Barry Garelick | The Atlantichttps://www.theatlantic.com/author/barry-garelick/2015-11-11T10:19:03-05:00Copyright 2019 by The Atlantic Monthly Group. All Rights Reserved.tag:theatlantic.com,2015:50-414924<p dir="ltr"><span><span>At a middle school in California, the state testing in math was underway via the Smarter Balanced Assessment Consortium (SBAC) exam. A girl pointed to the problem on the computer screen and asked “What do I do?” The proctor read the instructions for the problem and told the student: “You need to explain how you got your answer.”</span></span></p><p dir="ltr"><span><span>The girl threw her arms up in frustration and said, “Why can’t I just do the problem, enter the answer and be done with it?”</span></span></p><p dir="ltr"><span><span>The answer to her question comes down to what the education establishment believes “understanding” to be, and how to measure it. K-12 mathematics instruction involves equal parts procedural skills and understanding. What “understanding” in mathematics means, however, has long been a topic of debate. One distinction popular with today’s math-reform advocates is between “knowing” and “doing.” A student, reformers argue, might be able to “do” a problem (i.e., solve it mathematically) without understanding the concepts behind the problem-solving procedure. Perhaps he or she has simply memorized the method without understanding it and is performing the steps by “rote.”</span></span></p><p dir="ltr"><span><span>The Common Core math standards, adopted in 42 states and the District of Columbia and reflected in Common Core-aligned tests like the SBAC and the Partnership for Assessment of Readiness for College and Careers (PARCC), take understanding to a whole new level. “Students who lack understanding of a topic may rely on procedures too heavily,” </span><a href="http://www.corestandards.org/Math/"><span>states the Common Core website</span></a><span>. “But what does mathematical understanding look like?” And how can teachers assess it?</span></span></p><blockquote>
<p dir="ltr"><span><span>One way is to ask the student to justify, in a way that is appropriate to the student’s mathematical maturity, why a particular mathematical statement is true, or where a mathematical rule comes from.</span></span></p>
</blockquote><p dir="ltr"><span><span>The underlying assumption here is that if a student understands something, he or she can explain it—and that deficient explanation signals deficient understanding. But this raises yet another question: What constitutes a satisfactory explanation?</span></span></p><p dir="ltr"><span><span>While the Common Core leaves this unspecified, current practices are suggestive. Consider a problem that asks how many total pencils there are if five people have three pencils each. In the eyes of some educators, explaining why the answer is 15 by stating, simply, that 5 x 3 = 15 is not satisfactory. To show they truly understand why 5 x 3 is 15, and why this computation provides the answer to the given word problem, students must do more. For example, they might draw a picture illustrating five groups of three pencils. (And in some instances</span><a href="http://www.nbcwashington.com/news/local/Viral-Math-Problem-Reignites-Common-Core-Debate-338294782.html"><span>, as was the case recently in a third-grade classroom,</span></a><span> a student would be considered to </span><span>not</span><span> understand if he or she drew three groups of five pencils.)</span></span></p><p dir="ltr"><span><span>Consider now a problem given in a pre-algebra course that involves percentages: “A coat has been reduced by 20 percent to sell for $160. What was the original price of the coat?”</span></span></p><p dir="ltr"><span><span>A student may show the solution as follows:</span></span></p><blockquote>
<p dir="ltr"><span><span>x = original cost of coat in dollars</span><br class="kix-line-break"><span>100% – 20% = 80%</span><br class="kix-line-break"><span>0.8x = $160</span><br class="kix-line-break"><span>x = $200</span></span></p>
</blockquote><p dir="ltr"><span><span>Clearly, the student knows the mathematical procedure necessary to solve the problem. In fact, for years students were told not to explain their answers, but to show their work, and if presented in a clear and organized manner, the math contained in this work was considered to be its own explanation. But the above demonstration might, through the prism of the Common Core standards, be considered an inadequate explanation. That is, inspired by what the standards say about understanding, one could ask “Does the student know why the subtraction operation is done to obtain the 80 percent used in the equation or is he doing it as a mechanical procedure—i.e., without understanding?”</span></span></p><p dir="ltr"><span><span>In a middle school observed by one of us, the school’s goal was to increase student proficiency in solving math problems by requiring students to explain how they solved them. This was not required for all problems given; rather, they were expected to do this for two or three problems in class per week, which took up to 10 percent of total weekly class time. They were instructed on how to write explanations for their math solutions using a model called “Need, Know, Do.” In the problem example given above, the “Need” would be “What was the original price of the coat?” The “Know” would be the information provided in the problem statement, here the price of the discounted coat and the discount rate. The “Do” is the process of solving the problem.