
<p>It’s OK if you’re looking for a pencil to do this: </p>
<p> 24<br>
<u>x 6</u><br>
144</p>
<p>But what are some other ways you might you figure it out? </p>
<p>Multiply 20 x 6 to get 120. Now multiply 4 x 6 to get 24. Add the two figures: 120 + 24 = 144.</p>
<p>Another way would be to focus on 25 x 6 to get 150. Now subtract 6 and you’ve got 144. </p>
<hr>
<p>
<em>
<strong>
Read more:
<a href="https://theconversation.com/the-new-math-how-to-support-your-child-in-elementary-school-87479">The 'new math': How to support your child in elementary school</a>
</strong>
</em>
</p>
<hr>
<p>In all math problems (including addition or subtraction), your child can use their fingers and you can too. </p>
<figure class="align-right ">
<img alt="" src="https://images.theconversation.com/files/327316/original/file-20200412-10562-6mjs1u.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/327316/original/file-20200412-10562-6mjs1u.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=600&h=450&fit=crop&dpr=1 600w, https://images.theconversation.com/files/327316/original/file-20200412-10562-6mjs1u.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=600&h=450&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/327316/original/file-20200412-10562-6mjs1u.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=600&h=450&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/327316/original/file-20200412-10562-6mjs1u.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&h=566&fit=crop&dpr=1 754w, https://images.theconversation.com/files/327316/original/file-20200412-10562-6mjs1u.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=754&h=566&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/327316/original/file-20200412-10562-6mjs1u.jpg?ixlib=rb-4.1.0&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">Author Tina Rapke finds an occasion for everyday math in making cookies.</span>
<span class="attribution"><span class="source">(Shutterstock)</span></span>
</figcaption>
</figure>
<p>You can also look for opportunities to highlight math in daily activities. </p>
<p>One fun way is through baking. Arrange three rows of cookie dough with four cookies in each row. Ask how many cookies per batch or how many each family member will get if they share equally. </p>
<p>Being successful at <a href="https://www.jstor.org/stable/10.5951/mathteacher.108.7.0543?casa_token=53fYsdfs758AAAAA:iROqe6Bs17ufC1uUB1x_ToGBlxgh-LgCEmMqSXgYT9cfbcLkdq0BdhWUjkxEfmYM5aLT__nM3eJ2CBiRa7EIwNPcR9W5BhbYspgB1oC4YDJaM2LWdp4#metadata_info_tab_contents">mental math</a> (like the arithmetic you do at the store) happens gradually over time. </p>
<p>Try focusing on basic math skills with your child for 10 minutes or less, every other day. </p>
<h2>The takeaway</h2>
<p>Think of quality over quantity. </p>
<p>If you want to support math learning at home based on math research: talk with your child, see learning as a partnership and make sure to celebrate their ideas. Your child may teach you something new. </p>
<p>We’d love to hear about how math has provoked families to slow down, have fun, go with the flow and connect.</p><img src="https://counter.theconversation.com/content/135114/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Tina Rapke received funding from SSHRC: Partnership Engage Grants. </span></em></p><p class="fine-print"><em><span>Cristina De Simone 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>Your cheat sheet for best practices in teaching math at home. Keep it positive and mask your shock when your child tells you there are many ways to multiply numbers.Tina Rapke, Associate Professor of Mathematics Education, York University, CanadaCristina De Simone, Middle School Teacher. PhD Mathematics Education Student, York University, CanadaLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/1051092018-11-27T11:41:11Z2018-11-27T11:41:11ZThe key to fixing the gender gap in math and science: Boost women’s confidence<figure><img src="https://images.theconversation.com/files/246490/original/file-20181120-161633-uo02kp.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Girls who are more confident in their math skills are more likely to pursue math-intensive degrees. </span> <span class="attribution"><a class="source" href="https://www.shutterstock.com/image-photo/students-class-174265457?src=PZezSUO7ZmyHLj8f90AGZA-1-3">Areipa.lt/shutterstock.com</a></span></figcaption></figure><p>The gender gap in <a href="https://www.nctm.org/Publications/Teaching-Children-Mathematics/Blog/Current-Research-on-Gender-Differences-in-Math/">math</a> and <a href="http://uis.unesco.org/en/topic/women-science">science</a> isn’t going away. Women remain less likely to enroll in math-heavy fields of study and pursue math-heavy careers. This pattern persists despite major studies finding <a href="http://psycnet.apa.org/doi/10.1037/a0021276">no meaningful differences in mathematics performance</a> among girls and boys. </p>
<p>Among U.S. students who score the same on math achievement tests, girls are <a href="https://doi.org/10.3389/fpsyg.2017.00386">less confident in their math ability</a> than boys are. That confidence predicts who goes on to major in math-heavy fields like <a href="https://www.aauw.org/research/solving-the-equation/">engineering and computer science</a>. The <a href="http://www.tcrecord.org/Content.asp?ContentID=18026">gender gap varies</a> across STEM fields – science, technology, engineering and math. Women <a href="http://psycnet.apa.org/doi/10.1037/a0027020">remain underrepresented</a> in <a href="https://www.shrm.org/resourcesandtools/hr-topics/compensation/pages/graduates-pay-2017.aspx">high-earning</a> and <a href="https://www.ed.gov/stem">high-demand</a> fields that require the most math skills, such as engineering and physics. <a href="https://doi.org/10.1080/00221546.2018.1486641">My team’s recent study</a> finds women are 12 percent less likely to earn math-heavy STEM degrees than men. </p>
<p><a href="https://perezfelkner.com/research/reset/">My colleagues and I</a> have studied gender gaps in STEM for several years, examining <a href="https://nces.ed.gov/surveys/slsp/">U.S. data</a> on teenagers as they move from high school to and through college. Across our studies, we find a consistent pattern: Girls with strong mathematics ability in high school <a href="https://perezfelkner.files.wordpress.com/2012/12/perez-felkner_mcdonald_schneider_highachieving_females_stem_ebook.pdf">do not necessarily leave the sciences entirely</a>, but they <a href="https://doi.org/10.3389/fpsyg.2015.00530">major in math-heavy fields</a> at significantly lower rates than their otherwise identical male peers.</p>
<p>Here’s the good news: These patterns can change. <a href="https://doi.org/10.3389/fpsyg.2015.00530">In one study</a>, we found that 12th-grade girls with the highest levels of confidence in their mathematics ability with challenging material are more three times more likely to major in math-heavy STEM fields than girls with the lowest levels of confidence. </p>
<h2>Ability beliefs, girls and STEM</h2>
<p>Our findings build not only on our own prior work, but also on decades of research finding girls underrate their abilities on <a href="http://psycnet.apa.org/doi/10.1037/0022-3514.59.5.960">tasks</a> and <a href="https://sociology.stanford.edu/sites/default/files/publications/gender_and_the_career_choice_process-_the_role_of_biased_self-assessments.pdf">careers</a> that are culturally considered male. </p>
<p><a href="https://nces.ed.gov/surveys/els2002/">Contemporary data</a> on U.S. students who were 10th graders in 2002 and were followed through 2012 show that girls <a href="https://www.apa.org/news/press/releases/2014/04/girls-grades.aspx">do better in school</a> than boys do and are more likely to <a href="https://www.theatlantic.com/business/archive/2017/11/gender-education-gap/546677/">graduate from college</a>. Girls are increasingly prepared for college-level math, thanks to the fact that they take more <a href="https://files.eric.ed.gov/fulltext/ED509653.pdf">STEM courses in high school</a>, even in <a href="https://medium.com/@codeorg/girls-set-ap-computer-science-record-skyrocketing-growth-outpaces-boys-41b7c01373a5">computer science</a>.</p>
<p>In one of our case studies on computer science undergraduates at two research universities, we found that women were more likely to take further computer science courses if they <a href="https://doi.org/10.3389/fpsyg.2017.00602">perceived that they had high skills and felt challenged</a>. These findings complement those of our <a href="https://doi.org/10.3389/fpsyg.2017.00386">national study</a>, which showed that women with positive math ability beliefs were more likely to choose math-heavy STEM majors.</p>
<p>Girls are excelling at math. Still, boys think they can do better. Among those at the 90th percentile of mathematics ability in 12th grade, <a href="https://doi.org/10.3389/fpsyg.2017.00386">boys rate themselves higher</a> than do their female peers. </p>
<h2>Progress failures and promising interventions</h2>
<p>The push toward equity has not just been slow; it at times seems to go in reverse. Emerging research suggests gender gaps in STEM seem wider in <a href="http://doi.org/10.1126/science.aar2307">more economically developed countries</a> and <a href="https://cepa.stanford.edu/content/gender-achievement-gaps-us-school-districts">more affluent zip codes</a>. Since the personal computing and technology boom, women have been <a href="https://www.aauw.org/research/solving-the-equation/">losing representation among degree earners</a> in computer science.</p>
<p>Among U.S. universities, we found the gender gap in math-heavy fields was widespread, but <a href="http://www.tcrecord.org/Content.asp?ContentID=18026">worse at less selective institutions</a>. And, while the majority of community college students are female, after controlling for student and institutional characteristics, the gender gap in natural and engineering sciences at <a href="https://perezfelkner.files.wordpress.com/2018/11/perez-felkner_etal_2yearcollegesgendergapstemdegrees_2018.pdf">two-year colleges</a> is slightly worse – 12.4 percent more men – than at four-year institutions – 11.7 percent more men than women.</p>
<p>There are signs of promise as <a href="https://anitab.org/braid-building-recruiting-and-inclusion-for-diversity/">institutions collaborate on gender equity</a> and try other interventions, from introductory course redesign to curricular changes aimed at students’ <a href="https://ies.ed.gov/ncee/wwc/PracticeGuide/5#tab-summary">beliefs in their abilities</a>. While not directly focused on this issue, organizations may engage in confidence-raising to get girls and women into math-heavy fields like <a href="https://ed.ted.com/featured/16DCJILa">coding</a>.</p>
<p>As someone who has studied this issue closely, I believe those of us interested in gender equity should make female confidence a priority. This includes both directly building up girls’ and women’s confidence and educating influential actors in their lives. Socializing messages and support from mentors, teachers, peers and parents may help counter gendered stereotypes and create spaces for girls to build confidence in their ability to succeed in math and science.</p><img src="https://counter.theconversation.com/content/105109/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Lara Perez-Felkner receives research funding from the National Science Foundation, the Gates Foundation, and the ECMC Foundation.</span></em></p>High school girls who are more confident in their math abilities are more likely to pursue math in college and beyond.Lara Perez-Felkner, Assistant Professor of Higher Education and Sociology, Florida State UniversityLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/922492018-03-12T10:48:58Z2018-03-12T10:48:58ZCelebrating Marion Walter – and other unsung female mathematicians<figure><img src="https://images.theconversation.com/files/209790/original/file-20180310-30986-yhwoa0.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Searching for role models in the math world.</span> <span class="attribution"><a class="source" href="https://www.shutterstock.com/image-photo/rear-view-thoughtful-woman-who-tries-297702716">ImageFlow/shutterstock.com</a></span></figcaption></figure><p>When I was teaching mathematics in the 90s, before the internet, I had a book of “women mathematicians.” This was helpful for sharing inspirational stories with my middle school students, but there were just six women in this short book. </p>
<p>These days, we have the internet – and more stories about women in mathematics. For example, the 2016 blockbuster movie “Hidden Figures,” based on the book by <a href="http://margotleeshetterly.com/">Margot Lee Shetterley</a>, introduced the world to African-American women mathematicians Katherine Johnson, Mary Jackson, Dorothy Vaughan and Christine Darden. The world recently lost <a href="https://theconversation.com/maryam-mirzakhani-was-a-role-model-for-more-than-just-her-mathematics-81143">Maryam Mirzakhani</a>, the first woman to win a Fields Medal (sort of like the Nobel Prize, but in mathematics). </p>
<p>But we still need more stories about women in mathematics. While many mathematicians know of my colleague Marion Walter, she isn’t known well outside her field. And she should be, for her own story and the lessons she brings to our understanding of mathematics. </p>
<h2>Meeting Marion Walter</h2>
<p>Marion Walter turned 90 years old in July 2018, but when you ask her about growing up, she’ll tell you she hasn’t yet. </p>
<p>She was born in 1928 to a Jewish family in Berlin. She and her sister, Ellen, attended a Jewish boarding school 390 miles away, in Herrlingen, Germany. </p>
<p>In 1939, Marion and Ellen left Nazi Germany on a Kindertransport, the rescue operation that evacuated thousands of Jewish children to England before the outbreak of World War II. When the war began, her school – like many others on the south coast – was evacuated to the English countryside. </p>
<figure class="align-right zoomable">
<a href="https://images.theconversation.com/files/209610/original/file-20180308-30954-1r9mi3d.png?ixlib=rb-4.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/209610/original/file-20180308-30954-1r9mi3d.png?ixlib=rb-4.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/209610/original/file-20180308-30954-1r9mi3d.png?ixlib=rb-4.1.0&q=45&auto=format&w=600&h=574&fit=crop&dpr=1 600w, https://images.theconversation.com/files/209610/original/file-20180308-30954-1r9mi3d.png?ixlib=rb-4.1.0&q=30&auto=format&w=600&h=574&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/209610/original/file-20180308-30954-1r9mi3d.png?ixlib=rb-4.1.0&q=15&auto=format&w=600&h=574&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/209610/original/file-20180308-30954-1r9mi3d.png?ixlib=rb-4.1.0&q=45&auto=format&w=754&h=721&fit=crop&dpr=1 754w, https://images.theconversation.com/files/209610/original/file-20180308-30954-1r9mi3d.png?ixlib=rb-4.1.0&q=30&auto=format&w=754&h=721&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/209610/original/file-20180308-30954-1r9mi3d.png?ixlib=rb-4.1.0&q=15&auto=format&w=754&h=721&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Marion Walter is professor emerita of mathematics at the University of Oregon.</span>
<span class="attribution"><a class="source" href="http://pages.uoregon.edu/wmnmath/People/Biographies/MarionWalter.html">Marion Walter</a>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span>
</figcaption>
</figure>
<p>At Christmastime in 1944, the headmistress of Marion’s secondary school called her to ask if she knew yet what her plans were now that she had graduated. It was the middle of the winter, in the middle of the war, and the math teacher had just quit unexpectedly. So the headmistress offered Marion the position, for the salary of 10 shillings per week. This was enough for Marion to purchase fish and chips in the neighboring village, if she biked there. </p>
<p>Though Marion still had to sleep in the dormitory with her students, she was pleased to have access to the teachers’ lounge. At age 16, Marion taught math to students aged 5 to 16, and says that her graduating students all passed their school certificate examination. </p>
<h2>Learning and helping others learn</h2>
<p>Marion taught for two terms before moving on to her next adventure: academic studies in mathematics and education.</p>
<p>She arrived in London in the fall of 1945, where she began her formal study of mathematics at the Regent Street Polytechnic. After she attained her intermediate Bachelor of Science degree in mathematics, she moved with her family to New York City. While there, she earned a bachelor’s degree in mathematics and education from Hunter College, then a master’s degree in mathematics from New York University. </p>
<p>She studied at night because she did computing work during the day. Marion worked on computations for research professors at New York University, using a <a href="https://www.youtube.com/watch?v=nmwSmwNF9XY">Marchant calculator</a>. This mechanical computing machine required the user to physically move a cylinder over when adding numbers with multiple digits. </p>
<p>Marion went on to earn her doctorate in mathematics education from the Harvard Graduate School of Education, and was hired to teach at Simmons College. While in Massachusetts, she founded the <a href="https://www.bostonareamathspecialists.org/">Boston Area Mathematics Specialists</a>, a group focused on improving the teaching and learning of mathematics for school children. In 1977, she moved to the University of Oregon. I met her there in 2016. </p>
<p>Marion’s major mathematical line of study has been problem posing – the art of asking and refining mathematical questions. </p>
<p>Her most significant book is likely <a href="https://www.questia.com/library/7839630/the-art-of-problem-posing">“The Art of Problem Solving”</a>, co-authored with mathematics educator Stephen Brown. She has also written several articles and children’s books about mathematics. </p>
<figure class="align-left zoomable">
<a href="https://images.theconversation.com/files/209832/original/file-20180311-30975-1c7co7l.png?ixlib=rb-4.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/209832/original/file-20180311-30975-1c7co7l.png?ixlib=rb-4.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/209832/original/file-20180311-30975-1c7co7l.png?ixlib=rb-4.1.0&q=45&auto=format&w=600&h=452&fit=crop&dpr=1 600w, https://images.theconversation.com/files/209832/original/file-20180311-30975-1c7co7l.png?ixlib=rb-4.1.0&q=30&auto=format&w=600&h=452&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/209832/original/file-20180311-30975-1c7co7l.png?ixlib=rb-4.1.0&q=15&auto=format&w=600&h=452&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/209832/original/file-20180311-30975-1c7co7l.png?ixlib=rb-4.1.0&q=45&auto=format&w=754&h=568&fit=crop&dpr=1 754w, https://images.theconversation.com/files/209832/original/file-20180311-30975-1c7co7l.png?ixlib=rb-4.1.0&q=30&auto=format&w=754&h=568&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/209832/original/file-20180311-30975-1c7co7l.png?ixlib=rb-4.1.0&q=15&auto=format&w=754&h=568&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 visual depiction of Marion Walter’s Theorem.</span>
<span class="attribution"><span class="source">Jennifer Ruef</span>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span>
</figcaption>
</figure>
<p>Marion is also one of few people to have a <a href="https://girlsangle.wordpress.com/2014/07/15/marion-walters-theorem-via-mass-points/">theorem</a> named after her, based on <a href="http://www.jstor.org/stable/27968561">the following question</a>: If the sides of a triangle are trisected, what is the resulting area of the hexagon that’s created? </p>
<p>Like Marion, we can benefit from cultivating a sense of wonder. Wonder at how people make sense of mathematics, and wonder at how mathematics can describe the world. Wonder keeps us learning and growing.</p>
<h2>Representing math</h2>
<p>I asked Marion, who retired in 1993, if she had any advice to share. She was adamant on two counts. First, adults should never tell children, “I was not good at math.” And second, they should not tell children that they are wrong.</p>
<p>Research supports both claims. Telling students we’re not good at mathematics sends the <a href="https://www.youcubed.org/resources/parents-beliefs-math-change-childrens-achievement/">message</a> that they might not be either. And when children think they are not good at math, it hurts their ability to engage with it. On the second count, when someone has a “wrong” answer in mathematics, it’s often because they are thinking about a <a href="https://deepblue.lib.umich.edu/handle/2027.42/78024">different problem</a>, or the same problem in a different way. </p>
<p>That said, sometimes there are mathematically incorrect answers. So what do we say? Marion suggests asking, “How are you thinking about the problem?” When children are partners in problem-solving, they are invited into an <a href="https://www.sfchronicle.com/opinion/article/You-re-all-math-people-you-just-12236409.php">apprenticeship</a> as mathematicians.</p>
<p>The next time you have a reason to think about mathematicians, I hope you will remember Marion Walter. Women and girls have been told, in many ways, that there is no room in math and science for them. Representations matter. The more powerful women we see in mathematics, the more evidence we have that mathematics is for all people. Children who are learning about the world, and their potential place in it, benefit from visions of who they might become – perhaps a woman in mathematics.</p><img src="https://counter.theconversation.com/content/92249/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Jennifer Ruef does not work for, consult, own shares in or receive funding from any company or organization that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>Women’s History Month is a time to recognize female role models. In mathematics, when we think of powerful women, we should think of Marion Walter.Jennifer Ruef, Assistant Professor of Education Studies, University of OregonLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/825522017-09-12T02:18:56Z2017-09-12T02:18:56ZThese four easy steps can make you a math whiz<figure><img src="https://images.theconversation.com/files/184994/original/file-20170906-9830-3mf2kc.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Can you cut it in this math problem?</span> <span class="attribution"><a class="source" href="https://www.shutterstock.com/image-photo/camembert-cheese-brie-575682340">Sergey Lapin/shutterstock.com</a></span></figcaption></figure><p>Many people find mathematics daunting. If true, this piece is for you. If not, this piece is still for you.</p>
<p>What do you think of when you think about mathematics? Perhaps you think about x’s and y’s, intractable fractions, or nonsensical word problems. The cartoonist Gary Larson once depicted hell’s library as containing only giant tomes of word problems. You know, “If a train leaves New York…” </p>
<p>I was trained as a mathematician, and I will let you in on a trade secret: That is not what mathematics is, nor where it lives. It’s true that learning mathematics often involves solving problems, but it should focus on the joy of solving puzzles, rather than memorizing rules. </p>
<p>I invite you to see yourself as a problem solver and mathematician. And I’d like to introduce you to the man who once invited me to the study of problem solving: George Pólya. </p>
<h1>Math Pólya’s way</h1>
<p>For many reasons, not the least of which is that Pólya <a href="http://articles.latimes.com/1985-09-08/news/mn-2892_1_polya-george-mathematician">died</a> in 1985, you will meet him as I did – through his wildly successful <a href="http://press.princeton.edu/titles/669.html">“How to Solve It</a>.” Penned in 1945, this book went on to sell over one million copies and was translated into 17 languages. </p>
<p>As a mathematician, Pólya worked on a wide range of problems, including the study of heuristics, or how to solve problems. When you read “How to Solve It,” it feels like you’re taking a guided tour of Pólya’s mind. This is because his writing is metacognitive – he writes about how he thinks about thinking. And metacognition is often the heart of problem solving.</p>
<p>Pólya’s problem solving plan breaks down to four simple steps:</p>
<ol>
<li> Make sure you understand the problem.</li>
<li> Make a plan to solve the problem.</li>
<li> Carry out the plan.</li>
<li> Check your work to test your answer.</li>
</ol>
<p>There it is. Problem solving in the palm of your hand – math reduced to four steps.</p>
<p><a href="http://epltt.coe.uga.edu/index.php?title=Situated_Cognition">Here’s a classic problem</a> from research on mathematics education done by Jean Lave. A man, let’s call him John, is making ¾ of a recipe that calls for 2/3 cup of cottage cheese. What do you think John did? What would you do? </p>
<p>If you’re like me, you might immediately dive into calculations, perhaps struggling with what the fractions mean, working to remember the rules for arithmetic. That’s what John seemed to do, at first. But then he had a Eureka! moment.</p>
<p>John measured 2/3 cup of cottage cheese, then dumped it onto a cutting board. He patted the cheese into a circle and drew lines into it, one vertical, one horizontal, dividing the cheese patty into quarters. He then carefully pushed one quarter of the cottage cheese back into its container. Voilá! Three-quarters of 2/3 cup of cottage cheese remained. </p>
<p>John is a mathematician and problem solver. First, he understood the problem: He needed ¾ of what the recipe called for, which was 2/3 cup. Then, he made a plan, most likely visualizing in his head how he would measure and divide the cottage cheese. Finally, he carried out the plan. </p>
<p>Did he check his answer? That remains unclear, but we can check the validity of his work for him. Did he indeed end up with ¾ of 2/3 cup of cottage cheese? Yes, because the full amount was reduced by one-quarter, leaving three-quarters. </p>
<h1>Another approach</h1>
<p>Would this solution work with different foods or serving sizes? So long as a person could divide that serving into quarters, yes, the plan would work. </p>
