When Knowing How Isn't Enough

Barry Garelick and Katharine Beals discuss why knowing how to do math problems just isn't good enough under Common Core:

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.

I hear this silliness about "explaining" often.  I assert that a student who can solve a multi-step algebraic problem and get the correct answer shouldn't then have to explain each step--their comprehension is demonstrated already by the systematic steps taken!  If someone still disagrees with me, I give them this challenge:  "Divide 100 by 6 using long division, and explain to me why that algorithm works."  99% of people can't explain why the algorithm works, but does that really matter if they know the algorithm and can execute it flawlessly?  And why does it matter why the algorithm works?  After all, no one does division for its own sake but rather to solve a problem; the division itself is only a tool, not a goal in and of itself.  Yes, it would be nice if someone could explain it, but are they at all mathematically handicapped if they cannot?  Is someone handicapped at driving a car because they cannot explain the 4 strokes of a "4 stroke engine"?

But lets get back to Beals and Garelick:

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?

This is intuitively obvious.  And Garelick and Beals point out the greatest flaw in the "explain your answer" pedagogy:  requiring the types of explanations identified as good by Common Core undermines, and in fact is counter to, the conciseness of mathematics.

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.

Don't intentionally misunderstand what I'm saying.  I'm not saying that students shouldn't be able to justify their work or shouldn't have to explain anything.  I'm saying that what's put forth as "Common Core" is excessive, it's geared towards the more verbal and less mathematical among us, and is not good math.

The closing of the linked article says it all:

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.