After grading my exams from the first semester, I can recognize some definite highlights in my honors exams, which on the whole were the very best set of exams I’ve ever graded. I thought the exam was very challenging, and the students took the questions with great facility. I’m particularly impressed at how much they are growing in figuring out when a particular model (constant velocity, constant acceleration, etc) applies in a given situation. This is a very hard skill to master, and I don’t think I’ve ever had students who so clearly get the idea of what a model is, what the features of various models are, and how they can begin to ascertain when a model applies to a specific phenomenon. This is probably a result of me being much more explicit in sticking with the modeling curriculum and philosophy and practicing these skills.

My intro exams were a bit more of a mixed bag. My students are definitely developing a stronger ability to reason conceptually, but I’m concerned with how they’re seeing connections between conceptual and mathematical reasoning.

You can see this most clearly in two problems I gave on the exam. Here’s the first problem:

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This is a fairly straight forward data analysis question designed to see whether students can plot data, write out a linear equation to model the data, explain the meaning of the parts of the model, and be able to use the model to make a prediction. There is a slight twist, in that the units of the two things aren’t the same, and so jumping to the conclusion $C=\pi d$ would be incorrect (though most students didn’t even mention the relationship $\pi$ on this question).

Here’s a pretty typical piece of student work on this problem that exemplifies the struggle my students faced with this problem:

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For starters, you can see this student can plot data, even when it doesn’t perfectly line up on the axes, reasonably well (this is true of most of my students). Most of the struggle comes in the second question, part (b), where where he tries to wade in to the thorny world of mathematical notation. He starts by trying to calculate the slope, and right off the bat, you can see he mixes up writing out the formula for slope as $\frac{x_2-x_1}{y_2-y_1}$ rather than $\frac{y_2-y_1}{x_2-x_1}$, but if you look at the numbers, you can see that he sets up the calculation correctly. This is a clue to me that symbols in the formula don’t really have a lot of meaning for him, and are difficult to interpret, but he knows how to calculate slope in a procedural sense. You can also see that he isn’t really internalizing a deeper understanding of slope, because he didn’t start with $\frac{\textrm{rise}}{\textrm{run}}$ (do students just see this as babyish?), and he didn’t make his calculation specific to this data set with something like $\frac{C_2-C_1}{d_2-d_1}$, which is what we practice often. Furthermore, you see he isn’t ascribing a lot of meaning to the numbers themselves, since there are no units, and we talk all the time about not having naked numbers on the paper. My students know every number tells a story, and must be fully dressed with units, and a reasonable level of precision every time you write a number.

Things get a bit worse for symbolic understanding when the student tries to write an equation to describe this pattern. First, you should know my students are capable math students. They may not think this themselves, but they know slope intercept form, can recite $y=mx+b$ all day long, and even as 9th graders have a full year of algebra under their belts, and have SOHCAHTOA memorized perfectly. In fact, as I wrote earlier, I think they may be on the road to being in the top 1% of mathematical minds.

But clearly, when you look at typical response to this problem, something is a bit amiss

• Most of my students show little appreciation for the meaning of symbols. Our approach to writing out a model is to explicitly write out $y=mx+b$ first, and then ask “what is plotted along the vertical axis?” (circumference, C), what is plotted along the horizontal-axis?” (diameter, d), what is the meaning of slope?” and “what is the meaning of the vertical intercept?” I want them to make the connection between the variable typically plotted along the vertical and horizontal axis, y and x, and the measurements we are usually plotting in graphs (that are never y and x). But you can see here, I’ve failed to highlight this for this student, since there isn’t a single variable on the right hand side of the equation. This kid would never write the equation of a line as y=4 +2, so why can’t he see the flaw here. I think part of the reason is that they don’t really ascribe meaning to the symbols. To them y,x,C, d,m and cm, are all the same. Fuzzy things that live in math or physics land we don’t really care about.
• I think the other part of the reason why kids struggle with these questions is clutter. Here I plead guilty to requiring the kids to write units on every number, and I know that this can make reading an equation much harder. We also talk about why parenthesis would then be used as a grouping symbol, so you could reduce clutter and write something like $0.031\frac{\textrm{m}}{\textrm{cm}}$. All my students can say this, and seemingly claim to understand it, but they still make tons of mistakes like writing just the unit in parenthesis, or putting the variable in parenthesis, and this is only going to get worse when they see parenthesis in function notation. But as a teacher who wants them to use math to describe the real world, I think the units are essential, and I want the kids to learn to manipulate them, in exactly the same way they learn to solve those crazy exponent rule problems I had to teach like simplify $\frac{x^3y^2z^{-1}}{xy^4z}$.

I think my real worry comes in part (c) where I ask the students to explain the meaning of slope, using the words “for every.” I wrote previously about how I think “for every” is a wonderful thinking tool for kids that can be used to replace the slightly more opaque “per” and really give kids insight into the meaning of ratios of quantities and units. I think it is critical that students can look at a graph and interpret the slope as telling you how much the quantity on the vertical axis changes for every unit change in the horizontal quantity.

If you look at this student’s work, you’ll see I’ve earned a big fat “F” in teaching this idea. So what’s the problem here? Do they not see the implicit ‘1’ in the denominator of the fraction? (BTW, I’m with Joe Bower on renaming numerator and denominator, “part” and “whole” respectively—I could never keep those things straight when I was learning fractions). Or is it they don’t have enough practice interpreting slopes that represent ratios of real quantities (meters for every second, grams for every mL, etc), and are too used to slopes like 2, 1/2 and $1.\bar{3}$? Don’t even get me started on my distaste for the repeating bar, when making measurements about the real world. Are they too used to just finding slopes and not explaining what they mean?

