{"id":73,"date":"2015-11-18T23:17:49","date_gmt":"2015-11-19T03:17:49","guid":{"rendered":"http:\/\/www.bettypeters.com\/bettypeters\/?p=73"},"modified":"2019-04-18T07:00:55","modified_gmt":"2019-04-18T11:00:55","slug":"math-problems-knowing-doing-and-explaining-your-answer","status":"publish","type":"post","link":"http:\/\/www.bettypeters.com\/bettypeters\/2015\/11\/18\/math-problems-knowing-doing-and-explaining-your-answer\/","title":{"rendered":"Math Problems: Knowing, Doing, and Explaining Your Answer"},"content":{"rendered":"

Math Problems: Knowing, Doing, and Explaining Your Answer<\/a>
\nby Barry Garelick and Katharine Beals<\/em><\/p>\n

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 \u201cWhat do I do?\u201d The proctor read the instructions for the problem and told the student: \u201cYou need to explain how you got your answer.\u201d<\/p>\n

The girl threw her arms up in frustration and said, \u201cWhy can\u2019t I just do the problem, enter the answer and be done with it?\u201d<\/p>\n

The answer to her question comes down to what the education establishment believes \u201cunderstanding\u201d to be, and how to measure it. K-12 mathematics instruction involves equal parts procedural skills and understanding. What \u201cunderstanding\u201d in mathematics means, however, has long been a topic of debate. One distinction popular with today\u2019s math reform advocates is between \u201cknowing\u201d and \u201cdoing.\u201d A student, reformers argue, might be able to \u201cdo\u201d a problem (i.e., solve it mathematically), without understanding the concepts behind the problem solving procedure. Perhaps he has simply memorized the method without understanding it.<\/p>\n

The Common Core math standards, adopted in 45 states and reflected in Common Core-aligned tests like the SBAC and the PARCC, take understanding to a whole new level. \u201cStudents who lack understanding of a topic may rely on procedures too heavily,\u201d states the Common Core website. \u201c\u2026 But what does mathematical understanding look like?\u201d And how can teachers assess it?<\/p>\n

\u201cOne way \u2026 is to ask the student to justify, in a way that is appropriate to the student\u2019s mathematical maturity, why a particular mathematical statement is true or where a mathematical rule comes from.\u201d (http:\/\/www.corestandards.org\/Math\/<\/a>).<\/p><\/blockquote>\n

The underlying assumption here is that if a student understands something, she can explain it\u2014and that deficient explanation signals deficient understanding. But this raises yet another question: what constitutes a satisfactory explanation?<\/p>\n

While the Common Core leaves this unspecified, current practices are suggestive: Consider a problem that asks how many total pencils are there if 5 people have 3 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 5 groups of 3 pencils.<\/p>\n

Consider now a problem given in a pre-algebra course that involves percentages: A coat has been reduced by 20% to sell for $160. What was the original price of the coat?\u201d<\/p>\n

A student may show their solution as follows:<\/p>\n

x = original cost of coat in dollars
\n100% \u2013 20% = 80%
\n0.8x = $160
\nx = $200<\/p>\n

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 \u201cDoes the student know why the subtraction operation is done to obtain the 80% used in the equation or is he doing it as a mechanical procedure \u2014 i.e., without understanding?\u201d<\/p>\n

Providing instruction for explanations\u2014the road to \u201crote understanding\u201d<\/strong><\/p>\n

In a middle school observed by one of us, the school\u2019s 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 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 \u201cNeed, Know, Do.\u201d In the problem example given above, the \u201cNeed\u201d would be \u201cWhat was the original price of the coat?\u201d The \u201cKnow\u201d would be the information provided in the problem statement, here the price of the discounted coat and the discount rate. The \u201cDo\u201d is the process of solving the problem.<\/p>\n

Students were instructed to use \u201cflow maps\u201d 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 \u201cDo\u201d (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 \u201cDo\u201d section from the final narrative. But in order for their explanation to qualify as \u201chigh level,\u201d they couldn\u2019t simply state \u201c100% \u2013 20% = 80%\u201d; they had to explain what that means. For example, they might say, \u201cThe discount rate subtracted from 100% gives the amount that I pay.\u201d<\/p>\n

