tag:theconversation.com,2011:/ca/topics/teaching-math-18173/articlesTeaching math – The Conversation2018-09-17T21:32:31Ztag:theconversation.com,2011:article/1031922018-09-17T21:32:31Z2018-09-17T21:32:31ZNo matter what method is used to teach math, make it fun<figure><img src="https://images.theconversation.com/files/236683/original/file-20180917-158237-1lj212i.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">The Ontario government "back to basics" approach to the curriculum will not best serve children who need a mixture of traditional and discovery learning methods. </span> <span class="attribution"><span class="source">Shutterstock</span></span></figcaption></figure><p>There’s a change coming to Ontario’s elementary school math curriculum. The new provincial government says it is responding to a decline of standardized test scores and plans to recommend a return to “back-to-basics” teaching methods for mathematics teachers. </p>
<p><a href="https://news.ontario.ca/opo/en/2018/08/ontarios-government-for-the-people-respecting-parents-by-holding-unprecedented-consultation-into-education-reform.html">Premier Doug Ford and Minister of Education Lisa Thompson have told educators that directives for changes will be coming within a matter of weeks</a>. Teachers, whose lesson plans for the fall are already drawn, are possibly now scrambling to implement new curriculum guidelines.</p>
<p><a href="https://globalnews.ca/news/4416177/ontario-elementary-school-math-scores-down/">Education Quality and Accountability Office (EQAO) scores</a> for students in Grade 6 and Grade 3 in Ontario are both down one percentage point from last year. Previous efforts, <a href="https://www.huffingtonpost.ca/2017/08/30/eqao-results-show-60-million-strategy-hasnt-made-ontario-students-any-better-at-math_a_23191535/">including a $60-million initiative by the former Liberal government</a> to improve the EQAO scores, have not worked.</p>
<h2>What are ‘back-to-basics’ methods?</h2>
<p>What exactly are “back-to-basics” teaching methods for mathematics? These traditional methods of mathematics education include an emphasis on drills, formulas and memorization. If you are old enough, then this was how you were taught mathematics in grade school. In contrast, discovery-based methods spend less time on rules and puts more emphasis on problem solving and applications.</p>
<p>The two methods have somewhat opposing approaches. In traditional methods, rules are taught first and then drilled into students via memorization and solving problems. In discovery methods, problems and examples come first and are abstracted to rules and formulas.</p>
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<span class="caption">Doug Ford’s ‘back-to-basics’ approach to mathematics is a traditional method where math is drilled into students via memorization.</span>
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<p>For example, in a traditional math lesson, children are told the rule that the order of multiplication of two numbers doesn’t matter, and then they would work on problems related to that topic. In discovery math, children would work out examples such as 2 times 3 and 3 times 2, and then abstract this to the general case. Both approaches teach the same thing, but in different ways.</p>
<p>No one should be surprised by these changes after the Progressive Conservatives won a majority in the Ontario election in June. During the election campaign, Ford tweeted, “…We are going to scrap discovery math, and replace it with proven methods of teaching:”</p>
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<p><a href="https://www.cbc.ca/news/canada/toronto/ontario-sex-ed-curriculum-1.4794889">A similar change was announced regarding Ontario’s sex-ed curriculum</a> and elementary school teachers are now required to teach a curriculum essentially dating back to 1998.</p>
<h2>Which option is better?</h2>
<p>There are pros and cons to both traditional and discovery methods.</p>
<p>My issue with the debate about the “correct” way to teach mathematics to children is the way it is phrased as a binary, either-or approach. The choices we are given are:</p>
<p>1) Drill students on topics such as fractions and timetables.</p>
<p>2) Have children discover math rules and formulas from scratch.</p>
<p>Neither approach in isolation does justice to math education or reflects how people learn mathematics.</p>
<p>Learning rules and formulas in mathematics is an essential skill, as you need a foundation from which to build. Children need to know what the product of 6 and 8 is without having to rediscover it every time.</p>
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<span class="caption">Children gain critical skills through discovery.</span>
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<p>At the same time, children gain critical problem-solving skills via discovery. They get to think more deeply about the subject. No one would teach language skills by only teaching grammar. You teach children the rudiments of grammar to get them speaking, reading and writing.</p>
<p>In my university teaching, I employ a mixture of traditional and discovery approaches. For example, in a first-year Calculus course, I introduce a formula or rule at the beginning of a lecture but spend most of the class working out examples interactively with the class so they may figure out how things work. In a more advanced, upper-year mathematics course, I state a theorem or problem, but then break up the class into smaller groups and have the students discover the proof with hints from me along the way.</p>
<h2>A third path: Math specialists</h2>
<p>When I was in elementary school, our classes had the occasional visit from specialist teachers who focused solely on art or music. These teachers didn’t perform the regular, daily instruction in classes but instead floated between classes enriching the curriculum. It was always a treat when these specialist teachers came and it also was a break from the routine of everyday instruction.</p>
<p>Let’s imagine something like this with mathematics education. Math specialists could be teachers with a mathematics background in their university education, or even math professors or university students with the proper training to engage with elementary school classes. I can think of plenty of fun and engaging lessons in my research area of networks for a Grade 6 class, for example. Math specialists would float between classes with the sole goal of enriching math education for kids.</p>
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<span class="caption">Lisa Thompson, Ontario’s Minister of Education scrums with reporters following Question Period, at the Queens Park Legislature, in Toronto on Thursday, August 9, 2018.</span>
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<p>Math specialists may assist with the teaching of core material, but more importantly, they would help coach teachers and provide lesson plans that <em>complement</em> the material. Their goals would be to engage students and cultivate their interest in learning mathematics. </p>
<p>While mathematics educators like myself might point to the positive impact of math specialists, research on their effectiveness is still emerging. A study funded by a National Science Foundation <a href="http://www.vamsc.org/index.php/research-on-mathematics-specialists/">at Virginia Commonwealth University</a> found that math specialists have a significant, positive impact on student achievement. Similar results were reported by the <a href="https://www.nctm.org/Standards-and-Positions/Position-Statements/The-Role-of-Elementary-Mathematics-Specialists-in-the-Teaching-and-Learning-of-Mathematics/">National Council of Teachers of Mathematics</a>.</p>
<h2>Beyond test scores</h2>
<p>While test scores are important, there is much more to mathematics education. Despite the declining EQAO scores, our kids aren’t exactly flunking out of math in droves. <a href="https://ottawacitizen.com/news/local-news/reevely-ontario-students-math-scores-keep-slipping-but-new-government-is-changing-little">The EQAO standards require a 70 per cent or better to qualify as meeting the standard</a>, not the typical passing grade of 50 per cent. </p>
<p>Our children must be exposed to a rich, engaging mathematics curriculum, even if they don’t become mathematicians or have anything to do with STEM (science, technology, engineering, math) directly in their adult lives. Numeracy, like literacy, is an essential skill in our modern world.</p>
