# The Vicious Cycle of Teaching

Blah. Once you get behind in one class, it throws off the entire next class. On Monday I got behind — because I spent so much time going over homework — and that meant I only had 15 minutes to present new material. Which meant the students had trouble on homework. Which mean that on Tuesday I spent so much time going over homework — and that meant I only had 15 minutes to present new material.

A vicious cycle.

That’s happened in both my Algebra II classes and I’m going to put an end to it today. I’m going to spend a lot of time before we go over the homework taking a “mulligan” (a do-over) and re-presenting the material. And then hopefully their “ah haaah!” moment will happen and they’ll say “oh I get it now!” and be able to see how to do the problems they had difficulty with on the previous night’s homework. Well, they’re 10th graders, so hard-to-read, and the actual “oh I get it now!” verbal exclamation won’t actually happen.

This problem is actually forcing me to reconsider the content of presentation. Normally I try to ease into a formula or definition — and spend more time talking about it conceptually — and less doing problems. Not that my board is all theory and no practice — but perhaps it is too much theory and less practice. I don’t want them leaving my classroom thinking the equation for a circle (x-h)^2+(y-k)^2=r^2 is a circle “just because.” I want them to understand that:

1. the circle equation is an EQUATION, just like any other one they’ve been working with. So for a line y=mx+b, all (x,y) that lie on the line are solutions. Similarly, for the circle, all (x,y) that satisfy the equation lie on the circle. It’s pretty amazing. [It’s clear to me that some of them didn’t get this… they just see a lot of letters and symbols and squares and have no idea what they’re looking at…]

2. without knowing anything, they can easily plot a few points and get an intuitive sense it’s a circle. If the x-coordinate is h, they can find the two y-coordinates easily. And if the y-coordinate is k, they can find the two x-coordinates easily. And they at least show something that COULD be a circle.

3. they should know — but not have to rederive — that this formula for a circle comes from the pythagorean theorem. It is just a new way to look at what they already know.

But I spent all this time showing these things, and then we have little time to do simple problems like:

1. If the center of a circle is at (2,3) and the radius is 7, what is the equation of the circle (simple application of formula)
2. If the center of a circle is at (1,4) and a point on the circle is (-2,5), what is the equation of the circle (slight leap in understanding needed… have to get find the radius first…)

And so when they see those on the homework, they can usually only do a problem like #1 and #2 poses more difficulty. And importantly, something like

3. If you have the points (2,3) and (1,-7) lying at the endpoints of a diameter, what is the equation for the circle?

is super hard, because there’s a few concepts that have to be brought together (have to find the center first… then get the radius…)

It’s tricky stuff, but all this time on the conceptual level leaves them in the lurch in problems like that.

The other teacher I think does things more problem-based — where he spends the majority of the class presentation time working on problems from the book/homework, so they can do the homework. I have to find out how to plan a proper lesson plan that can do both of these things.

1. MJS says: