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.

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 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…


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.


“Sticky” Notes

This past week, I attended a less-than-inspiring AP conference for AP Calc, as I am teaching the course for the first time come September. Though some parts were helpful, the presenter spent almost all of the 8 hours every day just lecturing about Calculus and going through mediocre worksheets with us. He was a perfectly warm and friendly guy, but he was also sloppy, disorganized and often slightly incorrect, not to mention not creative at all. I was pretty disappointed. [Disclaimer: People have given me far better reviews about AP conferences in the past… I think it depends on the presenter organizing].

But, while watching the Calculus curriculum being presented methodically on the board (without any distractions because my wireless wasn’t working), I was struck by how confusing it must be to stare up at a mess of disorganized mathematical notation. I decided to brainstorm ways to improve the taking-notes-from-the-board aspect of my own course – to make my notes more “sticky” in my students mind and to make them more useful for the problem solving. We can all inspire some day to have a completely student centered, inquiry based, problem solving classroom, but even in those there is certainly room for (and a need for) teacher directed instruction… and that can always get better too.

Inspired by Square Root of Negative One Teach Math’s loop to convert logs to exponents to logs and Sam’s Riemann Sum setup, I tried to think of ways to use visual ways to connect conceptual math with notation (which is probably the biggest hangup with my students), to basically create a sort of intermediate form to help make the abstraction make more sense. Here are a few ideas I had… keep in mind I haven’t tried any of these with my students.

1. A Beefier Number Line for Graph Sketching

Problem: One of the things I noticed this past year is that my students would dutifully make number lines to test the derivatives but would sometimes totally forget what they were doing in the process. Also, many would mix up the first and second derivative.
Solution: Have the students immediately interpret their results with visual indications of increasing/decreasing and concave up/concave down. Make the separations on the second derivative number line be double lines instead of one to reflect the double prime part of the second derivative notation.

2. A Point-Slope Picture for Point-Slope Form

Problem: Anytime there are multi-step problems, many students either try to memorize algorithms or get completely overwhelmed calculating one thing that they lose other parts in their work.
Solution: Draw a picture of a tangent line and let the point be the O in POINT and the line be part of the L in slope. Then, finding these two items gets you everything you need to find the tangent line. Maybe arranging them vertically and carrying the final part of each step out to the side might keep students more organized. The bonus is that this is a picture that fits with the math and not just a forced acronym.

3. Enhancing Volume Integrals With Pictures of Cross Sections

Problem: The hardest part of figuring out the volume of solids is setting up the integral. Students have trouble figuring out what area equation to integrate and then which variable to use when integrating (i.e. which way to go).
Solution: Draw the cross-section near the solid and an arrow in the direction in which you are accumulating cross sections (or on the problem words if you skip the picture). Then draw the same shape next to the integral sign and an arrow. Inside the shape of the integral write the area equation as you would see it in geometry, and above the arrow write a d-whichever-way-the-other-arrow-goes. Then replace the area equation with something else that is in terms of the whatever in d-whatever. Works for the disk and washer methods in volumes of revolution too.

Okay, so maybe those aren’t all THAT helpful, but I personally prefer thinking about small changes when I have so much on my mind about the school year. Though these are obviously not replacements for deeper understanding, maybe they could be crutches to help students go from something that might make sense to them to the abstraction of notation. Main point: I’m going to pledge to sit down and try to think about how to make notes more “sticky” before every unit.

from @bowmanimal

Make it Better: Memory Modeling

“A monk weighing 170 lbs begins a fast to protest a war. His weight after t days is given by W = 170e^(-0.008t). When the war ends 20 days later, how much does the monk weigh? At what rate is the monk losing weight after 20 days (before any food is consumed)?” <– That’s an actual problem from our Calculus book, which I find very amusing. Though it doesn’t really fit Dan Meyer’s definition of psuedocontext, I just get a kick out my mental picture of a monk sitting in a dark room taking a break from protesting the war to scribble away on a notepad trying to make predictions with an exponential model… There are so many word problems that force “real-life” situations into the convenient framework of whatever math topic is being presented in that section. I guess these are supposed to demonstrate to students how useful and relevant math is, but I think we all know that students just find them to be tricky and unyielding disguises to math that they generally know how to do.

