# My Favorite Test Question of All Time

In Calculus, we just finished our limits unit. I gave a test. It had a great question on it, inspired by Bowman and his limit activity.

Which reads: “Scratch off the missing data. With the new information, now answer the question: What do you think the limit as x approaches 2 of the function is (and say “d.n.e.” if it does not exist)? Explain why (talk about what a limit is!).

So then they get this…

This is what I predicted. (And this was conjecture.) Almost all my kids are going to get part (a) right. I’ve done them well by that. However, with part (b), there are going to be two types of thinkers…

one kind of thinker, where they think “Mr. Shah wouldn’t give us this scratch off and this new data if the answer doesn’t change. So it has to change. What can it be? Clearly it has to be 2.5, because that’s the new information given to us. So I’m going to put 2.5 for the answer and then come up with some way to explain it, like saying since the function has a height of 2.5 when $x$ is 2, clearly it means the limit is 2.5.” (WRONG.)

the other kind of thinker, who will get the problem right and for the right reasons.

What’s the difference between the two kinds of thinkers? My guess: confidence. More than anything, this is a question that really gets at how confident kids are with the knowledge they have. You have to be pretty sure of yourself to come up with the right response, methinks.

My conjecture was pretty spot on. Let me tell you the responses are fascinating. So far my conjecture seems to be holding water. And it’s just the most intriguing thing to read the responses from students who got it wrong. The phrase that springs to mind is cognitive dissonance. There are a number of kids who are saying two totally contradictory things in their explanation, even from sentence to sentence, but they don’t recognize the contradiction. They’ll say “a limit is what y is approaching as x is approaching a number, and doesn’t have anything to do with the value of the function at the point” and then in the next sentence say “since the value of the function at x=2 is 2.5, we know the limit of the function as x approaches 2 must be 2.5.”

It’s a great question… my favorite test question of all time I think… but I wonder if that’s because of the scratch off.

I know we don’t tend to share student work often on blogs, but I asked and my kids were okay with me anonymously sharing their responses.

I don’t know exactly why I wanted to post student work. I don’t have anything specific I wanted to get out of it right now. But I know I was fascinated by it, and I figured y’all would be too.

But for me, it’d be interesting at the most basic level to just see the different ways our kids respond to questions in other classes… Even regular, basic non-writing problems! Just to see if anyone has ways to get kids to organize their work? Or if we could find a way to examine one student response to a question and throw around ideas about how to best proceed with the kid? Or talk about how we actually write feedback and what kind of feedback we give (and why)? Just a thought… Not for now, but something to mull over…

# A Positive and Healthy Approach to Learning

A few weeks ago in Algebra II I had students fill out a series of questions… questions which was going to lead to a discussion about mathematics and intelligence. I cribbed this sheet from my friend and teacher extraordinare Bowman Dickson.

I didn’t capitalize on it immediately, but I think I can still get some good mileage out of this. The thing that brought me back to this sheet was that yesterday in Algebra II, I gave an assessment that students didn’t fare as well as thought they would.

With one section, today, I had a heart to heart with them about what I saw, and this disconnect, and I talked a lot about the difference between active learning and passive learning. I think I got through to them. And I said: take what I said to heart. Be an active learner. And I’m going to give you an assessment on the same material next week. Show yourselves that you are capable. Because I know you are — but you just need to learn and implement the right strategies to be able to do it and make a lasting change.

Or something like that.

So we’ve had the talk about the concrete things… and I think next week, after kids take the reassessment (and hopefully — HOPEFULLY! — do much better), it would be worth it to have a talk about the more abstract side of things… attitude. The way students approach math, think about math, think about intelligence.

I’d love for any ideas about how to structure/have this discussion. I’ll throw my class data below, but if I’m going to do this, I want whatever I plan to be as powerful as possible. I want it to really get kids to think about what learning is, and how important having a growth mindset is. I have a few thoughts, but nothing great. So any brainstorming you might have, awesome.

As linked to on Sonata Mathematique:

Yeah, I want that DOUBLE POSTER SIZE in my classroom.

Without further ado, here’s my (fascinating) class data…

ALL DATA COMPILED

DATA REDUCED TO AGREE/DISAGREE

# The Messiness of Trying Something New

It’s now more than halfway through the first quarter, and things are … messy.