<p>A generation with weak math skills will not be competitive to tackle the next set of challenges in our knowledge-based economy. And a dislike of math tends to pass on from one generation to the next. No child should think it makes them cool to boast that they hate math.</p>
<p>While we are rethinking math education in Ontario, let’s use the best of both traditional and discovery methods and add in math specialists. Done correctly, this should not only increase test scores, but also bolster student engagement.</p>
<p>Let’s also take the time and effort to make math <em>fun</em>. Imagine if children were excited to learn mathematics? Isn’t that what we all want?</p><img src="https://counter.theconversation.com/content/103192/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Anthony Bonato 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>Premier Doug Ford and Minister of Education Lisa Thompson have told elementary school teachers to expect curriculum change directives for "back to basics" mathematics, a move that could hurt students.Anthony Bonato, Professor of Mathematics, Ryerson UniversityLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/1023162018-09-11T21:39:07Z2018-09-11T21:39:07ZLet's teach students why math matters in the real world<figure><img src="https://images.theconversation.com/files/234923/original/file-20180904-45181-10qk96e.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Teaching students about how ancient civilizations used geometry to build structures like the pyramids in Egypt is part of a new integrated approach to learning science, technology, engineering and math.</span> <span class="attribution"><span class="source">(AP Photo/Nariman El-Mofty)</span></span></figcaption></figure><p>“When will I ever use this?” It’s a question math and science teachers hear all the time from their high school students.</p>
<p>Teaching science, technology, engineering and math (STEM) skills is more important than ever, but it’s often difficult for students to understand the practical applications of such fundamental learning and how it will help them down the road.</p>
<p>Classroom activities should be relevant, meaningful and connected to students’ prior knowledge and experiences. Learning must be based on lived experiences within both formal and informal educational settings.</p>
<p>Increasingly, teacher educators are realizing that we must break away from traditional silos of courses, disciplines and formal schooling. Educators must lead by example and provide students with opportunities to explore interdisciplinary approaches to learning. </p>
<h2>Creative thinking</h2>
<p>The new <a href="https://curriculum.gov.bc.ca/">British Columbia curriculum</a> embraces these principles of learning. In the same spirit, I’m part of a new and unique Bachelor of Education program at Thompson Rivers University where <a href="http://inside.tru.ca/2018/06/12/love-for-teaching-stems-from-science/">teacher candidates are learning to teach STEM by actively engaging students</a>. The program promotes cross-curricular and interdisciplinary approaches to learning and is tied to the provincial curriculum <a href="https://curriculum.gov.bc.ca/competencies">core competencies</a> of communication, critical and creative thinking.</p>
<p>So how do you teach a subject like math differently in a way that can help students learn through creative thinking and experience, rather than rote memorization?</p>
<p>Let’s take, for example, Pi. </p>
<p>I often ask my teacher candidates: What is π? Many respond “3.14” and, if probed further, explain the meaning by merely stating an equation like A=πr² (where A is the area of a circle and r is the radius of a circle). Or they may tell me C=2πr (where C is the circumference of a circle).</p>
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<span class="caption">A door handle in the shape of Pi at the National Museum of Mathematics in New York,</span>
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<h2>Teaching through discovery</h2>
<p>I encourage these teacher candidates to think differently and to help students discover mathematical concepts for themselves. What better way to teach students that π is the ratio of a circle’s circumference to its diameter than to have them trace any circle and then measure it with a piece of string?</p>
<p>They will soon learn that regardless of the size of the circle, the ratio of circumference to diameter will always be 22/7, an approximation of π.</p>
<p>Innovative educators can integrate history, geography, math and science lessons by teaching a thematic unit on ancient civilizations.</p>
<p>For example, the Egyptians succeeded in building great pyramids with incredible precision and accuracy. These magnificent architectural accomplishments have stood the test of time, remaining largely intact after centuries — a tribute to their construction.</p>
<p>The ancient Egyptians understood the significance of mathematics through the very beauty and symmetry of nature. They used geometry to solve everyday problems.</p>
<h2>Tearing down silos</h2>
<p>Increasingly, teacher educators are realizing that we must break away from traditional silos of courses, disciplines and formal schooling — exactly the opposite of the <a href="https://www.ontariopc.ca/on_education_doug_ford_will_respect_parents_and_get_back_to_basics">“back to basics” approach suggested by populist politicians like new Ontario Premier Doug Ford</a>.</p>
<p>Students benefit from learning experiences that are meaningful, relevant and well-connected to their own experiences. For that to happen, the people teaching those students must be prepared to take on new attitudes of reflectiveness and inquisitiveness.</p>
<p>What is necessary is to follow in the footsteps of the great thinkers like Galileo and Newton, who questioned our perceptions of reality and sought answers from tactile experiences rather than textbooks or teachers.</p><img src="https://counter.theconversation.com/content/102316/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Edward R. Howe 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>To get more students interested in STEM subjects, teachers must break out of the traditional subject-matter silos and use an approach that helps kids understand how math is used in the real world.Edward R. Howe, Associate Professor, Thompson Rivers UniversityLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/874792017-11-29T22:41:35Z2017-11-29T22:41:35ZThe 'new math': How to support your child in elementary school<figure><img src="https://images.theconversation.com/files/197013/original/file-20171129-12027-1r1wfrk.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Parents find new methods for learning math challenging, as they are different. But they work for children, building upon what they have learned about numbers and reinforcing the strategy they use for reading.</span> <span class="attribution"><span class="source">(Shutterstock)</span></span></figcaption></figure><p>There is likely no topic in Canada at the moment that is more acrimonious than elementary school mathematics education. The entire country, it seems, is divided. </p>
<p>On one side, there are those who are enraged by the so-called “new math” that has been held simultaneously responsible for a) diminished achievement by students and b) frustration among parents who feel helpless in the face of unfamiliar strategies. </p>
<p>On the other side are those who insist that math must make sense to today’s students — children who have grown up in a digital age, are adept with multiple technologies and will likely never be required to perform long division.</p>
<p>As a researcher who is deeply committed to engaging parents as partners in mathematics education, I spend many evenings on the road. I work with school staff and school councils across the province of Ontario to support parents in their efforts to help their children learn and love mathematics. </p>
<p>In communities from Chesterville to Picton, Guelph to Thunder Bay and Courtice to Fort Frances, I have encountered the same question repeatedly: What are you teaching my child?</p>
<h2>Arithmetic from Mexico to Japan</h2>
<p>The question is always sincere. The rationale differs considerably, but in most cases, the question arises because the computational strategies that the child is using to perform multi-digit calculations look very different from those learned by the parents, resulting in confusion and mistrust.</p>
<p>Experience has taught me to give a quick mini-lesson on arithmetic around the world to emphasize that there is no one global set of rules for calculations.</p>
