U-shaped Development:

Sometimes, as we learn new skills, our performance drops before it rises again. Intuitively, this makes sense of sense, especially in the context of sports. Let’s say you intuitively learned how to shoot free-throws a certain way, and over time you were able to get up to 70% of them in. Unfortunately, you end up stuck at 70% accuracy for years. A friend comes along and shows you a better technique. At first, when you try his method, you can barely get in 25%. Your tempted to give up, because it seems like this method is never going to work. Your friend encourages you to keep trying. After a few weeks of putting in the practice, you end up getting in close to 90% in.

This kind of development shows up in children as well. Children’s performance balancing blocks declines as they start to develop theories of how objects balance. Children’s use of language declines as they start to develop generalizations about the structure of language. Sure enough, even through their performance dips in the short run, it comes back better than before as they develop more general, more powerful strategies for understanding the world and the languages they are immersed in.

Physics Makes You Worse at Physics?

The same kind of thing might be said about students’ performance in physics. Learning formal physics concepts and problem solving strategies can make students’ performance decline, at least in the short run. One example of this comes from Andrew Heckler’s work that I’ve written about before. Students who are prompted to use force diagrams do worse than students’ who aren’t. This can be accounted for because students who draw force diagram are more likely to use formal strategies (which they are novice in carrying out), whereas students who aren’t prompted are more likely to use intuitive strategies (in which they are more experienced). Another account is that students who are prompted to use force diagrams stop actively monitoring for whether what they are doing makes sense, because the method itself doesn’t really make sense to them. That is, force diagrams, are not a sense-making tool. They are just something your physics teacher asks you to do. Doing problems without trying to make sense of what your doing is a recipe for not being successful.

Some Recent Work by an Undergrad

Recently, an undergraduate student working with me asked physics students and non-physics students the following question:

Two cars are located 200 miles apart from each other on a long straight road. Both cars start driving toward each other. One car drives at 50 mph, while the other car drives at 30 mph. How long will it take for the cars to meet along the road?

The results: About 57% of the physics students we surveyed get the answer correct, while about 90% of non-physics majors surveyed get it right. Our numbers are small (about 25 per group), but that difference is quite large, and statistically significant enough for exploratory work (p < .05).

Why do Students do Worse?

Roughly, there are three categories of physics students who get the question wrong.

Category #1:  Students pursue an equations-based approach, and somewhere along the way they either give up or do something erroneous.

Category #2: Students explicitly state that the problem cannot be solved without kinematics equations, which they don’t have memorized.

Category #3:  Students take an intuitive approach that gives the wrong answer.

So what about non-physics students? THey almost always take an intuitive approach. 90% of the time they get the answer correct, and 10% they get answer incorrect. So, what are the intuitive approaches:

Intuitive Approaches that lead to high level of Success

Draw a sketch and use to figure out where the cars are hour by hour.

Consider the combined speed they have as 80 mph covered.

Set up an “algebra” problem (30x + 50x = 200)

Intuitive Approaches that lead to low levels of Success:

Calculate how long it would take each to travel 200 miles and then do “something”–sometimes average the times, but more often subtract the times,

Try to construct a ratio where the ratio of speeds equal some other ratio

Now, one might be tempted to think that non-physics majors do better because they use intuitive approaches, and that most of them use intuitive approaches that lead to high levels of success. But that isn’t entirely true. Many of the non-physics majors begin with an intuitive approach that doesn’t get them the right answer. But then something leads them to solve the problem another way, or to cross out that answer and try again. Some students explicitly write about how their first method was invalid and how it didn’t make sense. They end up with the right answer, because they are actively checking for whether their answer and their approach makes sense. They try some new way. Others students solve the problem multiples ways to check and see if they get the same answer. Almost none of the physics students try to solve the problem multiple ways, or check the reasonableness of their answer. Due to this, physics students who do take an intuitive approach are more likely to stop with that wrong answer.

