Good Teaching, Bad Results

The article by Alan J. Schoenfeld entitled “When Good Teaching Leads to Bad Results: The Disasters of Well-Taught Mathematics Courses” was very intriguing and a truly revelatory read that I would recommend to all teachers of mathematics. It has forced me to question what I have long considered effective mathematics teaching and I have a feeling I will remember the reading of this article as a turning point in my educational career.

In what other subject would students be asked to complete a task like the construction problem without having to provide any explanation as to why the construction makes sense? The emphasis on the physical skill of using a straight edge and compass, particularly the speed and precision with which the skill is performed, devalues the importance of the theoretical framework underlying such operations. In the case of the class where the six students take turns using the one compass to display their solutions instead of simultaneously free handing a sketch and explaining their reasoning is an explicit example of sacrificing the “why” for the “how”. In reflection, this is something I have surely been guilty of in the past, namely in the graphing section of Physics 2204. I consider myself lucky to have read this article and to be consciously aware of this issue.

From a previous course taken on the teaching of writing, I am familiar with the whole “form vs. function” debate in the area of language arts, and am disappointed to see it crop up in mathematics as well. I’m positive that while reading the section in Schoenfeld about the strict adherence to proper form in proof writing, my mouth was wide open, jaw touching the floor. Is this not ludicrous? Again, how can we so blatantly sacrifice understanding in lieu of some arbitrarily pre-determined format? I understand presenting an option that can help those in need of organization, but I just can’t justify taking away marks on an assessment for straying from said option.

I like that Schoenfeld does acknowledge that students need to learn basic facts and procedures through short fragmented exercises, and I certainly agree with him to an extent. But we are indeed in need of some balance in this highly lop-sided relationship between exercises and exploratory problems. Some problems should take longer than a few minutes to complete, and should involve complex mathematical thinking. It is quite sad to think that students would sooner give up with the impression of certain failure than to tackle a problem for longer than a few minutes.

The last point in the article reminded me of a class I was teaching on three-variable substitution method just a couple weeks ago. Of course when doing these problems one can choose any variable to start with and there are a multitude of paths to arrive at the correct answer. I chose a certain variable for my example on the board, and there was a student who expressed confusion because her workings looked different than mine but she had still obtained the right answer. She appeared to be in disbelief when I explained to her that there were several options when it comes to these types of problems. I feel like this exemplifies the issue at hand; that our students are under the impression math is mainly about memorizing what someone else has already figured out. When teaching physics several years ago a student came up with his own way of solving a gravitational potential energy problem. During final exam review I used this method in some of my examples, referring to it as the “Roberian” approach”, after his last name Robere. Reading this article has made me feel positive about that decision.

I feel like the most important point here is that we do not blame the teachers in these situations for any of this. Their intentions were always to help students succeed. Teachers are bound by the specifications of standardized assessments and objective-based curricula and are often wrongfully judged against the results of such assessments. Just when I thought my disdain for standardized testing couldn’t get any worse!