Jo Boaler | The Atlantichttps://www.theatlantic.com/author/jo-boaler/2018-05-10T12:34:13-04:00Copyright 2019 by The Atlantic Monthly Group. All Rights Reserved.tag:theatlantic.com,2016:50-478053<p>A few weeks ago I (Jo Boaler) was working in my Stanford office when the silence of the room was interrupted by a phone call. A mother called me to report that her 5-year-old daughter had come home from school crying because her teacher had not allowed her to count on her fingers. This is not an isolated event—schools across the country regularly ban finger use in classrooms or communicate to students that they are babyish. This is despite a compelling and rather surprising branch of neuroscience that shows the importance of an area of our brain that “sees” fingers, well beyond the time and age that people use their fingers to count.</p><p>In <a href="http://www.ncbi.nlm.nih.gov/pmc/articles/PMC4360562/">a study</a> published last year, the researchers Ilaria Berteletti and James R. Booth analyzed a specific region of our brain that is dedicated to the perception and representation of fingers known as the somatosensory finger area. Remarkably, brain researchers know that we “see” a representation of our fingers in our brains, even when we do not use fingers in a calculation. The researchers found that when 8-to-13-year-olds were given complex subtraction problems, the somatosensory finger area lit up, even though the students did not use their fingers. This finger-representation area was, according to their study, also engaged to a greater extent with more complex problems that involved higher numbers and more manipulation. <a href="http://www.ncbi.nlm.nih.gov/pubmed/16306017">Other</a> <a href="http://carleton.ca/cmi/wp-content/uploads/CSS07_pp740-penner-wilger.pdf">researchers</a> have found that the better students’ knowledge of their fingers was in the first grade, the higher they scored on number comparison and estimation in the second grade. Even university students’ finger perception predicted their calculation scores. (Researchers assess whether children have a good awareness of their fingers by touching the finger of a student—without the student seeing which finger is touched—and asking them to identify which finger it is.)</p><p>Evidence from both <a href="http://www.ncbi.nlm.nih.gov/pubmed/18387567">behavioral</a> and <a href="http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3886996/">neuroscience</a> studies shows that when people receive training on ways to perceive and represent their own fingers, they get better at doing so, which leads to higher mathematics achievement. The tasks we have developed for use in schools and homes (see below) are based on the training programs researchers use to improve finger-perception quality. Researchers found that when 6-year-olds improved the quality of their finger representation, they improved in arithmetic knowledge, particularly skills such as counting and number ordering. In fact, the quality of the 6-year-old’s finger representation was a better predictor of future performance on math tests than their scores on tests of cognitive processing.</p><p>Neuroscientists often debate why finger knowledge predicts math achievement, but they clearly agree on one thing: That knowledge is critical. As <a href="http://www.mathematicalbrain.com/author">Brian Butterworth</a>, a leading researcher in this area, has written, if students aren’t learning about numbers through thinking about their fingers, numbers “will never have a normal representation in the brain.”</p><p>One of the recommendations of the neuroscientists conducting these important studies is that schools focus on <i>finger discrimination</i>—not only on number counting via their fingers but also on helping students distinguish between those fingers. Still, schools typically pay little if any attention to finger discrimination, and to our knowledge, no published curriculum encourages this kind of mathematical work. Instead, thanks largely to school districts and the media, many teachers have been led to believe that finger use is useless and something to be abandoned as quickly as possible. Kumon, for example, an after-school tutoring program used by thousands of families in dozens of countries, tells parents that finger-counting is a “no no” and that those who see their children doing so should report them to the instructor.</p><p>Stopping students from using their fingers when they count could, according to the new brain research, be akin to halting their mathematical development. Fingers are probably one of our most useful visual aids, and the finger area of our brain is used well into adulthood. The need for and importance of finger perception could even be the reason that pianists, and other musicians, often <a href="http://www.livescience.com/51370-does-music-give-you-math-skills.html">display higher mathematical understanding</a> than people who don’t learn a musical instrument.</p><p>Teachers should celebrate and encourage finger use among younger learners and enable learners of any age to strengthen this brain capacity through finger counting and use. They can do so by engaging students in a range of classroom and home activities, such as:</p><hr><p>Give the students colored dots on their fingers and ask them to touch the corresponding piano keys:</p><div style="text-align:center">