<p>For example, I show a method that was used in Mexico, called “llevamos uno” — <em>we carry one</em>. Instead of noting ones or 10s to be “carried” at the top of the next column, students were taught to note those figures to the right side of the problem.</p>
<p> 1 9 4<br>
<u>+ 3 9¹¹</u><br>
2 3 3</p>
<p>I share a method that I learned from the Philippines, where students use dashes to indicate groups of 10. </p>
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<span class="caption">Elementary students in Baybay City, the Philippines, December 2015.</span>
<span class="attribution"><span class="source">(Shutterstock)</span></span>
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<p>Finally, I share a Japanese “scratch method” that is similar to the one used in the Philippines, but instead of dashes, overstrikes are used to keep track of groups of 10s. In addition, the leftover amounts are indicated by the use of subscripts.</p>
<p> 2 6<br>
<strike>7</strike>₁ <strike>6</strike>₂<br>
<u>+2 <strike>8</strike>₀</u><br>
1 3 0</p>
<p>Again, we begin at the right, at the top of the column: six plus six is 12, which is 10 (strike through the six) and two is left over (subscript two); two plus eight is 10, (strike through the eight) and zero (subscript zero). Write the zero under the ones column, and carry two groups of 10; two (10s) and two is four, plus seven (10s) is 11. Strike through the seven (to represent 100) and record one (subscript one). One plus two is three. Write the three in the 10s column and carry one group of 100. The answer is 130.</p>
<h2>We read left to right</h2>
<p>Having made the point that there is no universal set of rules to add multi-digit numbers and that all unfamiliar methods (including those used by their children) seem complex and incomprehensible at first glance, I am able to emphasize two important reasons to support new strategies for multi-digit addition.</p>
<p>When I ask parents to reflect on how they read to and with their toddlers, the answer is immediate and consistent: From left to right, using their index finger to trace the direction of the words.</p>
<p>Then I ask them what happens when we introduce children to the task of adding two-digit numbers. The light bulbs go on. We teach them to work right to left. </p>
<p>“Why?” I ask. </p>
<p>Dead silence or: “Because.”</p>
<p>In our number system, the value of a digit depends on its place, or position, in the number. So, for example, the number 4,276 is made up of 4,000 + 200 + 70 + six. Children who understand place value, i.e., that the value of a digit (zero to nine) depends on its position in a number, can easily decompose a number — an important strategy for mental math.</p>
<p>It’s ironic that after months of teaching the importance of place value, a fundamental concept in math, we do not apply that knowledge in practical ways to simplify multi-digit addition. As soon as we introduce questions like …</p>
<p> 8 7<br>
<u>+ 6 5</u></p>
<p>… we instruct students to begin at the right. This is in conflict with everything that children have been taught about reading from left to right and the importance of place value, i.e., that we read numbers from left to right, in order of magnitude. The algorithm, in fact, leads children to “unlearn” everything they know about place value.</p>
<h2>Building on children’s understandings</h2>
<p>Multi-digit arithmetic makes sense when we add from left to right, applying what we know about place value and reading.</p>
<p> 6 7<br>
+<u> 2 4</u><br>
8 0<br>
<u>1 1</u><br>
9 1</p>
<p>In this case, we add the 10s column first, 60 plus 20 to get 80. Next, we add seven to four to get 11. Add 80 and 11 to get the sum. This eliminates the need for “carrying” because the numbers align according to their value. </p>
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<p>Children respond positively to this strategy because it makes sense. It builds on their understanding of place value and how numbers are made. </p>
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<p>Why are parents so resistant to such strategies? The traditional algorithms are used by adults in their peer group and come from adults whom they respect. This may attach an aura to the traditional methods as the “real” or ultimately correct way to compute. </p>
<p>As mathematics education giant <a href="http://www.pearsoned.ca/highered/divisions/hss/vandewalle/index.html">John van de Walle</a> once noted, it’s difficult to ignore the power of adding “the way my dad taught me.”</p>
<p>But it’s time to ask: Are the traditional algorithms really necessary? Or are we holding our children back by our own fears and lack of understanding of the alternatives?</p><img src="https://counter.theconversation.com/content/87479/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Lynda Colgan receives funding from SSHRC, NSERC, The Council of Ontario Directors of Education, The Ministry of Education for the Province of Ontario, The Mathematics Knowledge Network (The Fields Institute for Research in the Mathematical Sciences) </span></em></p>You may not know it, but the elementary math wars are raging. Our expert explains the 'new math' - why it works for kids, and how to do it.Lynda Colgan, Professor of Elementary Mathematics, Queen's University, OntarioLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/786602017-06-21T10:31:01Z2017-06-21T10:31:01ZChallenging the status quo in mathematics: Teaching for understanding<figure><img src="https://images.theconversation.com/files/174303/original/file-20170618-28772-1vhqkpw.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">How can we change math instruction to meet the needs of today's kids?</span> <span class="attribution"><a class="source" href="https://flic.kr/p/97aGY8">World Bank Photo Collection / flickr</a>, <a class="license" href="http://creativecommons.org/licenses/by-nc-nd/4.0/">CC BY-NC-ND</a></span></figcaption></figure><p>Despite decades of <a href="http://files.eric.ed.gov/fulltext/ED372969.pdf">reform efforts</a>, mathematics teaching in the U.S. <a href="http://www.jstor.org/stable/20405948">has changed little</a> in the last century. As a result, it seems, American students have been left behind, now ranking <a href="https://nces.ed.gov/pubs2017/2017048.pdf#page=31">40th in the world</a> in math literacy. </p>
<p>Several state and national reform efforts have tried to improve things. The most recent <a href="http://www.corestandards.org/Math/">Common Core standards</a> had a great deal of promise with their focus on how to teach mathematics, but after several years, <a href="http://journals.sagepub.com/doi/full/10.3102/0013189X17711899">changes in teaching practices</a> have been minimal. </p>
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<p>As an education researcher, I’ve observed teachers trying to implement reforms – often with limited success. They sometimes make changes that are more cosmetic than substantive (e.g., more student discussion and group activity), while failing to get at the heart of the matter: What does it truly mean to teach and learn mathematics?</p>
<h2>Traditional mathematics teaching</h2>
<p>Traditional middle or high school mathematics teaching in the U.S. <a href="http://www.jstor.org/stable/20405948">typically follows this pattern</a>: The teacher demonstrates a set of procedures that can be used to solve a particular kind of problem. A similar problem is then introduced for the class to solve together. Then, the students get a number of exercises to practice on their own.</p>
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<span class="caption">The basics of math instruction have changed little since George Eaton taught at Phillips Academy (1880-1930).</span>
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<p>For example, when students learn about the area of shapes, they’re given a set of formulas. They put numbers into the correct formula and compute a solution. More complex questions might give the students the area and have them work backwards to find a missing dimension. Students will often learn a different set of formulas each day: perhaps squares and rectangles one day, triangles the next. </p>
<p>Students in these kinds of lessons are learning to follow a rote process to arrive at a solution. This kind of instruction is so common that it’s seldom even questioned. After all, within a particular lesson, it makes the math seem easier, and students who are successful at getting the right answers find this kind of teaching to be very satisfying.</p>
<p>But it turns out that teaching mathematics this way can actually <a href="http://www.jstor.org/stable/3696735">hinder learning</a>. Children can become dependent on <a href="http://www.jstor.org/stable/10.5951/teacchilmath.21.1.0018">tricks and rules</a> that don’t hold true in all situations, making it harder to adapt their knowledge to new situations.</p>
<p>For example, in traditional teaching, children learn that they should distribute a number by multiplying across parentheses and will practice doing so with numerous examples. When they begin learning how to solve equations, they often have trouble realizing that it’s not always needed. To illustrate, take the equation 3(x + 5) = 30. Children are likely to multiply the 3 across the parentheses to make 3x + 15 = 30. They might just as easily have divided both sides by 3 to make x + 5 = 10, but a child who learned the distribution method might have great difficulty recognizing the alternate method – or even that both procedures are equally correct.</p>
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<span class="caption">Students who learn by rote drilling often have trouble realizing that there are equally valid alternative methods for solving a problem.</span>
<span class="attribution"><span class="source">Kaitlyn Chantry</span></span>
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<h2>More than a right answer</h2>
<p>A key missing ingredient in these traditional lessons is conceptual understanding. </p>
<p>Concepts are ideas, meaning and relationships. It’s not just about knowing the procedure (like how to compute the area of a triangle) but also the significance behind the procedure (like what area means). How concepts and procedures are related is important as well, such as how the area of a triangle can be considered half the area of a rectangle and how that relationship can be seen in their area formulas. </p>
<p>Teaching for conceptual understanding has <a href="http://math.coe.uga.edu/Olive/EMAT3500f08/instrumental-relational.pdf">several benefits</a>. Less information has to be memorized, and students can translate their knowledge to new situations more easily. For example, understanding what area means and how areas of different shapes are related can help students understand the concept of volume better. And learning the relationship between area and volume can help students understand how to interpret what the volume means once it’s been calculated.</p>
<p>In short, building relationships between <a href="https://doi.org/10.1007/s10648-015-9302-x">how to solve a problem and why it’s solved that way</a> helps students <a href="https://doi.org/10.1037//0022-0663.91.1.175">use what they already know</a> to solve new problems that they face. Students with a truly conceptual understanding can see how methods emerged from <a href="https://doi.org/10.1037/0022-0663.91.1.175">multiple interconnected ideas</a>; their relationship to the solution goes deeper than rote drilling.</p>
<p>Teaching this way is a critical first step if students are to begin recognizing mathematics as meaningful. Conceptual understanding is a key ingredient to helping people think mathematically and use mathematics outside of a classroom.</p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/174193/original/file-20170616-537-p8ad2j.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip">
<figcaption>
<span class="caption">Procedural learning promotes memorization instead of critical thinking and problem solving.</span>
<span class="attribution"><a class="source" href="https://www.shutterstock.com/image-photo/math-study-exam-set-book-pencil-250606378">m.jrn/shutterstock.com</a></span>
</figcaption>
</figure>
<h2>The will to change</h2>
<p>Conceptual understanding in mathematics has been recognized as important for <a href="http://www.nctm.org/uploadedFiles/About/President,_Board_and_Committees/Board_Materials/MLarson-SF-NCTM-4-16.pdf">over a century</a> and widely discussed for decades. So why has it not been incorporated into the curriculum, and why does traditional teaching abound? </p>
<p>Learning conceptually can take longer and be more difficult than just presenting formulas. Teaching this way may require additional time commitments both in and outside the classroom. Students may have never been asked to think this way before.</p>
<p>There are systemic obstacles to face as well. A new teacher may face pressure from fellow teachers who teach in traditional ways. The <a href="https://www.thoughtco.com/high-stakes-testing-overtesting-in-americas-public-schools-3194591">culture of overtesting</a> in the last two decades means that students face more pressure than ever to get right answers on tests. </p>
<p>The results of these tests are also being <a href="https://tcta.org/node/13251-issues_with_test_based_value_added_models_of_teacher_assessment">tied to teacher evaluation systems</a>. Many teachers feel pressure to teach to the test, drilling students so that they can regurgitate information accurately.</p>
<p>If we really want to improve America’s mathematics education, we need to rethink both our education system and our teaching methods, and perhaps to <a href="http://www.nea.org/home/40991.htm">consider how other countries approach mathematics instruction</a>. Research has provided evidence that teaching conceptually has <a href="http://www.ascd.org/publications/educational-leadership/feb04/vol61/num05/Improving-Mathematics-Teaching.aspx">benefits</a> not offered by traditional teaching. And students who learn conceptually typically do <a href="https://doi.org/10.3102/0034654310374880">as well or better</a> on achievement tests. </p>
<p>Renowned education expert <a href="https://pasisahlberg.com/">Pasi Sahlberg</a> is a former mathematics and physics teacher from Finland, which is renowned for its world-class education. He <a href="http://www.smithsonianmag.com/innovation/why-are-finlands-schools-successful-49859555/">sums it up</a> well:</p>
<blockquote>
<p>We prepare children to learn how to learn, not how to take a test.</p>
</blockquote><img src="https://counter.theconversation.com/content/78660/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Christopher Rakes receives funding from the National Science Foundation. </span></em></p>Math instruction is stuck in the last century. How can we change teaching methods to move past rote memorization and help students develop a more meaningful understanding – and be better at math?Christopher Rakes, Assistant Professor of Mathematics Education, University of Maryland, Baltimore CountyLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/709252017-01-09T20:26:48Z2017-01-09T20:26:48ZSouth Africa can't compete globally without fixing its attitude to maths<figure><img src="https://images.theconversation.com/files/152080/original/image-20170109-23482-uehzdp.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Investing in pupils' maths skills is an investment in a country's economy.</span> <span class="attribution"><span class="source">Global Partnership for Education/Flickr</span></span></figcaption></figure><p>South Africa is not producing enough school leavers who are competent in maths and science. This is a fact borne out by international assessments such as the <a href="http://timssandpirls.bc.edu/publications/timss/2015-methods.html">Trends in International Mathematics and Science Study</a> (TIMMS) and the World Economic Forum’s <a href="https://www.weforum.org/reports/the-global-competitiveness-report-2016-2017-1">Global Competitiveness Report</a>. These show that South Africa is not making much headway when it comes to maths and science. </p>
<p>The 2016 Global Competitiveness Report ranked South Africa last among 140 countries for maths and science. This places it behind poorer African countries like Mozambique and Malawi.</p>
<p>In 2016 there was a <a href="https://businesstech.co.za/news/government/148875/matric-results-2016-maths-and-science-suffers/">marginal improvement</a> in the maths pass rate, from 49.1% the previous year to 51.1%. The country is moving at a glacial pace in an area that demands urgent attention. After all, science and maths are key to any country’s economic development and its competitiveness in the global economy. </p>
<p>The TIMMS study ranks Singapore, Hong Kong, South Korea and Japan among its top maths and science performers. It’s no coincidence that these countries feature among the <a href="http://www.wipo.int/pressroom/en/articles/2016/article_0008.html">top 20</a> on the Global Innovation Index. Good quality education fuels an economy. South Africa needs to increase its supply of science and technology university graduates, which at the moment constitute the bulk of <a href="http://www.dhet.gov.za/Gazette/Government%20Gazette%20No%2039604,%2019%20January%202016.%20List%20of%20Occupations%20in%20High%20Demand%202015.pdf">scarce skills</a> outlined by the Department of Higher Education and Training. </p>