I’m pretty much going through Calculus like I did last year, except for the fact that everything is so much easier because I have standards based grading down. [1] I know what works. While Calculus was hell for me first quarter last year, it’s cake for me now. So calculus is not messy. [2]

So while Calculus is going smoothly, I’m finding Algebra II to be messy. Not in terms of my kids. I love my Algebra II classes. But like last year — when I vowed to really focus on Calculus and leave my other courses well-enough alone — this year I vowed to focus on Algebra II and leave my other courses alone.

Specifically, I’m working on two major things: making groups and groupwork a norm, and having problem solving be a regular (and non-special) part of the curriculum. (As you can guess, the two go hand-in-hand.)

I haven’t written much about my inclusion of problem solving into the curriculum, but right now we’re doing a day of problem solving before each unit (related to the unit), I have slowly started including problem-solving problems in our home enjoyment (our supremely corny term for homework), I have been putting simple problem-solving problems on each assessment, and we so far have had a single problem set (something which I may or may not continue with). Still, I should be clear that most of my curriculum and my classes are traditional.

Now, if you’re a teacher who teaches more traditionally and uses a standard curriculum, you know that this a huge change. Because there’s a huge activation energy involved in switching teaching modes. For me, I kept on saying “next year, next year” and I never did. It’s daunting! And why screw around with something that works well?

And if you’re a teacher who teaches with lots of groupwork, and uses problem solving regularly, you probably remember the year you went through the transition. And how it got easier each subsequent year, as you picked up more tricks of the trade. Tacit knowledge.

And if you’re not a teacher, what the heck are you doing reading this blog? Seriously?!? GET OUTTA HERE!

Switching to this mode has played havoc with my emotions. You see, it’s not healthy and I try to avoid it, but my self-worth is tied up with how well I think I’m doing in the classroom. When I feel like I’m doing things well, I walk around like I own the world. I have confidence. My head is held high. And when I feel like I’m doing a poor job, my head hangs low. I question my desire to teach. I wonder what I’m doing in the classroom. And I’m depressed.

This year, I’m playing emotional ping-pong.

There are times when I feel like I’m killing it in Algebra II. These are usually days before each unit, where we spend the entire period working in groups and problem solving. I love watching kids think and discuss, and they’ve gotten how to work well in groups down. I’ve never had it work so seemlessly. It’s amazing. They’re independent. They’re identifying their own misconceptions and fixing them. I leave these classes wondering why it took me so long as a teacher to get to this point… I feel like my kids are finally and truly grappling, and I love that. (And I’m starting to do this successfully when we’re not problem solving… I made an “exponent lab” which was just 20 “simplify this” problems… and I was seeing great things when they worked together.)

And then there are times when I feel like I’m being killed. I have classes where I want to crawl under my desk and hide. Some of these classes happen the day after kids problem solve, and they present their solutions. Kids put their work on the board, or under the document projector, and present. Or if we don’t have time, I’ll have them put their work up, and I’ll talk through it. These classes have never worked for me. It’s like pulling teeth. Kids don’t know how to present. They don’t know how to engage if they’re in the audience. It takes forever. I don’t think anyone is getting much out of these days. [3] Or there are the more frequent regular classes (where we’re not doing problem solving), and I find I’m standing at the front of the classroom the entire class, cold calling and explaining. And it’s ugh. I feel ugh. There’s no spontaneity. It’s not fun. I don’t mix things up or have different ways of introducing/practicing material to break up class.

What’s interesting is that I feel my kids think that I’m doing a crappy job. I know they — in actuality — don’t think our classtime sucks. (I had my kids anonymously answer some questions, including the what two or three adjectives would you use to describe our classtime question.)

But even though intellectually I know that my kids don’t think I’m doing a crappy job teaching, it doesn’t change the fact that I feel they think I’m doing a crappy job teaching. It’s a slight distinction, but maybe others of you out there know what I’m talking about.

So as I said changing things is messy. Because you don’t know what works yet, and what doesn’t. It’s taking a risk. It requires more work. And you feel like you’re constantly flailing and failing. And that’s not a good feeling. Here’s a recent Facebook “convo”:

I know this is sort of rambling. I’m just trying to work through some things, but I still don’t know where things are going. Which is why there isn’t a real point to this. Just a state of affairs, from an emotional vantage point. I’m not looking for sympathy or advice. I just wanted to try to get my thoughts down — and just let you know that if you’re going through a similar transition, you’re not alone.