There was one word problem that fit an exponential decay model to someone forgetting information, so I decided that instead of just doing the word problem, we would test the model by recreating the experiment. The day after we had a midterm exam, instead of handing back their corrected test, I put them in groups and gave them the following list of 50 three-letter syllables that I generated with a random number generator:


I gave them 15 minutes to memorize as many as they could and then tested them by having them write down all that they remembered. Then, I handed out the midterms and we started going over them. About 5 minutes later, I had them write down as many of the syllables as they could again. Then, we went over a few problems on the midterm… then another memory test…. then more midterm… then another memory test. They had absolutely no idea why we were doing this, so each time they groaned and complained. And they groaned even more when I opened class the next day with another trial. And then again two days after that… And then a last time a week and a half later. All without studying the list after the original 15 minutes.

Finally, I revealed the purpose of the whole experiment. We collected data and used GeoGebra to fit various models to their data. There were four different mathematical models to choose from that I found from various psychological studies (which I had loaded into a GeoGebra file with sliders so that they could move the various models around to fit their data). Each student picked the one that they thought fit their data best (a function to calculate how many words they would remember over time), took the derivative of that to calculate their “forgetting function” (a function that tells them how fast they are forgetting words at any given time), and then used both to calculate how many words they will remember in a few weeks and how fast they will be forgetting them at the point.

We graphed all of their functions on the same axes (y-axis = number of words remembered, x-axis = time in hours) to analyze which model was best and analyze how their memories compared to their classmates. The results are below. The different colors correspond to the model that each student chose.

        CLASS 1 –

        CLASS 2 –

Now, the clean final result of that graph hides how messy the model fitting part was. Though some students’ data fit well, some didn’t, at all, which was actually really nice. They really struggled trying to fit the model and hopefully realized that a lot of these models that we are dealing with in cooked textbook problems aren’t as powerful as they purport to be. If I could do it again, I would have them use more mathematically sound ways of fitting the models than just eyeballing it (I hadn’t really considered this and realize now that, though it would be an investment in time, it would make the whole thing much better).

But besides doing some authentic math that was individually tailored to each student, my favorite part of the experiment was the followup meta-cognitive discussion. We ended up having a really great conversation on how best to memorize these random things, which then led to a great discussion about how to learn and study best (especially how you should go about studying math). We talked about how some people put the words in context by using a story, some people made patterns by grouping similar items together, and the ones that didn’t do very well talked about how they just tried to memorize these random unconnected things by rote memorization. Many also noticed that throughout the closely connected trials on the first day, their number memorized actually went up, so we talked about how assessment can actually help you learn something too (in addition, of course, to regular practice).

Make it Better.

I have one simple question this time: the thing that I really didn’t like about this experiment was that it was entirely teacher centered. They were in the dark about what was going on (for experimental purposes) until the day that we collected data, fit models and did some quick calculations. How can I make this more student-centered and add elements of inquiry? I have a few ideas, but I wanted to see what other people thought.


  1. Word document with list of random words
  2. Excel spreadsheet for collecting data
  3. GeoGebra file with various forgetting models, ready to drop data in
from @bowmanimal 

Fundamental Theorem of Calculus

This is my 500th post. I started writing something off the cuff about where I was and how far I’ve come as a teacher since I started blogging (which is when I started teaching). But then I realized that in the past two years (I’ve been teaching for four), I’ve stagnated in my evolution, and I got all depressed and wrote something that would probably resonate some of you, but also that would elicit pity comments. And I those depress me more.

So instead (DEFLECTION ALERT!), I thought I’d post something I came up with last year to deal with the Fundamental Theorem of Calculus, Part I.

Although it is easy enough to tell kids “this is what is says, this is how you apply it” (and they can do it), what I have always had a problem with is explaining: this is what the FTC PT I means, and this is why it works. [1] The reason? It’s freakin’ scary! There are lots of variables being thrown around… I thought I posted it last year after writing it, but apparently I did not.

I came up with a guided worksheet that breaks down the ideas into individual pieces, and helps students work through it:

Excuse the fact that it keeps on referring to FTC Part II… I always conflate which is which.