<figure style="display: inline-block;"><img alt="" class="bordered" height="233" src="https://cdn.theatlantic.com/assets/media/img/posts/2016/04/hand/e02586c07.jpg" width="350"><figcaption class="credit">youcubed.org</figcaption></figure></div><p></p><div style="text-align:center">
<figure style="display: inline-block;"><img alt="" class="bordered" height="279" src="https://cdn.theatlantic.com/assets/media/img/posts/2016/04/piano/20883da50.jpg" width="250"><figcaption class="credit">youcubed.org</figcaption></figure></div><p></p><hr><p>Give the students colored dots on their fingers and ask them to follow the lines on increasingly difficult mazes:</p><div style="text-align:center">
<figure style="display: inline-block;"><img alt="" class="bordered" height="265" src="https://cdn.theatlantic.com/assets/media/img/posts/2016/04/Screen_Shot_2016_04_13_at_11.01.30_AM-1/83cc2126e.jpg" width="630"><figcaption class="credit">youcubed.org</figcaption></figure></div><p>(The full set of activities is given <a href="https://www.youcubed.org/finger-activities/">here</a>.)</p><hr><p>The finger research is part of a larger group of studies on cognition and the brain showing the importance of visual engagement with math. Our brains are made up of “distributed networks,” and when we handle knowledge, different areas of the brain communicate with each other. When we work on math, in particular, brain activity is distributed among many different networks, which include areas within the ventral and dorsal pathways, both of which are visual. Neuroimaging has shown that even when people work on a number calculation, such as 12 x 25, with symbolic digits (12 and 25) our mathematical thinking is grounded in visual processing.</p><p>A striking example of the importance of visual mathematics comes from <a href="http://www.psy.cmu.edu/~siegler/sieg-ram09.pdf">a study</a> showing that after four 15-minute sessions of playing a game with a number line, differences in knowledge between students from low-income backgrounds and those from middle-income backgrounds were eliminated.</p><div style="text-align:center">
<figure style="display: inline-block;"><img alt="" height="96" src="https://cdn.theatlantic.com/assets/media/img/posts/2016/04/numberline/987c72e57.jpg" width="630"></figure></div><p>Number-line representation of number quantity has been shown to be particularly important for the development of numerical knowledge, and students’ learning of number lines <a href="http://www.cs.cmu.edu/~jlbooth/sieglerbooth-cd04.pdf">is believed to be</a> a precursor of children’s academic success.</p><p>Visual math is powerful for all learners. A few years ago Howard Gardner proposed a <a href="https://en.wikipedia.org/wiki/Theory_of_multiple_intelligences">theory of multiple intelligences</a>, suggesting that people have different approaches to learning, such as those that are visual, kinesthetic, or logical. This idea helpfully expanded people’s thinking about intelligence and competence, but was often used in unfortunate ways in schools, leading to the labeling of students as particular type of learners who were then taught in different ways. But people who are not strong visual thinkers probably need visual thinking more than anyone. Everyone uses visual pathways when we work on math. The problem is it has been presented, for decades, as a subject of numbers and symbols, ignoring the potential of visual math for transforming students’ math experiences and developing important brain pathways.</p><aside class="callout-placeholder" data-source="curated"></aside><p>It is hardly surprising that students so often feel that math is inaccessible and uninteresting when they are plunged into a world of abstraction and numbers in classrooms. Students are made to memorize math facts, and plough through worksheets of numbers, with few visual or creative representations of math, often because of policy directives and faulty curriculum guides. The Common Core standards for kindergarten through eighth grade pay more attention to visual work than many previous sets of learning benchmarks, but their high-school content commits teachers to numerical and abstract thinking. And where the Common Core does encourage visual work, it’s usually encouraged as a prelude to the development of abstract ideas rather than a tool for seeing and extending mathematical ideas and strengthening important brain networks.</p><p>To engage students in productive visual thinking, they should be asked, at regular intervals, how they <i>see</i> mathematical ideas, and to draw what they see. They can be given activities with visual questions and they can be asked to provide visual solutions to questions. When the youcubed team (a center at Stanford) created a free set of visual and open mathematics lessons for grades three through nine last summer, which invited students to appreciate the beauty in mathematics, they were downloaded 250,000 times by teachers and used in every state across the U.S. Ninety-eight percent of teachers said they would like more of the activities, and 89 percent of students reported that the visual activities enhanced their learning of mathematics. Meanwhile, 94 percent of students said they had learned to “keep going even when work is hard and I make mistakes.” Such activities not only offer deep engagement, new understandings, and visual-brain activity, but they show students that mathematics can be an open and beautiful subject, rather than a fixed, closed, and impenetrable subject.