Category Archives: Math

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.” (
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.
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 [1944]).

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 [1944]).
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

Common Core math standards encourage dubious inquiry-based approaches

“It’s important to note Common Core does not prohibit the teaching of the standard algorithms prior to the grade level in which they appear in the standards. This has been confirmed by the lead writers of the Common Core math standards, Jason Zimba and Bill McCallum. Zimba recommends the standard algorithm for addition and subtraction be taught sooner. Zimba would introduce the standard algorithm for addition late in 1st grade, with two-digit addends. He would then increase the complexity of its use in subsequent grades, continually providing practice toward fluency. The goal would be complete fluency by 4th grade. Zimba also would introduce the subtraction algorithm in 2nd grade and increase its complexity until 4th grade.”

Purveyors of the inquiry-based and student-centered math agenda, which has persisted for more than 20 years, argue traditionally taught math has never worked for the vast majority of the nation’s student population, and they say mental math is the key to “understanding,” as opposed to what they consider to be rote procedures.

The mantras of “students shall understand” and “explain” appear in the Common Core standards and are, according to Tom Loveless of the Brookings Institution, “dog whistles” that serve as a signal to proponents of the inquiry-based and student-centered math agenda to see Common Core’s math standards as requiring those approaches.

It is ironic how inquiry-based math approaches seem to spend more time showing students strategies they might discover on their own than on teaching the standard algorithms they almost certainly won’t learn on their own.

The analytic, problem-solving, and critical-thinking capabilities inquiry-based math proponents tout as the stated goal of Common Core only develop after students master facts and procedures. Understanding works in tandem with procedural fluency, but the current emphasis on conceptual understanding places the cart before the horse. Proponents of inquiry-based math point to students’ procedural fluency as evidence “they can do it but they don’t know what they’re doing,” but the reality is the students cannot do the math if they do not master the procedures.

Once Is Never Enough

The inquiry-based thinking that dominates interpretation of the Common Core standards holds one indication of “understanding” is if a student can solve a problem in multiple ways. Thus, the proponents of inquiry-based math insist on having students come up with more than one way to solve a problem. Forcing students to think of multiple ways to get the same answer, however, does not in and of itself cause understanding. It is as if the reformers are saying, “If we can just get them to do things that look like what we imagine a mathematician does, they will be real mathematicians.”

In my experience, and that of many math teachers I know, delays in understanding are normal and do not signal a failure of the teaching method. Students learn to do something; they learn to apply what they’ve mastered; they learn to do more; they begin to see why; and eventually the light comes on. The inquiry-based math community fails to understand that what they deride as “procedures devoid of meaning” is hard to accomplish with elementary math because the very act of learning procedures is itself informative, and the repetitious use of them conveys understanding to the user. Placing the cart before the horse by insisting on “deeper conceptual understanding” prior to teaching standard algorithms and procedures is simply not effective.

Standard Procedures Delayed

Unfortunately, the Common Core math standards delay the teaching of standard algorithms for addition and subtraction, as well as other key operations. The standard algorithm for multi-digit addition and subtraction does not appear until the 4th grade.

It’s important to note Common Core does not prohibit the teaching of the standard algorithms prior to the grade level in which they appear in the standards. This has been confirmed by the lead writers of the Common Core math standards, Jason Zimba and Bill McCallum. Zimba recommends the standard algorithm for addition and subtraction be taught sooner. Zimba would introduce the standard algorithm for addition late in 1st grade, with two-digit addends. He would then increase the complexity of its use in subsequent grades, continually providing practice toward fluency. The goal would be complete fluency by 4th grade. Zimba also would introduce the subtraction algorithm in 2nd grade and increase its complexity until 4th grade.

Despite admonitions from those in the know, such as Zimba, the word does not seem to have reached school districts, administrators, policymakers, publishers, or test makers. Nor do such recommendations appear in the Common Core background discussion on its website.

Morgan Polikoff, assistant professor at the University of Southern California’s Rossier School of Education, was quoted in a recent article in The Florida Times-Union, and his comments show the widespread disdain for teaching the standard algorithms.

“Common Core … put[s] the algorithm last and the conceptual understanding first,” Polikoff said. “They want you to understand how the tool works before they give you the tool. If you understand the concepts, then when you get to higher levels of math problems, you’ll be thinking like a mathematician and you can do it.”

Such thinking manifests itself in many schools and school districts where teachers have gone so far as to issue warnings to parents not to teach their children the standard method at home because it would interfere with the student’s learning. Many parents are choosing to ignore such warnings. More power to them.

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. Environmental Protection Agency and is teaching middle and high school math in California. Garelick is the author of Teaching Math in the 21st Century, which recounts his experiences as a long-term substitute in a high school and middle school in California.

Three years later, this article is more useful than ever

Since the Common Core State Standards Initiative (CCSSI) has been fully implemented, I have received more complaints from constituents about the CCSSI than any other subject. Most of these complaints have been about math, with parents of elementary school students being the most vocal. I recently reread this 2012 article by high school math teacher Barry Garlick. The entire article is excellent, but I want to quote for you the section that I think zeroes in on the gist of our problem:

As I’ve discussed elsewhere, the criticism of traditional math teaching is based largely on a mischaracterization of how it is/has been taught, and misrepresented as having failed thousands of students in math education despite evidence of its effectiveness in the 1940s, ’50s and ’60s. Reacting to this characterization of the traditional model, math reformers promote a teaching approach in which understanding and process dominate over content. In lower grades, mental math and number sense are emphasized before students are fluent with procedures and number facts. Procedural fluency is seldom achieved. In lieu of the standard methods for adding/subtracting, multiplying and dividing, in some programs students are taught strategies and alternative methods. Whole class and teacher-led explicit instruction (and even teacher-led discovery) has given way to what the education establishment believes is superior: students working in groups in a collaborative learning environment. Classrooms have become student-centered and inquiry-based. The grouping of students by ability has almost entirely disappeared in the lower grades—full inclusion has become the norm. Reformers dismiss the possibility that understanding and discovery can be achieved by students working on sets of math problems individually and that procedural fluency is a prerequisite to understanding. Much of the education establishment now believes it is the other way around; if students have the understanding, then the need to work many problems (which they term “drill and kill”) can be avoided.

The de-emphasis on mastery of basic facts, skills and procedures has met with growing opposition, not only from parents but also from university mathematicians. At a recent conference on math education held in Winnipeg, math professor Stephen Wilson from Johns Hopkins University said, much to the consternation of the educationists on the panel, that “the way mathematicians learn is to learn how to do it first and then figure out how it works later.” This sentiment was also echoed in an article written by Keith Devlin (2006). Such opposition has had limited success, however, in turning the tide away from reform approaches.
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