<p>Could we solve the problem another way with the same result? Sure — there are many ways to solve this problem, and they should all result in the same half-cup answer. Here is one. </p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/185181/original/file-20170907-9603-cugdd3.png?ixlib=rb-4.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/185181/original/file-20170907-9603-cugdd3.png?ixlib=rb-4.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/185181/original/file-20170907-9603-cugdd3.png?ixlib=rb-4.1.0&q=45&auto=format&w=600&h=265&fit=crop&dpr=1 600w, https://images.theconversation.com/files/185181/original/file-20170907-9603-cugdd3.png?ixlib=rb-4.1.0&q=30&auto=format&w=600&h=265&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/185181/original/file-20170907-9603-cugdd3.png?ixlib=rb-4.1.0&q=15&auto=format&w=600&h=265&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/185181/original/file-20170907-9603-cugdd3.png?ixlib=rb-4.1.0&q=45&auto=format&w=754&h=333&fit=crop&dpr=1 754w, https://images.theconversation.com/files/185181/original/file-20170907-9603-cugdd3.png?ixlib=rb-4.1.0&q=30&auto=format&w=754&h=333&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/185181/original/file-20170907-9603-cugdd3.png?ixlib=rb-4.1.0&q=15&auto=format&w=754&h=333&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">How to find ¾ of 2/3.</span>
<span class="attribution"><span class="source">Jennifer Ruef</span>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span>
</figcaption>
</figure>
<p>Notice that this solution uses pictures. <a href="https://www.youcubed.org/downloads/?file_id=1584&file_name=jacmaths-seeing-article&resource_name=visual-math-article-in-journal-of-applied-computational-mathematics">New brain research</a> validates what mathematics educators have been saying for decades: Pictures help us think. Drawing pictures also happens to be another of Pólya’s suggestions.</p>
<p>John probably made use of one of Pólya’s most important suggestions: Can you think of a related problem?</p>
<p>Of course, this is a cheesy problem – sorry, I really didn’t even try to fight that pun – which is a common complaint about story problems. I chose it because it has delighted math researchers for years, and because John is quite clever in his solution. He is also extremely mathematical.</p>
<p>I’ve taught mathematics, and how to teach mathematics, for nearly 30 years. For over a decade, it was my job to convince high school freshmen not only that algebra was meaningful, but that it was meant for them, and they for it. In my work, I’ve met many people who love mathematics and many who find it overwhelming and nonsensical. And so it’s an important part of my work to help people see the beauty and wonder of mathematics, and think of themselves as mathematicians. </p>
<p>These <a href="https://www.youcubed.org/resources/parents-beliefs-math-change-childrens-achievement/">messages</a> are especially important for parents helping children learn mathematics. If you understand the problem you’re trying to solve, you’re well on your way to solving it. And you, yes you, are a problem solver.</p>
<p>We all know it’s not always so simple to solve problems. Pólya did too. That’s the glory of it – the messy, wonderful, powerful adventure.</p><img src="https://counter.theconversation.com/content/82552/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Jennifer Ruef does not work for, consult, own shares in or receive funding from any company or organization that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>Dreading math class as you head back into school? Never fear: Try these tips from famed mathematician George Pólya.Jennifer Ruef, Assistant Professor of Education Studies, University of OregonLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/705962016-12-21T19:52:22Z2016-12-21T19:52:22Z20% maths decree sets a dangerous precedent for schooling in South Africa<figure><img src="https://images.theconversation.com/files/150744/original/image-20161219-24307-1e5x8xg.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Learning the fundamentals of maths can equip children with critical thinking and reasoning skills.</span> <span class="attribution"><span class="source">Reuters/Mohamed Nureldin Abdallah</span></span></figcaption></figure><p>The department of basic education in South Africa has reacted to pupils’ exceedingly low attainment rates in maths education in a controversial fashion. It has issued <a href="http://www.education.gov.za/Portals/0/Documents/Publications/Circular%20A3%20of%202016.pdf?ver=2016-12-07-154005-723">an urgent circular</a> to its heads of departments, principals, managers, directors and exam and curriculum heads outlining a “special condonation dispensation”. This applies to pupils completing grades 7, 8 and 9 in the 2016 academic year which has just ended.</p>
<p>Pupils who did not meet the 40% criteria in mathematics will now be able to progress to the next grade provided they met all other pass requirements and obtained more than 20% in mathematics. Only those who passed Grade 9 maths with 30% or more will be allowed to continue with the subject.</p>
<p>Those who achieved in the 20% band will have to take mathematical literacy in the last years of their school careers. This is a somewhat different and far less demanding subject.</p>
<p>The move has been widely condemned from most quarters. For instance, the education department in the Western Cape province, <a href="http://www.gov.za/af/node/759324">warned</a> that if no “drastic action” is taken, “we will be sitting in the same position next year”. Indeed. The national department claims that its directive constitutes “<a href="http://www.iol.co.za/news/south-africa/no-such-thing-as-a-20-percent-pass-mark-for-maths-7160039">an interim measure</a>”. But how does it hope to address the crisis in maths education in future years? What can be done to instil pupils with the valuable, relevant skills developed through good maths teaching?</p>
<h2>Building critical thinking skills</h2>
<p>South African pupils’ chronic under-performance in maths is not a one-off event. It has <a href="http://www.telegraph.co.uk/education/2016/11/29/revealed-world-pupil-rankings-science-maths-timss-results/">become entrenched</a>. An earlier <a href="http://www.amesa.org.za/TIMSSR.htm">study shows</a> that pupils’ basic maths abilities – calculating fractions, simple number sense, analysis and probability – have steadily declined. </p>
<p>All of this illustrates the debilitating burden that generations of South African children have had to endure, from the apartheid era until the present day: an education system that has failed them. It has not inducted pupils into the custom of thinking and reasoning on logical, rational and critical terms. Critical thinking is a vital skill. </p>
<p>Research <a href="http://www.doane.edu/facstaff/resources/cetl-home/31812">has shown</a> that a well-cultivated critical thinker raises vital questions and problems, formulating them clearly and precisely. They are able to gather and assess relevant information, using abstract ideas to interpret it effectively. They can reach well-reasoned conclusions and solutions, testing them against relevant criteria and standards.</p>
<p>The relationship between “mathematics education” and “more complex thinking” is typically symbiotic and mutually inclusive. Good, productive mathematics education can positively <a href="http://math-site.athabascau.ca/documents/HistoryMathematicsEducation.pdf">raise pupils’ skills</a> in diagnostic, methodical thinking.</p>
<p>Defective teaching and learning could consign them to a school life steeped in frustration, fear and failure. This is precisely the fate of those <a href="https://theconversation.com/boredom-alienation-and-anxiety-in-the-maths-classroom-heres-why-69570">perpetually struggling</a> in this crucial area of learning.</p>
<p>Primary school pupils’ gross underachievement in maths education suggests they may not have been equipped, at their relevant levels, with the skills needed to <a href="http://link.springer.com/referenceworkentry/10.1007%2F978-94-007-4978-8_35#page-1">think and reason</a> effectively and meaningfully. This is why the subject occupies such an eminent place in <a href="https://www.acer.edu.au/timss">global schooling assessment criteria</a> – not only because of its content, but for the skills that are transferred and developed alongside it.</p>
<p>The education department <a href="http://www.iol.co.za/news/south-africa/no-such-thing-as-a-20-percent-pass-mark-for-maths-7160039">argues</a> that some students are more inclined towards the arts while others are better with technical subjects. This is not well founded. David Pearson, a scholar of cognitive psychology, refers in <a href="https://theconversation.com/exploding-the-myth-of-the-scientific-vs-artistic-mind-57843">his writing</a> to the domain of neuroscience which has confirmed that “everyone uses both sides of the brain when performing any task”. Pearson argues that while certain impulses of brain activity have occasionally been associated with creative or cogent thinking,</p>
<blockquote>
<p>…it doesn’t really explain who is good at what – and why. Studies have actually revealed considerable overlap in the cognitive processes supporting both scientific and artistic creativity.</p>
</blockquote>
<p>Here’s another, fairly widespread <a href="https://theconversation.com/pressured-south-african-schools-had-no-choice-but-to-relax-maths-pass-mark-70289">fable about maths</a>: “The ability to factorise quadratic functions is not a prerequisite for an educated child”. Such standpoints devalue the subject’s more authentic meaning. Instead it’s important to ponder what happens when an education system continually fails to equip its students with the aptitudes required by so many positions or professions, even if those aptitudes are not explicitly mathematical.</p>
<p>Employers have <a href="https://www.prospects.ac.uk/careers-advice/what-can-i-do-with-my-degree/mathematics">long recognised</a> that applicants with maths credits are more inclined to succeed at jobs that call for logical reasoning, precise enquiry and careful deduction. Not only this, but a wide and protracted variety of job descriptions and professional occupations – in both the sciences and humanities – <a href="http://unesdoc.unesco.org/images/0019/001914/191425e.pdf">call</a> for <a href="https://www.gov.uk/government/news/every-pupil-needs-a-good-mathematics-education">maths education</a> to a lesser or greater degree.</p>
<h2>A dangerous precedent</h2>
<p>South Africa’s maths dilemma should not be perceived purely on narrow, technical grounds. The domain of mathematics education must be seen instead in its full complexity and for its potency to endow pupils to meet some of life’s most vital challenges. Children’s dismal failure at maths is a reflection of an education system that has continually quelled their capacities to arrive at sound answers based on accurate reasoning.</p>
<p>Drastically lowering standards – such as “condoning” a 20% mark – sets a dangerous precedent from which the country may not recover for years. The education department’s directive essentially diminishes the great and important role of maths in children’s general educational as well as their broader human development. Such a course, regrettably, will assuredly exacerbate an already dire situation.</p>
<p>Maths education can only really flourish and generate more fruitful outcomes within the context of a well-functioning national education system. It is here, arguably, that the real problem lies. South Africa’s education system is merely a reflection of its broader social system. This is generally characterised by high levels of economic and social inequality, poverty, violence and abuse, and dysfunctionality. </p>
<p>It stands to reason, then, that positive educational change is incumbent upon profound social change. In the absence of social change, however, what is the fate of the overwhelming majority of South Africa’s 12 million and more school-going children today? My own <a href="http://ruralreporters.com/research-schooling-in-rural-south-africa/">field studies</a> show the importance and usefulness of gaining a deeper understanding of how certain poor, isolated schooling communities have endeavoured to overcome the odds. </p>
<p>Such an approach presents a prospective framework – broadly defined – for others to emulate or, at the very least, to contemplate while real social change remains obscure and elusive.</p><img src="https://counter.theconversation.com/content/70596/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>In this piece Clive Kronenberg draws on his prior post-doctoral fellowship studies, funded by the Critical Thinking Group attached to CPUT's Faculty of Education. </span></em></p>Maths occupies an eminent place in global schooling assessment criteria not just because of its content, but for the skills that are taught and developed alongside it.Clive Kronenberg, NRF Accredited & Senior Researcher; Lead Coordinator of the South-South Educational Collaboration & Knowlede Interchange Initiative, Cape Peninsula University of TechnologyLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/702822016-12-12T11:52:27Z2016-12-12T11:52:27ZWhy it doesn’t help – and may harm – to fail pupils with poor maths marks<figure><img src="https://images.theconversation.com/files/149628/original/image-20161212-31402-1xeb6x.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Learning deficits in Maths compound over time.</span> <span class="attribution"><span class="source">Shutterstock</span></span></figcaption></figure><p>Many South Africans were <a href="https://businesstech.co.za/news/finance/146351/sa-government-lowers-pass-mark-for-maths-to-20/">outraged</a> by the <a href="http://www.education.gov.za/Portals/0/Documents/Publications/Circular%20A3%20of%202016.pdf?ver=2016-12-07-154005-723">recent announcement</a> that for 2016, pupils in Grades 7 to 9 could progress to the next grade with only 20% in Mathematics. </p>
<p>The usual minimum has been 40%, provided that all other requirements for promotion are met. Pupils with less than 30% in Mathematics in grade 9 must take <a href="http://www.education.gov.za/Portals/0/CD/National%20Curriculum%20Statements%20and%20Vocational/CAPS%20FET%20_%20MATHEMATICAL%20LITERACY%20_%20GR%2010-12%20_%20Web_DDA9.pdf?ver=2015-01-27-154330-293">Mathematical Literacy</a> (this involves what the Department of Basic Education calls “the use of elementary mathematical content” and is not the same as Mathematics) as a matric subject.</p>
<p>Public concern is understandable. South Africans should be deeply worried about the state of mathematics teaching and learning. The country was placed <a href="http://www.telegraph.co.uk/education/2016/11/29/revealed-world-pupil-rankings-science-maths-timss-results/">second from last</a> for mathematics achievement in the latest Trends in International Maths and Science Study. </p>
<p>Research closer to home has <a href="http://www.ekon.sun.ac.za/wpapers/2014/wp272014">shown</a> that pupils, particularly from poorer and less well resourced schools, are under performing in mathematics relative to the curriculum outcomes. These learning deficits compound over time, which makes it increasingly difficult to address learning difficulties in mathematics in the higher grades. </p>
<p>All of this means that children and young people may be in Mathematics classes but are not learning. But the answer to this problem does not lie with making pupils repeat an entire grade because of poor mathematical performance. There’s extensive research evidence to suggest that grade repetition does more harm than good.</p>
<h2>Repetition is not effective</h2>
<p>Grade repetition is practised worldwide – despite there being very little evidence for its effectiveness. In fact, it can be argued that its consequences are mainly negative for repeating pupils. Grade repetition is a predictor of <a href="http://www.create-rpc.org/pdf_documents/PTA16.pdf">early school leaving</a>, sometimes called “drop out”. </p>
<p>Pupils who repeat grades and move out of their age cohort become <a href="https://www.oecd.org/economy/grade-repetition-a-comparative-study-of-academic-and-non-academic-consequences.pdf">disaffected with school</a>. They disengage from learning. </p>
<p>Repeating a grade <a href="http://www.unesco.org/iiep/PDF/Edpol6.pdf">lowers motivation</a> towards learning and is <a href="https://books.google.co.za/books?id=yxDawksXxn0C&printsec=frontcover&dq=International+guide+to+student+achievement&hl=en&sa=X&redir_esc=y#v=onepage&q=grade%20retention%20is%20not%20associated%20with%20academic%20growth&f=false">seldom associated with improved learning outcomes</a>. </p>
<p>South Africa’s rates of grade repetition are high. Research by the Department of Basic Education <a href="http://www.education.gov.za/Portals/0/Documents/Publications/General%20Household%20Survey%202013.pdf?ver=2015-07-07-111309-287">shows</a> that on average, 12% of all pupils from grades one to 12 repeat a year. The grades with the highest repetition rates are grade 9 (16.3%), grade 10 (24.2%) and grade 11 (21.0%).</p>
<p>And grade repetition is an equity issue. <a href="http://www.socialsurveys.co.za/factsheets/AcessToEducation-TechnicalReport/29ff21.pdf">The Social Survey-CALS (2010)</a> report found that black children are more likely to repeat grades than their white or Indian peers. This reflects the fracture lines that signal socioeconomic disadvantage in South Africa.</p>
<p>Repetition rates decrease as the education level of the household head increases. Poor access to infrastructural resources, like piped water and flush toilets, are associated with higher rates of grade repetition. Boys are more likely to repeat than girls. There’s also an uncertain link between pupil achievement and grade repetition, particularly for black learners in high schools. </p>
<p>So why does grade repetition persist?</p>
<h1>Beliefs about the benefits of repetition</h1>
<p>Schools and societies <a href="http://www.journals.uchicago.edu/doi/abs/10.1086/667655?journalCode=cer">still believe</a> in the value of making children repeat grades, despite evidence to the contrary.</p>
<p>A recent survey of 95 teachers in Johannesburg – which is currently under review for publication in a journal – showed how teachers believe the additional time spent in a repeated year allows pupils to “catch up” and be better prepared for the subsequent grade. This view is reflected in recent <a href="http://www.groundup.org.za/article/teachers-oppose-20-pass-mark-maths/">reports</a> that teachers are against the new 20% concession which has stirred so much controversy. Their opposition is echoed by countless callers to talk shows, who all seem to assume that repeating subject content results in improved understanding.</p>
<p>But unless the reasons for a pupil’s misunderstanding of concepts are identified and addressed, any improvement is unlikely. Given that the deficits in mathematical understanding may stretch back to the foundation phase (Grades 1 - 3), it’s doubtful that merely repeating a grade in the senior phase is going to be sufficient for remediation. </p>
<p>And teachers may struggle to provide support to pupils repeating a grade. Research conducted in South Africa <a href="http://www.tandfonline.com/doi/abs/10.1080/13603116.2015.1095250?journalCode=tied20">reveals</a> that teachers lack confidence in their ability to teach pupils who experience learning difficulties. They would prefer to refer such pupils to learning support specialists and psychologists who are seen to have more expertise. </p>
<p>Many of the teachers we surveyed believe that grade repetition solves problems intrinsic to pupils. Immaturity is seen as one reason for learning difficulties and teachers expect that the repeated year compensates for this. Other teachers regard the threat of retention as a means to motivate pupils who are not sufficiently diligent or who are “slow” or “weak”. When learning difficulties are seen as being intrinsic to pupils, it is less likely that factors within the education system will be considered as the cause of barriers to learning.</p>
<h2>Failing pupils is not the solution</h2>
<p>Poor achievement in mathematics is not going to be solved by making pupils repeat their grade. Repetition effectively makes pupils and their families pay an additional – financial and emotional – cost for the system’s failure.</p>
<p>Repetition because of poor mathematics achievement during the senior phase compounds the bleak outlook for these pupils. They already have a minimal grasp of mathematics, which denies them access to Science, Technology, Engineering and mathematics (STEM) subjects and careers. Then they’re also at risk of leaving school early and joining the ranks of <a href="https://theconversation.com/how-two-crucial-trends-are-affecting-unemployment-in-south-africa-56296">the unemployed</a>.</p>
<p>The Department of Basic Education’s 20% concession indicates that it knows grade repetition won’t achieve much. The public outcry should not be that these learners are being given a “free pass” and don’t deserve to be promoted. Instead, civil society needs to hold the government accountable for addressing the crisis in mathematics teaching and learning across all grades – and particularly in the crucial primary school years.</p><img src="https://counter.theconversation.com/content/70282/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Elizabeth Walton does not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>There’s extensive research evidence to suggest that grade repetition does more harm than good.Elizabeth Walton, Associate professor, University of the WitwatersrandLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/695702016-12-01T15:40:09Z2016-12-01T15:40:09ZBoredom, alienation and anxiety in the maths classroom? Here’s why<figure><img src="https://images.theconversation.com/files/147889/original/image-20161129-10988-lfuft.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">A student revising class work in Kenya. The quality of maths teaching in schools can have a profound impact on learner interest.</span> <span class="attribution"><span class="source">Reuters/Thomas Mukoya</span></span></figcaption></figure><p>The quest for appropriate teaching and learning practices for children and young people is ongoing and ever present. A major challenge is to make maths teaching more inclusive and maths itself more accessible to a wider cross section of children and young people. This is essential if we are to successfully meet the <a href="http://www.un.org/sustainabledevelopment/education/">UN Sustainable Development Goal</a> of ensuring inclusive and equitable quality education. This in turn is intended to promote lifelong learning opportunities for all. </p>
<p>One important imperative is to develop active and participatory approaches that lead to the development of problem solving and thinking skills. This specific issue came to the foreground in two recent research projects – in Scotland and in Ghana – which sought to investigate the quality of mathematics learning. </p>
<p>The Scottish project on <a href="http://www.tandfonline.com/doi/abs/10.1080/00220272.2014.979233">developing mathematical thinking in the primary classroom</a> was carried out in collaboration with teachers and local authorities. It was set up within a design based research framework, with the aim of promoting classroom-based action research on the part of participants in a Masters level course. </p>
<p>In reporting this research and development work the authors refer to the “epistemic quality” of the mathematics being taught and learned in school. This is the quality of what pupils come to know, understand and become able to do.</p>
<p>In the parallel study, Evelyn Oduro reports similar findings from a <a href="http://dx.doi.org/10.13140/RG.2.1.2678.2165">study of teachers’ assessment practices in Ghana</a>. This study focused on “basic schools” – kindergarten through to junior high – and was conducted in 2011 with a sample of four classroom teachers and two head teachers. It involved a qualitative research design.</p>
<p>From the two sets of research in Scotland and Ghana, it’s clear how high epistemic quality in the classroom can be supported. One can also see how low epistemic quality can be reinforced by external vested interests. These often involve excessive emphasis on high stakes external testing, summative assessment and school league tables. </p>
<h2>Elephant in the classroom</h2>
<p>In the Scottish project the teachers involved experienced very powerful responses to one of the set readings: Jo Boaler’s book <a href="https://www.nationalnumeracy.org.uk/sites/default/files/eitc_character_development_-_updated_branding.pdf">The Elephant in the Classroom</a>. In it she writes:</p>
<blockquote>
<p>I have called this book <em>The elephant in the classroom</em> because there is often a very large elephant standing in the corner of maths classrooms. The elephant, or the common idea that is extremely harmful to children, is the belief that success in mathematics is a sign of general intelligence and that some people can do maths and some can’t. </p>
<p>Even maths teachers (the not so good ones) often think that their job is to sort out those who can do maths from those who can’t. This idea is completely wrong and this is why. In many maths classrooms a very narrow subject is taught to children, that is nothing like the maths of the world or the maths that mathematicians use. </p>
<p>This narrow subject involves copying methods that teachers demonstrate and reproducing them accurately over and over again … But this narrow subject is not mathematics, it is a strange mutated version of the subject that is taught in schools.</p>
</blockquote>
<h2>Fundamentalist approach</h2>
<p>It is argued in the Scottish findings that the process of “mutation” reflects the process of didactic transposition, which can;</p>
<ul>
<li><p>change the mathematical knowledge profoundly such that the knowledge in question is not knowledge for acting and solving problems in the social contexts in which it was created and; </p></li>
<li><p>lead to the epistemic quality of the subject becoming degraded as it is transposed into school mathematics. </p></li>
</ul>
<p>The findings describe this mutated or degraded version of mathematics as being of low epistemic quality. It is characterised by a fundamentalist approach that presents the subject as infallible, authoritarian, dogmatic, absolutist, irrefutable and certain. The degraded version involves right and wrong answers based on superficial imitative memorised and algorithmic reasoning. </p>
