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 “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.”
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?”
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 has simply memorized the method without understanding it.
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. “Students who lack understanding of a topic may rely on procedures too heavily,” states the Common Core website. “… But what does mathematical understanding look like?” And how can teachers assess it?
“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.” (http://www.corestandards.org/Math/).
The underlying assumption here is that if a student understands something, she can explain it—and that deficient explanation signals deficient understanding. But this raises yet another question: what constitutes a satisfactory explanation?
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.
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?”
A student may show their solution as follows:
x = original cost of coat in dollars
100% – 20% = 80%
0.8x = $160
x = $200
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% used in the equation or is he doing it as a mechanical procedure — i.e., without understanding?”
Providing instruction for explanations—the road to “rote understanding”
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 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.
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% gives the amount that I pay.”
An example of a student’s written explanation for this problem is shown in Figure 1: Example of student explanation.
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.
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.”
The “Need” and “Know” steps in the above process are not new and were advocated by Polya (1957) in his classic book “How to Solve It”. 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, 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’s thinking process — called “metacognition” — is part and parcel to problem solving (see Mayer, 1998).
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.
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?
Requiring explanations undoes the conciseness of math
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 — 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 “mathematical understanding,” in other words, impede the “doing” of actual mathematics.
“I can’t do this orally, only headily”
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 — whether orally or in written words — how they arrived at their answers.
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’s Syndrome (a high functioning subtype of autism), Attwood notes that “the personalities of some of the great mathematicians include many of the characteristics of Asperger’s syndrome.” (Attwood, 2007)
And yet, Attwood adds (ibid.), many children on the autistic 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” (Asperger, H., 1991 ).
Writing from Australia decades later, a few years before the Common Core took hold in America, Attwood adds 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” (Attwood, 2007, p. 241). Here in post-Common Core America, this inability has morphed into an unprecedented liability.
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?
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?
What is really being measured?
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.
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.
As Alfred North Whitehead famously put it about a century before the Common Core standards took hold:
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.
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.
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: “Teaching Math in the 21st Century”.
Anindito Aditomo. Cognitive load theory and mathematics learning: A systematic review, Anima, Indonesian Psychological Journal, 2009, Vol. 24, No. 3, 207-217
Hans Asperger. “Problems of infantile autism,” Communication: Journal of the National Autistic Society, London 13, 45-32), 1991 ).
Tony Attwood. The Complete Guide to Asperger’s Syndrome, Jessica Kingsley Publishers, Philadelphia, PA, 2007. (p. 240)
G. Pólya, How to Solve It: A New Aspect of Mathematical Method, Doubleday, Garden City, NY, 1957
Richard Mayer. Cognitive, metacognitive, and motivational aspects of problem solving, Instructional Science, 1998, March, Vol. 26, Issue 1-2, pp 49-63