</p><p>Some scholars note that it will be those who have developed visual thinking who will be “at the top of the class” in the world’s new high-tech workplace that increasingly draws upon visualization technologies and techniques, in business, technology, art, and science. Work on mathematics draws from different areas of the brain and students need to be strong with visuals, numbers, symbols and words—but schools are not encouraging this broad development in mathematics now. This is not because of a lack of research knowledge on the best ways to teach and learn mathematics, it is because that knowledge has not been communicated in accessible forms to teachers. Research on the brain is often among the most impenetrable for a lay audience but the knowledge that is being produced by neuroscientists, if communicated well, may be the spark that finally ignites productive change in mathematics classrooms and homes across the country.</p>Jo Boalerhttp://www.theatlantic.com/author/jo-boaler/?utm_source=feedLang Chenhttp://www.theatlantic.com/author/lang-chen/?utm_source=feedPhilippe Lissac / Godong / CorbisWhy Kids Should Use Their Fingers in Math Class2016-04-13T11:29:41-04:002016-04-13T11:29:41-04:00Evidence from brain science suggests that far from being “babyish,” the technique is essential for mathematical achievement.tag:theatlantic.com,2015:50-421710<p dir="ltr"><span><span>Why do so many students hate math, fear it, or both? </span></span></p><p dir="ltr"><span><span>If you ask most students what they think their role is in math classrooms, they will tell you it is to get questions right. Students rarely think that they are in math classrooms to appreciate the beauty of mathematics, to ask deep questions, to explore the rich set of connections that make up the subject, or even to learn about the applicability of the subject; they think they are in math classrooms to perform. This was brought home to me recently when a colleague, Rachel Lambert, told me that her 6-year-old son had come home saying he didn’t like math; when she asked him why, he said that math was “too much answer time and not enough learning time.” </span></span></p><p dir="ltr"><span><span>Students from an early age realize that math is different from other subjects. In many schools across the U.S., math is less about learning than it is about answering questions and taking tests—performing.</span></span></p><aside class="partner-box custom"><hr><h4>More From The Hechinger Report</h4>
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</ul><hr></aside><!-- END "MORE FROM" Custom --><p dir="ltr"><span><span>The testing culture in the U.S., which is more pervasive in math than other subjects, is a large part of the problem. When sixth-graders in my local district came home saying that they had a test on the first day of middle school, it was in one subject only: math. Most students and parents didn’t question whether a test was the right way to introduce a new year of mathematics. As one girl said to me, “Well, the teacher was just finding out what we know.” But why does this only happen in math? Teachers in history or English don’t give tests on the first day to find out what students know. And why do so many math teachers boil the subject down to producing short answers to narrow questions under pressure? It is no wonder that so many students decide mathematics is not for them.</span></span></p><p dir="ltr"><span><span>In fact, it is not surprising that teachers test math all the time</span>—<span>in the last decade teachers’ jobs have come to depend on student performance on narrow state tests. The Common Core promises an improvement in the types of tests used</span>—<span>with questions that are less narrow and require thinking, instead of choosing a letter (A, B, C or D)</span>—<span>but the bigger problem is the testing culture in classrooms. It is not unusual for high-school math teachers to test children every week, communicating to students that they are constantly being evaluated.</span></span></p><p dir="ltr"><span><span>Educators know that the most productive math-learning environments are those in which students receive positive messages about their unlimited potential and work on interesting and complex problems; in which they feel free to try ideas, fail, and revise their thinking. Students with a “growth” mindset are those who believe that their ability is not “fixed” and that failure is a natural part of learning. These are the students who perform at higher levels in math and in life. But students don’t get the opportunity to see math as a growth subject if they mainly work on short, closed questions accompanied by frequent tests that communicate to them that math is all about performance and there is no room for failure. When students inevitably struggle, most decide they are not a “math person.” The last decade has