Weird examples of transfer


It always amazes me to see what sorts of things students do or don’t transfer into learning or everyday experiences. We recently finished giving our 4th and final invention activity of the year to our first year students.

What is an Invention activity, you ask?

If you’re not familiar with invention activities, the idea is that you give students a problem to solve before they’ve learned about the concept. They have to creatively invent a solution, whereby an expert solving the problem would invent/use the correct, canonical formula. In Schwartz & