Daily Archives: July 13, 2010

So a while ago, I mentioned to some of you on twitter that I was getting really frustrated with a particular problem we were presented with. I have a conjecture that I’m almost certain is true, but I can’t prove it.

Consider the unit circle . Plot equally spaced points on the circle starting from . Now draw the chords from to the others. What is the product of the lengths of all of these chords?

(There is an extension problem, which is changing the unit circle to an ellipse , for those who already have seen or find the original problem too easy.)

So feel free to write your own blog post with your solution, or throw your solution in the comments (just write SPOILER at the top so we know…).

What I’m interested in is if we could get a precalculus class to get the solution to this problem. Where they actually understand it. So if you had, say, 15 non-honors precalculus students and one week to work on this problem, how would you design the lesson?

I guess you have to have solved it or have seen a solution to know how to design the lesson. But even if you didn’t solve it (a la me!)… if there’s a solution you’ve read that someone posted in the comments… what would you do?

UPDATE: Mr. Ho has a great GeoGebra applet at his site; Mimi has some nice colorful diagrams and some explanation up at her site. Also, for those who want to wording for the ellipse problem… This extension I haven’t seen before, so I am citing Bowen Kerins (see comments below!) or Darryl Yong: “Take the diagram you drew in [the unit circle problem] and stretch it vertically so that the circle becomes the ellipse . All the points for the chords scale too. What is the product of the lengths of all of these chords?”

Today I attended a session where three university profs — ed researchers — formed an informal panel. There was one important point that came up at the beginning, and became a riff for a few minutes. It was, as you prolly suspected from the really innovative title of this post, about the power of looking past teachers to teaching.

It’s a slight distinction, but crucial to the reorientation that I’m having about teaching.

Some points that came up in the conversation:

• Replacing teachers won’t change things; replacing teaching methods will.
• Focusing on teaching and not on teachers is the basis of lesson study (and the Seattle video club I talked about in the last post). It focuses the conversation on teaching/teacher moves.
• Changing the conversation from teachers to teaching more readily implies that teaching is learnable. So we have to look past individual teachers to the methods of teaching. That being a good teacher can be taught. Another way to think about it: teaching is a complicated activity, rather that something owned by a particular person.
• There are universal tasks to teaching that we can investigate (e.g. which ideas to privilege in a classroom).
• It gets us away from the “I taught it but they didn’t learn it” phenomenon. That phrase doesn’t really make any sense when focusing on teaching and not the teacher.
• The greatest untapped resource we can use in the classroom are our students and their insights. And by focusing less on the teacher and more on teaching moves, we can tap into that.
• This outlook shifts the conversation away from teacher bashing (but one should also be cautious of going in the other direction of student bashing).

Yes, I know. There are some inconsistencies, and worse, this is all very abstract. And I HATE THAT. But this all tapped into the idea I wrote about recently, about how teaching moves are something that one can pay attention to. One can learn. One can revise. And through this process, hone the craft of teaching.

In other words, the focus on teaching instead of teachers is that it puts the emphasis on the ways teachers can do their jobs by focusing on students and learning.

So that was one part of the talk. In another part of the talk, there was a question about the constant tension between the jam-packed curricula with a zillion micro-pico-standards and getting students to really grapple with big ideas.

One speaker said that we “need more effort and courage” from teachers. I drew a sad face in my notebook next to that.

The second speaker actually spoke articulately, in defense of having common standards in theory [1]. He also said that he doesn’t see the problem as having a zillion pico standards. It’s that we go through all these little ideas that never get added up to any big ideas. His suggestion for dealing with this is to outline learning trajectories, with big ideas as the landmarks on the way. I don’t know what precisely he had in mind, but I figured that it probably involves student drawing connections by working on unfamiliar problems that force relationships among mathematical ideas (e.g. systems of equations with matrices; asymptotes for the tangent graphs and asymptotes of rational functions; absolute value equations and absolute value inequalities; etc.).

The third person then finished up speaking about the Common Core Standards — and eloquently continued the second speaker’s defense of standards.

That’s about it for the maths stuff I want to write about. (It’s late and I have lots to do tomorrow.)

On the non-math side of things, I had a wonderful night BBQing with friends and watching the sky change hughes, from orange, to light blue, to dark blue, to black. As the air got colder and the light retreated, the stars starting coming out, first slowly then quickly. As people left, conversations got less frenetic and more personal. And I left, after being regaled with a shooting star, at peace with Utah.

[1] Having these standards gets us focused on teaching. It also promotes the sharing of ideas; if someone gets it/does it right, then those lessons and approaches will be in demand.