</span></span></p><p dir="ltr"><span><span>Students were instructed to use “flow maps” and diagrams to describe the thinking and steps used to solve the problem, after which they were to write a narrative summary of what was described in the flow maps and elsewhere. They were told that the “Do” (as well as the flow maps) explains what they did to solve the problem and that the narrative summary provides the why. Many students, though, had difficulty differentiating the “Do” section from the final narrative. But in order for their explanation to qualify as “high level,” they couldn’t simply state “100% – 20% = 80%”; they had to explain what that means. For example, they might say, “The discount rate subtracted from 100 percent gives the amount that I pay.”</span></span></p><p dir="ltr"><span><span>An example of a student’s written explanation for this problem is shown in Figure 1:</span></span></p><p dir="ltr"><span><span><img alt="Figure 1: Example of student explanation." height="488px;" src="https://lh5.googleusercontent.com/40bh5-qrjXriA5ZrzqpNhCG2HrgKCRNcYrmIa0unSjTbnNAfvORJLci68PaPKiRH2avhTJAb9XBD4S8gVbz7oLxBiUkEn4in-mPJImKDuDQ21z-vz9i0SZG6idUD7J485FuzgPqVffVxGO3z" width="506px;"></span></span></p><p dir="ltr"><span><span>For problems at this level, the amount of work required for explanation turns a straightforward problem into a long managerial task that is concerned more with pedagogy than with content. While drawing diagrams or pictures may help some students learn how to solve problems, for others it is unnecessary and tedious. As the above example shows, the explanations may not offer the “why” of a particular procedure.</span></span></p><p dir="ltr"><span><span>Under the rubric used at the middle school where this problem was given, explanations are ranked as “high,” “middle,” or “low.” This particular explanation would probably fall in the “middle” category since it is unlikely that the statement “You need to subtract 100- 20 to get 80” would be deemed a “purposeful, mathematically-grounded written explanation.”</span></span></p><p dir="ltr"><span><span>The “Need” and “Know” steps in the above process are not new and were advocated by George Polya in the 1950s in his classic book </span><a href="http://www.amazon.com/How-Solve-It-Mathematical-Princeton/dp/069111966X"><em><span>How to Solve It</span></em></a><span>. The “Need” and “Know” aspect of the explanatory technique at the middle school observed is a sensible one. But Polya’s book was about solving problems, not explaining or justifying how they were done. At the middle school, problem solving and explanation were intertwined, in the belief that the process of explanation leads to the solving of the problem. This conflation of problem solving and explanation arises from a complex history of educational theories. One theory holds </span><a href="http://rhartshorne.com/fall-2012/eme6507-rh/cdisturco/eme6507-eportfolio/documents/Mayer%201998.pdf"><span>that being aware of one’s thinking process—called “metacognition</span></a><span>”—is part and parcel to problem solving. Other theories that feed the conflation predate the Common Core standards and originated during the Progressive era in the early part of the 20</span><span>th</span><span> Century when “conceptual understanding” began to be viewed as a path to, and thus more important than, procedural fluency.</span></span></p><p dir="ltr"><span><span>Despite the goal of solving a problem and explaining it in one fell swoop, in many cases observed at the middle school, students solved the problem first and then added the explanation in the required format and rubric. It was not evident that the process of explanation enhanced problem solving ability. In fact, in talking with students at the school, many found the process tedious and said they would rather just “do the math” without having to write about it.</span></span></p><aside class="callout-placeholder" data-source="curated"></aside><p dir="ltr"><span><span>In general, there is no more evidence of “understanding” in the explained solution, even with pictures, than there would be in mathematical solutions presented in a clear and organized way. How do we know, for example, that a student isn’t simply repeating an explanation provided by the teacher or the textbook, thus exhibiting mere “rote learning” rather than “true understanding” of a problem-solving procedure?