<p>In contrast, high epistemic quality is characterised by an approach which presents mathematics as fallible, refutable and uncertain. It also promotes critical thinking, creative reasoning and the generation of multiple solutions. It encourages young people to learn from errors and mistakes. </p>
<p>Creative mathematical reasoning involves novelty, plausibility and mathematical foundation. Creativity is also seen as an orientation or disposition toward activity that can be fostered broadly in school. Such creative reasoning can have many functions in mathematics. These include verification, explanation, systematisation, discovery, communication, construction of theory and exploration.</p>
<p>Low epistemic quality has a dramatic impact on students. It leaves them bored and demotivated. They also experience fear and anxiety and feel alienated from maths as a subject. </p>
<h2>Ghana study findings</h2>
<p>The Ghana study was conducted in a context in which “assessment for learning” was being used as a policy driver to promote higher epistemic quality. The findings illustrate how teachers use both formal and informal assessments in mathematics classrooms. Formal assessment dominates practice. </p>
<p>Of particular relevance were the teachers’ views about the nature of mathematics and the ways in which these can impact so as to degrade the “epistemic quality” of school mathematics. </p>
<p>The findings also highlight the ways in which teachers’ views are related to their classroom practices. What’s very interesting is how teachers’ assessment practices are affected by contextual factors. These factors are related to institutional policies, professional development and classroom conditions.</p><img src="https://counter.theconversation.com/content/69570/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Brian Hudson received funding from the Scottish Government from 2010 to 2012 to support the project 'Developing Mathematical Thinking in the Primary Classroom'. </span></em></p>The quality of what pupils come to know, understand and are able to do has a big impact on students. Low quality leads to boredom as well as fear and anxiety about maths as a subjectBrian Hudson, Professor of Education, University of SussexLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/625082016-07-20T07:42:53Z2016-07-20T07:42:53ZMastery over mindset: the cost of rolling out a Chinese way of teaching maths<figure><img src="https://images.theconversation.com/files/131058/original/image-20160719-13854-1p5ih80.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">
</span> <span class="attribution"><span class="source">Oleksandr Berezko/www.shutterstock.com</span></span></figcaption></figure><p>Half of primary schools in England <a href="https://www.gov.uk/government/news/south-asian-method-of-teaching-maths-to-be-rolled-out-in-schools">will receive £41m over four years</a> to teach mathematics using a method called the “mastery approach” that is used in Chinese schools. Yet during the last ten years many primary schools in England have started embracing another method, called the “<a href="https://www.eescpdportal.org/_images/96/flyers/EES_for_Schools_Mindset_Flyer_DP.pdf">mindset approach</a>”, which is supported by 30 years of psychological research in the US and <a href="https://v1.educationendowmentfoundation.org.uk/uploads/pdf/Changing_Mindsets.pdf">more recently in the UK</a>. </p>
<p>While the <a href="https://theconversation.com/explainer-what-is-the-mastery-model-of-teaching-maths-25636">mastery model</a> breaks down learning into small goals which have to be achieved before moving on, the mindset maths model aims to get pupils to develop an intuitive understanding of mathematical concepts before learning formal procedures such as addition or multiplication. Can the two work together in schools to help young children learn maths?</p>
<h2>Mindset is about motivation</h2>
<p>Mindset theory has established that each person has ingrained beliefs regarding each subject, which pre-determine whether learning the subject will be successful. Carol Dweck, the most prominent researcher in this field, has demonstrated spectacularly that <a href="https://psychology.stanford.edu/sites/all/files/Implicit%20Theories,%20Attributions%20and%20Coping_0.pdf">a wrong kind of environment can destroy a learner’s motivation</a>, possibly with long-lasting effects. </p>
<p>Vice versa, it is possible to build learning environments which nurture a learner’s motivation and so <a href="https://psychology.stanford.edu/sites/all/files/cdwecklearning%20success_0.pdf">make learning possible</a>. Mathematics, a subject which is often portrayed as difficult in the media, has immediately <a href="http://www.youcubed.org/wp-content/uploads/14_Boaler_FORUM_55_1_web.pdf">attracted the attention of researchers</a>, and a <a href="http://www.growthmindsetmaths.com/">number of principles</a> have evolved for developing classroom environments which are conducive to learning mathematics.</p>
<p>One thing that proponents of both the mastery and mindset theory agree with is avoiding the idea that a child can have a <a href="https://psychology.stanford.edu/sites/all/files/cdweckmathgift_0.pdf">gift for maths</a> so that high expectations are set for all pupils. According to mindset theory, a learner’s belief that they are or are not talented is a dangerous illusion, which damages their progress.</p>
<p>Some of the suggestions for organising in-class activities are <a href="https://www.ncetm.org.uk/resources/45776">similar in the mastery and mindset approaches</a>. For example, all pupils spend the same period of time working on the same problem, rather than some pupils racing forward to the next problem. This may involve some pupils starting work, perhaps unsuccessfully, on some deeper or more general versions of the problem. </p>
<h2>A shortcut to maths concepts</h2>
<p>While on the surface the mastery approach talks of a curriculum designed to enable deep conceptual, as well as procedural, knowledge, a look at its principles and sample lessons reveals undue concentration on numerous rigid objectives, achieved in a strict linear order. </p>
<p>In one <a href="https://theconversation.com/explainer-what-is-the-mastery-model-of-teaching-maths-25636">implementation of the mastery approach</a>, the process of adding two numbers is split into 23 consecutive learning objectives, which all need to be mastered separately, and may be tested separately by the teacher. </p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/131144/original/image-20160719-8008-fxbx4p.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/131144/original/image-20160719-8008-fxbx4p.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=600&h=401&fit=crop&dpr=1 600w, https://images.theconversation.com/files/131144/original/image-20160719-8008-fxbx4p.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=600&h=401&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/131144/original/image-20160719-8008-fxbx4p.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=600&h=401&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/131144/original/image-20160719-8008-fxbx4p.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&h=504&fit=crop&dpr=1 754w, https://images.theconversation.com/files/131144/original/image-20160719-8008-fxbx4p.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=754&h=504&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/131144/original/image-20160719-8008-fxbx4p.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=754&h=504&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">The goal: avoid maths anxiety.</span>
<span class="attribution"><span class="source">J2R/www.shutterstock.com</span></span>
</figcaption>
</figure>
<p>Mindset approach is completely different. It aims to be more exploratory, and tends to start from the “complicated” concepts, rather than reaching them by a long path. This stems from the belief that most ideas of mathematics are not abstract, but directly correspond to what can be demonstrated by real-world models. For example, a mindset-approach teacher can introduce addition via joining two heaps of cardboard counters (or other props) together, <a href="https://www.youcubed.org/math-topic/addition/">explore properties of addition via activities</a>, and only then break the process of adding numbers into procedural steps.</p>
<h2>Developing number sense</h2>
<p>Mindset approach advocates that pupils develop what is called “number sense”: an ability to choose (or invent) those methods of working with numbers which are more convenient and efficient in a particular situation. </p>
<p>As a simple example, a person possessing number sense will never calculate 5x3 as 3+3+3+3+3, because the process of adding threes is slower, less convenient and more prone to error than adding fives, that is, 5+5+5. </p>
<p>A person with a developed number sense may have gaps in the knowledge they store in their memory (for example, they will not necessarily know the whole times table by heart), but they will be effective at using the principles they have internalised. If a certain mathematical fact needs to be rediscovered because they do not remember it, they will be able to do it effortlessly. Some argue that <a href="https://www.youcubed.org/fluency-without-fear/">a person possessing number sense will not have maths anxiety</a> in their later life and, therefore, will be better prepared for learning more mathematics.</p>
<h2>Too fast, too soon?</h2>
<p>What is concerning at this stage is that <a href="https://www.gov.uk/government/uploads/system/uploads/attachment_data/file/495939/Resilience_posters_FINAL.pdf">unlike the mindset approach</a> there is, as yet, no indication that the mastery approach is being systematically and empirically tested. Only one preliminary study in the UK has been <a href="https://www.gov.uk/government/publications/evaluation-of-the-maths-teacher-exchange-china-and-england">published</a>, and it expresses only cautious optimism regarding the mastery model. </p>
<p>There are striking differences between mindset-approach and mastery-approach classroom activities, and we are not sure that in the long term, the mastery approach is better. The new government focus on the mastery method may also be confusing for teachers, after they have been encouraged to teach with the mindset approach. </p>
<p>We are left with some worrying unanswered questions – particularly as half of UK schools are now being encouraged to move to a mastery approach before any empirical evaluation of its impact on learning has taken place.</p><img src="https://counter.theconversation.com/content/62508/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Sherria Hoskins has received funding from the Education Edowment Foundation (two grants) to carry out Randomised Controlled Trials into Mindset.</span></em></p><p class="fine-print"><em><span>Alexei Vernitski does not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>£41m will be spent on ‘mastery learning’ – will it improve learning in primary schools?Alexei Vernitski, Senior Lecturer in Mathematics, University of EssexSherria Hoskins, Head of Psychology, University of PortsmouthLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/591432016-05-16T10:48:53Z2016-05-16T10:48:53ZWhat makes a mathematical genius?<figure><img src="https://images.theconversation.com/files/121865/original/image-20160510-20731-150nn8p.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">An early understanding of numbers may be a sign of mathematical ability.</span> <span class="attribution"><span class="source">Oksana Kuzmina</span></span></figcaption></figure><p>The film <a href="http://www.imdb.com/title/tt0787524/">The Man Who Knew Infinity</a> tells the gripping <a href="https://theconversation.com/the-man-who-knew-infinity-a-mathematicians-life-comes-to-the-movies-50777">story of Srinivasa Ramanujan</a>, an exceptionally talented, self-taught Indian mathematician. While in India, he was able to develop his own ideas on summing geometric and arithmetic series without any formal training. Eventually, his raw talent was recognised and he got a post at the University of Cambridge. There, he worked with Professor G.H. Hardy until his untimely death at the age of 32 in 1920.</p>
<figure class="align-right ">
<img alt="" src="https://images.theconversation.com/files/121862/original/image-20160510-20721-1jgsbcn.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/121862/original/image-20160510-20721-1jgsbcn.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=600&h=822&fit=crop&dpr=1 600w, https://images.theconversation.com/files/121862/original/image-20160510-20721-1jgsbcn.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=600&h=822&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/121862/original/image-20160510-20721-1jgsbcn.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=600&h=822&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/121862/original/image-20160510-20721-1jgsbcn.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&h=1033&fit=crop&dpr=1 754w, https://images.theconversation.com/files/121862/original/image-20160510-20721-1jgsbcn.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=754&h=1033&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/121862/original/image-20160510-20721-1jgsbcn.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=754&h=1033&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Srinivasa Ramanujan.</span>
<span class="attribution"><span class="source">wikimedia</span></span>
</figcaption>
</figure>
<p>Despite his short life, Ramanujan made substantial contributions to number theory, elliptic functions, infinite series and continued fractions. The story seems to suggest that mathematical ability is something at least partly innate. But what does the evidence say?</p>
<h2>From language to spatial thinking</h2>
<p>There are many different theories about what mathematical ability is. One is that it is closely tied to the capacity for understanding and building language. Just over a decade ago, a study <a href="http://www.ncbi.nlm.nih.gov/pubmed/15319490">examined members of an Amazonian tribe</a> whose counting system comprised words only for “one”, “two” and “many”. The researchers found that the tribe were exceptionally poor at performing numerical thinking with quantities greater than three. They argued this suggests language is a prerequisite for mathematical ability. </p>
<p>But does that mean that a mathematical genius should be better at language than the average person? There is some evidence for this. In 2007, researchers scanned the brains of 25 adult students while they were solving multiplication problems. The study found that individuals with higher mathematical competence <a href="http://www.ncbi.nlm.nih.gov/pubmed/17851092">appeared to rely more strongly on language-mediated processes</a>, associated with brain circuits in the <a href="http://brainmadesimple.com/parietal-lobe.html">parietal lobe</a>. </p>
<p>However, recent findings have challenged this. One <a href="http://www.pnas.org/content/113/18/4909.abstract">study</a> looked at the brain scans of participants, including professional mathematicians, while they evaluated mathematical and non-mathematical statements. They found that instead of the left hemisphere regions of the brain typically involved during language processing and verbal semantics, high level mathematical reasoning was linked with activation of a bilateral network of brain circuits associated with processing numbers and space. </p>
<p>In fact, the brain activation in professional mathematicians in particular showed minimal use of language areas. The researchers argue their results support previous studies that have found that knowledge of numbers and space during early childhood can predict mathematical achievement.</p>
<p>For example, a <a href="http://www.sciencedirect.com/science/article/pii/S0022096515003057">recent study of 77 eight- to 10-year-old children</a> demonstrates that visuo-spatial skills (the capacity to identify visual and spatial relationships among objects) have an important role in mathematical achievement. As part of the study, they took part in a “<a href="http://www.tandfonline.com/doi/abs/10.1080/87565640801982361">number line estimation task</a>”, in which they had to position a series of numbers at appropriate places on a line where only the start and end numbers of a scale (such as 0 and 10) were given.</p>
<p>The study also looked at the children’s overall mathematical ability, visuospatial skills and visuomotor integration (for example, copying increasingly complex images using pencil and paper). It found that children’s scores on visuospatial skill and visuomotor integration strongly predicted how well they would do on number line estimation and mathematics. </p>
<h2>Hidden structures and genes</h2>
<p>An alternative definition of mathematical ability is that it represents the capacity to recognise and exploit hidden structures in data. This may account for an <a href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.581.27&rep=rep1&type=pdf">observed overlap</a> between mathematical and musical ability. Similarly, it could also explain why training in chess can benefit <a href="http://www.sciencedirect.com/science/article/pii/S1747938X16300112">children’s ability to solve mathematical problems</a>. Albert Einstein famously claimed that images, feelings and musical structures formed the basis of his reasoning rather than logical symbols or mathematical equations.</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/122689/original/image-20160516-15920-1624g0m.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/122689/original/image-20160516-15920-1624g0m.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=600&h=834&fit=crop&dpr=1 600w, https://images.theconversation.com/files/122689/original/image-20160516-15920-1624g0m.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=600&h=834&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/122689/original/image-20160516-15920-1624g0m.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=600&h=834&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/122689/original/image-20160516-15920-1624g0m.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&h=1048&fit=crop&dpr=1 754w, https://images.theconversation.com/files/122689/original/image-20160516-15920-1624g0m.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=754&h=1048&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/122689/original/image-20160516-15920-1624g0m.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=754&h=1048&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Albert Einstein playing the violin.</span>
<span class="attribution"><span class="source">E. O. Hoppe</span></span>
</figcaption>
</figure>
<p>However, the extent to which mathematical ability relies on innate or environmental factors remains controversial. A <a href="http://www.nature.com/ncomms/2014/140708/ncomms5204/full/ncomms5204.html">recent large scale twin and genome-wide analysis</a> of 12-year-old children found that genetics could explain around half of the observed correlation between mathematical and reading ability. Although this is quite substantial, it still means that the learning environment has an important role to play. </p>
<p>So what does all this tell us about geniuses like Ramanujan? If mathematical ability does stem from a core non-linguistic capacity to reason with spatial and numerical representation, this can help explain how a prodigious talent could blossom in the absence of training. While language might still play a role, the nature of the numerical representations being manipulated could be crucial. </p>
<p>The fact that genetics seems to be involved also helps shed light on the case – Ramanujan could have simply inherited the ability. Nevertheless, we should not forget the important contribution of environment and education. While Ramanujan’s raw talent was sufficient to attract attention to his remarkable ability, it was the <a href="https://theconversation.com/the-man-who-taught-infinity-how-gh-hardy-tamed-srinivasa-ramanujans-genius-57585">later provision</a> of more formal mathematical training in India and England that allowed him to reach his full potential.</p><img src="https://counter.theconversation.com/content/59143/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>David Pearson does not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>You may have got what it takes to be a mathematical genius without even being aware of it.David Pearson, Reader of Cognitive Psychology, Anglia Ruskin UniversityLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/497462015-11-20T11:12:17Z2015-11-20T11:12:17ZThe rush to calculus is bad for students and their futures in STEM<figure><img src="https://images.theconversation.com/files/101493/original/image-20151110-5460-1v3e63d.JPG?ixlib=rb-4.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">The author, teaching at the very front of his calculus class.</span> <span class="attribution"><span class="source">Kevin Knudson</span>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span></figcaption></figure><p>Two years ago I taught a section of Calculus I to approximately 650 undergrad students in a large auditorium. Perhaps “taught” isn’t the right word. “Performed,” maybe? Unsurprisingly, my student evaluation scores were not as high as they usually are in my more typical classes of 35 students, but I do remember one comment in particular: “This class destroyed my confidence.” According to a <a href="http://www.nctm.org/Publications/Mathematics-Teacher/2015/Vol109/Issue3/Insights-from-the-MAA-National-Study-of-College-Calculus/">new report</a> from the Mathematical Association of America (MAA), this outcome is common, even among students who successfully completed a calculus course in high school. So what is going on? </p>
<p>Former MAA president <a href="http://www.macalester.edu/%7Ebressoud/">David Bressoud</a> led this five-year comprehensive study funded by the National Science Foundation. He’s been thinking about this problem for many years and has synthesized a huge amount of data measuring high school and college calculus enrollments. I heard Bressoud <a href="http://www.macalester.edu/%7Ebressoud/talks/2010/UFL-transition.pdf">speak</a> about some preliminary results of the study a few years ago, and one piece of data stuck in my head: in the mid-1980s, when I was in high school, approximately 5% of high school students took an AP exam in calculus.</p>
<p>That aligns with my personal experience in which there were about 150 students in my entire North Carolina county taking calculus in any given year (out of roughly 3,000 high school seniors). Nationally, about 60,000 students took an AP calculus exam my senior year (1987). Today? That number has risen to nearly 350,000 students taking an AP exam in calculus in 2011 (roughly 15% of high school students). As one of my colleagues remarked after Bressoud’s talk, it’s not as if the talent pool has gotten that much deeper in the last 30 years. This tripling of the proportion of students taking these exams feels wrong somehow. </p>
<h2>Why the dramatic increase?</h2>
<p>There appear to be at least two driving forces behind the rush to calculus. </p>
<figure class="align-right zoomable">
<a href="https://images.theconversation.com/files/102074/original/image-20151116-4947-11gmow6.png?ixlib=rb-4.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/102074/original/image-20151116-4947-11gmow6.png?ixlib=rb-4.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/102074/original/image-20151116-4947-11gmow6.png?ixlib=rb-4.1.0&q=45&auto=format&w=600&h=464&fit=crop&dpr=1 600w, https://images.theconversation.com/files/102074/original/image-20151116-4947-11gmow6.png?ixlib=rb-4.1.0&q=30&auto=format&w=600&h=464&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/102074/original/image-20151116-4947-11gmow6.png?ixlib=rb-4.1.0&q=15&auto=format&w=600&h=464&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/102074/original/image-20151116-4947-11gmow6.png?ixlib=rb-4.1.0&q=45&auto=format&w=754&h=583&fit=crop&dpr=1 754w, https://images.theconversation.com/files/102074/original/image-20151116-4947-11gmow6.png?ixlib=rb-4.1.0&q=30&auto=format&w=754&h=583&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/102074/original/image-20151116-4947-11gmow6.png?ixlib=rb-4.1.0&q=15&auto=format&w=754&h=583&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Breakdown of all Advanced Placement exams taken in 2013.</span>
<span class="attribution"><a class="source" href="https://commons.wikimedia.org/wiki/File:AP_Exams_Taken_in_2013.svg">Ali Zifan</a>, <a class="license" href="http://creativecommons.org/licenses/by-sa/4.0/">CC BY-SA</a></span>
</figcaption>
</figure>
<p>One is college admissions. Students and their parents seek an advantage in the increasingly competitive admissions tournament, and the number of AP courses taken is a metric that is easy for students to boost. The increase in the number of AP exams taken is not unique to calculus; indeed, the total population of <a href="http://apreport.collegeboard.org/">students taking exams</a> doubled between 2003 and 2013, with the number of exams administered increasing by 150% over that period. As the name “Advanced Placement” suggests, these exams often yield college credit for students; this appeals to parents, as well, since it ostensibly lowers tuition costs later.</p>
<p>Another factor that must be considered is the <a href="http://www.nytimes.com/roomfordebate/2014/06/03/are-new-york-citys-gifted-classrooms-useful-or-harmful/americas-future-depends-on-gifted-students">overall decline in support</a> for enhanced education for gifted students. In an era of shrinking education budgets, school administrators find it tempting to conflate advancement with enrichment. Pushing gifted students ahead at a faster rate via AP courses is seen as a solution for meeting the needs of advanced students.</p>
<p>This approach may be dangerous in any discipline, but it is especially problematic in mathematics, where a strong foundation is key to success in upper division courses. The general strategy in high school is one of uniform advancement – taking advanced coursework in all disciplines under the assumption that gifted students are exceptional in every subject. In the drive to make it to calculus by the senior year, students often rush through algebra and geometry in lockstep with their gifted peers whether they are ready for it or not.</p>
<p>The end result is a group of students who have “succeeded” in high school calculus without really having the proper foundations, a tower built on sand. It is quite possible for students to learn the mechanics of many categories of calculus problems and to answer questions correctly on exams without really understanding the concepts. To quote the MAA’s report:</p>