<p>But instead of chasing improved results the government is lowering the bar for maths at school level. At the end of 2016 it set <a href="http://www.education.gov.za/Portals/0/Documents/Publications/Circular%20A3%20of%202016.pdf?ver=2016-12-07-154005-723">20% as a passing mark</a> for pupils in grades 7, 8 and 9. This lends credence to the common view of maths as a subject only the “gifted” can comprehend. </p>
<p>It’s time to place a premium on maths and to ensure that pupils – especially those from poorer backgrounds – receive the necessary support to excel at maths. This is critical if South Africa is to produce the human capital needed to drive economic growth and create new industries in the future. </p>
<h2>How maths and science boost economies</h2>
<p>Maths and science are a gateway to new industries. Mastery of them endows an economy with the human capital needed to ride the technological wave. In his work on the industries of the future Alec Ross, who advised Hillary Clinton on innovation during her term as US Secretary of State, <a href="http://www.simonandschuster.com/books/The-Industries-of-the-Future/Alec-Ross/9781476753652">points out</a> that sectors such as robotics, advanced life sciences, codification of money, big data and cybersecurity – all of which require mastery of technology and mathematical skills – are the pillars of the <a href="https://www.weforum.org/agenda/2016/01/the-fourth-industrial-revolution-what-it-means-and-how-to-respond/">fourth industrial revolution</a>. </p>
<p>Simply put, this “revolution” is the age of technology that’s already upon us.</p>
<p>More importantly, a grasp of maths and science boosts confidence and expands career possibilities for pupils. This ultimately gives them an edge in the labour market. </p>
<p>Many students drop out of maths not by choice but because they’re frustrated by a lack of adequate support. I speak from experience: I dropped the subject when I was 14 at the end of what’s now Grade 9 but used to be called Standard 7. Our maths teacher didn’t encourage those he called “slow learners” to continue with the subject and I was one of many intimidated into giving up on maths.</p>
<p>But succeeding in maths, or in any area of skill, isn’t entirely a matter of genetic endowment. Psychologist Anders Ericsson, <a href="http://www.goodreads.com/book/show/26312997-peak">in his book Peak</a>, draws on three decades of research to prove why natural talent and other innate factors have less of an impact than what he calls deliberate or purposeful practice.</p>
<p>He contends that</p>
<blockquote>
<p>a number of successful efforts have shown that pretty much any child can learn math if it is taught in the right way.</p>
</blockquote>
<p>South Africa should be focusing on how to teach maths in the right way rather than buying into the myth that it is an impossible subject. The current approach is robbing the economy of critical human capital.</p>
<h2>Radical interventions</h2>
<p>Some may argue, though, that any improvement or shift is impossible in an education system that’s plagued by weak infrastructure, a lack of teacher development and support and too few qualified maths and science teachers. Even if the numbers of teachers in these subjects were to increase, it’s crucial that the quality of education rises too.</p>
<p>Radical interventions are needed, now – or South Africa will never become a global player in the fourth industrial revolution. </p>
<p>The country must develop new teacher training methods and nurture a supportive environment for teachers. Innovative teaching tools should be introduced in the early phases to demystify maths and science for young pupils. If these subjects are more fun to learn, more pupils may be drawn to them as future career options.</p>
<p>Taking these steps will give South Africa a better chance in the future to harness the talent of its youth to powering the economy, and improve its global competitiveness.</p><img src="https://counter.theconversation.com/content/70925/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Mzukisi Qobo does not work for, consult, own shares in or receive funding from any company or organization that would benefit from this article, and has disclosed no relevant affiliations beyond their academic appointment.</span></em></p>Good quality education fuels an economy. South Africa needs to increase its supply of science and technology university graduates. But instead it's lowering the bar, especially when it comes to maths.Mzukisi Qobo, Associate Professor at the Institute for Pan African Thought and Conversation, University of JohannesburgLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/557402016-03-04T19:18:30Z2016-03-04T19:18:30Z'The Math Myth' fuels the algebra wars, but what's the fight really about?<figure><img src="https://images.theconversation.com/files/113910/original/image-20160304-17753-1dd1vyc.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">A confused student might not be leaving a math classroom....</span> <span class="attribution"><a class="source" href="http://www.shutterstock.com/pic.mhtml?id=295860566">Student image via www.shutterstock.com.</a></span></figcaption></figure><p>I discovered recently that my calculus students do not know the meaning of the word “quorum.” Since a course in American government is a high school graduation requirement in most states (including here in Florida), I was taken aback.</p>
<p>How should I react? Should I take to the editorial pages of <em>The New York Times</em>, bemoaning the state of civics education? Should I call out political scientists and high school history teachers for their failures?</p>
<p>Surely you’d admonish me to calm down a bit and perhaps not venture into disciplines where I’m not an expert.</p>
<figure class="align-right zoomable">
<a href="https://images.theconversation.com/files/113885/original/image-20160304-17753-1oywxos.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=1000&fit=clip"><img alt="" src="https://images.theconversation.com/files/113885/original/image-20160304-17753-1oywxos.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip"></a>
<figcaption>
<span class="caption"></span>
<span class="attribution"><a class="source" href="http://thenewpress.com/books/math-myth">The New Press</a></span>
</figcaption>
</figure>
<p>Yet Andrew Hacker, professor emeritus of political science at the City University of New York, recently <a href="http://www.nytimes.com/2016/02/28/opinion/sunday/the-wrong-way-to-teach-math.html">took this exact approach</a> to attack the teaching of algebra in American schools. He also <a href="http://thenewpress.com/books/math-myth">wrote a book</a>. And he’s <a href="http://www.nytimes.com/2012/07/29/opinion/sunday/is-algebra-necessary.html">done it before</a>.</p>
<p>Nor is he alone. Novelist Nicholson Baker <a href="https://harpers.org/archive/2013/09/wrong-answer/">wrote a piece</a> for <em>Harper’s</em> in 2013 that got the math community talking. The real target of Baker’s piece was the accountability movement and the associated standardized testing, but he chose mathematics as his straw man because it (a) is easy, and (b) will sell magazines. He manages to boil the modern course in Algebra II down to this:</p>
<blockquote>
<p>It’s a highly efficient engine for the creation of math rage: a dead scrap heap of repellent terminology, a collection of spiky, decontextualized, multistep mathematical black-box techniques that you must practice over and over and get by heart in order to be ready to do something interesting later on, when the time comes.</p>
</blockquote>
<p>At least Baker is an entertaining writer.</p>
<p>Hacker makes many of the same points in his <em>Times</em> articles, decrying algebra as a high school graduation requirement that holds back far too many students from having a productive life. He argues instead for “numeracy” and suggests what such a course should contain. It’s mostly statistics and financial mathematics, and lessons in visualizing and analyzing data.</p>
<p>To fight off the counterassertion that it’s possible to learn this material in a high school advanced placement statistics course, Hacker comes up with lists of obscure terminology: “The A.P. [Statistics] syllabus is practically a research seminar for dissertation candidates. Some typical assignments: binomial random variables, least-square regression lines, pooled sample standard errors.”</p>