</span></span></p><p dir="ltr"><span><span>Math learning is a progression from concrete to abstract. The advantage to the abstract is that the various mathematical operations can be performed without the cumbersome attachments of concrete entities—entities like dollars, percentages, groupings of pencils. Once a particular word problem has been translated into a mathematical representation, the entirety of its mathematically relevant content is condensed onto abstract symbols, freeing working memory and unleashing the power of pure mathematics. That is, information and procedures that have been become automatic frees up working memory. With working memory less burdened, </span><a href="http://repository.ubaya.ac.id/25249/7/Anindito_Cognitive_Abstract_2014.pdf"><span>the student can focus on solving the problem at hand</span></a><span>. Thus, requiring explanations beyond the mathematics itself distracts and diverts students away from the convenience and power of abstraction. Mandatory demonstrations of “mathematical understanding,” in other words, can impede the “doing” of actual mathematics.</span></span></p><p class="p1"><span class="s1">Advocates for math reform are <a href="http://www.wellesley.k12.ma.us/sites/wellesleyps/files/file/file/explainthinking.pdf">reluctant to accept</a> that delays in understanding are normal and do not signal a failure of the teaching method. Students learn to do, they learn to apply what they’ve mastered, they learn to do more, they begin to see why and eventually the light comes on. Furthermore, math reformers often fail to understand that conceptual understanding <a href="https://www.uni-trier.de/fileadmin/fb1/prof/PSY/PAE/Team/Schneider/Rittle-JohnsonEtAl2015.pdf">works in tandem with procedural fluency</a>. Doing a procedure devoid of any understanding of what is being done is actually <a href="http://www.psy.cmu.edu/~siegler/r-jhnsn-etal-01.pdf">hard to accomplish</a> with elementary math because the very learning of procedures is, itself, informative of meaning, and the <a href="http://repetitious%20use%20of%20them%20conveys%20understanding%20to%20the%20user">repetitious use of them conveys understanding to the user</a>.</span></p><p dir="ltr"><span><span>Explaining the solution to a problem comes when students can draw on a strong foundation of content relevant to the topic currently being learned. As students find their feet and establish a larger repertoire of mastered knowledge and methods, the more articulate they can become in explanations. Children in elementary and middle school who are asked to engage in critical thinking about abstract ideas will, more often than not, respond emotionally and intuitively, not logically and with “understanding.” I</span><span>t is as if the purveyors of these practices are saying: “If we can just get them to do things that </span><span>look</span><span> like what we imagine a mathematician does, then they will be real mathematicians.” </span><span>That may be behaviorally interesting, but it is not mathematical development and it leaves them behind in the development of their fundamental skills.</span></span></p><p dir="ltr"><span><span>The idea that students who do not demonstrate their strategies in words and pictures or by multiple methods don’t understand the underlying concepts is particularly problematic for certain vulnerable types of students. Consider students whose verbal skills lag far behind their mathematical skills—non-native English speakers or students with specific language delays or language disorders, for example. These groups include children who can easily do math in their heads and solve complex problems, but often will be unable to explain—whether orally or in written words—how they arrived at their answers.</span></span></p><p dir="ltr"><span><span>Most exemplary are children on the autism spectrum. As the autism researcher Tony Attwood has observed, mathematics has special appeal to individuals with autism: It is, often, the school subject that best matches their cognitive strengths. Indeed, writing about Asperger’s Syndrome (a high-functioning subtype of autism), Attwood in his 2007 book </span><em><span>The Complete Guide to Asperger’s Syndrome</span></em><span> notes that “the personalities of some of the great mathematicians include many of the characteristics of Asperger’s syndrome.”</span></span></p><p dir="ltr"><span><span>And yet, Attwood added, many children on the autism spectrum, even those who are mathematically gifted, struggle when asked to explain their answers. “The child can provide the correct answer to a mathematical problem,” he observes, “but not easily translate into speech the mental processes used to solve the problem.” Back in 1944, Hans Asperger, the Austrian pediatrician who first studied the condition that now bears his name, famously cited one of his patients as saying that, “I can’t do this orally, only headily.”</span></span></p><p dir="ltr"><span><span>Writing from Australia decades later, a few years before the Common Core took hold in America, Attwood added that it can “mystify teachers and lead to problems with tests when the person with Asperger’s syndrome is unable to explain his or her methods on the test or exam paper.” Here in Common Core America, this inability has morphed into an unprecedented liability.