I’ve been spending January and February teaching MATH1002 Linear Algebra at the Summer School at the University of Sydney. I’m trying a whole bunch of new things this year: the most significant change from previous years is the Wordpress blog that I’m running the whole site through.
But I’ve also been thinking a lot about a very common question that first-year maths students ask, and one that I remember having constantly in the back of my mind when I was in first-year. I’ll explain with an example.
In the course students are introduced to geometric vectors simply as objects possessing a length and a direction (a very abstract and difficult idea for most of these students), and then shown how to embed them in the plane or space and obtain the i, j and k components. The dot product of two vectors is then introduced: it’s an operation that takes in two vectors and spits out a number. But what actually is it? What is the dot product of two vectors?
The question is of course slightly misguided: students should ask exactly the same question of the length of a vector (or even the sum of two integers!), but don’t because length is something that is easily measured with a ruler and intuited concretely. But it is a fair question and I’m never really completely sure how to answer it.
So what is the dot product, philosophically speaking? The best way to answer this, in my opinion, is to consider the extreme cases and think about what happens in between. The dot product of two parallel vectors is just the product of their lengths, and the dot product of two perpendicular vectors is zero. So for two vectors of length one, as they move from pointing in opposite directions to perpendicular to coinciding, their dot product will follow the cosine curve between 0 and pi. So the dot product is precisely giving us the information of the angle between two vectors: this is what it is.
Pedagogically I think this is a sound idea: if you want to understand a quantity, first try to understand its extremes, and then try to understand how the general case fits in between these extremes. Would it have satisfied me in first year? Possibly: I certainly remember being very confused about defining a whole bunch of new operations without any mention of motivation or meaning and some sort of discussion of where the idea for this operation has come from wouldn’t have gone astray. But also I think this is just an unavoidable symptom of learning: you will feel lost, you will feel like new concepts are coming out of nowhere. It’s the job of teachers to try and guide students through this phase and make it seem like it has an end.