Lessons taught; Lessons learnt

The Seed

I recently found myself reflecting, while solving a mathematical problem that occurred to me on a car journey, on the differences between mathematics as I experience it, and the mathematics that my students experience.

This post introduces the problem, and gives some reflections on the problem solving process, as well as suggestions for ways to simplify or extend the problem.

The Problem

What’s the question?

Fix two points. Consider all the quadratics which go through those two points. What is the locus of the stationary points of these quadratics?

If the two fixed points are (2, 0) and (0, 2), for example, here are four quadratics passing through those points, with their stationary points highlighted in red:

To solve the problem we need to find what happens to the stationary points of all the quadratics which pass through the two fixed points.

The Motivation

Why is this interesting?

First, I think this is an interesting problem, with a result that isn’t immediately obvious.

Second, I have an emotional investment in it — the problem arose naturally in the course of an investigation I was doing, and I spent time thinking about and proving the result.

Third, I believe this is an example of a problem where dynamic geometry can be used to help make conjectures and build understanding.

However, the problem isn’t some new area of maths. It isn’t a significant result with many corollaries or consequences. It doesn’t shed light on other mathematical objects. It’s a mathematical chew toy.

The Answer

What’s the answer?

Really, then, the answer isn’t important. The point is the journey, and considering how to continue the journey once we’ve reached our initial destination, the answer.

And that’s why I’m not going to tell you what the answer is.

It’s actually harder for me to withhold than it would be to write up the page of working I have in front of me. The culture of the mathematics classroom is one where answers are all important. Throughout school and university, and not just in mathematics, the way we assess students is based on the following assumption:

“Good students are those who can answer questions on lots of unrelated topics in a set time limit.”

As someone who passed through this system with flying colours, I have an unhealthy need to demonstrate my facility in tests.

This buying into, and even enjoyment of tests as an end in themselves is a hallmark of the more ‘nerdy’/scientific subjects. And I can’t ignore it, because it’s been built into my personality, into my job, and into society.

So, no answer.

What I am going to do is to describe some of the ways which I investigated the problem, and some further avenues that suggested themselves to me.

The Journey

How do I solve it?

The first thing I did was to build a tool to help me explore the problem. In this case, it was the Geogebra worksheet which appears at the end of an earlier post on this problem.

I then picked two fixed points, and generated the following:

There are a number of features to notice in this example: the locus goes through both the fixed points; it seems to have a vertical asymptote roughly half-way between the fixed points; it seems to have another, oblique asymptote.

How many of these features are generic, and how many are artifacts of the particular example? I notice that the fixed points I’ve chosen are fairly symmetrical: (0,2) and (2,0). Does it alter the features if I destroy this symmetry?

Here’s a less symmetrical example, and all the features I noticed are preserved. So we have here a particular example which is serving as an exemplar for the general case.

We can be led astray, though, into thinking that the general case always happens. Are there any special cases where these features would break down? What happens, for example, when the chosen fixed points have the same x coordinate? There are no quadratics at all in this case. What about fixed points with the same y coordinate?

Something different is happening here. A moment’s reflection reveals that this particular example is equivalent to saying that a quadratic with roots at 0 and 1 always has a stationary point at 1/2. This isn’t interesting to me, because I’m well aware of the properties of quadratics — but even just investigating this restricted problem could be interesting to someone who hadn’t thought about quadratics in this way before.

Does it make a difference if the points are not on the x-axis?

No. So another feature of the locus should be: a translation of the fixed points leads to a translation of the locus (are you convinced? if not, what would make you convinced?).

So, just through playing with some fixed points, we’ve developed an intuition about the form and properties our answer should take and hold. How do we actually find this answer? It will almost inevitably involve algebra: we need to assign values to the two fixed points, and try to figure out the equation of the locus.

We could just call one point (p,q), the other point (r,s), and dive in from there… but once we’ve noticed the translational property of the locus, why not fix one of the points to be something simple, like (0,0)? To do this requires me to have the experience to know that fixing one of the points will make my life easier (I now have two variables instead of four), and that (0,0) is likely to be a good choice for the fixed point (do I have any non-intuitive reason for this?).

[Actually, the first time I looked at this problem, I didn't fix one of the points to be (0,0) -- I had one of the points on the x-axis, and the other on the y-axis.]

If I wasn’t comfortable with abstracting straight away, I could have been even more specific, and just looked at the problem for one particular set of fixed points.

Diving into algebra, then — what is it I actually want to do? I want to find a set of equations which represent all the quadratics that go through (0,0) and (p,q). This will be a one parameter family of equations. I then want, for each choice of parameter, to figure out where the stationary point of that quadratic is. This will give me a parametric equation for the locus. I could stop with that, but I’ll probably try to shake things around until the parameter disappears, and I have an equation for the locus in the standard ‘y=f(x)’ style.

That’s what I did, and you do get a ‘nice’ answer.

Nothing here should be beyond a decent A-level student, although it’s not a typical A-level question, so I’d be interested to see how many of my students would be able to do this.

The Destination

Once we’ve got an answer, what do we do with it?

This is a question which is outside the comfort zone of most students, across all age ranges, school types, and abilities. The answer is the goal — once we have something that matches what the back of the book says, it’s time to move on to the next question.

Some students are so focused on ‘the answer’ that they don’t even check whether the answer they get makes sense. In the context of this question, checking our answer involves verifying that it holds the properties that we found in our initial investigation: what are the asymptotes? does it really go through the fixed points? does it account for the special behaviour when the points are horizontal?

We should also look for any check we’ve actually answered our initial question. For example, I should make sure that I can give the formula for any two points, rather than requiring that one point is the origin.

The View

Where now?

This is the question that is almost never asked at a meaningful level in school mathematics. It’s true that discussing extensions and further work was supposed to be a key point of GCSE coursework, but that soon degenerated into teachers giving students fixed lists of phrases to use: “I’d try this with different shapes/higher numbers/different data, etc.”

Here are some things which occurred to me, after I answered the initial question:

• Perhaps I could introduce this to students by asking them to generate the equation of a quadratic which passes through two given points, with a stationary point at a given value of x.
• Even just the question: “How do we generate the quadratic which goes through three specific points?” is an interesting one for them to investigate, although quite a standard one.
• What if only one point was fixed? Would the locus be the whole plane?
• What about cubics? We’d have to fix three points to get the locus to be a line. Would it look similar to the quadratic case, or does new behaviour emerge?
• What about if I restrict the fixed points in some way?

These are a combination of thoughts on how to ’scaffold’ this question so that it would be accessible to others, and thoughts on how to alter the question to create further interesting questions.

(This post is featured in the 188th Carnival of Education. Check it out!)

“In preparing a lecture I find I always have to work hardest on the things I do not say. The things I am sure to say I can easily get up. They are obvious and generally accessible. But they, I find, are not enough. I must have a broad background of knowledge which does not appear in speech. I have to go over my entire subject and see how the things I am to say look in their various relations, tracing out connections which I shall not present to my class.

One might ask what is the use of this? Why prepare more matter than can be used? Every successful teacher knows. I cannot teach right up to the edge of my knowledge without a fear of falling off. My pupils discover this fear, and my words are ineffective. They feel the influence of what I do not say. One cannot precisely explain it; but when I move freely across my subject as if it mattered little on what part of it I rest, they get a sense of assured power which is compulsive and fructifying.”

The Teacher: Essays and Addresses on Education, page 17. Written in 1908. I find this quote deeply relevant and stimulating, as a classroom teacher who is currently preparing for the return of school next week!

I’m a maths teacher, and this next quote could easily have been written about the current trends in the teaching of my subject:

“Among the many changes in mathematical education during the last twenty years, and among the many and often conflicting ideals which have directed these changes, one element at least appears throughout; a desire to relate the subject to reality, to exhibit it as a living body of thought which can and does influence human life at a multitude of points… Our children must learn to think.”

This is from page 35 of Essays on Mathematical Education, written in 1913.

These are just two of more than 8000 results that appear when you search archive.org for texts mentioning ‘education’.

Places like archive.org allow us to correct the notion many people have about the way people were taught in the past. Vocational education; project-based teaching; differentiation; learning styles; curriculum content; the importance of the physical education of youngsters — all of these and more have been considered by teachers for many generations.