</span></span></p><p dir="ltr"><span><span>Is it really the case that the non-linguistically inclined student who progresses through math with correct but unexplained answers—from multi-digit arithmetic through to multi-variable calculus—doesn’t understand the underlying math? Or that the mathematician with the Asperger’s personality, doing things headily but not orally, is advancing the frontiers of his field in a zombie-like stupor?</span></span></p><p dir="ltr"><span><span>Or is it possible that the ability to explain one’s answers verbally, while sometimes a sufficient criterion for proving understanding, is not, in fact, a necessary one? And, to the extent that it isn’t a necessary criterion, should verbal explanation be </span><span>the</span><span> way to gauge comprehension?</span></span></p><p dir="ltr"><span><span>Measuring understanding, or learning in general, isn’t easy. What testing does is measure “markers” or byproducts of learning and understanding. Explaining answers is but one possible marker.</span></span></p><p dir="ltr"><span><span>Another, quite simply, are the answers themselves. If a student can consistently solve a variety of problems, that student likely has some level of mathematical understanding. Teachers can assess this more deeply by looking at the solutions and any work shown and asking some spontaneous follow-up questions tailored to the child’s verbal abilities. But it’s far from clear whether a general requirement to accompany all solutions with verbal explanations provides a more accurate measurement of mathematical understanding than the answers themselves and any work the student has produced along the way. At best, verbal explanations beyond “showing the work” may be superfluous; at worst, they shortchange certain students and encumber the mathematics for everyone.</span></span></p><p dir="ltr"><span><span>As Alfred North Whitehead famously put it about a century before the Common Core standards took hold:</span></span></p><blockquote>
<p dir="ltr"><span><span>It is a profoundly erroneous truism … that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilization advances by extending the number of important operations which we can perform without thinking about them.</span></span></p>
</blockquote>Katharine Bealshttp://www.theatlantic.com/author/katharine-beals/?utm_source=feedBarry Garelickhttp://www.theatlantic.com/author/barry-garelick/?utm_source=feedRogelio V. Solis / APExplaining Your Math: Unnecessary at Best, Encumbering at Worst2015-11-11T10:00:00-05:002015-11-11T10:19:03-05:00Common Core-era rules that force kids to diagram their thought processes can make the equations a lot more confusing than they need to be.tag:theatlantic.com,2013:50-274362<p><a href="http://cdn.theatlantic.com/static/mt/assets/national/honors.jpg"><img alt="honors.jpg" src="https://cdn.theatlantic.com/static/mt/assets/national/assets_c/2013/03/honors-thumb-570x335-116992.jpg" class="mt-image-none" style="" height="335" width="570"></a>
<span class="credit">solgas/<a href="http://www.shutterstock.com/">Shutterstock</a></span>
</p><p>Is it my imagination, or have you noticed that some public high school courses that are now called "honors" are equivalent to the regular "college prep" curriculum of earlier eras? And have you also noticed that what is now called "college prep" is aimed largely at students who are deemed low achievers or of low cognitive ability?</p><p>In fact, this trend is nobody's imagination. Over the past generation, public schools have done away with "tracking" -- a practice that began in the early 1900′s. By the 20′s and 30′s, curricula in high schools had evolved into four different types: college-preparatory, vocational (e.g., plumbing, metal work, electrical, auto), trade-oriented (e.g., accounting, secretarial), and general. Students were tracked into the various curricula based largely on IQ but sometimes other factors such as race and skin color. Children of immigrants, and children who came from farms rather than cities, were often assumed to be inferior in cognitive ability and treated accordingly.</p><!-- START "MORE ON" BOX WITH IMAGES v. 2 --><div style="margin: 10px;
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Mark Bowden on Being in the Slow Kids' Class</a>
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</div><!-- END "MORE ON" BOX WITH IMAGES v. 2 --><p>During the 60's and 70's, radical education critics such as Jonathan Kozol brought accusations against a system they found racist and sadistic. They argued that public schools were hostile to children and lacked innovation in pedagogy. Their goal -- which became the goal of the larger education establishment -- was to restore equity to students, erasing the lines that divided them by social class and race. The desire to eliminate inequity translated to the goal of preparing every student for college. The goal was laudable, but as college prep merged with the general education track, it became student-centered and needs-based, with lower standards and less homework assigned. </p><p>Some of the previous standards returned during the early 80's, when the "Back to Basics" movement reacted against the fads of the late 60's and the 70's by reinstituting traditional curricula. But the underlying ideas of Kozol and others did not go away, and the progressive watchword in education has continued