That Which is Always there... (part 2)

The past couple of days I have been thinking a lot about trigonometry. This has been spurred in part by the fact that I will be doing some lessons on it soon and in part by Anne Watson's excellent opening address at the recent ATM conference. The address forced me to think about how joined-up maths really is, but also about how disconnected it is often taught. Starting with this post I am going to work on ways of encouraging 'Relational Understanding' in my students, an ability to see these connections for themselves.

This post looks at the links between linear sequences, gradients, algebra and trigonometry.

I was struck yesterday by the image of someone using a right angle triangle to calculate the gradient of a line. "Hold on" I thought, "are they not calculating the Tangent of some angle? And more to the point isn't that the angle that defines the steepness of the line? And more to the point do we not call this the gradient?" This was quite striking to me. Stop me if this is painstakingly obvious, but here are two processes completed to varying degrees of success by thousands of students around Britain that I have not once seen taught as interconnected.

Now let me re-assure those of you who are reading this, mouth agape, ready to fire off an email telling me that you've been teaching this for years, I am a mere NQT my experience is sorely lacking. However this was an important realisation for me. The fact that I had students using the same skill in several areas but who were unaware of it was an exciting prospect!

My mind then moved from gradients and tangents to linear sequences. How could this be connected to both trigonometry and gradients? Think about the connection between similar triangles and trigonometry... any thoughts? Leave a comment.
Now when introducing tangent to my class I presented a set of right-angled triangles all with a 21º angle and let them work out opposite over adjacent...surprise!!! They all give the same thing! I'm starting to wonder now whether surprise was the best emotion to connect with this fact however. Based upon the experience that they already have (linear sequences, similar triangles) could they not have been scaffolded towards an understanding that things cannot be otherwise? That the tangent of an angle is necessarily unique? I suppose that this is something to think about.

That Which is Always there... (part 1)

Thinking about mathematics leads inevitably to generalities and it is these generalities that imbue maths with the power that allows us to talk of ideas "shot through with infinity".


The title of the post above was intended to refer to algebra but the more I write the more I feel the presence of infinity within and throughout my thinking. Could it be that part of our use of algebra is to allow us to deal with the infinite? What does this mean to you?

The reliance upon some sort of algebra in all aspects of our lives is quite phenomenal but it takes a lot to become conscious of when exactly we are doing it. Try and think of a time when you may have applied algebra (it will be a lot more meaningful than any example from me) were you aware of applying algebra? Could you generalise based on that application? Could you deal with infinity? Would that even be meaningful in this context?

What is it about algebra that makes for it being such a stumbling block? That makes it a point of which so many adults look back on as some alien language? Try to recall your experience of algebra at school and compare it to a time you feel that you have used it informally in your life. What is different?