All of these things are subtle details about how we write math with precision to communicate meaning. This is probably akin to an English teacher getting frustrated at students not seeing the difference between “Let’s eat, Grandma!” and “let’s eat Gramdma!” Small symbols have deep meanings, and using them incorrectly can obliterate your ability to communicate understanding.

Here’s the second exam question where my students struggled to use their mathematical understanding to gain physical insight into a situation.

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And here’s the same student’s work:

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Notice first the answer to (b). This student nails it. In fact, almost every one of my students nailed it. I let each student bring 1 page of notes into the exam, and almost all of them put this equation on the sheet. This shows that they are world-class at plugging and chugging. Notice also, this isn’t your basic plug and chug, students have to make a unit conversion from meters to cm in order to get the right answer. This is good, but is this how they see math?

If students have this formula so well under their belts, you’d think (a) would be a cinch, right? Not true. This is a far more difficult question, since it asks them to think carefully about the relationship the equation is describing. Not one student got this question, and most couldn’t even write down something halfway relevant (like the equation—is it possible they could not see that it would be useful here?). And this is despite the fact that we’d tried problems like this, and even discovered this relationship through modeling, always paying careful attention to visualizing the relationship using graphs.

And you can also see the trouble students have when it comes to describing the relationship with words in (c). Long ago, I thought we had started to overcome the wolverine, when we first make this graph. But you can see that mostly, all my students were left with were some stock phrases that they don’t really understand. They remember this looked like a parabola (but not that it was a sideways parabola). They remember that a parabola is $y=x^2$, but can’t tell if that really has any relevance to this picture. They remember that this isn’t a direct relationship, but can’t describe in general terms what this relationship is beyond saying “it’s a parabola.” These mistakes are typical, and many of my other favorites come up, like calling this an exponential relationship, simply because it looks curvy.

So all of this leaves me perplexed. I don’t want my students to come away thinking that the connection between math and physics is as shallow as manipulating a bunch of formulas without being about to understand the meaning of symbols and describe that meaning using their own words. I’m not satisfied with accepting the name dropping of stock terms they don’t fully understand, either. Is this too big of a goal for my students? Is there a better way to prepare them for these sorts of questions? I’m not sure. And furthermore, with intro students, I’m wondering if it is worth the struggle. These are very difficult ideas, and while I think mathematical reasoning is essential to their lives, I’m not so sure that it wouldn’t be better to tone down the math even further and get my kids on developing a conceptual mastery of physics. Given the choice between having my students be able to understand something about radiation so that they can properly assess the danger of “cell-phone radiation” or being able to really own the relationship between period and length for a pendulum, which do I prefer? Must I make the choice?

1. January 6, 2011 7:40 am

The trick here, I think, involves a few things, but one is working memory. I think that my “regular” physics students would have about the same issues and successes on both of these problems, and that not all of it is about what or how we teach them. Cognitive ability has a big dependence on working memory, and most of these students that I have always had in these classes simply can’t hold several concepts in their minds at once. They can remember the facts or even apply the concepts to different problems, but only one at a time (or two). Having to juggle competing or complementary concepts and synthesize them is the hardest thing for them, and even though we push (and should push) them towards this, a significant fraction simply can’t hold all of those ideas in their heads at once. Probably as a result of this, I’d answer your question “is this the way that my kids see math?” with a “yes,’ because that’s all of the concept flow that they can really get into their heads at once. The trap that lots of conceptual/regular/intro classes fall into is to remove lots of the symbolic manipulation (good idea) and to replace it with more complex reasoning that’s really dependent on understanding and applying the heart of the math all at once. In lots of ways, that pendulum “graph” is really really hard. Even though it’s elegant and awesome, lots of them can’t appreciate the argument, because they can’t understand the whole thing, but only pieces at a time.

• January 6, 2011 9:54 am

JG,
Thanks so much for this feedback. I think you are very right, and I need to think more about working memory and the effects is has on my student’s abilities to show understanding on assessments. I had not really thought too deeply about how I often just substitute challenging symbolic manipulation with challenging conceptual reasoning, that is dependent on mathematical understanding, logical reasoning, and synthesizing multiple ideas at once, but I think are skills that I very much want to help my students improve. I’m also working on a post about how to teach long chains of reasoning, which I think really are critical in science, and I’ve blogged a bit about before. https://quantumprogress.wordpress.com/wp-admin/edit-comments.php#comments-form

2. January 6, 2011 8:33 am

I’m really impressed with your exam questions and I think you *should* continue to both teach and expect this level of mathematical understanding. How do you ask questions like 5a during the modeling sections?

For me, whenever students and/or I develop an algebraic equation (one with symbols and not numbers), I try to remember to point to the various terms and ask questions like 5a. For lower level students my goal is to have them understand what numerator vs denominator terms do (“d” for down, “d” for denominator is a phrase you hear a lot in my classroom). I’ll ask things like “what would happen if you double the mass” or “what would you have to do to triple the force”. When I remember to do this, we spend an awful lot of time on it, so much that the numerical problem solving takes a hit. I’m not sure what the best balance is.

• January 6, 2011 10:00 am

Andy,
Both of these are good questions—I often specifically ask what happens to this when you double that in our modeling discussions, but I think part of the problem is that we started with a relationship that is hard and unfamiliar (the pendulum) and then moved away to relationships in constant velocity that seem simple, and even trivial to some students. In fact some students might see them as so trivial that they might not develop the rigorous habits of thinking through about proportional reasoning they’ll need for future relationships or be able to go back and fully reason about hard relationships they’ve seen before.

Also, I’m with you that these types of questions are the key to getting kids to understanding the relationships themselves, and move beyond plugging and chugging. As a former devotee of PSSC, with many memories of the agonies all of its proportional reasoning caused students, I don’t think I could fully let this type of thinking go, but I do need to think of ways to scaffold it better for introductory students.