to be "equality." <br></p><p>As a result, grouping students according to ability -- a practice viewed by many in the education establishment as synonymous with tracking -- has been almost completely eliminated in K-8. Instead, most schools practice full inclusion, which means educators are expected to teach students with diverse backgrounds and abilities in the same classroom using a technique known as "differentiated instruction." </p><p>Unfortunately, the efforts and philosophies of otherwise well-meaning individuals have attempted to eliminate the achievement gap by eliminating achievement. Exercises in grammar have declined to the point that they are virtually extinct. Book reports are often assigned in the form of a book jacket or poster instead of a written analysis. Essays now are "student-centered" -- even history assignments often call upon students to describe how they <i>feel</i> about past events rather than apply factual analysis. Math classes are now <a href="http://www.theatlantic.com/national/archive/2012/10/its-not-just-writing-math-needs-a-revolution-too/263545/?utm_source=feed">more about math appreciation</a> and being able to explain how a procedure works rather than the mastery of skills and procedures necessary to solve problems. </p><p>An exception can be found in gifted and talented programs. These programs -- some of which begin as early as third grade -- are a reaction against the low expectations brought on by full inclusion. These programs often make matters worse. They can be rigid, excluding late bloomers by testing them into a "non-honors" track early in life. In other words, the elimination of ability grouping has become a tracking system in itself that leaves many students behind. </p><p>Interestingly, new reports suggest that ability grouping may be making a comeback. The National Bureau of Economic Research has released <a href="http://www.nber.org/papers/w18848">a study that examines the effects</a> of sorting students by ability. The study looked at data from the Dallas Independent School District and found that sorting by previous performance "significantly improves students' math and reading scores" and that the "net effect of sorting is beneficial for both high and low performing students." The same benefits were found among gifted and talented students, special education students, and those with limited English proficiency. </p><p>The 2013 Brown Center Report on American Education contains <a href="http://www.brookings.edu/research/reports/2013/03/18-tracking-ability-grouping-loveless?cid=em_brown032013">a study by Tom Loveless of the Brookings Institution</a> that also looks at ability grouping and tracking in schools. The study found a recent increase in both practices -- a trend Loveless finds noteworthy, given the long-term resistance to each. He suggests a few possible reasons for the reversal: The emphasis on accountability, started by No Child Left Behind, may have motivated teachers to group struggling students together. The rise of computer-aided learning might make it easier for them to instruct students who learn at different rates. And a 2008 <a href="http://www.eric.ed.gov/PDFS/ED504457.pdf">MetLife Survey of the American Teacher</a> reports that many teachers simply find mixed-ability classes difficult to teach. </p><p>If ability grouping is indeed making a comeback, education may benefit significantly. Instead of relying on "gifted" and "honors" programs -- which are, in effect, tracking systems -- the ability grouping now being practiced in some schools allows for greater flexibility. When properly executed (as it was in my school during the 1960s), this enables students placed in lower-ability classes to advance to higher-ability classes based on their performance and progress. </p><p>Ability grouping by itself, however, will not be enough. Schools also need better K-8 curricula and more academic extracurricular opportunities at all levels. Most importantly, they need to hold students to specific expectations rather than leaving achievement up to some vague "natural process," like the student-centered assignments that essentially protect children from learning. <br></p><p>If implemented correctly, the new ability grouping will allow the curriculum to be better tailored to meet the needs of students at all levels. Such classes would help students get up to speed more effectively. This would not only make tutors much less necessary but would also have the advantage of making advancement easier for students whose parents cannot afford tutors or learning centers. Otherwise, it will be nobody's imagination if students continue to fulfill the low expectations that have been set for them. </p>Barry Garelickhttp://www.theatlantic.com/author/barry-garelick/?utm_source=feedLet's Go Back to Grouping Students by Ability2013-03-26T10:20:56-04:002013-03-29T14:42:22-04:00Since the late 1960s, well-meaning educators have shied away from placing kids in "faster" and "slower" classes. Now that trend is reversing—and for good reason. tag:theatlantic.com,2012:50-265444<p><i>A set of guidelines adopted by 45 states this year may turn children into "little mathematicians" who don't know how to do actual math.