<h2>It’s not just happening in math</h2>
<p>Every subject in school has been broken down into a string of often unrelated facts or tasks, not just mathematics.</p>
<p>I recall an episode from my own son’s experience in ninth grade while taking “Honors Pre-AP English I” (yes, that’s the real name of the course, not some Orwellian nightmare). His teacher led the class through the “CD/CM method” of essay writing, which goes like this. Fill out a worksheet with the “funnel” (4-7 sentence introduction), the thesis statement, and then for each of three paragraphs create 11 (!) sentences – the topic sentence (fine) and then CD#1, CM#1, CD#2,CM#2,…,CD#5,CM#5. What is a CD, you ask? Concrete Detail. A CM? Comment, of course.</p>
<p>Now, this is really just a superextended outline for an essay, but my son was extremely frustrated by this, eventually exclaiming, “I just want to write the damn paper!”</p>
<p>Is this example from the humanities really any different from what Hacker and Baker complain about?</p>
<p>Hacker is not completely wrong, however. School mathematics <em>has</em> largely been drained of context and beauty. University mathematicians complain about this, too.</p>
<p>For example, my son has also brought home worksheets full of dozens of polynomials with the simple instruction: Factor. But why?</p>
<figure class="align-left ">
<img alt="" src="https://images.theconversation.com/files/113797/original/image-20160303-13754-1e63a0r.png?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip">
<figcaption>
<span class="caption">Light rays striking a parabolic mirror reflect to a common point called the focus (point F above).</span>
<span class="attribution"><span class="source">created in Geogebra by the author</span></span>
</figcaption>
</figure>
<p>There is no context given for why we care about polynomial equations, no discussion of why parabolas (graphs of quadratic equations) are useful things. Maybe we should explain that without parabolas, we wouldn’t have good headlights on our cars or all those pretty pictures of deep space from the Hubble telescope. But just as mathematicians would not argue for the elimination of English or civics from the high school curriculum, Hacker shouldn’t be arguing for the elimination of algebra.</p>
<p>Let’s be honest. Mostly because of the accountability movement and high-stakes testing, K-12 education suffers from these types of problems in every subject. Picking on math alone because it’s particularly vexing for some people is unsporting.</p>
<h2>Credibility gap</h2>
<p>Of course, Hacker and Baker have proposals for how to fix this mess. The problem is that the major prerequisite for much of what Hacker proposes is, ironically, algebra. Not so much the grinding, symbol-driven form of algebra taught in school today, but algebra nonetheless. Reading bar graphs in the newspaper is a skill that we should expect high school graduates to be able to do, but nontrivial calculations with data require at least some facility with algebra. Hacker surely knows this, but it would undermine his argument to admit it.</p>
<p>He’s certainly not wrong that some students fall by the wayside, and the way we teach algebra and geometry in the middle grades is largely to blame. Stanford mathematician Keith Devlin wrote a <a href="http://www.huffingtonpost.com/dr-keith-devlin/andrew-hacker-and-the-cas_b_9339554.html">wonderful response</a> to Hacker’s recent piece, pointing out how his ideas may actually be correct but misguided:</p>
<blockquote>
<p>Not only did that suggestion [the elimination of algebra from the high school curriculum] alienate accomplished scientists and engineers and a great many teachers – groups you’d want on your side if your goal is to change math education – it distracted attention from what was a very powerful argument for introducing the teaching of algebra into our schools, something I and many other mathematicians would enthusiastically support.</p>
</blockquote>
<p>Unfortunately, Hacker undermines his credibility by stating falsehoods. For example, he claims “Coding is not based on mathematics … Most people who do coding, programming, software design, don’t do any mathematics at all.” It may be true that these individuals are not crunching numbers all day (that’s what software is for, of course), but the algorithmic processes underlying coding are the very essence of mathematics. To say otherwise is just delusional.</p>
<p>Hacker also asks, “Would you go to a mathematician to tell us what to do in Syria? It just defies comprehension.” Actually, it shouldn’t. The Central Intelligence Agency and other national security groups <a href="https://www.cia.gov/careers/opportunities/analytical">employ thousands of mathematicians to analyze data</a> associated with foreign affairs, looking for patterns amid the chaos. So, Hacker is just plain wrong about some things, even if his overall idea has merit. </p>
<h2>We’re all on the same team</h2>
<p>You see, college math professors <em>know</em> there is a problem with K-12 mathematics. We see the results in our classrooms on campus. As much as Hacker would like to believe his <em>ad hominem</em> assertions about math faculties at high schools and colleges, we really just want our students well-prepared for the beautiful, fascinating and, yes, useful material we have to offer.</p>
<p>Algebra is a beautiful baby; it would be a shame to throw it out with some dirty bathwater.</p><img src="https://counter.theconversation.com/content/55740/count.gif" alt="The Conversation" width="1" height="1" />
A new book criticizes how and what American math classes are teaching. Singling out math instruction in this age of high-stakes testing and accountability is unsporting.Kevin Knudson, Professor of Mathematics, University of FloridaLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/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-1.1.0&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-1.1.0&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-1.1.0&q=45&auto=format&w=754&fit=clip"></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-1.1.0&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-1.1.0&q=45&auto=format&w=754&fit=clip"></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-1.1.0&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-1.1.0&q=45&auto=format&w=754&fit=clip"></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/459102015-08-20T04:31:05Z2015-08-20T04:31:05ZIntense after-school tutoring holds many lessons – for learners and teachers<figure><img src="https://images.theconversation.com/files/92384/original/image-20150819-10873-1346s3l.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">Pre-service teachers can learn a great deal from their young pupils.</span> <span class="attribution"><span class="source">Shutterstock</span></span></figcaption></figure><p>Pre-service teachers must spend a few months working in schools to practice their craft and learn from qualified educators. This is an important part of their training, but it doesn’t allow pre-service teachers to work for an extended period with the same group of learners.</p>
<p>The absence of such a sustained, intense interaction deprives pre-service teachers of an important opportunity to understand their learners’ challenges – and their own shortcomings – before entering a classroom full time.</p>
<p>A project in a peri-urban area about 75 kilometres from Cape Town is exploring what happens when trainee teachers are able to spend a full year tutoring the same one or two children. The early results are extremely encouraging for both the pre-service teachers and their 52 learners. It is also reaching a much wider pool as learners share their experiences with their peers.</p>
<h2>Immersing pre-service teachers to learn lessons</h2>
<p>The project, initiated by the Wellington campus of the Cape Peninsula University of Technology, is part of the curriculum for pre-service teachers training at the campus. <a href="http://www.wellington.co.za/our-history">Wellington</a> is a picturesque small town. As with many places in South Africa, it is home to both <a href="http://www.enca.com/south-africa/extreme-inequality-continues-bring-sa-down-says-oxfam-report">great wealth and terrible poverty</a>. </p>