<blockquote>
<p>In some sense, the worst preparation a student heading toward a career in science or engineering could receive is one that rushes toward accumulation of problem-solving abilities in calculus while short-changing the broader preparation needed for success beyond calculus.</p>
</blockquote>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/102073/original/image-20151116-4976-1clnf19.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/102073/original/image-20151116-4976-1clnf19.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/102073/original/image-20151116-4976-1clnf19.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=600&h=399&fit=crop&dpr=1 600w, https://images.theconversation.com/files/102073/original/image-20151116-4976-1clnf19.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=600&h=399&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/102073/original/image-20151116-4976-1clnf19.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=600&h=399&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/102073/original/image-20151116-4976-1clnf19.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&h=501&fit=crop&dpr=1 754w, https://images.theconversation.com/files/102073/original/image-20151116-4976-1clnf19.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=754&h=501&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/102073/original/image-20151116-4976-1clnf19.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=754&h=501&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Calc students’ favorite friend: the graphing calculator.</span>
<span class="attribution"><a class="source" href="https://www.flickr.com/photos/cdnphoto/4537872477">Gene Wilburn</a>, <a class="license" href="http://creativecommons.org/licenses/by-nc-nd/4.0/">CC BY-NC-ND</a></span>
</figcaption>
</figure>
<h2>College versus high school calculus</h2>
<p>There are two flavors of AP calculus, AB and BC. The former is equivalent to a typical first-semester college course, while the latter covers the first two semesters. Exams are scored from 1 to 5; most universities grant credit for a score of 3 and up.</p>
<p>Many students take Calculus I again at their universities, even if they have a passing score on the AP exam. There are many reasons for this: some colleges insist (engineering programs in particular) and many medical schools <a href="http://www.cse.emory.edu/sciencenet/additional_math_reqs.pdf">demand</a> that applicants take the course at a university. Or students may not feel particularly confident about their abilities. In my own experience, the number of students retaking the calculus course is very high – in a typical section of engineering calculus, up to 90% of my students have taken it in high school. While there are some positive aspects to retaking the course, there are downsides, the most notable of which is overconfidence and a student’s misplaced certainty that he or she already knows the material.</p>
<p>A typical first-semester calculus course consists of 45 lectures delivered three times per week over a 15-week term. The pace is quick. Contrast that with a typical high school Calculus AB course, which meets five days per week for 180 class meetings. The college course covers the same material in a quarter of the time; students must therefore have solid skills in algebra and geometry along with good study and work habits to succeed.</p>
<p>So this is the crux of the problem: students lacking the requisite foundational abilities may not succeed because the college faculty member expects them to be at ease with these more basic ideas, freeing them to absorb and understand the new, more conceptual material. The rush to AP Calculus has instructed students in the techniques for solving large classes of standard calculus problems rather than prepare them for success in higher mathematics.</p>
<p>It’s precisely this disconnect that causes students to lose their confidence if they don’t do well in university calculus. All through high school, the evidence suggested that they were “good at math” because they succeeded in parroting what they saw demonstrated in class. Parroting is not learning, however, and may hide a student’s true abilities.</p>
<h2>What to do?</h2>
<p>The authors of the MAA report sum it up best:</p>
<blockquote>
<p>Students are better prepared for post-secondary mathematics when they have developed an understanding of the undergirding principles which, when accompanied by fluent and flexible application of the concepts and procedures of precalculus mathematics, enable them to understand calculus as a coherent and broadly applicable body of knowledge.</p>
</blockquote>
<p>Like so many issues in K-12 education, the reasons that we have gotten to the current state are manifold, and reversing trends is difficult. But if we want to advance STEM education and continue to produce a high-quality technical workforce we must confront this issue. We need to stop the rush to calculus and focus instead on a thorough grounding in algebra, geometry and functions.</p>
<p>Calculus is one of the great intellectual achievements of the last 400 years; shortchanging it by reducing its beauty and utility to a list of problems to be checked off a rubric does a disservice to everyone.</p><img src="https://counter.theconversation.com/content/49746/count.gif" alt="The Conversation" width="1" height="1" />
More students are taking Advanced Placement calculus in high school. They may be learning techniques for solving certain problems at the expense of the mathematical foundations they need to advance.Kevin Knudson, Professor of Mathematics, University of FloridaLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/465852015-09-09T10:25:07Z2015-09-09T10:25:07ZThe Common Core is today’s New Math – which is actually a good thing<figure><img src="https://images.theconversation.com/files/94197/original/image-20150908-4358-zdmhft.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Change can be a good thing – really.</span> <span class="attribution"><a class="source" href="http://www.shutterstock.com/pic-182868605/stock-photo-frustrated-father-throws-up-his-hands-in-despair-frustrated-elementary-age-boy-lays-his-head-on.html">Homework image via www.shutterstock.com.</a></span></figcaption></figure><p>Math can’t catch a break. These days, people on both ends of the political spectrum are lining up to deride the <a href="http://www.corestandards.org/">Common Core standards</a>, a set of guidelines for K-12 education in reading and mathematics. The Common Core standards outline what a student should know and be able to do at the end of each grade. States don’t have to adopt the standards, although many did in an effort to receive funds from President Obama’s <a href="http://www2.ed.gov/programs/racetothetop/index.html">Race to the Top</a> initiative.</p>
<p><a href="http://www.usnews.com/news/special-reports/a-guide-to-common-core/articles/2014/02/27/who-is-fighting-against-common-core">Conservatives</a> oppose the guidelines because they generally dislike any suggestion that the federal government might have a role to play in public education at the state and local level; these standards, then, are perceived as a threat to local control.</p>
<p><a href="https://www.laprogressive.com/fighting-common-core/">Liberals</a>, mostly via teachers’ unions, decry the use of the standards and the associated assessments to evaluate classroom instructors.</p>
<p>And parents of all persuasions are panicked by their sudden inability to help their children with their homework. Even <a href="http://www.newyorker.com/news/daily-comment/louis-c-k-against-the-common-core">comedian Louis CK got in on the discussion</a> (via Twitter; he has since deactivated his account). </p>
<blockquote>
<p>My kids used to love math. Now it makes them cry. Thanks standardized testing and common core!
— Louis CK (@louisck) April 28 2014</p>
</blockquote>
<p>In the middle are millions of American schoolchildren who are often confused and frustrated by these “new” ways of teaching mathematics.</p>
<p>Thing is, we’ve been down this path before.</p>
<h2>The old New Math</h2>
<p>When the Soviets launched Sputnik in 1957, the United States went into panic mode. Our schools needed to emphasize math and science so that we wouldn’t fall behind the Soviet Union and its allegedly superior scientists. In 1958, President Eisenhower signed the <a href="http://www.britannica.com/topic/National-Defense-Education-Act">National Defense Education Act</a>, which poured money into the American education system at all levels. </p>
<p>One result of this was the so-called New Math, which <a href="https://en.wikipedia.org/wiki/Secondary_School_Mathematics_Curriculum_Improvement_Study#Curriculum">focused more on conceptual understanding of mathematics</a> over rote memorization of arithmetic. Set theory took a central role, forcing students to think of numbers as sets of objects rather than abstract symbols to be manipulated. This is actually how numbers are constructed logically in an advanced undergraduate mathematics course on real analysis, but it may not necessarily be the best way to communicate ideas like addition to schoolchildren. Arithmetic using number bases other than 10 also entered the scene. This was famously spoofed by <a href="https://en.wikipedia.org/wiki/Tom_Lehrer">Tom Lehrer</a> in his song “New Math.”</p>
<figure>
<iframe width="440" height="260" src="https://www.youtube.com/embed/UIKGV2cTgqA?wmode=transparent&start=0" frameborder="0" allowfullscreen=""></iframe>
<figcaption><span class="caption">This 60’s song about New Math gives us a glimpse of what the ‘old math’ was like.</span></figcaption>
</figure>
<p>I attended elementary school in the 1970s, so I missed New Math’s implementation, and it was largely gone by the time I got started. But the way Lehrer tries to explain how subtraction “used to be done” made no sense to me at first (I did figure it out after a minute). In fact, the New Math method he ridicules is how children of my generation – and many of the Common Core-protesting parents of today – learned to do it, even if some of us don’t really understand what the whole borrowing thing is conceptually. Clearly some of the New Math ideas took root, and math education is better for it. For example, given the ubiquity of computers in modern life, it’s useful for today’s students to learn to do binary arithmetic – adding and subtracting numbers in base 2 just as a computer does. </p>
<p>The New Math fell into disfavor mostly because of complaints from parents and teachers. Parents were unhappy because they couldn’t understand their children’s homework. Teachers objected because they were often unprepared to instruct their students in the new methods. In short, it was the <em>implementation</em> of these new concepts that led to the failure, more than the curriculum itself.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/94201/original/image-20150908-14047-1q4g6nd.JPG?ixlib=rb-4.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/94201/original/image-20150908-14047-1q4g6nd.JPG?ixlib=rb-4.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/94201/original/image-20150908-14047-1q4g6nd.JPG?ixlib=rb-4.1.0&q=45&auto=format&w=600&h=427&fit=crop&dpr=1 600w, https://images.theconversation.com/files/94201/original/image-20150908-14047-1q4g6nd.JPG?ixlib=rb-4.1.0&q=30&auto=format&w=600&h=427&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/94201/original/image-20150908-14047-1q4g6nd.JPG?ixlib=rb-4.1.0&q=15&auto=format&w=600&h=427&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/94201/original/image-20150908-14047-1q4g6nd.JPG?ixlib=rb-4.1.0&q=45&auto=format&w=754&h=536&fit=crop&dpr=1 754w, https://images.theconversation.com/files/94201/original/image-20150908-14047-1q4g6nd.JPG?ixlib=rb-4.1.0&q=30&auto=format&w=754&h=536&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/94201/original/image-20150908-14047-1q4g6nd.JPG?ixlib=rb-4.1.0&q=15&auto=format&w=754&h=536&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Give us our New Math!</span>
</figcaption>
</figure>
<h2>Those who ignore history…</h2>
<p>In 1983, President Reagan’s National Commission on Excellence in Education released its report, <a href="http://www2.ed.gov/pubs/NatAtRisk/index.html">A Nation at Risk</a>, which asserted that American schools were “failing” and suggested various measures to right the ship. Since then, American schoolchildren and their teachers have been bombarded with various reform initiatives, privatization efforts have been launched and charter schools established.</p>
<p>Whether or not the nation’s public schools are actually failing is a matter of serious debate; indeed, many of the claims made in A Nation at Risk were <a href="http://eric.ed.gov/?id=EJ482502">debunked</a> by statisticians at Sandia National Laboratories a few years after the report’s release. But the general notion that our public schools are “bad” persists, especially among politicians and business groups. </p>
<p>Enter Common Core. Launched in 2009 by a consortium of states, the idea sounds reasonable enough – public school learning objectives should be more uniform nationally. That is, what students learn in math or reading at each grade level should not vary state by state. That way, colleges and employers will know what high school graduates have been taught, and it will be easier to compare students from across the country. </p>
<p>The guidelines are just that. There is no set curriculum attached to them; they are merely a list of concepts that students should be expected to master at each grade level. For example, here are the <a href="http://www.corestandards.org/Math/Content/3/NBT/">standards</a> in Grade 3 for Number and Operations in Base Ten:</p>
<ul>
<li><p>Use place value understanding and properties of operations to perform multi-digit arithmetic.</p></li>
<li><p><a href="http://www.corestandards.org/Math/Content/3/NBT/A/1/">CCSS.Math.Content.3.NBT.A.1</a>
Use place value understanding to round whole numbers to the nearest 10 or 100.</p></li>
<li><p><a href="http://www.corestandards.org/Math/Content/3/NBT/A/2/">CCSS.Math.Content.3.NBT.A.2</a>
Fluently add and subtract within 1,000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.</p></li>
<li><p><a href="http://www.corestandards.org/Math/Content/3/NBT/A/3/">CCSS.Math.Content.3.NBT.A.3</a>
Multiply one-digit whole numbers by multiples of 10 in the range 10-90 (eg, 9 × 80, 5 × 60) using strategies based on place value and properties of operations.</p></li>
</ul>
<p>There is a footnote that “a range of algorithms may be used” to help students complete these tasks. In other words, teachers can explain various methods to actually accomplish the mathematical task at hand. There is nothing controversial about these topics, and indeed it’s not controversial that they’re things that students should be able to do at that age.</p>
<p>However, some of the new methods being taught for doing arithmetic have caused confusion for parents, causing them to take to social media in frustration. Take the 32 - 12 problem, for example:</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/93956/original/image-20150904-14609-18h7ni1.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/93956/original/image-20150904-14609-18h7ni1.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/93956/original/image-20150904-14609-18h7ni1.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=600&h=800&fit=crop&dpr=1 600w, https://images.theconversation.com/files/93956/original/image-20150904-14609-18h7ni1.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=600&h=800&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/93956/original/image-20150904-14609-18h7ni1.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=600&h=800&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/93956/original/image-20150904-14609-18h7ni1.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&h=1005&fit=crop&dpr=1 754w, https://images.theconversation.com/files/93956/original/image-20150904-14609-18h7ni1.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=754&h=1005&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/93956/original/image-20150904-14609-18h7ni1.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=754&h=1005&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Just because you didn’t learn it that way doesn’t make it inscrutable or wrong.</span>
</figcaption>
</figure>
<p>Once again, it’s the <em>implementation</em> that’s causing the problem. Most parents (people age 30-45, mostly), remembering the math books of our youth filled with pages of exercises like this, immediately jump to the “Old Fashion” (sic) algorithm shown. The stuff at the bottom looks like gibberish, and given many adults’ <a href="https://theconversation.com/when-parents-with-high-math-anxiety-help-with-homework-children-learn-less-46841">tendency toward math phobia/anxiety</a>, they immediately throw up their hands and claim this is nonsense.</p>
<p>Except that it isn’t. In fact, we all do arithmetic like this in our heads all the time. Say you are buying a scone at a bakery for breakfast and the total price is US$2.60. You hand the cashier a $10 bill. How much change do you get? Now, you do <em>not</em> perform the standard algorithm in your head. You first note that you’d need another 40 cents to get to the next dollar, making $3, and then you’d need $7 to get up to $10, so your change is $7.40. That’s all that’s going on at the bottom of the page in the picture above. Your children can’t explain this to you because they don’t know that you weren’t taught this explicitly, and your child’s teacher can’t send home a primer for you either.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/94200/original/image-20150908-15659-ep1zt2.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/94200/original/image-20150908-15659-ep1zt2.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/94200/original/image-20150908-15659-ep1zt2.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/94200/original/image-20150908-15659-ep1zt2.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/94200/original/image-20150908-15659-ep1zt2.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/94200/original/image-20150908-15659-ep1zt2.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/94200/original/image-20150908-15659-ep1zt2.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/94200/original/image-20150908-15659-ep1zt2.jpg?ixlib=rb-4.1.0&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">New ways to learn can be better for students – if rolled out appropriately.</span>
<span class="attribution"><a class="source" href="https://www.flickr.com/photos/departmentofed/9610695698">US Department of Education</a>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span>
</figcaption>
</figure>
<h2>Better intuition about math, better problem-solving</h2>
<p>As an instructor of college-level mathematics, I view this focus on conceptual understanding and multiple strategies for solving problems as a welcome change. Doing things this way can help build intuition about the size of answers and help with estimation. College students can compute answers to homework problems to 10 decimal places, but ask them to ballpark something without a calculator and I get blank stares. Ditto for conceptual understanding – for instance, students can evaluate <a href="https://en.wikipedia.org/wiki/Integral">integrals</a> with relative ease, but building one as a limit of <a href="https://en.wikipedia.org/wiki/Riemann_sum">Riemann sums</a> to solve an actual problem is often beyond their reach.</p>
<p>This is frustrating because I know that my colleagues and I focus on these notions when we introduce these topics, but they fade quickly from students’ knowledge base as they shift their attention to solving problems for exams. And, to be fair, since the K-12 math curriculum is chopped up into discrete chunks of individual topics for ease of standardized testing assessment, it’s often difficult for students to develop the problem-solving abilities they need for success in higher-level math, science and engineering work. Emphasizing more conceptual understanding at an early age will hopefully lead to better problem-solving skills later. At least that’s the rationale behind the standards.</p>
<p>Alas, Common Core is in danger of being abandoned. Some states have already <a href="http://academicbenchmarks.com/common-core-state-adoption-map/">dropped the standards</a> (Indiana and South Carolina, for example), looking to replace them with something else. But these actions are largely a result of mistaken conflations: that the standards represent a federal imposition of curriculum on local schools, that the <a href="http://www.parcconline.org/about">standardized tests</a> used to evaluate students <em>are</em> the Common Core rather than a separate initiative.</p>
<p>As the 2016 presidential campaign heats up, support for the Common Core has become a political liability, possibly killing it before it really has a chance. That would be a shame. The standards themselves are fine, and before we throw the baby out with the bathwater, perhaps we should consider efforts to implement them properly. To give the Common Core a fair shot, we need appropriate professional development for teachers and a more phased introduction of new standardized testing attached to the standards.</p>
<p>But, if we do ultimately give in to panic and misinformation, let’s hope any replacement provides proper coherence and rigor. Above all, our children should develop solid mathematical skills that will help them see the beauty and utility of this wonderful subject.</p><img src="https://counter.theconversation.com/content/46585/count.gif" alt="The Conversation" width="1" height="1" />
Both have been much maligned by parents who felt like they couldn’t help their kids with basic math homework. But the Common Core could help with conceptual understanding and math intuition.Kevin Knudson, Professor of Mathematics, University of FloridaLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/463602015-09-01T13:54:40Z2015-09-01T13:54:40ZHow to get children to want to do maths outside the classroom<figure><img src="https://images.theconversation.com/files/93297/original/image-20150828-19940-qs4qnv.JPG?ixlib=rb-4.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Let's all go on a maths walk.</span> <span class="attribution"><span class="source">Steve Humble</span>, <span class="license">Author provided</span></span></figcaption></figure><p>Ask adults about maths and they’ll often say: “I was never very good at maths at school”. How can we stop young children growing up today saying the same thing? </p>
<p>It’s important to <a href="http://www.taylorandfrancis.com/education/articles/routledge_education_author_of_the_month_march_2015_steve_humble_aka_dr_math/">be inventive when teaching children</a> maths in primary school, in order to get them hooked early and want to keep doing more maths when they get home. In art and English lessons it’s easy for children to have ownership – “this is my piece of art” or “these are my thoughts in this essay”. It might seem that in maths this is harder to do, if not impossible. But ownership of maths is important so that children and adults can also say – “this is my maths”.</p>
<p>One way to develop ownership is to take children on a “maths walk”, opening their eyes up to the world around them. It’s like a treasure hunt, with the treasures hidden all around us waiting to be observed. </p>
<h2>Going on a maths walk</h2>
<p>A typical walk consists of a sequence of designated sites along a planned route where students stop to explore maths in the environment. This makes maths come alive for children. </p>
<p>Anyone can create a walk that targets a range of mathematical understanding. Some questions that can be used to help switch on your “mathematical eyes” include:</p>
<ul>
<li> Find three objects which have one line of symmetry</li>
<li> Find an object with rotational symmetry</li>
<li> Find a repeating pattern</li>
<li> Find an object that is approximately one metre tall</li>
</ul>
<p>Alternatively, the maths walk’s purpose could be to introduce children to places where mathematics can be found in our everyday world. These can include locations which have scientific, historical, literary, engineering, or business significance. These walks should invite all children, irrespective of their achievement level, to participate successfully in problem-solving activities and gain a sense of pride in the mathematics they create. </p>
<p>I recently developed a <a href="http://art.tfl.gov.uk/labyrinth/wp-content/uploads/2013/04/MW_MATH_TRAIL_TUBE_TRAIL_AW_REV.pdf">maths walk</a> to celebrate 150 years of the London Underground, that could be used at any of the 270 tube stations in London. Here is one of the questions from the “Tube puzzle walk” called “number cruncher”:</p>
<blockquote>
<p>Each underground train carriage has an identification number. These numbers can be three, four or five digits long. Similarly, each labyrinth artwork is numbered at the bottom right-hand corner. Your challenge is to make a calculation equalling the labyrinth number using only the digits of your train carriage number. </p>
<p>For example, if your carriage number is 5547 and the labyrinth number is 22 then you could create the following calculation: 4 × 5 + 7 – 5 = 22 </p>
<p>If your carriage number is 21060 and the labyrinth number is 63 then you could create either one of the following calculations:
60 + 2 + 1 or 2<sup>6</sup> –1 = 2 × 2 × 2 × 2 × 2 × 2 – 1 </p>
</blockquote>
<h2>Making maths stick</h2>
<p>Motivating learning is one of the keys to creating a positive experience for all learners. It was not until I came across research that looked at the benefits of “<a href="http://eric.ed.gov/?id=EJ876166">transfer triggers</a>” in education that I realised the power of teaching mathematics using real world examples. </p>
<figure class="align-right ">