</i></p><p><img alt="math-scribbles.jpg" src="https://cdn.theatlantic.com/static/mt/assets/national/math-scribbles.jpg" class="mt-image-none" style="" height="354" width="615"></p><p class="credit">zhu difeng/shutterstock</p><p>A few weeks ago, I wrote <a href="http://www.theatlantic.com/national/archive/2012/10/its-not-just-writing-math-needs-a-revolution-too/263545/?utm_source=feed">an article</a> for TheAtlantic.com describing some of the problems with how math is currently being taught. Specifically, some math programs strive to teach students to think like "little mathematicians" before giving them the analytic tools they need to actually solve problems. <br></p><p>Some of us had hoped the situation would improve this school year, as 45 states and the District Columbia adopted the new Common Core Standards. But here are two discouraging emails I received recently. The first was from a parent:</p><blockquote>They implemented Common Core this year in our school system in Tennessee. I have a third grader who loved math and got A's in math until this year, where he struggles to get a C. He struggles with "explaining" how he got his answer after using "mental math." In fact, I had no idea how to explain it! It's math 2+2=4. I can't explain it, it just is. </blockquote><p>The second email came from a teacher in another state:</p><blockquote>I am teaching the traditional algorithm this year to my third graders, but was told next year with Common Core I will not be allowed to. They should use mental math, and other strategies, to add. Crazy! I am so outraged that I have decided my child is NOT going to public schools until Common Core falls flat.</blockquote><p>So just what are the Common Core Standards for math? They are a set of guidelines written for both math and English language arts under the auspices of National Governors Association and the Council of Chief State School Officers. Where they are adopted, the Common Core standards will replace state standards in these subject areas, establishing more common ground for schools nationwide. </p><p>To read newspaper coverage of the new standards, you'd think they were raising the bar for math proficiency, not lowering it. "More is expected of the students," <a href="http://articles.mcall.com/2012-11-02/news/mc-whitehall-common-core-web-20121101_1_common-core-curriculum-math-whitehall-coplay-school-district-students">one article</a> declares. "While they still have to memorize or have fluency in key math functions and do the math with speed and accuracy, they will have to demonstrate a deeper understanding of key concepts before moving on." </p><p>But what does this mean in practice? Another <a href="http://www.minutemannewscenter.com/articles/2012/10/31/fairfield/news/doc50914e526689d826895135.txt">recent article</a> explains, "This curriculum puts an emphasis on critical thinking, rather than memorization, and collaborative learning." In other words, instead of simply teaching multiplication tables, schools are adopting "an 'inquiry method' of learning, in which children are supposed to discover the knowledge for themselves." An educator quoted in the article admits that this approach could be frustrating for students: "Yes. Solving a problem is not easy. Learning is not easy." </p><p>With 100 pages of explicit instruction about what should be taught and when, one would expect the Common Core Standards to make problem-solving easier. Instead, one father quoted in the aforementioned article complains, "For the first time, I have three children who are struggling in math." Why?</p><p>Let's look first at the 97 pages of what are called "Content Standards." Many of these standards require that students to be able to explain why a particular procedure works. It's not enough for a student to be able to divide one fraction by another. He or she must also "use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9, because 3/4 of 8/9 is 2/3." </p><p>It's an odd pedagogical agenda, based on a belief that conceptual understanding must come before practical skills can be mastered. As this thinking goes, students must be able to explain the "why" of a procedure. Otherwise, solving a math problem becomes a "mere calculation" and the student is viewed as not having true understanding. </p><p>This approach not only complicates the simplest of math problems; it also leads to delays. Under the Common Core Standards, students will not learn traditional methods of adding and subtracting double and triple digit numbers until fourth grade. (Currently, most schools teach these skills two years earlier.) The standard method for two and three digit multiplication is delayed until fifth grade; the standard method for long division until sixth. In the meantime, the students learn alternative strategies that are far less efficient, but that presumably help them "understand" the conceptual underpinnings. </p><p>This brings us now to the final three pages of the 100-page document, called <a href="http://www.corestandards.org/Math/Practice">"Standards for Mathematical Practice."