Welcome

Post date: Jan 8, 2017 2:26:51 PM

This new year (2017), I am going to try to start a blog on teaching mathematics. My aim is to write on (hopefully weekly) thoughts I have on the teaching and learning of mathematics. As I'm currently teaching at an undergraduate level, most of my posts will be on that topic, but I am also interested in thinking about outreach work and how mathematics is learned at all level of life.

This week, I will talk about the article "Ideas from Mathematics Education - An introduction for mathematicians: Lara Alcock, Adrian Simpson" (available here). I read this over the winter break, and I would definitely recommend it to any mathematician teaching at an undergraduate level. It is easy to read (took me no more than a few hours), and raises interesting questions.

Alcock and Simpson break their books into three points (which they elaborate in three chapters):

  1. Definitions: students need to understand the importance of formal definition, and not just learn mathematical concepts informally.
  2. Mathematical objects: at high level of mathematics, mathematicians are used to seeing mathematical objects on different level. For example x^2+2 can be thought as a function (taking an element of a ring to another, and hence is a process), or it can be thought as an element of the polynomial ring (so it interacts with other elements of the polynomial ring, and hence is an object). Students need to reach this level of understanding.
  3. Two reasoning strategies. Very broadly speaking, there are two approaches that students approaches proofs.

I'm going to focus on the first point this week.

Formal definition have a very important place in mathematics. To name an object we show that it satisfies all the properties of the given definition, and having named an object we know a lot about its properties (and can often deduce a lot more). When first hearing a definition students (and mathematicians) attached an image to this definition. This is a useful first step: when I hear a new definition I try to picture what this means to me, I also try to point out mental pictures that my students might find useful with every new definitions. But Alcock and Simpson points out that this is not enough. We mathematicians always fall back on the formal definition, but students can often missed that subtle point. As educators, we need to not only get students to have an intuition of a definition, but also to get them to internalise the definition and use it for their formal work.

To achieve this we should get students to work with the definition as much as possible. Currently one way we try this is to have students deduce properties of the object using the definition. Unfortunately, some students see the properties as "obvious and part of the definition" and do not understand the point of the exercise. Another approach (this is my thoughts, and research should go on to see if it helps), is to get students to explore boundary conditions. That is, give them loads of objects and see if they satisfy the definition.