seen a nation of children emerge from our schools terrified of failing in math and believing that only some students can be good at it</span>—<span>those who can effortlessly achieve on narrow tests. </span></span></p><p dir="ltr"><span><span>Teachers see some of the damage caused by our nation’s procedural and over-tested math classrooms in the ideas students hold about math. When asked what math is, students typically give descriptions that are very different from those given by experts in the field. Mathematicians define their subject as the study of patterns. They say it is an aesthetic, creative, and beautiful subject (for example, Keith Devlin, “</span><a href="http://www.amazon.com/Mathematics-Patterns-Universe-Scientific-Paperback/dp/0716760223/ref=sr_1_1?ie=UTF8&qid=1449509099&sr=8-1&keywords=devlin+the+study+of+patterns"><span>Mathematics: The Science of Patterns</span></a><span>”</span><span>; and Steven Strogatz, “</span><a href="http://www.amazon.com/Joy-Guided-Tour-Math-Infinity/dp/0544105850/ref=sr_1_1?ie=UTF8&qid=1449509149&sr=8-1&keywords=strogatz+the+joy+of+x"><span>The Joy of x</span></a><span>”). Knowledge of mathematical patterns has helped people navigate oceans, chart missions to space, develop technology that powers cellphones and social networks, and create new scientific and medical knowledge. But students will typically say that math is a subject of calculations, procedures, and rules. They believe that the best mathematical thinkers are those who calculate the fastest</span>—<span>that you have to be fast at math to be good at math. Yet mathematicians are often slow with math. I work with many mathematicians and they are simply not fast math thinkers. I don’t say this to be disrespectful to mathematicians. They are slow because they think carefully and deeply about mathematics. </span></span></p><p dir="ltr"><span><span>Laurent Schwartz won the Fields Medal in mathematics and was one of the greatest mathematicians of his time. But when he was in school he was one of the slowest in his class. In his autobiography, </span><em><span>A Mathematician Grappling with His Century</span></em><span>, he reflects on his school days and how he felt “stupid” because his school valued fast thinking:</span></span></p><blockquote>
<p dir="ltr" style="line-height: 30.0001px;">I was always deeply uncertain about my own intellectual capacity; I thought I was unintelligent. And it is true that I was, and still am, rather slow. I need time to seize things because I always need to understand them fully. Towards the end of the eleventh grade, I secretly thought of myself as stupid. I worried about this for a long time.</p>
<p dir="ltr" style="line-height: 30.0001px;">I’m still just as slow.... At the end of the eleventh grade, I took the measure of the situation, and came to the conclusion that rapidity doesn’t have a precise relation to intelligence. What is important is to deeply understand things and their relations to each other. This is where intelligence lies. The fact of being quick or slow isn't really relevant.</p>
</blockquote><p dir="ltr"><span style="line-height: 1.66667;">Yet, more than any other subject, mathematics continues to be presented as a speed race: Teachers take answers from the first student to shoot up their hand in class, parents and teachers give timed math tests and drill with flash cards, and math apps race against the clock. It is no wonder that students who think slowly and deeply are put off by mathematics.</span></p><p dir="ltr"><span><span>The fact that a narrow and impoverished version of mathematics is taught in many school classrooms cannot be blamed on teachers. Teachers are usually given long lists of content to teach, with hundreds of topics and no time to go into depth on any ideas. When teachers are given these lists, they see a subject that has been stripped down to its bare parts</span>—<span>like a dismantled bike</span>—<span>a collection of nuts and bolts that students are meant to shine and polish all year. Such lists not only take away the connections that weave all through mathematics, but present math as though the connections do not even exist. </span></span></p><p dir="ltr"><span><span>I don’t want students polishing disconnected bike parts all day. I want them to get onto the bikes and ride freely, experiencing the pleasure of math, the joy of making connections, the euphoria of real mathematical thinking. </span></span></p><p dir="ltr"><span><span>When teachers open up mathematics and teach broad, visual, creative math</span><span>, then they teach math as a learning subject, instead of a performance subject. It is very hard for students to develop a growth mindset if they are only ever answering short questions with right and wrong answers. Such questions themselves transmit fixed messages about math: that you can do it or you cannot. When educators teach open math and ask questions that have many solutions or pathways through them, and give students the opportunity to discuss different mathematical ideas, then students see that learning is possible. To put it simply, math questions should have space inside them for learning, for students to discuss and think about ideas; questions should not simply ask for answers that often require calculations or procedures with no encouragement for broader, engaging thought.