</a> While this discussion is short, the points it includes are often the focus of webinars and seminars on the new Common Core methods: </p><ol start="1" type="1"><ol><li>Make sense of problem solving and persevere in solving them</li><li>Reason abstractly and quantitatively</li><li>Construct viable arguments and critique the reasoning of others</li><li>Model with mathematics</li><li>Use appropriate tools strategically</li><li>Attend to precision</li><li>Look for and make use of structure</li><li>Look for and express regularity in repeated reasoning</li></ol></ol><p>These guidelines seem reasonable enough. But on closer inspection, these things are essentially habits of mind that ought to develop naturally as a student learns to do actual math. For example, there's nothing wrong with the first point: "Make sense of problem solving and persevering in solving them." But these standards are being interpreted to mean that students "make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution." </p><p>This is a rather high expectation for students in K- 6. True habits of mind develop with time and maturity. An algebra student, for instance, can take a theoretical scenario such as "John is 2 times as old as Jill will be in 3 years" and express it in mathematical symbols. In lower grades, this kind of connection between numbers and ideas is very hard to make. The Common Core standards seem to presume that even very young students can, and should, learn to make sophisticated leaps in reasoning, like little children dressing in their parents' clothes. </p><p>As the Common Core makes its way into real-life classrooms, I hope teachers are able to adjust its guidelines as they fit. I hope, for instance, that teachers will still be allowed to introduce the standard method for addition and subtraction in second grade rather than waiting until fourth. I also hope that teachers who favor direct instruction over an inquiry-based approach will be given this freedom. <br></p><p>Unfortunately, the emails and newspaper articles I've been seeing may herald a new era where more and more students are given a flimsy make-believe version of mathematics, without the ability to solve actual math problems. After all, where the Common Core goes, textbook publishers are probably not too far behind. </p>Barry Garelickhttp://www.theatlantic.com/author/barry-garelick/?utm_source=feedA New Kind of Problem: The Common Core Math Standards 2012-11-20T12:03:23-05:002012-11-21T11:48:31-05:00A set of guidelines adopted by 45 states this year may turn children into "little mathematicians" who don't know how to do actual math.tag:theatlantic.com,2012:50-263545<p><i>Today's fashion is to throw away the textbook and to teach kids to </i>think<i> like mathematicians. The problem? They're not learning how to do actual math. </i></p><p><img alt="math-doodles.jpg" src="https://cdn.theatlantic.com/static/mt/assets/national/math-doodles.jpg" class="mt-image-none" style="" height="275" width="615"></p><p class="credit">EtiAmmos/<a href="http://www.shutterstock.com/">Shutterstock</a></p><p>In <i>The Atlantic</i>'s ongoing debate about how to teach writing in schools, Robert Pondiscio wrote an eye-opening piece called <a href="http://www.theatlantic.com/national/archive/2012/09/how-self-expression-damaged-my-students/262656/?utm_source=feed">"How Self-Expression Damaged My Students."</a> In it, he tells of how he used modern-day techniques for teaching writing--not teaching rules of grammar or correcting errors but treating the students as little writers and having them write. He notes, however that "good writers don't just do stuff. They know stuff. ... And if this is not explicitly taught, it will rarely develop by osmosis among children who do not grow up in language-rich homes."</p><!-- START "SPECIAL REPORT" BUG CODE v. 1 --><div style="margin: 10px;
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</div><!-- END "SPECIAL REPORT" BUG CODE v. 1 --><p>What Pondiscio describes on the writing front has also been happening with mathematics education in K-6 for the past two decades. I first became aware of it over 10 years ago when I saw what passed for math instruction in my daughter's second grade class. I was concerned that she was not learning her addition and subtraction facts. Other parents we knew had the same concerns. Teachers told them not to worry because kids eventually "get it."</p><p>One teacher tried to explain the new method. "It used to be that if you missed a concept or method in math, then you were lost for the rest of the year. But the way we do it now, kids have a lot of ways to do things, like adding and subtracting, so that math topics from day to day aren't dependent on kids' mastering a previous lesson."