Mathematics Teaching

by Leland McInnes

As a university mathematics instructor I have found that there are some skills a lot of modern students seem to lack. This is not the usual complaint about poor algebra skills, a lack of decent calculus preparation, or any other specific item that might appear on a curriculum checklist, this is a deeper complaint. What many of the students I deal with seem to lack is a good understanding of formal reasoning and logic. This is troubling to me because I had always felt that this was one of the primary skills that mathematics was supposed to teach.

It is quite common these days to come across articles and books about the apparent decline in critical thinking. There is significantly less analysis that proposes explanations for this apparent decline. More often than not the question will simply be written off by noting that "we don't teach critical thinking anymore". Where exactly are we supposed to be teaching critical thinking? I would contend that its roots should be firmly planted in the mathematics classroom.

Critical thinking depends, ultimately, on reasoning and logic. It requires the ability to analyse a situation with an eye to determining what can be safely assumed, what must be demonstrated, and how to logically derive a conclusion from the assumptions. These are precisely the formal reasoning skills required for axiomatic mathematics; precisely the skills that so many of the students I deal with seem to lack. It's not that the students are not intelligent or capable, it is simply that they have never specifically been taught these skills - it is something we apparently expect students to somehow pick up implicitly.

It wasn't always this way; students used to spend considerable time doing constructive geometric proofs for Euclidean geometry. There were complaints about the applicability of such studies, and a great deal of hatred for deductive geometry from students. The topic was phased out and eventually largely dropped altogether. To be fair I have a lot of sympathy for these complaints. While it did tend to teach formal proof and axiomatic systems, in practice it did so in a rather indirect way, instead focusing on tedious ruler and compass constructions. The problem is that nothing has been offered to replace it.

Over the years mathematics, particularly at the high school level, has become increasingly applied; increasingly focused on the mechanical skills required for the everyday application of mathematics. While these is a valuable skills and certainly shouldn't be ignored, it has meant that mathematics teaching has completely lost sight of other deeper skills of logic and reasoning that underpin both advanced mathematics and basic critical thinking. From the mathematical point of view it means students who are unable to deal with the increasing abstraction at higher levels, and incapable of sufficient rigour for anything but the most applied topics. From a broader perspective it means students with little experience at careful and rigorous reasoning and analysis.

It is true that some students will simply pick these concepts up on their own as an implicit underlying structure to the mathematics that they are taught. These are, of course, the students who will excel at high school mathematics and have little difficulty with any mathematics they go on to do at university. The majority of students, however, are apparently not reading between the lines to find the hidden lesson and instead struggle with even basic predicate logic. Perhaps then, we should be looking to make these lessons explicit and to actually spend time teaching mathematics as a formal logical system. Perhaps it is time to see teaching students how to think and reason as important a goal in mathematics instruction as basic numeracy.