Back in November, 2015, online Atlantic published an article that Katharine Beals and I wrote on explaining your answer in math. It generated many comments (which no longer are available) as well as some discussion in math blogs. (Here and here, among others.) Apparently some people were highly annoyed at what we had to say. (On the other hand, there were also many who knew exactly what we were talking about.)

I’m revisiting this because the “controversial” opinions that Katharine and I raised are apparently still alive and well, and do not appear to be going away.

One particular physics professor was annoyed with our article. So annoyed in fact that he ranted about it in an article he wrote for Forbes. He stated:

*There’s a lot in the Garelick and Beals piece that I intensely dislike, but their core argument boils down to “The real point of math is being able to get the right answer, so as long as students get the right number, nothing else should matter.”*

Actually that’s not what we were saying. We were saying that in lower grades, requiring explanations of problems so simple that they defy explanation confuses rather than enlightens–“enlightens” as in providing “deeper understanding”.

The author claims that he “hated being required to ‘show work’ for math problems too…” Again, Katharine Beals and I have nothing against showing one’s work, and in fact the main premise of our article is that showing the math that one did to arrive at an answer provides an explanation in and of itself.

We also have nothing against teachers asking students questions about how they arrived at their answers–such questioning technique provides guidance to students in learning what their reasoning actually was, and how to verbalize it.

Our objection is the emphasis on written explanations, particularly in lower grades (K-6). Using diagrams as a means of explaining concepts has its use, particularly in teaching place value. Context is absolutely necessary to introduce why a procedure works as it does, but one needs to move beyond that quickly. Some students pick up on the underlying concept, but most do not. The insistence on students using inefficient methods/drawing of pictures to “show understanding” before students are allowed to use standard algorithms (which is advocated by purveyors of the “students must understand or they will die” camp) acts as a “barrier to entry” that holds students up when they are ready to move on.

It has been my experience as a teacher (as well as other teachers I know and respect) that when teaching students a new procedure, one needs to keep it as clean and accessible as possible. The people who propose these ideas that images and explanations lead to “deep understanding” do so because 1) they have forgotten that they themselves benefitted from the methods that they now hold in disdain, and 2) they are viewing the world as an adult who has mastered the procedure and practiced it for many years so that they implicitly understand the link between numbers and their representations in the real world. Young students largely do not. Most anyone who has been teaching for a while knows (or should know) that problems are much harder when put into the explanatory frameworks that are designed to show understanding.

Anna Stokke, a math professor at University of Winnipeg and is an advocate for better math education in Canada puts it this way:

*The understanding piece is a lot more difficult for students. They generally don’t like it and it’s something that really comes with much experience and mathematical maturity. It won’t make students like math more if we spend more time on understanding…it will just confuse and frustrate them more. In my experience, I’ve found that students like step-by-step procedures and algorithms more than anything else.*

The fetish towards understanding predates Common Core and has been going on for three decades with the advent of the 1989 NCTM standards and subsequent revisions—and finally, Common Core, which pushed these ideas. Many of these ideas and ideals are embedded through what Tom Loveless of Brookings calls the “dog whistles” of math reform that appear in Common Core. I strongly suspect that the reason that students arrive at high school profoundly confused is because far too much emphasis is put on “understanding” before the students are ready to do that.

Understanding, critical thinking, and problem solving ability come when students can draw on a strong foundation of domain 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. Put in neuroscience terms: The pre-frontal cortex (where critical thinking takes place) is underdeveloped in early and middle school years. It undergoes rapid development through teen years (where self-concept is growing) and this is where students should be challenged to more sophisticated reasoning, explanation of meaning and so on. It is not fully developed until one reaches early adulthood, sometime in one’s 20’s.

When a small child is asked to engage in critical thinking about abstract ideas, they will produce a response that may look like independent reasoning to an untrained adult, but it will involve more of a limbic response. That is, they are responding emotionally and intuitively, not logically and with “understanding”. That may be behaviorally interesting, but it is not mathematical development and it leaves them behind in the development of their fundamental skills. In many cases, the child develops a “rote understanding”.

“Rote understanding” may have some advantages. Since students figure out what the teacher is looking for them to say, they are implicitly being taught to memorize an explanation that will satisfy the teacher. So maybe the math reformers will eventually see the value of memorization and extend it to learning number facts and other essentials of math – a process that has been held in disdain for years.

I’ve run out of time to make a thoughtful comment, but I’m excited to share this around as a conversation starter. There’s a ton in here that really hit on a lot of thoughts and instincts I have been able to express. Thanks for sharing this!