# The Mysteries of Students’ Thinking Processes

A busy week of writing letters for advisees, writing a letter of rec for a former colleague, and pulling weekend dorm duty. Back on duty again tonight, so it is three out of four nights now!

Last week was the first time in quite a while that I found myself largely disappointed by my students and I have a couple of questions I want to air out. Trying to understand what students understand through assessment is, of course, one of our big challenges as teachers. People much smarter than I am have been hashing this out for a long, long time. So, I have two stories to share that are each nagging at me.

In AP Stats we are wrestling with probability. Most of my students have had very little, if any, exposure to probability before this class so this tends to be a tough unit. We had a problem on our last quiz that went like this:

Mr. Felps has 28 students in his AP Calculus BC class and 8 of them are left handed. We know that approximately 10% of the population is left handed. Can this situation in Mr. Felps’ class happen by chance?

A number of my students felt that this could not happen by chance. It seemed too unlikely to them. This bothered me a bit since we had looked at some simulations and talked about runs of short duration. We had discussed the law of large numbers and looked at a decent EXCEL simulation. I thought I had covered our bases on this one. But what really flustered me was that the follow up question asked for the probability of 8 out of 28 left handers under this condition. Every one of my students attempted this computation. Almost all got it right. BUT – a number who got it right had just told me that it was impossible for this to happen by chance. Somehow in the span of two minutes they seemed to forget that it was impossible and instead gave me the small percentage chance of it happening. What happens? Why do such good students have these kind of hiccups, especially in assessment situations? Man, it feels as if this is THE golden treasure to find as a teacher. How can we help our students step back and be metacognitive enough to sidestep these mistakes?

The second situation involves my Calc BC crew. We had a test last week and I try not to have too few questions on these tests so that each question does not feel so overwhelmingly significant. i have settled on feeling comfy with 7 questions in a 45 minute or 50 minute class test. Our recent unit on arc lengths and surface areas involve some problems that take a bit of time. To compensate for this while still having 7 questions I threw in what I thought was a gift wrapped set of points. Here is the question I tossed in as a softball for them.

I realize that if I increase my cycling speed by 3 MPH it will take me 40 seconds less time to cover each mile. What is my original speed?

I had students who left this problem completely blank. AP Calculus BC students who were so stymied by this that they did not even write an equation relating the information presented to them. I’ve been wrestling with this for days on a number of levels. It feels like this was an easy gift to them, one that my competent Alg II kids can easily solve. However, this was clearly not the way the problem was received by my students. They felt tricked or ambushed. They feel like it is unfair to lose points on a Calculus test on a problem that does not feel like it has anything to do with Calculus. I sort of sympathize on some level, but I feel that it is absolutely essential for these kids – kids who want to pursue serious, high powered technical degrees and futures – to be able to synthesize and recall old ideas with ease. Man, I am frustrated by this one. I felt I was tossing them a bone and it got stuck in their throats.

I have so much thinking to do (still!) about assessments and understanding what my kids understand.

## 8 thoughts on “The Mysteries of Students’ Thinking Processes”

1. Mr. D-
So sympathetic to this, especially the BC calculus question issue. I think we’ve done such a good job training students to take very narrowly-defined tests on such narrow topics that the moment we step outside that box, even a bit, or even back to stuff they know, they lock up. Heck, I bet you could have asked a question like “A line passes through (5, 2) and has a slope of -3. What is y if it passes through (2, y)?” and they would have yelped.

I don’t know what the solution to this is. Everyone talks about the importance of metacognition, but we just don’t teach it. Or maybe it’s better to say that we don’t provide students the opportunities to use metacognitive techniques in a variety of settings.

1. Thanks for dropping in, Mike!
I have long had a habit of announcing things like ‘The test is through section 4.7’ to try and make explicit my intentions. Where I doubt myself is with dealing with kids like this BC crowd who are in a pretty high stakes position now with regard to college placement and they have rarely had teachers hold them to this position. I am convinced it’s the right position to hold, but I wonder about how fair it is to spring this on a student in senior year who has been trained to succeed in a different way. My department is making moves – slower than I really wish – in this direction but it is still a shock to the system for my students in these classes. I feel differently in my Geometry class. They haven’t been trained quite as long in the process of learning small chucks and then dispensing with them.