<p>This project focuses on after-school tutoring in Maths and English for 52 primary school learners from disadvantaged backgrounds. The learners are all in grades 5 to 7, aged between 10 to 12, and are chosen based on their academic performance. They are the best students in their grades. The tutoring happens for an hour each week after school.</p>
<p>The project started in January 2015. The pre-service teachers were prepared for it during their English and Maths lectures. Two visiting scholars from the US who are working on community engagement projects like this one as part of their Fulbright scholarships came and shared their experiences.</p>
<p>Research <a href="http://quod.lib.umich.edu/m/mjcsl/3239521.0014.205?rgn=main;view=fulltext">shows</a> that this kind of after-school engagement has many benefits. It offers pre-service teachers who haven’t yet started working permanently in a classroom setting a real insight into the challenges that learners face when studying Maths and English.</p>
<p>It can also greatly develop the creativity and critical thinking skills of both the pre-service teachers and their young learners.</p>
<h2>Loving English and making Maths count</h2>
<p>The English leg of the project aims to improve teachers’ and learners’ proficiency in the language. <a href="http://www.omniglot.com/writing/afrikaans.htm">Afrikaans</a> is the most commonly spoken language in the Wellington area. All but one of the 30 pre-service teachers in the English component and all 52 learners from the four schools participating in the project speak English as a second language.</p>
<p>The lessons allow pre-service teachers to practice and develop their own communication and teaching skills. They enhance the learners’ love for English, develop their reading skills and give them a space in which to grapple with grammar problems.</p>
<p>This course provides a great opportunity for academic growth within a <a href="https://books.google.co.za/books?id=BAizTqaFT5kC&pg=PR8&lpg=PR8&dq=-Pamela+and+James+Toole+service+learning&source=bl&ots=V1Xbdqi9zH&sig=yKUMThs_fk6Hf0lc3KYB1FYD6gY&hl=en&sa=X&ved=0CC0Q6AEwA2oVChMIn4TQ7a6yxwIVQ7QaCh3pNA26#v=onepage&q&f=false">service-learning</a> context. </p>
<p>A non-profit organisation, Help2Read, sponsored containers full of new books, games, stickers, stationery and activities for each of the participating schools. These resources helped pre-service teachers to make their sessions with the learners <a href="http://teacher.scholastic.com/professional/bruceperry/pleasure.htm">more fun</a>.</p>
<figure class="align-left ">
<img alt="" src="https://images.theconversation.com/files/92275/original/image-20150818-12443-271lp7.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=237&fit=clip">
<figcaption>
<span class="caption">A pre-service teacher and one of her learners get down to work in Wellington.</span>
<span class="attribution"><span class="source">Rolene Liebenberg</span></span>
</figcaption>
</figure>
<p>For the maths component, learners are asked to bring school work they’ve been struggling with in class. At the start of each tutoring session, the pre-service teachers try to get a sense of whether learners are participating in their school maths classes.</p>
<p>For instance, learners will be asked what maths they were taught, how it was taught and whether they struggled with how concepts were explained. They are also asked to identify highlights from their classes. Engaging learners in this way gives them <a href="http://www.pythagoras.org.za/index.php/pythagoras/article/view/163">a voice</a> and, hopefully, teaches them that their experiences are valued.</p>
<p>They are also offered agency: in one session, two pre-service teachers reversed the power dynamic by giving learners the chance to structure questions and to question <em>them</em> rather than the other way around.</p>
<h2>Pre-service teachers get to learn, too</h2>
<p>Pre-service teachers at each of the four schools <a href="http://www.sciencedirect.com/science/article/pii/S0742051X12000819">form</a> a professional learning community. They meet each week after tutoring to share their experiences and talk about what they have learned. They are encouraged to <a href="http://link.springer.com/article/10.1007/s10857-005-1223-z">critically examine</a> what their learners are battling with and to ask tough questions about their own teaching methods.</p>
<p>Many pre-service teachers have reported that they struggled to address their learners’ challenges and questions during tutorial sessions. They needed to go away and think about their own understanding of the mathematical ideas being discussed and how best to develop this.</p>
<p>Teachers at the participating schools have seen a difference in their learners. During a progress meeting in July, one reported:</p>
<blockquote>
<p>My learners in the project … their self-confidence has increased … the way they analyse each other’s work … identifying errors and supporting other learners in class … (it) was not like this before</p>
</blockquote>
<h2>A ripple effect</h2>
<p>The project is having a positive impact beyond just the 52 learners who are taking part. Some learners have started tutoring their peers, using the skills they’ve developed during sessions with the pre-service teachers. </p>
<p>Learners who aren’t yet involved with the project have told their peers they want to work harder so they have a chance of taking part.</p>
<p>There has also been great interaction between pre-service teachers, who support and encourage each other through the project. Though these are still early days, it is shaping up as a wonderful professional collaboration opportunity for all involved and is set to become a permanent fixture in the curriculum.</p><img src="https://counter.theconversation.com/content/45910/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Rolene Liebenberg received funding from CHEC for the service learning project for 2015.</span></em></p><p class="fine-print"><em><span>Hanlie Dippenaar receives funding from CHEC</span></em></p>Pre-service teachers are learning a great deal about their own skills and the challenges of the job through a part of their curriculum that approaches training differently.Rolene Liebenberg, Mathematics Education Lecturer, Cape Peninsula University of TechnologyHanlie Dippenaar, Senior lecturer Dept of English, service-learning , Cape Peninsula University of TechnologyLicensed as Creative Commons – attribution, no derivatives.tag:theconversation.com,2011:article/450852015-08-18T10:06:28Z2015-08-18T10:06:28ZIn the push for marketable skills, are we forgetting the beauty and poetry of STEM disciplines?<figure><img src="https://images.theconversation.com/files/92112/original/image-20150817-5110-17nt3z4.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=496&fit=clip" /><figcaption><span class="caption">There is beauty in mathematical ideas and proofs.</span> <span class="attribution"><a class="source" href="https://www.flickr.com/photos/lucapost/694780262/in/photolist-24oVY3-diC14-4f3Jaz-4JqiwG-a5sw5-2RsTt1-geDfL-agNTbS-bz6igw-4f7GCA-aZLKD4-acJS5w-zdTJr-o8nVHc-6GsoZ-A3oZS-cd1WBC-8BMbiL-jXn1k8-jy4a28-4ikigj-usq3wD-6zjnBu-oo7TWg-anDsYW-2RsUqE-rzSR2m-pktg1Y-6aBPfC-qzzDXg-akeS8f-LfcF1-wdC58y-fkp13e-e9XnEF-73kFqy-d4AxJs-97N2Vr-baxAc-ugXsf-oqbq-8hDsUX-acJS9E-cVnd-pnMLBq-acJS7o-vhEJ3-6Mbpka-pDrCn8-5XbUTK">lucapost</a>, <a class="license" href="http://creativecommons.org/licenses/by-nc-nd/4.0/">CC BY-NC-ND</a></span></figcaption></figure><p>Thousands of students are preparing to begin their job searches with newly earned STEM (science, technology, engineering and mathematics) degrees in hand, eagerly waiting to use the logical, analytical and practical skills they’ve acquired.</p>
<p>However, as qualified as they might be, they could be missing one critical component of the STEM field – art.</p>
<p>I pursued an education and career in computer science and mathematics. And I know only too well that in the field of computer science, there is often an emphasis on elegance and beauty alongside sheer practicality. Indeed, programming itself is sometimes referred to as an <a href="http://ruben.verborgh.org/blog/2013/02/21/programming-is-an-art/">art</a>.</p>
<p>It is the same in related fields. The discipline of mathematics has long championed beauty as an important quality of ideas and proofs. And, of course, many engineers value elegance and beauty as important components in their designs and solutions.</p>
<h2>Poetry is at the heart of technology</h2>
<p>As many seasoned programmers and mathematicians will tell you, there is <a href="http://www.i-programmer.info/news/200-art/6808-writing-code-as-poetry-poetry-as-code.html">poetry in technology</a>. In fact, some regard such poetry as being at the heart of what they do. </p>
