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.
Without further ado, it reads:
Then I ask part (b)…
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 
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…
Continuous Everywhere but Differentiable Nowhere 
Yay student thinking, confidence, stuff. Okay I really only want to ask: how did you make the scratch off part??
http://www.easyscratchoffs.com/red-scratch-off-stickers.html
I love this question! And not just for the scratch off (although that is AWESOME). Questions that really make the kids think about what they’ve learned are what it’s all about. Limits problems can so easily get bogged down with algebraic manipulations, it’s great how you brought it back to basics in a way that highlights the relationship (or lack thereof) between limits and continuity. Yay for all the kids who drew pictures! Hope you don’t mind if I steal this for my calc class….
Obvi! Beg, borrow, steal. A teacher’s motto.
(Also, why we put stuff online!)
Nicely done. The scratch-off is something truly unique I haven’t seen before, and students will get the problem right for the right reasons, and wrong for the right reasons.
A couple of quick caviats…
– The problem says it’s a “table of limits”, but really it’s the input-output table of a function.
– The problem asks to find the limit of f(x), but the table gives “x” and “y” values. The table should should maybe just have labeled rows x and f(x) then the direct data.
– Is there a reason you have the kids say “DNE” and not “does not exist”? Just curious!
Good stuff and thanks! My favorite test problem was a bonus question: “What is the probability that you get this question right?”
Thanks for pointing things out. I’ll fix them on my electronic copy.
As for DNE, it’s just something everyone in my school does. I think I did it too when I was in high school, but I’m not 100% sure.
I really like these types of questions as well. I teach 9th graders, so often I hear “That’s a trick question!”, when really it’s a simple matter of, as you said, being confident in your understanding. If you really understand it, then the answer is pretty straight forward. Keep on being awesome Sam!
Oooh! If you have a bunch of good questions, would you share?
Very nice question. Just not sure why the table is described as “a table of limits”? Shouldn’t that read “a table of values”?
I like the spirit of this problem and what you’re getting at (and I LOVE the scratch off, thanks for that idea), but I think we always have to be careful about making it right for students to automatically assume that patterns continue in math, which just adds fuel to many of their assumptions that say six data points or examples constitutes a proof. I only give pattern problems with a caveat, “assuming the pattern continues,” but of course, your pattern does not continue.
So, you are asking students to make assumptions about x=1.9999999 and 2.000001, but not about x=2. After all, you could have x=1.99999999, and y=-456.2345, there’s no reason why not.
Now a REALLY good answer to the question would address case 1 that the pattern continues for all but x=2, and then case 2, where it doesn’t, and does not exist could be the right answer (as could 2.5 or any other number……)
We talk about how a table of values don’t give you 100% confidence in your answer for a limit… But you want to use the evidence you’ve been given and come up with the MOST sensible answer WITH WHAT YOU HAVE.
But I agree with you — a truly wonderful answer would say “I have very good confidence that the answer is 2, and this is why… however it could in theory be something else because…”
You could do parts a, b, and c to tease that out. Part a as is, w/the wording changes others have suggested. Part b could say that assuming that the pattern you saw in part a continues for all points except x=2, what is the limit, which is the conceptual question you were going for, and then part c (maybe extra credit, maybe not) could be more open ended and say given only the information in parts a and b, what values are possible for the limit, explain. Part c could also be a very nice class discussion topic that gets into a lot of thinking about infinity and small epsilons (even if you’re not doing epsilon delta proofs)
Yes, that might make sense to do! I would have had to have more class discussion about these matters to feel comfortable about putting that part (c) on there, but I might next year.
Might be fun just to ask them in class to see what they say….. not for testing purposes, but I’d bet you’d get a lot of information about their thinking, and could imagine some debate happening……
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I love reading your written comments because I can picture you in your apartment shouting YES and NO at your students’ quizzes…. I say this because I do this and feel like a crazy person. Now I feel a little less crazy.
My favorite thing is that earlier, on a few kids’ tests, I wrote “I just threw up a little bit in my mouth.” Of course, I was only joking (I try not to shut down my kids with judgment… but I felt I had a rapport with them)…
Sometimes I write a lot, sometimes a little. I’ve gotten to writing less, lately, because I want kids to figure out what they’re doing wrong independently.
One of the reasons I find value in your posting the student work is that when we plan future lessons on this topic we have student work to refer to so that we can anticipate student responses and misconceptions. It’s a great deal easier to plan questions in advance that will help students come to see their misunderstandings on their own than it is to come up with a response “in the moment.” In the moment it’s too easy just to “explain” the misunderstanding but I don’t think that’s as effective as when they come to the realization on their own. So, thanks!
Stolen.
I’m thinking I am going to steal this for my upcoming exam for my college Calc. I classes. I’ll have to blog about the experiences when all is said and done. Thanks for the idea!
Thank you for sharing such an excellent question. I used this question on my second AP Calculus AB quiz covering limits, and found that for many students, despite discussing the approach vs at-the-value issue, they still had conceptual confusion. Or possessed a lack of confidence concerning their answer to part (a). It was amazing how many students could correctly address part (a), but change their minds for part (b).
Thank you again!