Archives for the month of: March, 2011

Once each week since February I have been giving my Calculus students one section of an AP test for practice, because that seemed like a good idea. We’re at the point in the year when they’ve seen most of the material. Yesterday they got some open response problems (2003 part 1), and today they looked at the scoring guide, because that seemed like a good idea too.

What they are taking away from the experience is that they got everything wrong. They are all diligent students, but after a while they have grown unfocused in their activity of comparing what they did to the scoring rubric. They are discouraged and I feel their intention and energy draining away from test prep and into the pool of “Oh, well. Whatever.”

What I am taking away from the experience is that even with fairly mature students, debriefing their practice test needs some structure to it, something that will channel their energy through the activity, help them focus on what they did well, and point out in a way that feels useful and encouraging what they need to do differently next time. A pep talk along those lines helped, but next time I want to do it differently.

“Focus on what you did well and on what you could do differently, not on whether or not you “got it” or “didn’t get it,” I say. “Keep doing that ’till you die, and you’ll have a happy life.”

“Do you do that?” a student asks.

Yep. Doin’ it now.

Multiple choice. A student looks at the graph on his calculator and concludes the minimum is at x= “a little past negative 8″, so he chooses (C) -8 because “it’s the closest one so it’s the best answer.” General consensus among the vocal. They wait for me to move us on to the next thing, I wait for them to tell me they’re done.

Some whispering. One or two students have noticed choice (D) -25/3. Three students turn to me separately and mouth, “Isn’t it D?” I deadpan. A minute goes by. Three minutes go by. Finally, my quietest student makes a noise I can’t make out. The student at the board hears her and repeats out loud, “Is it D?” Now they all know they’re done.

Questions:

1. What am I going to do next year to build number sense?
2. What am I going to do with my underclassmen this year to build number sense?
3. Most importantly, all year I’ve been deliberately emphasizing the importance of putting ideas and objections out for the group, but somehow I must be unknowingly communicating something different, because they are still afraid to do that.

That last may be too harsh. I think they feel safer questioning each other on the calculus than on fractions – if you reveal yourself as confused on calculus, hey join the club, but if you’re confused on fractions, that’s embarrassing. Hmm.

One more note – as I moved us on to the next thing, I definitely articulated my impatience, not with their not recognizing -25/3, but that three people had a different idea and didn’t put it out there. I wonder if showing my frustration when they don’t take a risk is the best way to encourage them to take a risk. I suspect not.

Yesterday: a student tells me he solved a 3×3 system by letting x, y & z be (3, 1, 1) in the first equation, (2, 2, 1) in the second, and (1, 2, 2) in the third. It’s cool, I say, that he found solutions for each equation, but we’re looking for x, y & z that are the same in all three equations and still work. “That’s the game,” I say, as his eyes widen and he really takes in the nature of the puzzle he’s facing.

Today: the puzzle is to find the equation of a parabola that passes through three given points. The aim is for them to discover a strategy. The minutes tick by. Many students, not knowing how to do this, have given up. I call them together, we go through finding the equation of a line. “So, how are we going to find b?” “Plug in x and y,” says J. “Good. Does that help with our problem?”

I tour the groups. It has not helped. Energy in the room: zero. Less than 10 minutes to go. Call them back together. “Let’s try J’s solution on our problem. If I plug in the first point, what do I get?” Two or three voices chant out: “0 equals a times 1 squared plus b times 1 plus c”. We continue and soon there’s a 3×3 system on the board. How do we solve that? “Like we learned yesterday.” “Ok, go to it.” Energy is still zero – there isn’t a problem anymore. Assiduous students plug away. The rest wait for the bell.

In inquiry classes, what do you do when they stop looking? Leaving them stuck deflates them. Revealing the answer deflates them. What’s the right thing in between, and what do you do if you don’t have it? That’s the game.

I teach AP Calculus to students who, last year, did well in a rather limited Algebra II course* and who have not had pre-calculus. When faced with a question like (2007 AB FRQ4):

A particle moves along the x-axis with position at time t given by $x(t)=e^{-t}\sin t$ for $0\le t\le 2\pi$. Find the time t at which the particle is farthest from the left,

they get hung up on things like:

• Is $e^{-t}$ like $e^x$?
• If the particle is on the $x$-axis, then is $t$ on the $y$-axis?

They know how to differentiate the function and how to find a minimum, they just don’t know that this is what they need to do.

* The Algebra II course I am teaching this year is similarly limited – more on that another time.

PS A thousand blessings on whoever decided to put $\LaTeX$ into wordpress.

PPS Anyone know how I can get the inline formulas more inline?