<img alt="" src="https://images.theconversation.com/files/92542/original/image-20150820-7231-18gtayo.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/92542/original/image-20150820-7231-18gtayo.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=600&h=800&fit=crop&dpr=1 600w, https://images.theconversation.com/files/92542/original/image-20150820-7231-18gtayo.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=600&h=800&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/92542/original/image-20150820-7231-18gtayo.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=600&h=800&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/92542/original/image-20150820-7231-18gtayo.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&h=1005&fit=crop&dpr=1 754w, https://images.theconversation.com/files/92542/original/image-20150820-7231-18gtayo.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=754&h=1005&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/92542/original/image-20150820-7231-18gtayo.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=754&h=1005&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Numbers underground.</span>
<span class="attribution"><span class="source">Steve Humble</span></span>
</figcaption>
</figure>
<p>Emotion is also very valuable for young children’s mathematical memory. One engaging technique is called “episodic learning” – creating a moment in a child’s mind upon which to build learning. <a href="http://www.tandfonline.com/doi/abs/10.1080/00958964.2014.905431#.VeBSnlZj71o">Research shows</a> that these types of learning moments can help to develop stronger and longer lasting memories, which help cultivate learning.</p>
<p>There are lots of other things you can do with children outside school that can help their understanding of mathematics. Most of these episodes do not even seem mathematical to the children, as maths is so deeply linked to our daily life. </p>
<p>Card, board and dice games develop number skills, logic and strategy. Games <a href="http://www.kappancommoncore.org/it-all-adds-up-learning-early-math-through-play-and-games/">help develop mathematical understanding</a> and are fun for the whole family. Cooking together also teaches children about weights and measures. If a recipe is for two and you want to make it for five, you need to think about ratios. Cooking also involves all sorts of non-standard units of measure such as teaspoons, tablespoons, millilitres, pints and cups. Having conversations as you cook enhances children’s understanding of these units. </p>
<p>Travelling in the car can also bring possibilities for mathematical games such as counting cars of a certain age, colour or type that you pass. Children can learn about probability with this activity, as you may be more likely to spot silver cars than yellow.</p>
<p>At a young age, mathematics learning is about giving children confidence and a willingness to try. It is also about explaining mathematical ideas from an inventive perspective that encourage children to take part, think about maths differently and associate what they are learning with their everyday lives.</p><img src="https://counter.theconversation.com/content/46360/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Steve Humble 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>Maths is all around us. Let children hunt for it.Steve Humble, Mathematics Education Primary and Secondary PGCE, Newcastle UniversityLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/430532015-06-25T10:36:38Z2015-06-25T10:36:38ZDon’t freak if you can’t solve a math problem that’s gone viral<figure><img src="https://images.theconversation.com/files/86322/original/image-20150624-31526-1jbqvaz.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Are you smarter than a third grader in Vietnam?</span> <span class="attribution"><a class="source" href="http://www.shutterstock.com/pic.mhtml?id=252676549&src=lb-29877982">Woman image via www.shutterstock.com</a></span></figcaption></figure><p>It’s been quite a year for mathematics problems on the internet. In the last few months, three questions have been online everywhere, causing consternation and head-scratching and blowing the minds of adults worldwide as examples of what kids are expected to know these days.</p>
<p>As a mathematician, I suppose I should subscribe to the “no such thing as bad publicity” theory, except that problems of this ilk a) usually aren’t that difficult once you get the trick, b) sometimes aren’t even math problems and c) fuel the defeatist “I’m not good at math” fire that pervades American culture. The inability to solve such a problem quickly is certainly not indicative of a person’s overall math skill, nor should it prompt a crisis of confidence about the state of American math aptitude.</p>
<h2>When is Cheryl’s birthday?</h2>
<p>In April, the internet erupted with shock that 10-year-olds in Singapore were asked to answer the following question on an exam.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/86311/original/image-20150624-31504-1omznvi.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/86311/original/image-20150624-31504-1omznvi.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/86311/original/image-20150624-31504-1omznvi.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=600&h=436&fit=crop&dpr=1 600w, https://images.theconversation.com/files/86311/original/image-20150624-31504-1omznvi.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=600&h=436&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/86311/original/image-20150624-31504-1omznvi.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=600&h=436&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/86311/original/image-20150624-31504-1omznvi.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&h=548&fit=crop&dpr=1 754w, https://images.theconversation.com/files/86311/original/image-20150624-31504-1omznvi.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=754&h=548&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/86311/original/image-20150624-31504-1omznvi.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=754&h=548&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">The logic puzzle from the Singapore and Asian Math Olympiads.</span>
</figcaption>
</figure>
<p>Except that it wasn’t for elementary school students at all; rather it appeared on an Asian Olympiad exam designed for mathematically talented high school students. What’s more, this isn’t even a math problem, but a logic problem. It’s true that students tend to learn formal logic via mathematics (plane geometry in particular), so it is common to see problems of this type in mathematics competitions. When I was in junior high, we spent a good deal of time on these puzzles in my language arts class, and I met them again when taking the GRE prior to entering graduate school (the test contains a whole section of them). </p>
<p>If you’re stumped, check out a <a href="http://www.independent.co.uk/news/world/asia/singapore-maths-problem-can-you-solve-the-viral-maths-question-that-was-set-to-children-in-singapore-10173090.html">solution to the problem</a>.</p>
<h2>Vietnamese eight-year-olds do arithmetic</h2>
<p>A month later, we heard about a third grade teacher in Vietnam who set the following puzzle for his students. Place the digits from 1 to 9 in this grid, using each only once (the : represents division).</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/85280/original/image-20150616-5829-129m39.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/85280/original/image-20150616-5829-129m39.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=600&h=433&fit=crop&dpr=1 600w, https://images.theconversation.com/files/85280/original/image-20150616-5829-129m39.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=600&h=433&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/85280/original/image-20150616-5829-129m39.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=600&h=433&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/85280/original/image-20150616-5829-129m39.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&h=545&fit=crop&dpr=1 754w, https://images.theconversation.com/files/85280/original/image-20150616-5829-129m39.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=754&h=545&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/85280/original/image-20150616-5829-129m39.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=754&h=545&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">A puzzle for Vietnamese children.</span>
<span class="attribution"><span class="source">VN Express</span></span>
</figcaption>
</figure>
<p>This reminds me of the (probably apocraphyl) story of one of the greatest mathematicians in history, <a href="https://en.wikipedia.org/wiki/Carl_Friedrich_Gauss">Carl Friedrich Gauss</a>. Legend has it that when Gauss was seven or eight, his teacher, wanting to occupy his students for a while, told the class to add up the numbers from 1 to 100. Gauss thought about it for 30 seconds or so and wrote the correct answer, 5,050, on his slate and turned it in.</p>
<p>The puzzle above has a similar feel. It’s really a question about knowing the order of arithmetic operations (multiplication/division, addition/subtraction, in that order). Beyond that, it just takes trial and error; that is, it’s kind of just busy work. Someone who knows some algebra might be able to generate some equations to gain insight into how you might find a <a href="http://www.theguardian.com/science/alexs-adventures-in-numberland/2015/may/20/can-you-do-the-maths-puzzle-for-vietnamese-eight-year-olds-that-has-stumped-parents-and-teachers">solution</a>.</p>
<p>Another approach would be to open up a spreadsheet program and just try all the possibilities. Since there are nine choices for the first box, then eight choices for the second, and so on, there are only (9)(8)(7)(6)(5)(4)(3)(2)(1) = 362,880 possible configurations, of which only a few will give a valid equation. This can be programmed with very little effort.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/86327/original/image-20150624-31495-14gxmoi.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/86327/original/image-20150624-31495-14gxmoi.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/86327/original/image-20150624-31495-14gxmoi.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=600&h=480&fit=crop&dpr=1 600w, https://images.theconversation.com/files/86327/original/image-20150624-31495-14gxmoi.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=600&h=480&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/86327/original/image-20150624-31495-14gxmoi.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=600&h=480&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/86327/original/image-20150624-31495-14gxmoi.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&h=603&fit=crop&dpr=1 754w, https://images.theconversation.com/files/86327/original/image-20150624-31495-14gxmoi.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=754&h=603&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/86327/original/image-20150624-31495-14gxmoi.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=754&h=603&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Yellow or orange, students didn’t find the problem sweet.</span>
<span class="attribution"><a class="source" href="http://www.shutterstock.com/pic-275334341/stock-photo-orange-and-yellow-jelly-candies-closeup-sweet-background.html">Candy image via www.shutterstock.com</a></span>
</figcaption>
</figure>
<h2>Hannah’s sweets</h2>
<p>Just a couple of weeks ago, students in the UK vented their frustration via social media about a problem on the Edexcel GCSE (General Certificates of Secondary Education) mathematics exam. It is a probability question: Hannah has a bag containing <em>n</em> candies, six of which are orange and the rest of which are yellow. She takes two candies out of the bag and eats them. The probability that she ate two orange candies is 1/3. Given this, show that <em>n² - n - 90 = 0</em>. The students’ complaint? It’s too difficult.</p>
<p><div data-react-class="Tweet" data-react-props="{"tweetId":"606837485239463937"}"></div></p>
<p>I’ve taught math long enough to recognize the pitfalls of setting this problem. The students actually have the knowledge to do it, if they know basic probability, but it is unlike problems they would have practiced. A typical question would indicate the total number of candies in the bag and ask students to compute the probability of a certain outcome. This question gives the probability and asks for a condition on the number of candies. It’s just algebra. You may read the solution (and some humorous memes about the question) <a href="http://www.telegraph.co.uk/education/11652918/Students-vent-their-frustration-at-Edexcel-GCSE-maths-exam.html">here</a>.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/86326/original/image-20150624-31495-1wy27l9.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/86326/original/image-20150624-31495-1wy27l9.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/86326/original/image-20150624-31495-1wy27l9.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=600&h=450&fit=crop&dpr=1 600w, https://images.theconversation.com/files/86326/original/image-20150624-31495-1wy27l9.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=600&h=450&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/86326/original/image-20150624-31495-1wy27l9.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=600&h=450&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/86326/original/image-20150624-31495-1wy27l9.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&h=566&fit=crop&dpr=1 754w, https://images.theconversation.com/files/86326/original/image-20150624-31495-1wy27l9.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=754&h=566&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/86326/original/image-20150624-31495-1wy27l9.jpg?ixlib=rb-4.1.0&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">What does his lifelong future with math look like?</span>
<span class="attribution"><a class="source" href="https://www.flickr.com/photos/grahams__flickr/360774920">Prisoner 5413</a>, <a class="license" href="http://creativecommons.org/licenses/by-nc/4.0/">CC BY-NC</a></span>
</figcaption>
</figure>
<h2>A nation at risk?</h2>
<p>Mathematicians dread cocktail parties because we inevitably have to endure the response we receive when asked what we do: “Oh, I hated (or am terrible at) math.” No other subject in school receives such scorn, nor would we find it acceptable for an adult to admit they are terrible at reading or writing. So when these “unsolvable” problems pop up, they simply reinforce our culture’s math anxiety. </p>
<p>And that’s a real shame, because everyone likes math when they’re young. We all like to count. We like playing with blocks and shapes. We all use math daily whether we realize it or not – reading maps, planning routes, calculating tips. I once had a flooring installer tell me he was bad at math <em>while I watched him lay tile</em>. <a href="http://www.theatlantic.com/education/archive/2013/10/the-myth-of-im-bad-at-math/280914/">It’s a myth</a> that all these people can’t do math. When people say they are “bad at math,” they usually mean that they had trouble with algebra, although if you corner them and ask the right questions you can usually make them realize that they use algebra all the time without noticing it. This leads to <a href="https://grantwiggins.wordpress.com/2013/04/10/my-100th-post-so-why-not-bash-algebra/">valid criticisms</a> of how we teach math, but it doesn’t mean we’re a nation of math idiots.</p>
<p>So, the next time one of these outrageous problems comes along, instead of giving in to anxiety, why not think about it for a few minutes and try to find a solution? You might be surprised how satisfying it can be.</p><img src="https://counter.theconversation.com/content/43053/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Kevin Knudson does not work for, consult, own shares in or receive funding from any company or organization that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>People shouldn’t let these tricky puzzlers reinforce their misguided notion that they stink at math.Kevin Knudson, Professor of Mathematics, University of FloridaLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/402162015-04-16T01:58:18Z2015-04-16T01:58:18ZIt’s often the puzzles that baffle that go viral<figure><img src="https://images.theconversation.com/files/78033/original/image-20150415-24615-x1edmo.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">A love of puzzles.</span> <span class="attribution"><a class="source" href="https://www.flickr.com/photos/jypsygen/3732589905">Flickr/jypsygen</a>, <a class="license" href="http://creativecommons.org/licenses/by-nc-nd/4.0/">CC BY-NC-ND</a></span></figcaption></figure><p>Many of us are devoted to our morning crossword, acrostic, anagram or Sudoku puzzle. Quite a few religiously listen to the Sunday <a href="http://www.npr.org/people/2101852/will-shortz">Puzzlemaster</a> <a href="http://willshortz.com/">Will Shortz</a> (who also sets puzzles for the New York Times) on National Public Radio.</p>
<p>So perhaps it is not surprising – even though many of us did not like school maths – that every so often a logical puzzle or maths problem goes viral. The most recent example is “<a href="http://www.9news.com.au/world/2015/04/14/11/09/can-you-solve-this-singaporean-maths-problem">Cheryl’s birthday</a>”.</p>
<p>The puzzle was originally posted on the <a href="https://www.facebook.com/kennethjianwenz/photos/a.173663129479243.1073741827.167504136761809/385751114937109/">Facebook page</a> of Singapore media personality Kenneth Hong, who said it was causing some debate with his wife.</p>
<blockquote>
<p>Albert and Bernard just became friends with Cheryl. and they want to know when her birthday is. Cheryl gives them a list of 10 possible dates.</p>
<div class="highlight"><pre class="highlight plaintext"><code> May 15 May 16 May 19
June 17 June 18
July 14 July 16
August 14 August 15 August 17
</code></pre></div>
<p>Cheryl then tells Albert and Bernard separately the month and the day of her birthday respectively.</p>
<p>Albert: I don’t know when Cheryl’s birthday is, but I know that Bernard does not know too.</p>
<p>Bernard: At first I don’t know when Cheryl’s birthday is, but I know now.</p>
<p>Albert: Then I also know when Cheryl’s birthday is.</p>
<p>So when is Cheryl’s birthday?</p>
</blockquote>
<p>Many people have tried their hand at solving the puzzle, including mathematician and writer <a href="http://www.alexbellos.com/">Alex Bellos</a>. Alex runs through it <a href="http://www.theguardian.com/science/alexs-adventures-in-numberland/2015/apr/13/how-to-solve-albert-bernard-and-cheryls-birthday-maths-problem">line by line</a>, showing how he gets to the solution: the key is to ask what each of Bernard and Albert learn from the other’s statements.</p>
<p>Knowing what information is superfluous is often helpful, explains Alex:</p>
<blockquote>
<p>The only way that Bernard could know the date with a single number, however, would be if Cheryl had told him 18 or 19, since of the ten date options only these numbers appear once, as May 19 and June 18.</p>
</blockquote>
<p>Proceeding in like fashion (read the rest of Alex’s explanation <a href="http://www.theguardian.com/science/alexs-adventures-in-numberland/2015/apr/13/how-to-solve-albert-bernard-and-cheryls-birthday-maths-problem">here</a>) we are led to:</p>
<blockquote>
<p>The answer, therefore is July 16.</p>
</blockquote>
<p>It has been suggested that it would have been easier to consult Cheryl’s Facebook page!</p>
<p><div data-react-class="Tweet" data-react-props="{"tweetId":"587846560685297665"}"></div></p>
<h2>Where did Cheryl’s birthday problem arise?</h2>
<p>The problem was set as a hard question on a regional competition for bright high school students: the Singapore and Asian Schools Math Olympiad (<a href="http://mathsolympiads.org/">SASMO</a>).</p>
<p>There is another in <a href="APSMO%20Maths%20Olympiad">Australasia</a>, and they culminate in the annual <a href="https://www.imo-official.org/">International Mathematical Olympiad</a>, with the questions getting progressively harder as the regions grow. The world champion Maths Olympians (or “mathletes”) are incredibly gifted young problem solvers.</p>
<p>Cheryl’s birthday problem was inadvertently originally described as a grade 5 question for ordinary school kids in Singapore. This probably helped it go viral.</p>
<p>But why do certain things go viral in the first place? By convenient coincidence a new scientific study on <a href="http://www.scientificamerican.com/article/the-secret-to-online-success-what-makes-content-go-viral/">The Secret to Online Success: What Makes Content Go Viral</a> has just appeared – as described this week in Scientific American.</p>
<p>The researchers looked at what it was that made certain things “spread like wildfire” and whether it was possible to deliberately make content that would make something achieve viral status.</p>
<p>They found there were a number of things that could increase the chances of any content being widely shared.</p>
<blockquote>
<p>Make it emotional – ideally triggering emotions like anger, anxiety or awe that tend to make our hearts race; and if you can, make it positive. This may be more even effective than other methods that are currently in wide use like targeting “influentials,” or opinion leaders. Crafting contagious content, as this research suggests, may provide more bang for your buck and create more reliably viral content.</p>
</blockquote>
<p>You can tick off the emotional positives that helped Cheryl’s birthday problem go viral. It was unlikely and curious, but reassuring and nonthreatening. It did not tell you you were a dummy if you could not figure it out, far from it! A media personality helped it get going, and so on.</p>
<p>You can also inoculate against things going viral. My own worst-read blog in the Conversation had the unappetising title <a href="https://theconversation.com/danger-of-death-are-we-programmed-to-miscalculate-risk-4598">Danger of death: are we programmed to miscalculate risk?</a>. This seemed to be bringing only bad news when in fact we discussed how to calculate relative risk better and be less alarmed.</p>
<p>A <a href="http://www.mathresources.com/">software company</a> I helped set up 20 years ago saw the sales of one product sky-rocket when the name was changed from MathProbe (too medical?) to the more inviting Let’s do Math.</p>
<h2>What makes a puzzle easy or hard</h2>
<p>Cultural differences matter. For many years students in the French South Pacific were given exactly the same mathematics exams as those in Paris or Bordeaux.</p>
<p>A famous question asked students to “consider a dairy cow looking at the pole star on a snowy night”. This was served up for kids in the French South Pacific who in a far-ago pre-internet world had never seen snow, the northern sky, or even a cow.</p>
<p>Likewise numeric puzzles, such as Sudoku or nonograms, often arise in Japan or Korea whose ideogram based scripts make crosswords and anagrams a non-starter.</p>
<p>Keith Devlin, a mathematician who is a very gifted expositor, has a book <a href="http://www.amazon.com/Math-Instinct-Mathematical-Genius-Lobsters/dp/1560256729/ref=la_B000APRPC6_1_12?s=books&ie=UTF8&qid=1429061413&sr=1-12">The Math Instinct</a> in which, among other things he describes how a change in language can make a seeming impervious problem easy.</p>
<p>For example, doing arithmetic in bases other than base ten sounds formidable. But whenever you watch a 50-50 cricket match and read on the screen that it is over 42.3, the ‘42’ are in base ten and the ‘3’ is in base six. Easy-peasy?</p>
<p>Similarly, you might discover that an abstract problem about probability can be expressed in terms of <a href="https://theconversation.com/how-betting-works-and-why-the-melbourne-cup-skews-the-odds-33357">horse races</a> or lotteries, or something else you know lots about.</p>
<h2>Some other problems that went viral (or should have)</h2>
<p>Conditional, sometimes counter-factual, thinking of the kind needed to determine Cheryl’s birthday is not something most humans find easy. Though there is a large subset of humanity who find some or all puzzles both enticing and accessible.</p>
<p>I had a friend who could usually do the notoriously hard cryptic crossword in the London Times in about ten minutes. He could not really explain how, he just saw the answers. I have another friend who is an expert crossword puzzle setter, as of course is Will Shortz. </p>
<p>Setting good puzzles or just good maths exam questions is an art in itself.</p>
<p>One of the most popular viral puzzles is known as the Monty Hall or “three door problem”. It has already been explained in <a href="https://theconversation.com/the-monty-hall-problem-going-with-your-gut-will-get-your-goat-14195">The Conversation</a> and makes a surprising story. It first went viral in large part because the correct answer seemed so unintuitive even to professional logicians.</p>
<p>To those who want to follow up on some other mind tickling examples I conclude with:</p>
<ol>
<li><p>The paradox of the <a href="http://docserver.carma.newcastle.edu.au/209/">unexpected hanging</a> and its gentler version the <a href="http://www-math.mit.edu/%7Etchow/unexpected.pdf">surprise examination</a> paradox.</p></li>
<li><p>The paradox of the <a href="https://terrytao.wordpress.com/2008/02/05/the-blue-eyed-islanders-puzzle/">blue eyed islanders</a>.</p></li>
</ol><img src="https://counter.theconversation.com/content/40216/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Jonathan Borwein (Jon) receives funding from the Australian Research Council.</span></em></p>Have you heard the one about Cheryl’s birthday? It’s the latest puzzle that’s baffling people across the world.Jonathan Borwein (Jon), Laureate Professor of Mathematics, University of NewcastleLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/289472014-07-14T05:07:20Z2014-07-14T05:07:20ZBetter at reading than maths? Don’t blame it all on your genes<figure><img src="https://images.theconversation.com/files/53626/original/8xf2sq8v-1405076039.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Many of the genes for maths and reading overlap. </span> <span class="attribution"><a class="source" href="http://www.shutterstock.com/pic.mhtml?id=123441943&src=nJFIF-zLUgF3lijbyuh8eA-1-32">Businessman via alphaspirit/Shutterstock</a></span></figcaption></figure><p>I disliked and feared maths for most of my school career and dropped it as soon as I possibly could. My mother recalls me crying as a five-year-old because: “I can’t do the people-on-the-bus sums”. If the bus has 12 passengers and three get off, how many are left? English, by contrast, was a breeze. At seven, I stood on a chair with a microphone and read my version of Sleeping Beauty aloud to the entire school. Reading and writing already ranked high among my passions.</p>
<p>Mine isn’t an unfamiliar tale. Many people label themselves as “not a maths person” or “not much of a reader”, often while they are still children. And yet, in a <a href="http://www.nature.com/ncomms/2014/140708/ncomms5204/full/ncomms5204.html">recent study published in Nature Communications</a>, scientists showed that around half of the genes that affect how well 12-year-olds in the UK perform in maths also affect how good they are at reading. And they showed this in a new and important way.</p>
<p>For the first time ever, this study – led by UCL’s Oliver Davis, Chris Spencer at Oxford and Robert Plomin at King’s College London – was able to estimate genetic influences on learning abilities using DNA alone. The implications of this for future genetically sensitive research in the behavioural and social sciences are highly significant. It is certainly much easier to get hold of DNA than it is to get hold of a results from a large <a href="https://theconversation.com/explainer-what-is-twin-research-26468">twin sample</a> – another good way of researching this area.</p>
<h2>Overlapping genes for maths and reading</h2>
<p>The researchers analysed millions of DNA variations from almost 3,000 people and found that more than half of the differences between how well 12-year-olds performed in reading and maths could be explained by differences in their genes. But they also found that reading and maths are correlated partly for genetic reasons: many genes appear to operate in both domains.</p>
<p>These results are not in themselves new. Twin studies <a href="http://onlinelibrary.wiley.com/doi/10.1111/j.1469-7610.2009.02114.x/full">have previously reached</a> very similar conclusions. However, the fact that such findings have now been confirmed by a genome-wide association study – where many common genetic variants are explored for association with a particular trait – is a very significant development. </p>
<p>It is interesting that in spite of using DNA data no particular genes emerged as significant influences on reading or maths. Instead, the researchers found collections of subtle DNA variations. </p>
<p>This ties in with <a href="http://www.nature.com/mp/journal/v16/n10/abs/mp201185a.html">other research</a> that has led to an understanding that many genes of small effect combine to influence complex traits and that the effects are mostly too small to pick up reliably, even with large samples. With genes it seems we’re more likely to find teams than star players.</p>
<p>The research supports the <a href="http://psycnet.apa.org/psycinfo/2005-08334-006">Generalist Genes Hypothesis</a>, the idea that genes are generalists and environments are specialists. Twin studies have found a great deal of evidence that important environments are likely to be specific to particular traits or learning outcomes. For example, a good English teacher might have a slightly different profile to a good maths teacher. The two subjects might benefit from different approaches to homework or different classroom organisation. They may, in short, need to offer different ways of drawing out genetic potential.</p>
<h2>Environment still hugely important</h2>
<p>What we have here is compelling evidence that genes influence maths and reading abilities and that at least half of the genes influencing one ability also influence the other. So, in that case, how can it be that some kids are so much better at reading than maths, and vice versa? </p>
<p>The first answer is simply that genes do not determine behaviour. Genes offer probabilities rather than prophecies. They represent “what is” rather than “what could be”. Even if you have the capacity to do well in a subject this does not automatically translate to high achievement. Motivation, confidence and interest all have a role to play too. </p>
<p>The second answer is that the genetic overlap between the two skill-sets is not 100%. Although there is evidence for shared genetic effects across reading and maths there is evidence of some genetic specificity too. </p>
<p>Most importantly, this new study highlights the role of the environment. It would appear that our life experiences have a particularly important part to play in making some people better at one subject than the other. This may happen through a variety of mechanisms including sparking an interest, inspiring future aspirations or nurturing appetites as well as aptitudes. Both genes and experiences influence the choices young people make about further education and careers and cannot be considered in isolation from each other.</p>
<p>The UK government has been pushing recently to <a href="http://news.tes.co.uk/b/news/2014/06/30/government-names-schools-which-will-integrate-shanghai-maths-methods-in-england.aspx">improve national mathematics performance</a>, and bring them up to levels seen in East Asia. Shared genetic influence on reading and maths suggests that if average maths performance lags behind average reading performance then this should, from a biological point of view, be possible. </p>
<p>The key to achieving this, however, lies in our environment rather than in our DNA. We need a better understanding of which aspects of the maths learning environment promote traits such as self-efficacy, interest and effort as well as achievement – and which don’t. We also need to understand that the same approach will not work for everyone. Drawing out individual potential requires at least a degree of personalisation.</p>
<p>And, just for the record, yes I can do the “people-on-the-bus” sums now.</p><img src="https://counter.theconversation.com/content/28947/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Kathryn Asbury does not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>I disliked and feared maths for most of my school career and dropped it as soon as I possibly could. My mother recalls me crying as a five-year-old because: “I can’t do the people-on-the-bus sums”. If…Kathryn Asbury, Lecturer, Psychology in Education, University of YorkLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/289272014-07-09T05:16:59Z2014-07-09T05:16:59ZWhat is 7x8? You’ll need confidence to answer correctly<figure><img src="https://images.theconversation.com/files/53293/original/hc4j7ykr-1404817259.jpg?ixlib=rb-4.1.0&rect=308%2C252%2C5641%2C3630&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">The answer... 56.</span> <span class="attribution"><a class="source" href="http://www.shutterstock.com/pic-174873203/stock-photo-set-of-simple-multiplication-tables-on-whiteboard-numbers-vector-version-also-available.html?src=oN8kJW91gNKchiDJBrNTiA-4-12">Multiplication table by extender_01/Shutterstock</a></span></figcaption></figure><p>The British Chancellor George Osborne <a href="http://www.telegraph.co.uk/news/politics/georgeosborne/10942559/George-Osborne-stalls-over-7x8-times-table-question-from-a-seven-year-old.html">recently refused to answer</a> a simple times table question posed to him by seven-year-old school boy Samuel Reddings. Osborne was asked the question 7x8, but declined, stating that he had “made it a rule in life not to answer”. As Osborne studied mathematics up to A-level, it seems his reluctance was more about confidence than competence. </p>
<p>Unfortunately, it is socially acceptable for well-educated adults to comment openly that they have no self-confidence in their own mathematical abilities. Conversely, it is socially unacceptable for well-educated adults to say openly that they cannot spell. One wonders how the chancellor would have responded to a simple spelling question.</p>
<p>Teachers have a range of strategies at their disposal to boost their pupils’ confidence in times tables. The use of songs, memorable chants and tricks are increasingly popular in the classroom. Many young people are taught a trick with their fingers for the 9 times tables or to recite chants like “I ate and I ate, so I was sick on the floor” to remember that 8x8=64. </p>
<p>In the same interview, Osborne admitted to being a fan of US musician Pharrell Williams. But if only he had been a fan of the Steps song 5678 instead, he might have felt more confident that 56=7x8. Such strategies can be useful when children do not have the confidence or developmental readiness to understand times tables.</p>
<h2>Rote learning has limitations</h2>
<p>In 2012, then schools minister, Nick Gibb, <a href="http://www.telegraph.co.uk/education/educationnews/9390294/Schools-Minister-rote-learning-vital-to-boost-maths-skills.html">stated</a>: “learning times tables by heart should become a fundamental part of primary education for all pupils”. Some teachers believe that the only way for children to achieve this is to learn the times tables by rote, often emulating their own learning experiences. But rote learning has a public perception of being archaic and boring, meaning some teachers use rote learning behind closed doors (for example, not when inspectors are around). </p>
<p>There is a concern that while children who learn by rote may be able to reproduce the correct answers in tests, they may not able to apply their skills in other contexts. But this has been challenged by <a href="http://books.google.fr/books/about/Thinking_Fast_and_Slow.html?id=ZuKTvERuPG8C&redir_esc=y">Israeli-American psychologist Daniel Kahneman</a> who describes two systems: “thinking fast” (system one) and “thinking slow” (system two). His argument is that the rapid recall of times tables facts using system one provides the necessary input and conceptual thinking space for slower and more in-depth system two, resulting overall in a more efficient use of cognitive resources.</p>
<p>The spectrum of approaches for <a href="http://homepages.warwick.ac.uk/staff/David.Tall/pdfs/dot1994a-gray-jrme.pdf">learning times tables ranges</a> from processes at one end of the scale to conceptual understanding at the other, with no real consensus of opinion across the education sector about the best methodology. The discussion centres on whether the knowledge of times tables will be used as a tool to access the wider curriculum or as mathematical concepts in their own right. </p>
<p>For example, many people own a DVD player and can use it to play DVDs with significant confidence and competence, but very few people have a thorough conceptual understanding of how and why the DVD player works. This is not perceived as a particular problem, since the DVD player is only a tool – a process by which the required outcome of playing a DVD is achieved. </p>
<p>Some would conclude that this is comparable to learning times tables. It is fine to learn the processes initially and to then develop conceptual understanding over time as it becomes more important.</p>
<h2>Calculators do it for you</h2>
<p>But an increasing number of children in schools are questioning why times tables need to be learnt at all when calculators and smart phones are so readily available. This is a logical argument, but the irony is that increased accessibility to technology makes it even more important to know the times tables. <a href="http://www.bbc.com/news/education-20259382">Politicians have argued</a> that blindly trusting calculator outputs can lead to an over-reliance on technology and an under-development of cognitive instincts. Calculators are <a href="https://theconversation.com/in-the-classrooms-of-singapore-calculators-are-not-crutches-26789">now being banned</a> in maths exams for most 11-year-olds. </p>
<p>Children ought to have a feel for the reasonableness of answers provided by calculators and a sense of how numbers within the number system fit together. Such judgements require self-confidence, a willingness to take mathematical risks and the capacity to develop conceptual understanding by learning from mistakes. </p>
<p>To get the answer to 7x8 right, both children and adults need to have confidence in their own competence. Celebrities, politicians and other role models should lead the way in setting that example.</p><img src="https://counter.theconversation.com/content/28927/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Adam Boddison 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 British Chancellor George Osborne recently refused to answer a simple times table question posed to him by seven-year-old school boy Samuel Reddings. Osborne was asked the question 7x8, but declined…Adam Boddison, Director of the Centre for Professional Education and Academic Principal for IGGY, University of WarwickLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/230652014-02-13T02:59:49Z2014-02-13T02:59:49ZA lack of maths just doesn’t add up for a career in science<figure><img src="https://images.theconversation.com/files/41352/original/6jmwsybz-1392191127.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">A better grounding in mathematics is needed for a career in science.</span> <span class="attribution"><span class="source">Shutterstock</span></span></figcaption></figure><p>Our future in science, technology and engineering relies on a foundation and understanding of mathematics.</p>
<p>And while it is pleasing to see a growth in interest in our advanced mathematics course at the University of Sydney, it is also worrying to see an increase in the number of highly capable students who come from high school with a limited background in mathematics.</p>
<p>This seriously limits their ability to undertake university degrees in science, technology, engineering and mathematical (STEM) areas, even allowing for the availability of bridging courses.</p>
<p>How we support these students entering STEM degree programs with a lower level of mathematical knowledge than is needed is being addressed at a two-day <a href="http://sydney.edu.au/news/iisme/1875.html?eventid=10507">national conference</a> on Assumed Knowledge in Maths: Its Broad Impact on Tertiary STEM Programs, being hosted at the University of Sydney.</p>
<p>The event acknowledges ongoing concern from teachers and academics about the inadequate mathematical preparation of Australian school students, especially when compared to their <a href="http://www.abc.net.au/lateline/content/2013/s3904427.htm">international counterparts</a>. The event will also look at what this means both for their career opportunities and the Australian economy.</p>
<h2>Critical in all science study</h2>
<p>Maths is critical to STEM training and careers in these areas because of the way it develops our abilities to conceptualise and solve challenging problems.</p>
<p>It is an essential tool in almost every area of science. This is perhaps easy to understand in physics and chemistry which fundamentally rely on maths, and for psychology which is critically dependent on statistics. But mathematics is used extensively in all the sciences.</p>
<p>The <a href="http://www.wired.com/wiredscience/2010/03/genome-at-10/">genomic revolution</a>, for example, has changed the nature of the biological sciences and resulted in the dramatic growth of the area of bioinformatics.</p>
<p>Vast amounts of data are now available and only mathematically based approaches are able to extract the patterns from this data. These patterns are informing our understanding of evolution and why different populations have different susceptibilities to diseases such as <a href="https://www.diabetesaustralia.com.au/Understanding-Diabetes/What-is-Diabetes/Type-2-Diabetes/">Type 2 Diabetes</a> and <a href="http://www.fightdementia.org.au/understanding-dementia/alzheimers-disease.aspx">Alzheimer’s disease</a>.</p>
<p>In almost every area of science, technology and engineering some computational modelling is now used to test theories and to develop predictions.</p>
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<img alt="" src="https://images.theconversation.com/files/41328/original/d6mc3y95-1392179000.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/41328/original/d6mc3y95-1392179000.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=600&h=385&fit=crop&dpr=1 600w, https://images.theconversation.com/files/41328/original/d6mc3y95-1392179000.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=600&h=385&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/41328/original/d6mc3y95-1392179000.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=600&h=385&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/41328/original/d6mc3y95-1392179000.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&h=484&fit=crop&dpr=1 754w, https://images.theconversation.com/files/41328/original/d6mc3y95-1392179000.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=754&h=484&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/41328/original/d6mc3y95-1392179000.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=754&h=484&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption">Generic shot of a climate change map.</span>
<span class="attribution"><a class="source" href="http://www.flickr.com/photos/ciat/6917871707/sizes/o/">Neil Palmer (CIAT)</a>, <a class="license" href="http://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA</a></span>
</figcaption>
</figure>
<p>Climate modelling is being used to develop predictions on how the earth might respond to the vast amounts of extra energy being trapped in our environment.</p>
<p>The more we learn about cancers, the more we realise that computational models are likely to be the best chance we have of understanding how this complexity functions and how to treat it. This understanding will also inform personalised medicine which guides us in how best to treat an individual’s cancer.</p>
<p>Scientists at our university analyse massive data sets to describe international trade relationships and supply chains in previously unattainable detail.</p>
<h2>In everyday lives</h2>
<p>We rely on mathematics in many other aspects of our everyday lives. Most of us use smartphones to exchange large amounts of data, some of it confidential. This depends on mathematics, and the field of cryptography, which allows accurate and private sharing of information is growing rapidly.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/41336/original/j435qwbn-1392181408.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/41336/original/j435qwbn-1392181408.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/41336/original/j435qwbn-1392181408.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=600&h=399&fit=crop&dpr=1 600w, https://images.theconversation.com/files/41336/original/j435qwbn-1392181408.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=600&h=399&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/41336/original/j435qwbn-1392181408.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=600&h=399&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/41336/original/j435qwbn-1392181408.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&h=501&fit=crop&dpr=1 754w, https://images.theconversation.com/files/41336/original/j435qwbn-1392181408.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=754&h=501&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/41336/original/j435qwbn-1392181408.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=754&h=501&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Smartphones are getting smarter thanks to mathematics.</span>
<span class="attribution"><a class="source" href="http://www.flickr.com/photos/samsungtomorrow/8660105014/sizes/h/">Flickr/Samsung Tomorrow</a>, <a class="license" href="http://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA</a></span>
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</figure>
<p><a href="https://theconversation.com/quantum-computers-could-crack-existing-codes-but-create-others-much-harder-to-break-21807">Quantum computing</a>, which we hope will power future developments in these areas, will require even higher levels of mathematics than current systems.</p>
<p>Mathematical skills are needed from the beginning of any degree in the STEM areas and having to catch up makes the task more challenging than it should be. It takes time to absorb mathematical concepts so for most students it would be preferable to develop that knowledge over years rather than months.</p>
<h2>Impact on careers</h2>
<p>The growth in interest in STEM courses is not keeping pace with the growth in careers in these areas and this is further undermined by a lack of proficiency in mathematics. Australia will suffer as a consequence if this disparity is not addressed soon.</p>
<p>Students who undertake sufficient mathematics study in their high school years will find it much easier to undertake STEM courses at university. They will then be able to contribute to, and benefit from, the opportunities that the increasing importance of STEM areas represent for the Australian economy.</p>
<p>What is needed are strategies that address the shortfall in the number of high school teachers actually trained to teach mathematics and science. We also need to address the reality of increasing numbers of underprepared students in STEM courses at university.</p>
<p>How universities contribute to the delivery of these strategies will be one of the topics at this national conference.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/41351/original/yzbnzqp2-1392190859.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/41351/original/yzbnzqp2-1392190859.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/41351/original/yzbnzqp2-1392190859.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=600&h=400&fit=crop&dpr=1 600w, https://images.theconversation.com/files/41351/original/yzbnzqp2-1392190859.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=600&h=400&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/41351/original/yzbnzqp2-1392190859.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=600&h=400&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/41351/original/yzbnzqp2-1392190859.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&h=503&fit=crop&dpr=1 754w, https://images.theconversation.com/files/41351/original/yzbnzqp2-1392190859.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=754&h=503&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/41351/original/yzbnzqp2-1392190859.jpg?ixlib=rb-4.1.0&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">Adam Spencer, new Mathematics and Science Ambassador.</span>
<span class="attribution"><span class="source">University of Sydney</span></span>
</figcaption>
</figure>
<p>As a first step we’re announcing today that <a href="http://adamspencer.com.au/">Adam Spencer</a> has agreed to serve as the University of Sydney’s Mathematics and Science Ambassador.</p>
<p>He’s a well known media personality with an Honours degree in pure mathematics. So it’s hoped he will help us to inspire students to realise the enjoyment and possibilities that mathematics has to offer.</p><img src="https://counter.theconversation.com/content/23065/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Trevor Hambley is the Dean of Science at the University of Sydney. He receives funding from the ARC. </span></em></p>Our future in science, technology and engineering relies on a foundation and understanding of mathematics. And while it is pleasing to see a growth in interest in our advanced mathematics course at the…Trevor Hambley, Dean of Science, University of SydneyLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/202002013-11-14T03:15:54Z2013-11-14T03:15:54ZA good move to master maths? Check out these chess puzzles<figure><img src="https://images.theconversation.com/files/35265/original/hwgydzn2-1384396967.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Chess can be applied to maths education – you just have to think outside the box.</span> <span class="attribution"><span class="source">practicalowl</span></span></figcaption></figure><p>In the spirit of the current <a href="http://chennai2013.fide.com/fide-world-chess-championship-2013-live/">world championship bout</a> between Norwegian grandmaster <a href="http://magnuscarlsen.com/">Magnus Carlsen</a> and Indian grandmaster <a href="http://en.wikipedia.org/wiki/Viswanathan_Anand">Viswanathan Anand</a>, we should seriously consider the role of chess in how young students learn mathematics.</p>
<p>The two activities have plenty in common. In either, one’s success relies strongly on the ability to be creative under some set of rules. </p>
<p>Beginners in both maths and chess seem to play only for the rules, for they don’t really understand much else yet. In maths, this means swinging the algebraic sword blindly in the hope of making progress. In chess, making any legal move is enough for a beginner, so long as their piece can’t be immediately taken.</p>
<p>Playing either game this way seems fine at first, for if the teacher has the right experience then the newbie will be punished or rewarded accordingly, and will shape their ideas and strategy for the next time around.</p>
<p>Though while chess has maintained huge popularity worldwide, the allure of doing maths seems <a href="https://theconversation.com/yes-theres-a-numeracy-crisis-so-whats-the-solution-6386">lower than ever</a>.</p>
<figure class="align-right zoomable">
<a href="https://images.theconversation.com/files/35264/original/sp5gzft8-1384396881.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/35264/original/sp5gzft8-1384396881.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/35264/original/sp5gzft8-1384396881.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=600&h=902&fit=crop&dpr=1 600w, https://images.theconversation.com/files/35264/original/sp5gzft8-1384396881.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=600&h=902&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/35264/original/sp5gzft8-1384396881.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=600&h=902&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/35264/original/sp5gzft8-1384396881.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&h=1134&fit=crop&dpr=1 754w, https://images.theconversation.com/files/35264/original/sp5gzft8-1384396881.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=754&h=1134&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/35264/original/sp5gzft8-1384396881.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=754&h=1134&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption"></span>
<span class="attribution"><span class="source">Brian Auer</span></span>
</figcaption>
</figure>
<p>And if we think about the reasons for which we bellow the importance of maths – critical thinking, decision making, mental agility – it seems surprising that chess isn’t routinely taught in maths classrooms across the country. </p>