</span></span></p><p dir="ltr"><span><span>Teachers can see the difference between fixed-math and growth-math questions with elementary- and secondary-school mathematics content. For example, they could ask elementary students to calculate one divided by two-thirds</span>—<span>a fixed question. Or they could encourage them to think creatively and ask them to visually represent one divided by two-thirds and then convince each other why their solution works. Teachers could ask students to find the area of a 12 x 2 rectangle. Or they could pose a more growth-oriented question, asking them to find and draw as many rectangles as you can with an area of 24, which encourages students to think about the relationships between length and width, and represent them visually, instead of simply performing a calculation. At the secondary level, teachers could ask students to prove that the sum of the first n-positive integer cube numbers is the square of the sum of the first n-positive integer numbers. Or they would ask students to make sense of the visual below. </span></span></p><figure><img alt="" height="568" src="https://cdn.theatlantic.com/assets/media/img/posts/2016/11/Screen_Shot_2016_11_15_at_11.50.37_AM/08774ea1a.png" width="611"><figcaption class="credit">Jo Boaler / YouCubed.org</figcaption></figure><p dir="ltr"><span><span>In the first, fixed version of each of these examples, students perform a calculation or move around algebraic symbols. In the second, they are using their own ideas, thinking deeply about math. One version is about performance, the other is about learning.</span></span></p><p dir="ltr"><span><span>When educators teach real mathematics—a growth subject of depth and connections—the opportunities for learning increase and classrooms become filled with happy, excited, and engaged math students. </span><span>Although news sites are filled with opposition to the Common Core, the new curriculum is at least a step in the right direction, as it asks students to engage in in the most mathematical of acts—reasoning. Mathematicians prove ideas by reasoning and justifying their thinking. Those who oppose the Common Core often do so because they want to keep the traditional mathematics approach in classrooms, even though this has turned off millions of students.</span></span></p><p dir="ltr"><span><span>Changing classrooms to teach growth-mindset mathematics has a transformative effect on students. Society urgently needs to free our young people from the crippling ideas that they cannot fail, that they cannot mess up, that only some students can be good at math, and that success should be easy and fast, and not involve effort. School teachers and leaders need to introduce students to creative, beautiful mathematics that allows them to ask questions that have not been asked, and to think of ideas that go against traditional and imaginary boundaries.</span><span> </span><span> </span></span></p><p dir="ltr"><span><span>When instructors encourage open, growth mathematics and the learning messages that support it, they develop our own intellectual freedom, as teachers and parents, and inspire that freedom in others. Now is the time to invite young people onto growth mindset pathways, encouraging them to be the people they should be, free from artificial rules and inspired by the knowledge that they have unlimited mathematics potential. For when school systems open mathematics, and give students the chance to ask their own questions and bring their own natural creativity and curiosity to the foreground as they learn, then they change them as people and the ways they interact with the world. When teachers set students free, beautiful mathematics follows.</span></span></p><hr><p><small><i>This post appears courtesy of </i><a href="http://hechingerreport.org/">The Hechinger Report</a>.<i> It </i><i>has been excerpted from Jo Boaler’s new book,<a href="http://www.amazon.com/Mathematical-Mindsets-Unleashing-Potential-Innovative/dp/0470894520"> </a></i><a href="http://www.amazon.com/Mathematical-Mindsets-Unleashing-Potential-Innovative/dp/0470894520"><span><span>Mathematical Mindsets</span></span></a><span>.</span></small></p>Jo Boalerhttp://www.theatlantic.com/author/jo-boaler/?utm_source=feedMike Groll / APThe Math-Class Paradox2015-12-31T08:00:00-05:002016-11-15T12:01:40-05:00Mastering the subject has become less about learning and more about performance.tag:theatlantic.com,2013:50-281303<p>Mathematics education in the United States is broken. Open any newspaper and stories of math failure shout from the pages: low international rankings, widespread innumeracy in the general population, declines in math majors. Here’s <a href="http://www.achievingthedream.org/sites/default/files/resources/PathwaysToImprovement_0.pdf">the most shocking statistic</a> I have read in recent years: 60 percent of the 13 million two-year college students in the U.S. are currently placed into remedial math courses; 75 percent of them fail or drop the courses and leave college with no degree.