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.

Dan it turns out that you are a great writer.

I have some thoughts for you but I can’t write them down now. Email me or harrass me via the internet if I don’t get back to you soon.

> Dan it turns out …

Thanks!

> I have some thoughts for you but I can’t write them down now ….

Didn’t Fermat say something similar?

[…] now he’s back, and I feel that my initial impression last summer, while quite positive, was […]

Here by recommendation.

I’ve got one class that feels like this some days. I’ve been thinking about how I did (or didn’t) establish with them that the class needs something from them – like there still just waiting for me to tell them.

Dan, this is a first time visiting your blog and so far – wow!

My thoughts on this: inquiry is not all the same. Some inquiry is solved through one insight, other inquiry need several. What you had on your hands was a single-step inquiry, and it’s no wonder that when you do give some help it’s all or nothing.

However even in this case I would have liked to give support a little sooner, and a little more than you did. I would have gathered their attention to me and the board, and, like you, developed “0 equals a times 1 squared plus b times 1 plus c”. OK, but from there I would rather have prodded them to consider what this is – an equation with three variables – and that they don’t have enough information to solve it. “Maybe there is more information somewhere? What will that information look like and what will you do with it? Predict! Then do it.”

It’s still lame, I know, but it does leave some inquiry to the students.

Another option would be to just have another task prepared after they finished the first one. “Work backwards – create your own 3×3 system and from there figure out what the three points are”. That kind of thinking on my feet sometimes works and sometimes flops but is always fun and exciting and the students can feel it.

One of the promised thoughts:

The parabola challenge was too hard. Solving a 3×3 system was still a new skill, enough so that a student yesterday didn’t understand you had to satisfy all three equations simultaneously. This is evidence that that skill is still half-way formed. Not until it’s all-the-way formed does it take its place in their utility belt in such a way that they’ll be able to call upon it flexibly in a different-looking context.

Also, the idea that the coefficients of a parabola can be

variablesis its own leap in abstraction. Normally x is the one you think of as the variable; but solving the system to find the parabola that fits the points demands you think of the coefficients that way. This is a hard shift. You supported that shift with the finding-the-line example, but I’m just pointing out the hill they had to climb. Too much at once, it turned out.Brainstorming what to do about this:

If you recognize that the challenge is inappropriately hard during class, my thought is, change gears but leave the problem open: “This is an awesome problem; let’s let it simmer on your back burners while we work on something else. But if you think of a strategy before we come back to it, then let us know.” Have them turn to a problem that will help consolidate the simultaneous-equations skills.

In terms of planning, I have two thoughts: (a) when you’re thinking about the difficulty of the challenges you’re posing, bear in mind that new skills are not going to get deployed flexibly, and (b) hedge against unanticipated difficulties by bringing in more than one inquiry challenge. This is also a hedge against some folks getting done before others (you have something else awesome for them to do while the others finish).

I’ve taken a lot of inspiration on this latter tip from the PCMI problem sets. (The intended audience is high school teachers.) They do a really beautiful job of bringing in a spectrum of challenges, so that everyone will be able to handle at least some of the work and everyone will be deeply challenged by at least some of the work. (I’ve totally appropriated the framework of coming to class every day with at least one problem that’s “important,” one that’s “interesting,” and one that’s “tough.” Of course this involves an egregious amount of planning esp. if you teach multiple preps daily, but it’s something to aspire to.)

[…] Ben’s right: the problems are too hard at this stage. But not grasping the nature of the puzzle being posed is only half the reason. The other half I want them to keep looking, but “looking” at this stage usually means rewriting things many different ways without knowing in advance how it’s going to work out, and they are unfamiliar with that whole mode of exploration. […]

Thanks, everyone, for stopping by to read.

@Marty – Word, bro.

@Julia – Yep. I tend to get in the trap of feeling like “they’re not getting it, so they should keep trying” instead of “they’re not getting it, so how can I adjust the problem into one they might take a shot at?”

@Ben – Aspirational for sure. Right now it’s “here’s today’s problem sequence” and I adjust how much direction I do/don’t give. I like the concept of bringing a menu of problems that have a wider range – I’d love an example that distinguishes interesting from important from tough for, say, the introductory few days of logarithms. (And, P.S.: thanks for the support of this blog!)