<p>Perhaps other mathematicians would want my blood for suggesting such a gambit, but learning chess could actually have a two-fold effect. Not only could we impart the aforementioned skills through something which more people seem to enjoy, but we might able to transition students to maths through chess.</p>
<p>Students of chess use symbolic notation to record their moves, arithmetic to add up their points and creativity to win position and pieces. And plenty of new ideas in maths could be first taught under the framework of chess.</p>
<p>Let’s have a look at a couple of excellent examples.</p>
<h2>Fewest queens, please</h2>
<p>In high school, advanced students who undertake a course in calculus will get to see the idea of <a href="http://en.wikipedia.org/wiki/Constrained_optimization">constrained optimisation</a>. A typical problem they will be (gently) forced to answer is this:</p>
<p><em>A farmer has 100 metres of fencing and wishes to build a rectangular paddock. What are the dimensions that will maximise the area of this paddock?</em></p>
<p>The point is to maximise the area under the constraint of having a fixed amount of fencing. At any rate, some techniques from calculus can be used to solve this (for those who don’t know, the farmer should build a square paddock with lengths of 25 metres).</p>
<p>But you don’t have to be a student of calculus to appreciate (or even attempt) the following constrained optimisation problem:</p>
<p><em>What is the minimum number of queens that can be placed on a chess board so that every square is being attacked by at least one queen?</em></p>
<p>Here we are trying to minimise the number of queens, under the constraint that every square must be attacked. (Queens are able to move any number of squares vertically, horizontally and diagonally.) After some experimentation, one solution that students might propose is the following:</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/35197/original/287g4dd7-1384383047.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/35197/original/287g4dd7-1384383047.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/35197/original/287g4dd7-1384383047.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=600&h=429&fit=crop&dpr=1 600w, https://images.theconversation.com/files/35197/original/287g4dd7-1384383047.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=600&h=429&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/35197/original/287g4dd7-1384383047.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=600&h=429&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/35197/original/287g4dd7-1384383047.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&h=539&fit=crop&dpr=1 754w, https://images.theconversation.com/files/35197/original/287g4dd7-1384383047.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=754&h=539&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/35197/original/287g4dd7-1384383047.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=754&h=539&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 attempt to solve the constrained optimisation problem.</span>
</figcaption>
</figure>
<p>But other students will soon point out that this isn’t the minimal number of queens. In fact, we only need five queens to do this:</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/35199/original/5hsnf2rq-1384383337.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/35199/original/5hsnf2rq-1384383337.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/35199/original/5hsnf2rq-1384383337.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=600&h=429&fit=crop&dpr=1 600w, https://images.theconversation.com/files/35199/original/5hsnf2rq-1384383337.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=600&h=429&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/35199/original/5hsnf2rq-1384383337.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=600&h=429&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/35199/original/5hsnf2rq-1384383337.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&h=539&fit=crop&dpr=1 754w, https://images.theconversation.com/files/35199/original/5hsnf2rq-1384383337.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=754&h=539&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/35199/original/5hsnf2rq-1384383337.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=754&h=539&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption">Ta da! The solution.</span>
</figcaption>
</figure>
<p>Sure, they won’t go straight to the solution of five queens. This will be achieved through the same means by which most high level maths is done: mathematicians will play with the problem to gain a deeper intuition before working towards and proposing a solution.</p>
<p>Not only can mathematical concepts be illustrated through the game of chess, but sometimes experience with chess can solve mathematical problems.</p>
<h2>The mutilated chessboard</h2>
<p>The following is one of my favourite questions to ask high school students:</p>
<p><em>Consider an 8 x 8 grid with the top-left and bottom-right corners removed:</em></p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/35234/original/h73t8f98-1384389002.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/35234/original/h73t8f98-1384389002.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/35234/original/h73t8f98-1384389002.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=600&h=340&fit=crop&dpr=1 600w, https://images.theconversation.com/files/35234/original/h73t8f98-1384389002.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=600&h=340&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/35234/original/h73t8f98-1384389002.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=600&h=340&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/35234/original/h73t8f98-1384389002.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&h=427&fit=crop&dpr=1 754w, https://images.theconversation.com/files/35234/original/h73t8f98-1384389002.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=754&h=427&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/35234/original/h73t8f98-1384389002.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=754&h=427&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>
</figcaption>
</figure>
<p><em>Is it possible to tile this mutilated grid perfectly using 2 x 1 dominoes?</em></p>
<p>If you haven’t seen this problem before, then you should have a good think about it. It’s very rewarding.</p>
<p>High school students draw up the grid and try all sorts of tilings. At any rate, they struggle to find a tiling, and so propose that there isn’t one. But the point is for them to convince me through sound argument of their proposition. </p>
<p>After letting them suffer for some time, I ask them to consider a chessboard instead of a grid. It’s actually the exact same problem; we’ve just given it some colour.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/35233/original/r9mdp7rg-1384388911.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/35233/original/r9mdp7rg-1384388911.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/35233/original/r9mdp7rg-1384388911.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=600&h=361&fit=crop&dpr=1 600w, https://images.theconversation.com/files/35233/original/r9mdp7rg-1384388911.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=600&h=361&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/35233/original/r9mdp7rg-1384388911.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=600&h=361&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/35233/original/r9mdp7rg-1384388911.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&h=453&fit=crop&dpr=1 754w, https://images.theconversation.com/files/35233/original/r9mdp7rg-1384388911.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=754&h=453&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/35233/original/r9mdp7rg-1384388911.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=754&h=453&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>
</figcaption>
</figure>
<p>And then it all falls into place. It’s definitely impossible to tile the chessboard (and thus the grid) and here’s why.</p>
<p>By removing the top-left and bottom-right squares, we have deleted two white squares, and so there are now two more black squares than white ones. But notice that every time we place a domino on the board, we cover up exactly one white square and one black square. So there will always be two more black squares.</p>
<p>If we get to the end of any tiling, we will have two black squares left over, and it will not be possible to place a single domino to cover these together. Therefore, a tiling is impossible.</p>
<p>The beauty of this problem is that it remains the same, regardless of whether it’s on a grid or chessboard. The latter object, however, allows an elegant proof to be given.</p>
<h2>Your say</h2>
<p>At any rate, one can not deny the polar popularity of maths and chess, nor the strong connection between them. But it would be interesting to hear what the readers think: would chess be a useful thing for young unconvinced students of maths to learn and play?</p><img src="https://counter.theconversation.com/content/20200/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Adrian Dudek's PhD is supported by an Australian Postgraduate Award and an ANU Supplementary Scholarship.</span></em></p>In the spirit of the current world championship bout between Norwegian grandmaster Magnus Carlsen and Indian grandmaster Viswanathan Anand, we should seriously consider the role of chess in how young students…Adrian Dudek, PhD Candidate, Australian National UniversityLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/169882013-08-29T20:19:11Z2013-08-29T20:19:11ZPutting maths on the map with the four colour theorem<figure><img src="https://images.theconversation.com/files/30181/original/znj2mq3p-1377745337.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Forget the calculator: maths is about ideas, not arithmetic.</span> <span class="attribution"><span class="source">celestehopkins</span></span></figcaption></figure><p>Recently, as a community ambassador for <a href="http://drss.anu.edu.au/student_equity/index.php">ANU Student Equity</a>, I took to a local secondary school to talk maths with a small group of students. </p>
<p>The goal? To give them an enjoyable mathematical experience and a glimpse at what maths is all about.</p>
<p>I started by asking them what they thought about maths. The response was that they didn’t much care for sums - a calculator was more likely to pursue such a career.</p>
<p>I knew I had to snap this misconception that “maths” was another way to say “dull arithmetic” and I had to do it fast - iPhones were starting to appear under the desks.</p>
<h2>A colouring exercise</h2>
<p>“Draw a map of Australia,” I told them. “Outline the states, and then colour the states in. Oh, but don’t let any two adjacent states be the same colour - it wouldn’t look very nice.”</p>
<p>They coloured happily, naively thinking themselves free of having to do any maths. </p>
<p>After walking around the classroom admiring their handiwork, I asked them the following question:</p>
<blockquote>
<p>What is the least number of colours needed to colour in Australia this way?</p>
</blockquote>
<p>They were pretty quick here, even though some had gone mad with the visible spectrum whereas others had been a bit more conservative. After a short amount of time, the class agreed on the answer as three: </p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/29818/original/9bgvbqw6-1377245353.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/29818/original/9bgvbqw6-1377245353.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/29818/original/9bgvbqw6-1377245353.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=600&h=558&fit=crop&dpr=1 600w, https://images.theconversation.com/files/29818/original/9bgvbqw6-1377245353.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=600&h=558&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/29818/original/9bgvbqw6-1377245353.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=600&h=558&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/29818/original/9bgvbqw6-1377245353.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&h=701&fit=crop&dpr=1 754w, https://images.theconversation.com/files/29818/original/9bgvbqw6-1377245353.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=754&h=701&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/29818/original/9bgvbqw6-1377245353.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=754&h=701&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>
</figcaption>
</figure>
<p>I congratulated them on their work before giving them the following two exercises.</p>
<p><em>1. Invent a country (with states) where the minimal number of colours needed is four. Swap with a classmate and get them to colour it in.</em></p>
<p><em>2. Invent another country where the minimal number of colours needed is five. Swap your map with a classmate and get them to colour it in.</em></p>
<p>Well, the students had a blast creating and naming their own countries. They challenged their friends in the first exercise, as the way to colour was not always obvious. </p>
<p>The following is the simplest example of a map that requires four colours:</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/29819/original/2vvc5bt2-1377245776.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/29819/original/2vvc5bt2-1377245776.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/29819/original/2vvc5bt2-1377245776.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=600&h=325&fit=crop&dpr=1 600w, https://images.theconversation.com/files/29819/original/2vvc5bt2-1377245776.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=600&h=325&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/29819/original/2vvc5bt2-1377245776.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=600&h=325&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/29819/original/2vvc5bt2-1377245776.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&h=409&fit=crop&dpr=1 754w, https://images.theconversation.com/files/29819/original/2vvc5bt2-1377245776.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=754&h=409&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/29819/original/2vvc5bt2-1377245776.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=754&h=409&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption"></span>
<span class="attribution"><span class="source">howmanycolours</span></span>
</figcaption>
</figure>
<p>The second exercise went exactly as I had hoped. There was some conflict: it turned out that even though the minimal number of colours was intended to be five, students were colouring in the maps correctly using only four.</p>
<p>One by one, we drew up each student’s map on the board and took turns showing that, in fact, each of them could be filled correctly using only four colours. Madness! I let them have another failed attempt, before letting them in on a well known mathematical theorem.</p>
<h2>The four colour theorem</h2>
<p><em>You never need more than four colours to colour in the regions of a map, such that any two adjacent regions are differently coloured.</em></p>
<p>We also have to stipulate that adjacent regions are those that share an edge, so regions that meet at a point are not deemed to be adjacent.</p>
<figure class="align-center zoomable">
<a href="https://images.theconversation.com/files/30178/original/3rb64pvx-1377744889.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/30178/original/3rb64pvx-1377744889.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/30178/original/3rb64pvx-1377744889.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=600&h=371&fit=crop&dpr=1 600w, https://images.theconversation.com/files/30178/original/3rb64pvx-1377744889.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=600&h=371&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/30178/original/3rb64pvx-1377744889.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=600&h=371&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/30178/original/3rb64pvx-1377744889.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&h=467&fit=crop&dpr=1 754w, https://images.theconversation.com/files/30178/original/3rb64pvx-1377744889.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=754&h=467&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/30178/original/3rb64pvx-1377744889.jpg?ixlib=rb-4.1.0&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"></a>
<figcaption>
<span class="caption"></span>
<span class="attribution"><span class="source">Wikimedia Commons</span></span>
</figcaption>
</figure>
<p>“This is probably the first example of real mathematics you’ve ever seen,” I told them. “Maths is about ideas, not arithmetic.” They wanted to know a bit more about it. </p>
<p>We talked about how, in 1852, a mathematician called <a href="http://en.wikipedia.org/wiki/Francis_Guthrie">Francis Guthrie</a> was colouring in the different counties of England, and noticed that only four colours were needed. He wrote this in a letter to his brother Frederick, who passed it on to another mathematician.</p>
<p>For over a century, different mathematicians would fail in their attempts to prove the four colour theorem. But in 1976, <a href="http://en.wikipedia.org/wiki/Kenneth_Appel">Kenneth Appel</a> and <a href="http://en.wikipedia.org/wiki/Wolfgang_Haken">Wolfgang Haken</a> finally succeeded. </p>
<p>Upon asking the students how one might prove this mighty theorem, there was a suggestion we could draw every possible map and then colour them in using only four colours. </p>
<p>I shut them down quickly with the following fact: there are infinitely possible maps. So how did they prove it?</p>
<h2>The idea behind the proof</h2>
<p>The four colour theorem serves as the first major mathematical theorem to be proved using a computer. Of course, there are some stunning ideas behind the computation.</p>
<p>To show that there are no maps that need more than four colours, Appel and Haken turned to <a href="http://www.thefreedictionary.com/reductio+ad+absurdum"><em>reductio ad absurdum</em></a> (reduction to absurdity), the greatest weapon the mathematician has. Here’s how it works:</p>
<p>If we want to prove that something is true, we instead assume that it is false and show that the rest of maths goes bad. That is, by assuming falsity, we encounter contradictions to already known truths. </p>
<figure class="align-left zoomable">
<a href="https://images.theconversation.com/files/30183/original/hr2xw4js-1377745616.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/30183/original/hr2xw4js-1377745616.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/30183/original/hr2xw4js-1377745616.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=600&h=971&fit=crop&dpr=1 600w, https://images.theconversation.com/files/30183/original/hr2xw4js-1377745616.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=600&h=971&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/30183/original/hr2xw4js-1377745616.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=600&h=971&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/30183/original/hr2xw4js-1377745616.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&h=1220&fit=crop&dpr=1 754w, https://images.theconversation.com/files/30183/original/hr2xw4js-1377745616.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=754&h=1220&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/30183/original/hr2xw4js-1377745616.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=754&h=1220&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption"></span>
<span class="attribution"><span class="source">marfis75</span></span>
</figcaption>
</figure>
<p>This tells us that our original assumption is incorrect, and that what we want to prove must be true. </p>
<p>Appel and Haken used this idea as follows. They assumed to the contrary, that there was some map out there which required more than four colours. They then showed that this rogue map was not allowed to contain within it one of 1,936 special, smaller maps. </p>
<p>Appel and Haken then showed that <em>any</em> map has to contain one of these special smaller maps and so appeared the contradiction.</p>
<p>There was a lot of checking to be done in the proof, so Appel and Haken wrote a computer program to do most of the working. As such, the four colour theorem was the first major mathematical theorem to be proved using a computer.</p>
<h2>The doubts</h2>
<p>The computer plays the biggest role in the proof and so there were concerns about the truth of this theorem, as it’s essentially impossible to verify by hand. As such, there were many sceptics.</p>
<p>In 1975, as an April Fool’s joke, the American mathematics writer <a href="http://en.wikipedia.org/wiki/Martin_Gardner">Martin Gardner</a> spread around a <a href="http://mathworld.wolfram.com/Four-ColorTheorem.html">proposed counterexample</a> to the four colour theorem. It took 24 years (and a lot of computer time) to show that only four colours were needed.</p>
<p>Even today, despite enjoying widespread academic acceptance, there are still sceptics who doubt the four colour theorem. </p>
<p>Can you come up with a counter-example? </p><img src="https://counter.theconversation.com/content/16988/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Adrian Dudek 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>Recently, as a community ambassador for ANU Student Equity, I took to a local secondary school to talk maths with a small group of students. The goal? To give them an enjoyable mathematical experience…Adrian Dudek, PhD Candidate, Australian National UniversityLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/131912013-04-09T20:40:06Z2013-04-09T20:40:06ZThe RSA algorithm (or how to send private love letters)<figure><img src="https://images.theconversation.com/files/21971/original/7wh22vgr-1364946431.jpg?ixlib=rb-4.1.0&rect=0%2C91%2C2048%2C1195&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Sending secure information? You could do a lot worse than employing the RSA algorithm.</span> <span class="attribution"><span class="source">Seq</span></span></figcaption></figure><p>A couple of days ago on The Conversation, I <a href="https://theconversation.com/your-numbers-up-a-case-for-the-usefulness-of-useless-maths-11799">set myself up with a task</a>: to defend the usefulness of so-called “useless” maths. Today, that defence continues, with a look at the RSA algorithm. </p>
<p>I finished last time by pointing out that three mathematicians – Ron Rivest, Adi Shamir and Leonard Adleman
– <a href="http://searchsecurity.techtarget.com/definition/RSA">created the RSA algorithm</a> in 1977, in one fell swoop establishing a practical use for <a href="http://mathworld.wolfram.com/NumberTheory.html">number theory</a> in the modern world. </p>
<p>So, to look at this idea more closely, let’s take a detour into the lives of Alice and Bob, two fictional characters who are infatuated with each other, despite never having met. </p>
<p>Suppose Alice wants to send Bob a private note outlining her affections. She could place this in a box, snap a padlock on it and ship it off. The problem is, obviously, that this would be useless to Bob without the key. </p>
<p>Alice could send this key across, but if both were intercepted (possibly by Bob’s jealous ex-wife, Eve) there would be trouble – and the fact this could happen means the method isn’t really secure.</p>
<figure class="align-right zoomable">
<a href="https://images.theconversation.com/files/21970/original/tnyfybhf-1364946074.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/21970/original/tnyfybhf-1364946074.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/21970/original/tnyfybhf-1364946074.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=600&h=819&fit=crop&dpr=1 600w, https://images.theconversation.com/files/21970/original/tnyfybhf-1364946074.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=600&h=819&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/21970/original/tnyfybhf-1364946074.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=600&h=819&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/21970/original/tnyfybhf-1364946074.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&h=1029&fit=crop&dpr=1 754w, https://images.theconversation.com/files/21970/original/tnyfybhf-1364946074.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=754&h=1029&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/21970/original/tnyfybhf-1364946074.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=754&h=1029&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption"></span>
<span class="attribution"><span class="source">Nina Matthews Photography</span></span>
</figcaption>
</figure>
<p>One solution to this problem runs in parallel with the RSA algorithm.</p>
<p>Bob knows people (Alice, in particular) want to send him secret messages, so he goes out and buys a stack of identical padlocks, all of which open with a single key he keeps hidden in his left shoe. Bob unlocks all of these padlocks and makes them available at, say, Bunnings.</p>
<p>If Alice wants to send Bob a secret message, she simply needs to go to Bunnings and get one of these open padlocks, then use it on the box she wants to send Bob. This will be a safe transmission, seeing as Bob is the only one with the key. </p>
<p>But what a complicated way of securing information: for people to receive secret messages, they need to have public padlocks available to everybody!</p>
<p>These days, of course, more people opt for email rather than snail mail. So how do we know Alice and Bob’s online communications aren’t being monitored by Eve?</p>
<p>Now we get into the <a href="http://mathworld.wolfram.com/RSAEncryption.html">RSA algorithm</a>, which is the strongest possible way to encrypt and decrypt information online.</p>
<h2>Feel the rhythm</h2>
<p>The algorithm’s design and strength are the work of historic results in number theory, and its security is guaranteed by the following fact:</p>
<p>It’s easy for a computer to multiply two large <a href="http://mathworld.wolfram.com/PrimeNumber.html">prime numbers</a> together (Google will do this without breaking a sweat). </p>
<figure class="align-left zoomable">
<a href="https://images.theconversation.com/files/21967/original/rw5h73jg-1364945512.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/21967/original/rw5h73jg-1364945512.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/21967/original/rw5h73jg-1364945512.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=600&h=600&fit=crop&dpr=1 600w, https://images.theconversation.com/files/21967/original/rw5h73jg-1364945512.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=600&h=600&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/21967/original/rw5h73jg-1364945512.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=600&h=600&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/21967/original/rw5h73jg-1364945512.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&h=754&fit=crop&dpr=1 754w, https://images.theconversation.com/files/21967/original/rw5h73jg-1364945512.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=754&h=754&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/21967/original/rw5h73jg-1364945512.jpg?ixlib=rb-4.1.0&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"><span class="source">slightly everything</span></span>
</figcaption>
</figure>
<p>But let’s say you multiply two large prime numbers together to get a resulting number: if you gave this new number to a computer and asked it to tell you what prime numbers you multiplied to construct it, the computer would struggle.</p>
<p>This is called a <a href="http://mathworld.wolfram.com/TrapdoorOne-WayFunction.html">trapdoor</a>, meaning it’s easy to go one way, but very hard to go the other.</p>
<p>Which two prime numbers did I multiply together to get 194477? A computer can probably unlock this number easily, but not if the prime numbers I use are much larger.</p>
<p>The mathematical difficulty of the above problem is what ensures the strength of our encryption (or lock).</p>
<h2>Crunching the numbers</h2>
<p>The RSA algorithm works as follows: </p>
<p>First, I find two huge (at least 100 digits each!) prime numbers <em>p</em> and <em>q</em>, and then I multiply them together to get the even bigger number <em>N</em>. I also combine <em>p</em> and <em>q</em> in a different way to generate another number <em>e</em> (details of this below).</p>
<p>I publish these two numbers <em>(N,e)</em> as my “public key”, with the knowledge that there is enormous difficulty with even the world’s fastest computers breaking <em>N</em> into its constituent prime atoms <em>p</em> and <em>q</em>.</p>
<figure class="align-right zoomable">
<a href="https://images.theconversation.com/files/22232/original/np89n4tg-1365487054.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/22232/original/np89n4tg-1365487054.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/22232/original/np89n4tg-1365487054.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=600&h=901&fit=crop&dpr=1 600w, https://images.theconversation.com/files/22232/original/np89n4tg-1365487054.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=600&h=901&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/22232/original/np89n4tg-1365487054.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=600&h=901&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/22232/original/np89n4tg-1365487054.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&h=1132&fit=crop&dpr=1 754w, https://images.theconversation.com/files/22232/original/np89n4tg-1365487054.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=754&h=1132&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/22232/original/np89n4tg-1365487054.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=754&h=1132&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption"></span>
<span class="attribution"><span class="source">zebble</span></span>
</figcaption>
</figure>
<p>It turns out the numbers <em>N</em> and <em>e</em> can be used by people to encrypt a message to send to me, which I can use my secret primes <em>p</em> and <em>q</em> to decrypt.</p>
<p>To illustrate this beautiful piece of mathematics, we can return to our fictional lovers.</p>
<p>Bob wants to encrypt and send Alice his age – 42. Of course, the RSA algorithm deals with sending numbers, but seeing as any text can be converted to digits in a variety of ways, we can securely transmit information of any type. </p>
<p>Also, I will use small prime numbers in this example, just to ease the calculations.</p>
<p>1) Alice knows that Bob wants to send her a message, so she selects two prime numbers. Let’s say she picks <em>p=17</em> and <em>q=29</em> (though in reality they would be much larger so as to ensure better security).</p>
<p>Alice then multiplies <em>p</em> and <em>q</em> together to get the number <em>N</em>:</p>
<blockquote>
<p>p x q = 17 x 29 = 493</p>
</blockquote>
<p>So Alice now has that <em>N=493</em>. </p>
<p>2) Alice also needs to generate another number <em>e</em>. She creates this number first by subtracting 1 off of each of her prime numbers <em>p</em> and <em>q</em>. This gives her:</p>
<blockquote>
<p>p - 1 = 16<br>
q - 1 = 28<br></p>
</blockquote>
<p>Alice then multiplies these two together to get:</p>
<blockquote>
<p>(p-1) x (q-1) = 448.</p>
</blockquote>
<p>This number is not <em>e</em>. The number <em>e</em> is allowed to be any number, which has no factors in common with this new number 448. To see what I mean, we can break 448 into a bunch of prime numbers multiplied together:</p>
<blockquote>
<p>448 = 2 x 2 x 2 x 2 x 2 x 2 x 7</p>
</blockquote>
<p>Then <em>e</em> is just any number which, when broken down into primes, does not possess a 2 or a 7 as a factor. So there are lots of possibilities. Let’s suppose Alice chooses <em>e=5</em>.</p>
<p>3) Alice then gives the numbers <em>N=493</em> and <em>e=5</em> to Bob. She could even place them on the local billboard if she wanted to. The important part is that the number <em>N</em> should be hard for anybody to break down into <em>p</em> and <em>q</em>. </p>
<p>We will just assume the number 493 scares people and so nobody tries to decompose it.</p>
<p>4) With these two numbers <em>N</em> and <em>e</em>, Bob can now encrypt his secret message, which, in this case is his age – 42. He starts by putting 42 to the power of <em>e</em>, which he knows is <em>5</em>. This just means he multiplies his age by itself five times:</p>
<blockquote>
<p>42 x 42 x 42 x 42 x 42 = 130,691,232</p>
</blockquote>
<p>5) Bob has now half-locked his message. At this point, he just needs to use <em>N</em> to finish the encryption process. He takes the above number 130,691,232 and divides it by the number <em>N=493</em>.</p>
<p>He is interested in the remainder upon performing this division, which is 383. So Bob has encrypted 42 as 383, which is the number that he sends to Alice.</p>
<p>6) Now, the messy bit. Alice has received the number 383 from Bob, and she needs to decrypt it to get his age. Her first step, is to use her secret prime numbers <em>p</em> and <em>q</em> and the public number <em>e</em> to form another number <em>d</em>, which she can use to decrypt Bob’s message. </p>
<p>Alice once again considers the number 448, which she obtained in step 3 by multiplying <em>(p - 1)</em> by <em>(q - 1)</em>. Alice needs to find a multiple of <em>e=5</em> which is exactly one more than a multiple of 448. </p>
<p>Number theory grants that this is always possible and we will show a tedious but straightforward way of doing this.</p>
<p>Alice lists the multiples of 5:</p>
<blockquote>
<p>5, 10, 15, 20, 25, 30, 35, 40 …</p>
</blockquote>
<p>… and the multiples of 448:</p>
<blockquote>
<p>448, 896, 1344, 1792, 2240, 2688 …</p>
</blockquote>
<p>She needs to find a number in the first sequence which is exactly one more than a number in the second sequence. If Alice goes far enough along in the first sequence – to the 269th term which is 1345 – she can see that this is exactly one more than 1344, the third term in the second sequence. Success!</p>
<p>Note that it was finding the number <em>d=269</em>, the place of the relevant multiple of <em>e</em>, which was the whole point of this step. </p>
<p>7) Finally, Alice can decode the message by doing one big calculation. She has Bob’s encrypted number, 383, and her new number, <em>d=269</em>. It turns out Bob’s age can be decoded by calculating 383 to the power of 269, and then finding the remainder upon dividing by <em>N=493</em>.</p>
<p>This looks like it will be quite a nasty calculation, but some tools from number theory can be employed here. Once Alice completes this calculation, the remainder will be 42, which is Bob’s age.</p>
<figure class="align-left zoomable">
<a href="https://images.theconversation.com/files/22233/original/zwxx2y7d-1365487161.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/22233/original/zwxx2y7d-1365487161.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/22233/original/zwxx2y7d-1365487161.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=600&h=795&fit=crop&dpr=1 600w, https://images.theconversation.com/files/22233/original/zwxx2y7d-1365487161.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=600&h=795&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/22233/original/zwxx2y7d-1365487161.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=600&h=795&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/22233/original/zwxx2y7d-1365487161.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&h=999&fit=crop&dpr=1 754w, https://images.theconversation.com/files/22233/original/zwxx2y7d-1365487161.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=754&h=999&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/22233/original/zwxx2y7d-1365487161.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=754&h=999&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption"></span>
<span class="attribution"><span class="source">Judy **</span></span>
</figcaption>
</figure>
<p>RSA is the best possible type of public key cryptography, yet due to the high computation involved it is often not used to encrypt/decrypt simple messages.</p>
<p>Instead, it’s used for signatures and protocol negotiations to allow two sides to receive private keys to use for their communication. </p>
<p>The RSA algorithm is but one of many systems where a set of mathematical theorems, often from number theory, can be synthesised to construct an encryption scheme.</p>
<p>There is some elegant mathematics going on behind the scenes ensuring the algorithm’s success. Of course, in reality, large prime numbers guarantee secure encryption, while the calculations are performed by computers. </p>
<p>Importantly, the algorithm parallels the physical situation where one makes open padlocks available to the public.</p>
<p>Mathematically speaking, our numbers <em>N</em> and <em>e</em> form the open padlock, and our secret primes <em>p</em> and <em>q</em> combine to form a key that we can use to retrieve the locked information. </p>
<p>But if somebody came up with a brilliant formula for breaking numbers into their prime factors (turning padlocks into keys), there would indeed be security issues around the world.</p>
<h2>I rest my case</h2>
<p>The inherent randomness of the primes is the force that currently prevents such a formula from existing. </p>
<p>Number theorists are at the forefront of this knowledge, but are still a while away from coming up with such a tool. </p>
<p>The world should rest assured that, as problems are solved, many more are created, and this progress not only allows the advancement of technology, but also endows the mathematician with endless motivation.</p>
<p>Just because something doesn’t have real-world applications yet doesn’t mean it won’t have in the future. </p>
<p>And so concludes my defence of the usefulness of useless maths. Did it work? Are you convinced? I’d be interested to hear your views. </p>
<p><strong>Read Adrian’s previous article:</strong><br>
<a href="https://theconversation.com/your-numbers-up-a-case-for-the-usefulness-of-useless-maths-11799">Your number’s up – a case for the usefulness of useless maths</a></p><img src="https://counter.theconversation.com/content/13191/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Adrian Dudek 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>A couple of days ago on The Conversation, I set myself up with a task: to defend the usefulness of so-called “useless” maths. Today, that defence continues, with a look at the RSA algorithm. I finished…Adrian Dudek, PhD Candidate, Australian National UniversityLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/117992013-04-07T20:15:33Z2013-04-07T20:15:33ZYour number’s up – a case for the usefulness of useless maths<figure><img src="https://images.theconversation.com/files/21969/original/rggyxzmr-1364945666.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">As all mathematicians know, the rift between useful and useless can change with time.</span> <span class="attribution"><span class="source">Manu gomi</span></span></figcaption></figure><p>I once made the mistake of asking a mathematician why he devoted his whole life to maths. “Because it’s fun!” he replied wildly, his flabby cheeks beaming with childlike excitement. </p>
<p>“Ah, of course,” I thought to myself. “It is fun.”</p>
<p>But what can it <em>do</em> for society? It’s a question I’ve been asked by various people on scholarships committees and one that, over the course of this article and another to follow, I hope to answer.</p>
<p>I spend my time in an area of maths called <a href="http://www.math.niu.edu/%7Erusin/known-math/index/11-XX.html">number theory</a>, which essentially looks at the shape of the <a href="http://mathworld.wolfram.com/CountingNumber.html">counting numbers</a>. </p>
<p>Other number theorists might encapsulate it differently, but it’s this perspective that keeps me on the edge of my seat. At the heart of this theory, we look at the distribution of the prime numbers.</p>
<h2>Prime numbers</h2>
<p>As one usually learns in their first few years of primary school, a prime number has a <a href="http://mathworld.wolfram.com/PrimeNumber.html">mathematical description</a> (to do with factors) which, if stated here, might alienate more readers than it attracts. Let’s instead consider the following light-hearted definition:</p>
<p>Let’s say you have a pile of biscuits. The number of biscuits you have is a prime number if there is no way to arrange those biscuits into a neat rectangle.</p>
<p>So if I gave you seven biscuits (think of it as a very early Christmas present) then you could arrange them like this:</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/19608/original/299gxh4p-1359068858.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/19608/original/299gxh4p-1359068858.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=600&h=73&fit=crop&dpr=1 600w, https://images.theconversation.com/files/19608/original/299gxh4p-1359068858.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=600&h=73&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/19608/original/299gxh4p-1359068858.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=600&h=73&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/19608/original/299gxh4p-1359068858.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&h=91&fit=crop&dpr=1 754w, https://images.theconversation.com/files/19608/original/299gxh4p-1359068858.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=754&h=91&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/19608/original/299gxh4p-1359068858.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=754&h=91&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption"></span>
</figcaption>
</figure>
<p>We think of the above as a line, and not a rectangle. We could also arrange them like this:</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/22109/original/yzxwwb49-1365132831.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/22109/original/yzxwwb49-1365132831.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=600&h=167&fit=crop&dpr=1 600w, https://images.theconversation.com/files/22109/original/yzxwwb49-1365132831.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=600&h=167&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/22109/original/yzxwwb49-1365132831.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=600&h=167&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/22109/original/yzxwwb49-1365132831.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&h=210&fit=crop&dpr=1 754w, https://images.theconversation.com/files/22109/original/yzxwwb49-1365132831.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=754&h=210&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/22109/original/yzxwwb49-1365132831.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=754&h=210&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption"></span>
</figcaption>
</figure>
<p>But no matter how hard you try, you just won’t be able to arrange them into a rectangle! Therefore, seven is a prime number. </p>
<p>On the other hand, 12 is not a prime number, for 12 biscuits can be nicely arranged into a rectangle as follows:</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/22110/original/c66znr7w-1365133013.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/22110/original/c66znr7w-1365133013.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=600&h=177&fit=crop&dpr=1 600w, https://images.theconversation.com/files/22110/original/c66znr7w-1365133013.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=600&h=177&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/22110/original/c66znr7w-1365133013.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=600&h=177&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/22110/original/c66znr7w-1365133013.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&h=222&fit=crop&dpr=1 754w, https://images.theconversation.com/files/22110/original/c66znr7w-1365133013.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=754&h=222&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/22110/original/c66znr7w-1365133013.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=754&h=222&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption"></span>
</figcaption>
</figure>
<p>Or even like this:</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/22107/original/x4tkztfv-1365132566.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&fit=clip" srcset="https://images.theconversation.com/files/22107/original/x4tkztfv-1365132566.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=600&h=266&fit=crop&dpr=1 600w, https://images.theconversation.com/files/22107/original/x4tkztfv-1365132566.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=600&h=266&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/22107/original/x4tkztfv-1365132566.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=600&h=266&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/22107/original/x4tkztfv-1365132566.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&h=334&fit=crop&dpr=1 754w, https://images.theconversation.com/files/22107/original/x4tkztfv-1365132566.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=754&h=334&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/22107/original/x4tkztfv-1365132566.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=754&h=334&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px">
<figcaption>
<span class="caption"></span>
</figcaption>
</figure>
<p>If you wanted to, you could arrange 12 biscuits in a way that wasn’t a rectangle. The point is, if you can arrange them in a rectangle in at least one way, then the number is not a prime.</p>
<p>So there is something shape-y about the prime numbers. The really delightful thing is that prime numbers seem to just pop their head up in the counting numbers wherever they feel like it. Not only that, but they go on forever.</p>
<p>This fact – that there are infinitely many prime numbers – is an ancient theorem and was first proved to be true by the grand mathematician <a href="http://en.wikipedia.org/wiki/Euclid">Euclid</a> around 300 BC. The sequence of prime numbers starts as follows:</p>
<blockquote>
<p>2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31 …</p>
</blockquote>
<p>Primes are often called the atoms of arithmetic, and there is a good reason for this. Using only the primes, we can, through multiplication, build all of the composite (i.e. not prime) numbers.</p>
<p>Likewise, if we choose any counting number (except for 1, which is neither prime nor composite), then this number is either a prime or a bunch of primes multiplied together.</p>
<blockquote>
<p>60 = 2 x 2 x 3 x 5<br>
34 = 2 x 17</p>
</blockquote>
<p>So why would the world’s mathematicians be interested in a subject that’s routinely covered in primary school?</p>
<p>The driving force here, as in most mathematics, is natural curiosity. As mentioned earlier, prime numbers, despite being very simple to describe, are tenacious in the way that they hide in the counting numbers. </p>
<p>That is, there is no identifiable pattern with which they appear.</p>
<p>In fact, questions concerning the distribution of prime numbers are among the hardest unsolved problems in all of mathematics.</p>
<p>A example from the top tier would be the <a href="http://mathworld.wolfram.com/TwinPrimeConjecture.html">twin prime conjecture</a>, which asks if there are infinitely many primes which differ by only 2.</p>
<p>There are many instances of these twin prime pairs – such as 11 and 13, and 29 and 31 – but it is unknown whether there are infinitely many of these.</p>
<h2>Number crunching</h2>
<p>Some people believe the primes have been scattered randomly but others know better.</p>
<p>There are a stack of <a href="http://primes.utm.edu/">certified results</a> that give order to the chaotic structure of the primes. One such result guarantees you <a href="http://mathworld.wolfram.com/BertrandsPostulate.html">can always find a prime between any number bigger than 1 and its double</a>.</p>
<p>Another <a href="http://eprints.maths.ox.ac.uk/1106/1/NA-06-17.pdf">result</a> ensures that given any fixed string of digits, there are infinitely many primes which contain this string. If you were to choose the string of digits “1234”, then there are infinitely many prime numbers which contain this string. Here are a few:</p>
<blockquote>
<p>12343, 12347, 112349, 123401, 123407 …</p>
</blockquote>
<p>The result does not tell you how to <em>find</em> these “string-containing primes”, only that infinitely many of them do exist. What an odd result! What if we were to choose, say, your mobile phone number for a string of digits? </p>
<p>Well, then there are infinitely many prime numbers which contain your phone number within their digits. Indeed, the prime numbers are rather intrusive!</p>
<p>It’s not at all surprising that many people have become mesmerised by the interplay between structure and randomness that presents itself within the sequence of prime numbers.</p>
<p>Some, they say, have even gone crazy trying to solve the difficult problems in the area. For most people, a more important problem seems to be the following: why should we care and is this useful maths?</p>
<h2>The rift</h2>
<p>Some people believe maths can be cleanly divided into two areas: useful maths and useless maths.</p>
<p>You could also be told that, as soon as a result has been used to model or measure some real-life scenario, it has been deemed useful. </p>
<figure class="align-right zoomable">
<a href="https://images.theconversation.com/files/21982/original/s8mf8j6j-1364953212.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/21982/original/s8mf8j6j-1364953212.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=237&fit=clip" srcset="https://images.theconversation.com/files/21982/original/s8mf8j6j-1364953212.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=600&h=800&fit=crop&dpr=1 600w, https://images.theconversation.com/files/21982/original/s8mf8j6j-1364953212.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=600&h=800&fit=crop&dpr=2 1200w, https://images.theconversation.com/files/21982/original/s8mf8j6j-1364953212.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=600&h=800&fit=crop&dpr=3 1800w, https://images.theconversation.com/files/21982/original/s8mf8j6j-1364953212.jpg?ixlib=rb-4.1.0&q=45&auto=format&w=754&h=1005&fit=crop&dpr=1 754w, https://images.theconversation.com/files/21982/original/s8mf8j6j-1364953212.jpg?ixlib=rb-4.1.0&q=30&auto=format&w=754&h=1005&fit=crop&dpr=2 1508w, https://images.theconversation.com/files/21982/original/s8mf8j6j-1364953212.jpg?ixlib=rb-4.1.0&q=15&auto=format&w=754&h=1005&fit=crop&dpr=3 2262w" sizes="(min-width: 1466px) 754px, (max-width: 599px) 100vw, (min-width: 600px) 600px, 237px"></a>
<figcaption>
<span class="caption"></span>
<span class="attribution"><span class="source">*Bitch Cakes*</span></span>
</figcaption>
</figure>
<p>In 1940, the renowned mathematician G. H. Hardy wrote <a href="https://www.google.com.au/url?sa=t&rct=j&q=&esrc=s&source=web&cd=2&ved=0CDcQFjAB&url=http%3A%2F%2Fwww.math.ualberta.ca%2Fmss%2Fmisc%2FA%2520Mathematician's%2520Apology.pdf&ei=77MBUZqNGoSjigfLtYCACw&usg=AFQjCNFLJnp5kmsrjf9GWcRubchS6h7GfQ&bvm=bv.41524429,d.aGc">A Mathematician’s Apology</a>, an essay he hoped would deliver justification of his life’s work in mathematics.</p>
<p>Within this essay, Hardy labels <a href="http://mathworld.wolfram.com/NumberTheory.html">number theory</a> (his own area of mathematics) as useless. Indeed, at the time, number theory did not have any known uses, so I would have struggled even more with the scholarship committees of Hardy’s time.</p>
<p>But as all mathematicians know, the rift between useful and useless can change with time. In 1977, a trio of mathematicians found a permanent place for number theory in our modern world.</p>
<p>Ron Rivest, Adi Shamir and Leonard Adleman created the <a href="http://searchsecurity.techtarget.com/definition/RSA">RSA algorithm</a> (the three letters are taken from the initials of their surnames), as a way of transmitting secret information between two parties. </p>
<p>Why did they do this and what function does that algorithm play in the real world? That’s a question for the second, and final, part of this argument.</p>
<p><strong>Read Adrian’s next article:</strong><br>
<strong><a href="https://theconversation.com/the-rsa-algorithm-or-how-to-send-private-love-letters-13191">The RSA algorithm (or how to send private love letters)</a></strong></p><img src="https://counter.theconversation.com/content/11799/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Adrian Dudek does not work for, consult, own shares in or receive funding from any company or organisation that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>I once made the mistake of asking a mathematician why he devoted his whole life to maths. “Because it’s fun!” he replied wildly, his flabby cheeks beaming with childlike excitement. “Ah, of course,” I…Adrian Dudek, PhD Candidate, Australian National UniversityLicensed as Creative Commons – attribution, no derivatives.