2. So, I will do my best to keep this reply as an actual comment, rather than a full-blown blog post reply!

I see this with my college students at practically all levels. Heck, I even see this with myself! But less so with mathematics and more so with something else that is relatively new (like piano). I think the narrow focus is natural. There is this famous video of “how many times did the team in white pass the ball”. It is awareness test about aimed at raising awareness for bicyclers on the road. Here is a link. https://www.youtube.com/watch?v=Ahg6qcgoay4

We also do this when we speak or read. Elephants! Were you expecting to see the word “elephants”? Almost certainly not. It’s out of the *anticipated* context. And this is exactly what happens on exams and assessments. A student’s expectation for problems are Calculus-based and they are befuddled because what you asked was not Calculus. They are still maturing mathematically.

I think asking the question you asked was great! It’s a good way to shock the system and remind them that what we are doing is trying to teach them mathematics as a whole, even if the classes they are taking may feel like piecewise isolated topics.

As for the probability / stats question. I’ve had similar issues. I’ve found that it boils down to two things: a) students are still getting developing a sense of probability and b) students are getting used to the calculations. They haven’t been able to merge (a) and (b). It’s an unfortunate side effect of the pace at which we have to teach course. Getting a sense of probability takes some time and a lot of fiddling. But we are also expecting that they be able to carry out computations on something they have no intuitive sense for.

Well, this is a quarter of a blog post. I’ll stop here and perhaps finish it up on my blog. 🙂

1. And I’ll clean up all my ridiculous typos! Sorry!

2. Manan – thanks for dropping by to share your thoughts. To me the key sentence in your reply is ‘But we are also expecting that they be able to carry out computations on something they have no intuitive sense for.’ I think you nailed the issue right there. As a classroom teacher who believes, deeply believes, that this kind of past knowledge question should be fair game, how can I better develop this intuitive sense that my students can rely on. When I answer that distance question I am DEFINITELY relying on an intuitive sense about the relationship between distance rate and time. I hope that my Calc BC kids have internalized this, but if they’ve succeeded time and again through sheer computational prowess and ability to study hard in small chunks, then they have no reason to work on an intuitive sense for a problem like the bicycling one. How do we disrupt this pattern?

3. Re: BC. I’m not so sure this is about student thinking as culture setting. Once you establish the culture of anything is fair game they’ll get over the ‘not fair’ reaction, which is really ‘this is different than I’m used to.’ In so far as it is about their thinking, have you done a think aloud for them about how you write a test? Might help them study in a broader fashion, focusing on the question of how you can tell if they understand.

The probability one is fascinating and deep. The way students in a physics class still think heavier objects fall faster. I’d love to hear about the conversation when you point out the contradiction. Should be some classic cognitive dissonance!

1. John – thanks for dropping by to share your thoughts!
I have to admit I dropped the ball on the Stats question. When I returned the quizzes I pointed out the obvious problem rather than probing for their reaction to it. In my defense, I suspect that the response would have mostly been along the lines of a self-administered head slap when they saw what they had done. As a teacher, I want to be able to help that head slap reaction occur in real time so that they catch themselves at this sort of mistake. Any brilliant teacher moves for developing this internal alarm system?

1. Not sure if this would have been appropriate for an AP stats level, but what if you had a side comment (or another question) after the second one asking, “does this make you more or less confident in your answer to #1?” Maybe that would have done some head slapping for them (because I think they do look at each question separately and not as a whole test like teachers do).

I am also having the same issue in preAP Precal with recalling previous information. I asked them to find the intersection of two lines and it was like I asked them to build a rocket to the moon. Exact quote: “I didn’t start because I didn’t know what formula to use.” !!!! But also like you, I don’t know how to fix it.