An example of a student\u2019s written explanation for this problem is shown in Figure 1: Example of student explanation.<\/p>\n

\"\"<\/a><\/p>\n

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 \u201cwhy\u201d of a particular procedure.<\/p>\n

Under the rubric used at the middle school where this problem was given, explanations are ranked as \u201chigh\u201d, \u201cmiddle\u201d or \u201clow.\u201d This particular explanation would probably fall in the \u201cmiddle\u201d category since it is unlikely that the statement \u201cYou need to subtract 100 -20 to get 80\u201d would be deemed a \u201cpurposeful, mathematically-grounded written explanation.\u201d<\/p>\n

The \u201cNeed\u201d and \u201cKnow\u201d steps in the above process are not new and were advocated by Polya (1957) in his classic book \u201cHow to Solve It\u201d. The \u201cNeed\u201d and \u201cKnow\u201d aspect of the explanatory technique at the middle school observed is a sensible one. But Polya\u2019s book was about solving problems, not explaining or justifying how they were done. At the middle school, however, 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 is based on a popular educational theory that being aware of one\u2019s thinking process \u2014 called \u201cmetacognition\u201d \u2014 is part and parcel to problem solving (see Mayer, 1998).<\/p>\n

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 \u201cdo the math\u201d without having to write about it.<\/p>\n

In general, there is no more evidence of \u201cunderstanding\u201d 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\u2019t simply repeating an explanation provided by the teacher or the textbook, thus exhibiting mere \u201crote learning\u201d rather than \u201ctrue understanding\u201d of a problem-solving procedure?<\/p>\n

Requiring explanations undoes the conciseness of math<\/strong><\/p>\n

Math learning is a progression from concrete to increasingly abstract. The advantage to the abstract is that the various mathematical operations can be performed without the cumbersome attachments of concrete entities \u2014 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 handled by available schemas frees up working memory. With working memory less burdened, the student can focus on solving the problem at hand (Aditomo, 2009). Thus, requiring explanations beyond the mathematics itself distracts and diverts students away from the convenience and power of abstraction. Mandatory demonstrations of \u201cmathematical understanding,\u201d in other words, impede the \u201cdoing\u201d of actual mathematics.<\/p>\n

\u201cI can\u2019t do this orally, only headily\u201d<\/strong><\/p>\n

The idea that students who do not demonstrate their strategies in words and pictures must not understand the underlying concepts assumes away a significant subpopulation of students whose verbal skills lag far behind their mathematical skills, such as non-native English speakers or students with specific language delays or language disorders. These groups include children who can easily do math in their heads and solve complex problems, but often will be unable to explain \u2014 whether orally or in written words \u2014 how they arrived at their answers.<\/p>\n

Most exemplary are children on the autistic spectrum. As 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\u2019s Syndrome (a high functioning subtype of autism), Attwood notes that \u201cthe personalities of some of the great mathematicians include many of the characteristics of Asperger\u2019s syndrome.\u201d (Attwood, 2007)<\/p>\n

And yet, Attwood adds (ibid.), many children on the autistic spectrum, even those who are mathematically gifted, struggle when asked to explain their answers. \u201cThe child can provide the correct answer to a mathematical problem,\u201d he observes, \u201cbut not easily translate into speech the mental processes used to solve the problem.\u201d 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, \u201cI can\u2019t do this orally, only headily\u201d (Asperger, H., 1991 [1944]).<\/p>\n

Writing from Australia decades later, a few years before the Common Core took hold in America, Attwood adds that it can \u201cmystify teachers and lead to problems with tests when the person with Asperger\u2019s syndrome is unable to explain his or her methods on the test or exam paper\u201d (Attwood, 2007, p. 241). Here in post-Common Core America, this inability has morphed into an unprecedented liability.<\/p>\n

Is it really the case that the non-linguistically inclined student who progresses through math with correct but unexplained answers \u2014 from multi-digit arithmetic through to multi-variable calculus \u2014 doesn\u2019t understand the underlying math? Or that the mathematician with the Asperger\u2019s personality, doing things headily but not orally, is advancing the frontiers of his field in a zombie-like stupor?<\/p>\n

Or is it possible that the ability to explain one\u2019s answers verbally, while sometimes a sufficient criterion for proving understanding, is not, in fact, a necessary one?<\/p>\n

What is really being measured?<\/strong><\/p>\n

Measuring understanding, or learning in general, isn\u2019t easy. What testing does is measure \u201cmarkers\u201d or byproducts of learning and understanding. Explaining answers is but one possible marker.<\/p>\n

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\u2019s verbal abilities. But it\u2019s 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 \u201cshowing the work\u201d may be superfluous; at worst, they shortchange certain students and encumber the mathematics for everyone.<\/p>\n

As Alfred North Whitehead famously put it about a century before the Common Core standards took hold:<\/p>\n

It is a profoundly erroneous truism \u2026 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.<\/p><\/blockquote>\n

Katharine Beals is a lecturer at the University of Pennsylvania Graduate School of Education and an adjunct professor at the Drexel University School of Education. She is the author of Raising a Left-Brain Child in a Right-Brain World: Strategies for Helping Bright, Quirky, Socially Awkward Children to Thrive at Home and at School.<\/p>\n

Barry Garelick has written extensively about math education in various publications including The Atlantic, Education Next, Educational Leadership, and Education News. He recently retired from the U.S. EPA and is teaching middle and high school math in California. He has written a book about his experiences as a long term substitute in a high school and middle school in California: \u201cTeaching Math in the 21st Century\u201d.
\n—<\/p>\n

References<\/strong><\/p>\n

Anindito Aditomo. Cognitive load theory and mathematics learning: A systematic review, Anima, Indonesian Psychological Journal, 2009, Vol. 24, No. 3, 207-217<\/p>\n

Hans Asperger. \u201cProblems of infantile autism,\u201d Communication: Journal of the National Autistic Society, London 13, 45-32), 1991 [1944]).<\/p>\n

Tony Attwood. The Complete Guide to Asperger\u2019s Syndrome, Jessica Kingsley Publishers, Philadelphia, PA, 2007. (p. 240)<\/p>\n

G. P\u00f3lya, How to Solve It: A New Aspect of Mathematical Method, Doubleday, Garden City, NY, 1957<\/p>\n

Richard Mayer. Cognitive, metacognitive, and motivational aspects of problem solving, Instructional Science, 1998, March, Vol. 26, Issue 1-2, pp 49-63<\/p>\n","protected":false},"excerpt":{"rendered":"

Math Problems: Knowing, Doing, and Explaining Your Answer by Barry Garelick and Katharine Beals 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 \u201cWhat do I do?\u201d The proctor read the […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_et_pb_use_builder":"","_et_pb_old_content":"","_et_gb_content_width":"","footnotes":""},"categories":[5],"tags":[17,18],"class_list":["post-73","post","type-post","status-publish","format-standard","hentry","category-math","tag-mathematical-understanding","tag-need-know-do"],"_links":{"self":[{"href":"http:\/\/www.bettypeters.com\/bettypeters\/wp-json\/wp\/v2\/posts\/73","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/www.bettypeters.com\/bettypeters\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.bettypeters.com\/bettypeters\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.bettypeters.com\/bettypeters\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.bettypeters.com\/bettypeters\/wp-json\/wp\/v2\/comments?post=73"}],"version-history":[{"count":10,"href":"http:\/\/www.bettypeters.com\/bettypeters\/wp-json\/wp\/v2\/posts\/73\/revisions"}],"predecessor-version":[{"id":149,"href":"http:\/\/www.bettypeters.com\/bettypeters\/wp-json\/wp\/v2\/posts\/73\/revisions\/149"}],"wp:attachment":[{"href":"http:\/\/www.bettypeters.com\/bettypeters\/wp-json\/wp\/v2\/media?parent=73"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.bettypeters.com\/bettypeters\/wp-json\/wp\/v2\/categories?post=73"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.bettypeters.com\/bettypeters\/wp-json\/wp\/v2\/tags?post=73"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}