</p><p>We need to change the way we teach math in the U.S., and it is for this reason that I support the move to Common Core mathematics. The new curriculum standards that are currently being rolled out in 45 states do not incorporate all the changes that this country needs, by any means, but they are a necessary step in the right direction.</p><p>I have spent years conducting research on students who study mathematics through different teaching approaches—in England and in the U.S. All of <a href="http://www.amazon.com/Whats-Math-Got-Do-It/dp/0143115715">my research studies</a> have shown that when mathematics is opened up and broader math is taught—math that includes problem solving, reasoning, representing ideas in multiple forms, and question asking—students perform at higher levels, more students take advanced mathematics, and achievement is more equitable.</p><p>One of the reasons for these results is that mathematical problems that need thought, connection making, and even creativity are more engaging for students of all levels and for students of different genders, races, and socio-economic groups. This is not only shown by my research but by <a href="http://gse.berkeley.edu/sites/default/files/users/alan-h.-schoenfeld/Schoenfeld_2002%20Making%20Math%20Work%20ER.pdf">decades</a> of <a href="http://uex.sagepub.com/content/30/4/476.abstract">research</a> in <a href="http://www.jstor.org/discover/10.2307/749551?uid=3739704&uid=2&uid=4&uid=3739256&sid=21102904488727">our field</a>. When all aspects of mathematics are encouraged, rather than procedure execution alone, many more students contribute and feel valued. For example, some students are good at procedure execution, but may be less good at connecting methods, explaining their thinking, or representing ideas visually. All of these ways of working are critical in mathematical work and when they are taught and valued, many more students contribute, leading to higher achievement. I refer to this broadening and opening of the mathematics taught in classrooms as <em>mathematical democratization</em>. When we open mathematics we also open the doors of math achievement and many more students succeed.</p><p>In mathematics education we suffer from the widespread, distinctly American idea that only some people can be “<a href="http://www.theatlantic.com/education/archive/2013/10/the-myth-of-im-bad-at-math/280914/?utm_source=feed">math people</a>.” This idea has been <a href="http://www.fil.ion.ucl.ac.uk/Maguire/Maguire2006.pdf">disproved</a> by scientific research showing the incredible potential of the brain to grow and adapt. But the idea that math is hard, uninteresting, and accessible only to “nerds” persists. This idea is made even more damaging by harsh stereotypical thinking—mathematics is for select racial groups and men. This thinking, as well as the teaching practices that go with it, have provided the perfect conditions for the creation of a math underclass. Narrow mathematics teaching combined with low and stereotypical expectations for students are the two main reasons that the U.S. is in dire mathematical straights.</p><!-- START "MORE ON" SINGLE STORY BOX v. 2 --><aside class="callout-placeholder" data-source="curated"></aside><!-- END "MORE ON" SINGLE STORY BOX v. 2 --><p>This summer <a href="https://class.stanford.edu/courses/Education/EDUC115N/How_to_Learn_Math/about">I taught a course</a> through Stanford’s open online platform explaining research evidence on ability and the brain and on good mathematics teaching, for teachers and parents. The course had a transformative effect. It was taken by <a href="http://www.telegraph.co.uk/education/universityeducation/10414989/University-education-maturing-of-the-Mooc.html">40,000 people</a>, and 95 percent said they would change their teaching or parenting as a result. Hundreds wrote telling me that the ideas in the course had been life-changing for them. Teachers and parents are open to research, and new technologies are finally providing a way that important research evidence, on mathematics, learning, and the brain, can reach the audiences that need them.</p><p>Conrad Wolfram, cofounder of Wolfram-Alpha, one of the world’s most important mathematical companies, has <a href="http://www.ted.com/talks/conrad_wolfram_teaching_kids_real_math_with_computers.html">spoken widely</a> about the mismatch between the math that people need in the 21st century and the math they spend most of their time on in classrooms: computing by hand. The Common Core helps to correct this problem by embracing broader mathematics and requiring the use of advanced technology, such as dynamic geometry software. Students in the Common Core will spend less time practicing isolated methods and more time solving applied problems that involve connecting different methods, using technology, understanding multiple representations of ideas, and justifying their thinking.</p><p>For example, consider the following two published test questions. The first comes from California’s old standards, the second from the Common Core.</p><p>1. Which of the following <strong>best</strong> describes the triangles shown below?</p><p><img alt="" height="186" src="https://cdn.theatlantic.com/assets/media/img/posts/boaler_math2.jpg" width="208"></p><p><strong>A</strong> both similar and congruent</p><p><strong>B</strong> similar but not congruent</p><p><strong>C</strong> congruent but not similar</p><p><strong>D</strong> neither similar nor congruent</p><p><small>California Standards Test, released test questions, geometry, 2009</small></p><hr><p>2. Triangle ABC undergoes a series of some of the following transformations to become triangle DEF:</p><ul><li>Rotation</li>