Regardless, I was pleased at how much better my kids last year understood the theorem. They understood the idea of the dummy variable. And that the integral was simply giving an accumulated area from some starting value to some indefinite, variable value in the future.

I’m hopefully going to start this tomorrow, so keep your fingers crossed.

[1] The cop out way is to only explain: “it’s the derivative of an integral, and since they undo each other, you’re left with the original function.” I feel doing so elides the mechanics of what’s going on. It’s a surface-y (and useful to some degree) way to think about it, but it lacks depth.

Disp “Riemann Sums” — Programming the TI-83/84

I just finished teaching Riemann Sums, using the patented Shah Technique. I’ve always had my kids enter a program in their calculator which automatically does Left Handed and Right Handed Riemann Sums (actually it also can do midpoint!). And last year we used this program to estimate how the number of rectangles was related to the error to the true area. (That came out of me just playing around.)

The program we enter is here:

(If you want to use this, this is what you need to know. If you want the Riemann Sum of 20 left handed rectangles of y=\sqrt{x} from [2, 14], you enter A=2, B=14, N=20, and R=0. If you want right handed rectangles, you enter R=1.)

This year I decided to not go into the whole error thing like I did last year. This year I wanted students to really and more fully understand how the program worked. I always explained it, but I never really was convinced that they got it. Me up there lecturing how the program worked wasn’t really effective. So I whipped up this worksheet.

I tried to do less talking and them do more thinking (in pairs). I felt like there were a number of students who had this “OMG!” and “this is crazy” moments. Some were awed that the program worked and it gave them the answers we had been calculating by hand. Some had this amazing moment when they figured out what the variable S stood for — and how it actually calculated the Riemann Sum. And my favorite was when a couple students figured out how the R variable worked — and why R=0 gave left handed rectangles, and R=1 gave right handed rectangles.

I really enjoyed this. I think the worksheet could be tweaked to be clearer, but it’s something I see myself doing again. Well, I guess I will be doing it again tomorrow with the other calculus class. But I mean: next year.

Calculus Mottos

There is one motto I like to bring up in calculus:

Take what you don’t know, and turn it into what you do know.

We also have a few extra mottos floating around:

Calculus above all.

Don’t be a hero.

(The last one refers to doing all the calculation in a problem at once, in one line of work. Too much room for horrible horrible error, they aren’t communicating their ideas, and a million other problems.)

On their midterm, I gave two easy bonus questions. I do this as a way to give a little “error correction” to my grading… With midterms, I have to grade a lot of long tests with a lot of questions, and I suspect that I probably make a small error on one or two of the tests. I figure giving a little bonus boost to everyone helps me feel better about any inadvertent errors.

This year the questions were:

A. Make up a new calculus motto and give examples of using it in this course.
B. Draw a picture of Mr. Shah. Make it snazzy.

I loved these bonuses. Reading them after grading the midterm (which kids did awesome on, b.t.dubs) was hilarious.

I am going to share the mottos with you:

  • Calculus is not just math, it’s a way of life.
  • Use yourself, then some others. After that, calculus won’t be a bother.
  • Practice makes perfect!
  • Take what your friends know, and turn it into what you know!
  • “Crazy for Calculus!”
  • Calculus: I find its commentary insightful and would like to subscribe to its newsletter.
  • Calculus: I promise it’s not as hard as you think it is
  • D.A.N.C.E. — Derive Awesome Nifty Calculus Equations
  • Build the basement before you shingle the roof.
  • Don’t forget the old skills from 1st semester or you won’t be able to register.
  • Calculus is life, life is calculus.
  • History repeats itself: Even if you won’t be tested on a skill, the concepts will come up later on.
  • Don’t freak out.
  • Break it down to its most basic form, to understand it.
  • Go to your limits, reach for infinity, and always aim for the other side of the curve!
  • Write it right and you’ll get it right.
  • “Zero is okay when you’re at the top, but terrible when you are at the bottom.” [re: dividing by zero]
  • (Almost) anything can be simplified in calculus.
  • Calculus above all, and even above senioritis!

Loves me some mottos. Their descriptions of them using it were priceless and spot on. Wondering if I should be ordering some new buttons soon? #methinkso!