And Another Thing, Dept.

Back in November, 2015, online Atlantic published an article that Katharine Beals and I wrote on explaining your answer in math.  It generated some controversy in the comment section, as well as some discussion in math blogs.

As I’ve indicated in other posts, the article was selected for inclusion in the anthology “Best Writing on Mathematics, 2016” published by Princeton University Press.

Apparently, one particular physics professor got annoyed with it first time around, and annoyed again when it made the rounds a second time after publication in the anthology. So annoyed in fact that he ranted about it in an article he wrote for Forbes. He states:

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–“englightens” as in providing “deep 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, although admittedly the article uses an example from middle school).

Using diagrams as a means of explaining concepts has its use, particularly in teaching place value, but the insistence on using it, and insistence on requiring an “understanding” before students are allowed to use standard algorithms acts as a “barrier to entry” that does more harm than good. In teaching students a new procedure you need to keep it as clean as possible. Some context is good to introduce why the 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. Insisting on introducing visual representations and explanations into the mix will more than likely confuse most students, who now have to try and link mentally the skill of doing the procedure and linking that to the much harder task of identifying what it might mean in a physical way.

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) have viewed the world through an adult and expert lens for many years so that they implicitly understand the link between numbers and their representations in the real world. Young students largely do not. Anyone who teaches knows that problems are much harder when put into contexts rather than just left as math, yet they suggest that putting new concepts into contexts makes understanding easier.

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 28+ years with the advent of the NCTM standards 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, problem solving 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 20s. 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.