<li>Reflection</li>
<li>Translation</li>
<li>Dilation</li>
</ul><p>Is DEF always, sometimes, or never <strong>congruent</strong> to ABC? Provide justification to support your conclusion. </p><p><img alt="" height="86" src="https://cdn.theatlantic.com/assets/media/img/posts/boaler_math.png" width="570"></p><p><small>Common Core Smarter Balanced Grade 8 Sample Item, 2013</small></p><p>The second question, from one of the Common Core assessment teams, does not simply test a mathematical definition, as the first does. It requires that students visualize a triangle, use transformational geometry, consider whether different cases satisfy the mathematical definition, and then justify their thinking. It combines different areas of geometry and asks students to problem solve and justify. It does not offer four multiple-choice options. Common Core mathematics is more challenging than the mathematics it will replace. It is also more interesting for students and many times closer to the mathematics that is needed in 21st-century life and work.</p><p>An important requirement in the Common Core is the need for students to discuss ideas and justify their thinking. There is a good reason for this: Justification and reasoning are two of the acts that lie at the heart of mathematics. They are, in many ways, the essence of what mathematics is. Scientists work to prove or disprove new theories by finding many cases that work or counter-examples that do not. Mathematicians, by contrast prove the validity of their propositions through justification and reasoning.</p><p>Mathematicians are not the only people who need to engage in justification and reasoning. The young people who are successful in today’s workforce are those who can discuss and reason about productive mathematical pathways, and who can be wrong, but can trace back to errors and work to correct them. In our new technological world, employers do not need people who can calculate correctly or fast, they need people who can reason about approaches, estimate and verify results, produce and interpret different powerful representations, and connect with other people’s mathematical ideas.</p><p>Another problem addressed by the Common Core is the American idea that those who are good at math are those who are fast. Speed is revered in math classes across the U.S., and students as young as five years old are given timed tests—even though these have been shown to create <a href="http://joboaler.com/timed-tests-and-the-development-of-math-anxiety/">math anxiety</a> in young children. Parents use flash cards and other devices to promote speed, <a href="http://www.amazon.com/Einstein-Never-Used-Flashcards-Learn/dp/1594860688">not knowing</a> that they are probably damaging their children’s mathematical development. At the same time mathematicians point out that speed in math is irrelevant. One of the world’s top mathematicians, Laurent Schwartz, reflected <a href="http://www.amazon.com/A-Mathematician-Grappling-His-Century/dp/3764360526">in his memoir</a> that he was made to feel unintelligent in school because he was the <em>slowest</em> math thinker in his class. But he points out that what is important in mathematics “is to deeply understand things and their relations to each other. This is where intelligence lies. The fact of being quick or slow isn't really relevant.” It is fortunate for Schwartz, and all of us, that he did not grow up in the speed- and test-driven classrooms of the last decade that have successfully dissuaded any child that thinks deeply or slowly from pursuing mathematics or even thinking of themselves as capable. </p><aside class="callout-placeholder" data-source="curated"></aside><p>The new Common Core curriculum gives more time for depth and exploration than the curricula it has replaced by removing some of the redundant methods students will never need or use. Sadly it does not go far enough in this regard, and the high-school grades in particular are still packed with obsolete content. But educational progress is rarely fast and the changes implemented in the Common Core are a step in the right direction.</p><p>The U.S. does not need fast procedure executors anymore. We need people who are confident with mathematics, who can develop mathematical models and predictions, and who can justify, reason, communicate, and problem solve. We need a broad and diverse range of people who are powerful mathematical thinkers and who have not been held back by stereotypical thinking and teaching. Common Core mathematics, imperfect though it may be, can help us reach those goals.</p>Jo Boalerhttp://www.theatlantic.com/author/jo-boaler/?utm_source=feedAPThe Stereotypes That Distort How Americans Teach and Learn Math2013-11-12T11:51:00-05:002018-05-10T12:34:13-04:00Speed doesn't matter, and there's no such thing as a "math person." How the Common Core's approach to the discipline could correct these misperceptions.