<p>In the 1980s, <a href="http://www.tracykidder.com/">Tracy Kidder</a> wrote The Soul of a New Machine, a book about the <a href="https://www.nytimes.com/books/99/01/03/specials/kidder-soul.html%22">pressure and effort</a> of building a next-generation computer. But more importantly, that account opened a lot of people’s eyes to the passion and beauty in creating these machines.</p>
<p>Many of the engineers in the book repeatedly explained that they didn’t work for the money, but rather for the gratification of invention and design – in essence, the beauty of it.</p>
<p>Indeed, reading that book was especially meaningful to me as I began my own studies in computing. </p>
<p>As I know through experience, constructing something poetic/beautiful is very fulfilling to the practitioner. Computer scientists value elegance and beauty in the creation of algorithms and computer programs. </p>
<h2>Mathematical beauty around us</h2>
<p>Similarly, ideas of beauty and poetry have always been important in mathematics.</p>
<p>Prominent mathematicians and computer scientists have long embraced elegance, beauty, poetry and literacy in the code that they write and the theorems that they prove. </p>
<figure class="align-center ">
<img alt="" src="https://images.theconversation.com/files/92115/original/image-20150817-25727-jkcswp.jpg?ixlib=rb-1.1.0&q=45&auto=format&w=754&fit=clip">
<figcaption>
<span class="caption">Beauty is important in programming as well.</span>
<span class="attribution"><a class="source" href="https://www.flickr.com/photos/seeminglee/8921779798/in/photolist-eAosiC-eM7Nt3-8fUHJ5-dJJaMa-75vxwE-7YmQp6-nVVCfx-7FiU1M-88RAmR-qcSVGW-73zoNZ-o9rumE-o9jEvD-o9CnNi-kJz7fa-75tWag-6sr9f1-74T8P4-9fK69A-pVwTQC-8zPfCv-75vGtW-6h7DLk-ao4yLs-6KosGR-9NnUjX-75y7TJ-9NqG1N-8KuSmz-9hYhPp-9NqGR3-9bavnP-qXct22-imxhyi-9NnXnt-4qfL61-byidYK-9wCAyR-e9VDki-8w1M8R-9fK64C-gDR3aL-bCL9xk-dDNWKm-73Arz1-81JzLb-j7vKq2-mocVVz-9s18nz-bBYzuW">See-ming Lee</a>, <a class="license" href="http://creativecommons.org/licenses/by/4.0/">CC BY</a></span>
</figcaption>
</figure>
<p>These ideas, in fact, have been around for millennia. Indeed, the extreme separation of the disciplines is relatively new in Western history.</p>
<p>Those doing science (natural philosophy) and mathematics were also often doing poetry and music. Many of today’s disciplines were subsumed as philosophy. So contemporary surprise at the idea that science and mathematics could be poetic is a somewhat recent phenomenon.</p>
<p>For example, Pythagoras was a philosopher/scientist/mystic/mathematician who <a href="http://www.pbs.org/wgbh/nova/physics/great-math-mystery.html">explored beauty</a> in art and music.</p>
<p>This attention to beauty and pattern continued through Fibonacci and beyond.</p>
<p>Fibonacci (13th century), considered to be the leading mathematician in the Middle Ages, is probably best known for the <a href="https://www.mathsisfun.com/numbers/fibonacci-sequence.html">Fibonacci Sequence</a> named after him: a number in the sequence is the sum of the previous two numbers (eg, start with 1, 2; then add to get 3. Then add 2, 3 to get 5, and it goes on: 1,2,3,5,8,13,21,34,…). Fibonacci <a href="http://io9.com/5985588/15-uncanny-examples-of-the-golden-ratio-in-nature">discovered</a> that much else that we regard as beautiful follows this elegant pattern.</p>
<p>This technical, mathematical beauty is evident in all of nature – from flower petals and shells to spiral galaxies and hurricanes,</p>
<h2>Discoveries come through intuition</h2>
<p>Intuition and discovery, rather than a kind of routine analysis, are important in computing as well as mathematics and science. Significant insights are known to come through intuition.</p>
<p>Intuition is deemed to be so important that developing “computer intuition” is one of the goals in the subfield of artificial intelligence.</p>
<p>So, in computing, there is really no “standard” way to write complex, interesting and aesthetically pleasing programs. Little surprise then that Stanford Professor Emeritus <a href="http://www-cs-faculty.stanford.edu/%7Euno/taocp.html">Donald Knuth’s</a> four-volume masterpiece is titled The Art of Computer Programming.</p>
<p>Similarly, years ago a colleague in the arts told me about the <a href="http://www.pbs.org/wgbh/nova/physics/andrew-wiles-fermat.html">PBS show</a> on Andrew Wiles’s proof of <a href="http://mathworld.wolfram.com/FermatsLastTheorem.html">Fermat’s Last Theorem</a>. Wiles, a British mathematician, devoted much of his career to proving Fermat’s Last Theorem, a problem that no one had been able to solve for 300 years.</p>
<p>My colleague confided that she was moved to tears during the program. Until that show she had thought that mathematics was cold, dry, absolute and passionless. That show completely changed her view so that she could finally see the passion and the poetry that permeates the STEM fields.</p>
<h2>STEM versus liberal arts?</h2>
<p>Many STEM graduates today spend their college years enrolling only in courses they believe will benefit them in their field, zeroing in on skills that will make them more marketable in the digital age, while overlooking social sciences, humanities and the arts. Of course, likewise, many humanities students try to avoid taking science and mathematics courses. </p>
<p>And it shouldn’t be this way; but that’s a discussion for another day.</p>
<p>It’s projected that <a href="http://www.bls.gov/emp/ep_table_101.htm">685,000 new employment opportunities</a> will be created by 2022 in computer and mathematical occupations.</p>
<p>But today’s students need to remember that technology is not just a matter of rote procedure – completing the task according to set protocol; that would not be particularly elegant. </p>
<p>As sciences, technology and computing become ever more powerful forces in the world, it’s important that the people piecing these things together are ethical and bring in the human attributes that are central to a liberal arts education.</p>
<p>We need thinkers, visionaries and creative minds. As the technology industry grows – and with it, employment opportunities – we need more candidates who are rooted in thought and fewer who can simply carry out a task.</p>
<p>For those students graduating with a liberal arts degree, who are unsure where their job hunt will take them, we welcome you with open arms to technology, mathematics and computing.</p>
<p>And for those in technology, celebrate your humanity and the the available cultural riches; become aware of the intuition and the poetry in what you do. Bring with you your love for beauty, passion and artistry, and be prepared to use them.</p><img src="https://counter.theconversation.com/content/45085/count.gif" alt="The Conversation" width="1" height="1" />
<p class="fine-print"><em><span>Paul Myers 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>Poetry is at the heart of technology. Did not Pythagoras find the connections between beautiful music and mathematics?Paul Myers, Chair of Computer Science , Trinity 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-1.1.0&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-1.1.0&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-1.1.0&q=45&auto=format&w=754&fit=clip"></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-1.1.0&q=45&auto=format&w=754&fit=clip">
<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>
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<a href="https://images.theconversation.com/files/86327/original/image-20150624-31495-14gxmoi.jpg?ixlib=rb-1.1.0&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-1.1.0&q=45&auto=format&w=754&fit=clip"></a>
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<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>
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<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>
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<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>
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<a href="https://images.theconversation.com/files/86326/original/image-20150624-31495-1wy27l9.jpg?ixlib=rb-1.1.0&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-1.1.0&q=45&auto=format&w=754&fit=clip"></a>
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<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>
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<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 organisation 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.