[1] I have a list of standards I can choose from, I have good exemplars of problems for each standard, I learned how to effectively introduce it, and I know how to set it up so I don’t die with all the extra work that comes along with reassessments.

[2] But yes, there are lots of things I could do to improve it. Always, always…

[3] I’ve talked with a teacher who does a lot of group work and presentations, and she gave me some excellent suggestions (revolving around using giant whiteboard) which I’m going to take on board.

# When you get too lost in the algebra…

I was hunting for a book on my bookshelves when I got distracted and started browsing. In one book, I came across this great idea that I didn’t want to lose. So I thought I’d type it here in an attempt to remember.

One of the hard things about working with derivatives, for me, is that I can easily get caught up in the wonderful (to me, annoying to my kids) algebra. We have the chain rule, the product rule, the quotient rule, and strange and funky derivatives like the derivatives of the inverse trig functions. And I admit it. I love going overboard with these sorts of questions. There’s something really cool about being able to have an answer to a problem take up the length of a page. It looks cool, darnit! And when we get to this point in the curriculum, I often lose sight of the meaning of the derivative. The process takes precedence. And for weeks, we’re swimming (drowning?) in a sea of equations.

When I get to that point, I hope to remember to give my kids this problem:

Find the derivative of $\log(\log(\sin(x)))$. I’m confident that by the time I’m done with them, my kids will get $\frac{\cos(x)}{\sin(x)\log(\sin(x))}$.

But then I have to ask them to sketch a graph of $\log(\log(\sin(x)))$.

This great setup is on pages 64 and 65 of Ian Stewart’s Concepts of Modern Mathematics. He continues, describing what happened when he gave this problem to his class:

This caused great consternation, because it revealed that the formula didn’t make any sense. For any value of $x$, $\sin(x)$ is at most equal to 1, so $\log(\sin(x)) \leq 0$. Since logarithms of negative numbers cannot be defined, the value $\log(\log(\sin(x)))$ does not exist; the formula is a fraud.

On the other hand, the ‘derivative’ … does make sense for certain values of $x$

Some people might enjoy living in a world where one can take a function which does not exist, differentiate it, and end up with one that does exist. I am not one of them.

There’s a great moral here, about remembering that taking the derivative of a function means something. Yes, you can talk about composition of functions and domains and ranges and all that stuff, but that’s not the enduring understanding I would pull from this. It is: divorcing calculus from meaning and focusing on routine procedures is a dangerous road to travel — so one must always be vigilant.

It actually reminds me of one of my most favorite calculus problems, which to solve it needs one to stop focusing on procedure and start thinking. I would never give this to my calculus kids, but for the very high achieving AP Calculus BC kid, this might throw them for a loop (in a good way):

$\int_{0}^{\pi/2} \frac{dx}{1+(\tan x)^{\sqrt{2}}}$

I first saw this problem in Loren C. Larson’s Problem-Solving Through Problems (pages 32-33). I don’t quite want to share the solution in case you want to try it yourself. After the jump, I’ll throw down the answer (but not solution) so you can see if you got it right.

# How Much and How Little: Exit Slips

This year I’ve been trying to use “exit slips” at least once a week in my classes. They’re to let me know whatever I want to know. Do my kids have any questions? Did they take away what I hoped they’d take away from class? Can they actually solve a problem we were supposed to have mastered a couple days ago?

What they’ve really done for me is highlighted how little my kids learn — especially my tenth and eleventh graders — in class. And how much more time I need to build into my plans to have them practice problems and ask each other questions and, well, basically go through that time to struggle in front of me. Of course where that time will come from, I don’t know. But I need to find it.

But I’m amazed and horrified that I have never done this before. It’s the most eye-opening thing to be able to know exactly what your kids can do. And even when I felt like I did a bang up job in a class, and I thought my kids were getting it from walking around and watching them work, how there were a good number that didn’t. A good number that I wouldn’t be aware of before the summative assessment.

Anyway, I thought I’d share with you the exit slips I’ve used in my Algebra II and Calc classes. I repeat to my students that these aren’t a test in any way. They’re for ME to know what they’re understanding and what they’re not so I can better help them. And they’re for THEM to get a sense of how well they understand stuff we’ve done in class, so they know if they’re on top of the material or not. Plus I get to identify and address misconceptions, and also bad notation!