</p><p>This was my initiation into the world of reform math. It is a world where understanding takes precedence over procedure and process trumps content. In this world, memorization is looked down upon as "rote learning" and thus addition and subtraction facts are not drilled in the classroom--it's something for students to learn at home. Inefficient methods for adding, subtracting, multiplying, and dividing are taught in the belief that such methods expose the conceptual underpinning of what is happening during these operations. The standard (and efficient) methods for these operations are delayed sometimes until 4th and 5th grades, when students are deemed ready to learn procedural fluency. </p><p>The idea is to teach students to "think like mathematicians." They are called upon to think critically before acquiring the analytic tools with which to do so. More precisely, they are given analytic tools for "understanding" problems and are then forced to learn the actual procedural skills necessary to solve them on a "just in time" basis. Such a process may eliminate what the education establishment views as tedious "drill and kill" exercises, but it results in poor learning and lack of mastery. Students generally work in groups with teachers who "facilitate" rather than providing direct instruction.</p><p>What I've described isn't the case in every K - 6 classroom in the U.S. but it is happening with enough frequency that it is becoming more the norm than the exception. The effect has been noticeable to high school math teachers whose algebra students do not know how to do simple mathematical procedures. At education schools, these reform techniques are taught to young and eager students who think the theory sounds wonderful. They learn that textbooks are bad and that teachers should veer from them as much as possible, supplementing with inquiry-based and student-centered assignments. </p><p>This is where reform texts come in. In the early 1990's, the National Science Foundation awarded grants to various universities and institutions to develop math programs that embodied reform math philosophies. Two of the most popular programs are Everyday Math (developed at University of Chicago, and now distributed by McGraw Hill) and Investigations in Number, Data and Space (developed by TERC, Inc. and distributed by Pearson/Scott Foresman). To get an inkling of the type of lessons students are doing, you might be interested in a little girl demonstrating how she was taught to add multidigit numbers using the Investigations program:
</p><div style="text-align: center;"><iframe width="420" height="315" src="https://www.youtube.com/embed/1YLlX61o8fg?wmode=transparent" frameborder="0" allowfullscreen=""></iframe></div><p>It is obvious to many parents that these reform math programs are largely ineffective. We have seen our children faced with ill-posed and open-ended problems, asked to develop "strategies" for solving them before gaining proper instruction. We have to see our children's textbooks, only to be told (in the case of Everyday Math and Investigations) that there is no textbook, only worksheets, and no worked examples.</p><p>Many of us are scientists, mathematicians, engineers, and teachers, who understand the necessity of starting out with a solid foundation, with topics presented in a logical sequence that builds upon itself. It is obvious to parents that children do not learn what they haven't been taught. </p><p>Parents have objected to these programs at school board meetings. For a period now spanning more than two decades, we have been told that traditional math may have worked for some people, but it also failed large numbers of students. School boards usually don't bother to define what they mean by fail, or specify how many students in fact "failed," or even clarify what specific era they're talking about. They just say that traditional math doesn't teach <i>all</i> students, but this new program does. </p><p>Many of these parents are then forced to teach their children what they are not being taught in school, hire tutors, or enroll their children in learning centers like Sylvan, Huntington, or Kumon. At my daughter's school, Huntington would put on an infomercial meeting every fall (somehow the principal allowed this), ostensibly to discuss how parents can help their children study effectively. I went to one of them. The presenter explained that the reason our kids weren't doing well in math is that schools no longer teach the math facts or standard procedures. "At Huntington, we do!" she said. The light went on in many parents' minds: The learning center uses the traditional methods decried by school board methods as having failed. </p><p>Schools across the U.S. still persist in using these reform programs, even as parents protest. The school boards trot out test scores from other schools that use the program, showing how they prove its effectiveness. No one ever acknowledges that the test scores may in fact reflect the effectiveness of outside help from centers like Huntington. Parents in affluent communities know the game that's being played. In poorer communities, there isn't any protest. The scores are not as good, but the school boards have an answer for that one too: What can you do? It's the poverty.</p>Barry Garelickhttp://www.theatlantic.com/author/barry-garelick/?utm_source=feedIt's Not Just Writing: Math Needs a Revolution, Too2012-10-12T08:32:30-04:002012-10-12T11:07:22-04:00Today's fashion is to throw away the textbook and to teach kids to think like mathematicians. The problem? They're not learning how to do actual math.