Why would physics students be less likely to check for reasonableness of an approach or an answer?

We ask students to engage in problem-solving methods and techniques that don’t make sense to them. They get so used to things not making sense, that they begin to stop trying to make sense of the things they do and the answers they get. In other words, learning physics teaches them not to check for sensibility.

So What?

And and all, I want to say that there are two reasons why physics students do worse. First, they are less likely to draw on informal methods for solving problems, which are often successful. Second, when they draw on informal methods for solving the problem, they are less likely to monitor their approach and their answers for whether or not it makes sense.

Sp? Is this a case of “Normal” U-shaped Development? One argument would say, “Yes. Don’t worry that students are doing worse. In the long run, student performance will rebound. Their intuitive approaches won’t get them much farther than easy problems. As they get better at these more powerful and generalizable methods, the benefits will be seen.” There is a part of me that thinks this argument makes sense.

One counter-argument to this is the following, expressed by my friend Sam McKagan. Normal U-shaped development might be fine if students went on to take more physics courses, but it looks like we are dropping students off at the bottom of the curve.

A second counter-argument is the following. It would be OK for students to do worse if they were just making mistakes in implementing expert strategies. That would be like the basketball player getting worse when learning the new free-throw technique. But imagine if the basketball player started trying to the new technique with no regard for whether or not the ball went in. What if they thought the goal was to do the method, and not even check if the ball went in, or adapting the new method to improve their game. That would seem bad, right? And that’s what it looks like students are doing–they stopped aiming for the basket.

All and all, a decline in performance isn’t necessarily a bad thing. We have to look the causes of that performance decline. My hypothesis is that this particular decline in performance is less about healthy/normal U-shaped development than it is about the effect that instruction has on students’ sense of the game they are trying to play. In grand picture, I’d like to collect more data and to continue the arguments going.





6 thoughts on “

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  1. U-shaped development is really important for teaching, as well. I think people who are really comfortable, competent lecturers are like your 70% free-throw shooters. Sadly many professors try something new once, fail miserably, and conclude (completely unreasonably) that the new technique can’t work and/or better performance is impossible.

  2. At first I wasn’t liking Sam’s point — because it seems to me that one could use it to say “when students think for themselves, they initially do worse than if they just implement a set of rules… so if they aren’t going further in science, they should then just be taught rules. The ‘think for yourself’ should be reserved for the special few who are going on in science.” It’s this kind of claim that makes people okay with crappy formulaic science in K-12.

    But I can see the other point to her claim — which is to say “traditional instruction — which is, employing a set of rules — drops students off at the bottom of the U-shaped curve.” (That is, are they trying to “own” the rules or are they just bad at employing those rules?)

    I like talking about U-shaped development because I want a reason to say that when my students start making “mistakes” that you just don’t see in a traditional class, this is a sign of *real* learning. I first saw U-shaped curves in Karmiloff-Smith and representational redescription and I loved it. – So I can used U-shaped to say that they’re doing something unique, powerful, and ultimately BETTER than students who show no such dip.

    But it also says to me: performance alone isn’t a measure of learning. I think the dip you see *isn’t* representational redescription, where students have been employing rules rotely and are now making mistakes as they start to “own” it — but making that claim seems hard. Though I think your 3 categories help to make that claim.

    I really really love this study and data!!

  3. Yep — this is where I’m at with my teaching. The good news: I continue to aim for the basket. The bad news: my (recognition of my) failure to hit it is making me crazy and exhausted.

    Back to students: in an inquiry-based physics class, it seems like we are asking them to abandon their intuitive approaches to physics and their intuitive approaches to learning as well. They are likely to get worse at both. Does this make it a “double-U curve”?

    I had an interesting conversation the other day about the “grief” students sometimes experience when expected to transition from memorization to self-direction. My interlocutor enthused that we’re killing off their old self to allow a new, more metacognitive self to be born. And yes, that’s exactly what I’m afraid of.

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