An Education Week article titled “How Teacher Prep Programs Can Help Teachers Teach Math Conceptually”, starts out right away making the usual uncontested claim about how mathematics is taught:

“Future teachers are likely to teach as they were taught—which can be problematic, researchers wrote in a recent study, “because most teachers experienced school mathematics as a set of disconnected facts and skills, not a system of interrelated concepts.” But even when prospective teachers are taught to teach math conceptually, a good content knowledge base is still important, the study found.”

In other words, it helps to be able to do mathematics in order to teach it. The research study referenced in the article was one in which teachers at an ed school who were taught how to teach math “conceptually” were then observed in a classroom. They found that of four conceptual techniques, the teachers tended to use two of them: “use of mathematical language to support students’ sense making and use of visual representations”.

Also, I wonder how true it is that the teachers were taught math as a set of disconnected facts and skills, totally unconnected with each other. The research study focused on six first-year teachers, so I am assuming that these teachers were fairly young–say in their 20’s. Being in their 20’s, these means that they received their elementary math education starting about 15 years ago, taking us to the 2000’s. Of note, is that many elementary math programs in the 2000’s and late 1990’s had been influenced by the math reform ideologies starting with NCTM’s 1989 standards which gradually became incorporated in ed school programs and in textbooks–not to mention the burgeoning of textbooks written under grants from the National Science Foundation including Everyday Math, Investigations in Number, Data and Space, Connected Math Program, and others, so that maybe, just maybe, any shortcomings in the teachers’ mathematical proficiency and understanding could be attributed to the weaknesses associated with such programs. And even though such programs purported to “teach the concepts”, spiralling and beating about the bush rather than utltimately telling students what they need to know to solve the problems might be at issue here. Just guessing.

The article summarizes the research study:

“During the classroom observations of the six first-year teachers, the researchers found that the teachers were more equipped at teaching math conceptually when they had learned the topic in their preservice classes (which incorporated all four of those instructional practices). When they hadn’t been taught a topic in teacher prep, they focused on procedural talk rather than using academic language and conceptual meanings. They also weren’t sure of what appropriate visual representations to use to illustrate the concepts.”

I wonder what textbook these teachers using from which they were teaching the lessons. Based on what I’ve seen being used, explanations of what’s going on with math concepts seem to be in short supply. In particular, the “Big Ideas” series of textbooks that I’ve had the misfortune of having to use, concepts are embedded within discovery-based activities, and some hidden within the problems at the end of the lesson.

Also of interest is that one of the conceptual strategies taught to these teachers was:

“Pressing students for explanations. Doing so allows students to further develop their understanding by working through obstacles and contradictions and reaching for connections across strategies. Teachers should establish classroom norms, researchers say, where a good explanation is a mathematical argument and not simply a description of the procedures, and errors are further opportunities to learn.”

I guess it would depend on what level of explanation we’re dealing with, wouldn’t it? There are different levels of understanding. Also, there is no simple path of understanding first and then skills; an idea which pervades a lot of modern math education pedagogy. Adding to this, words can get in the way. A student may know how to do something but won’t know how to put it into words, except in those cases where he/she has been given the correct mathematical vocabulary. But is that parotting back the words the student thinks might make a teacher happy–i.e, “rote understanding”? I’ll go out on a limb here and say as I have many times: understanding is not tested by words, but by whether the student can do the problems.

It becomes harmful when students are expected to focus too much on the “process” and less on the subject matter of the problem solving. Problem solving skill is highly contextual, so domain knowledge mastery is really the critical thing. If that means they know the procedures but are weak on the conceptual understanding, there are worse things that can happen–and are happening.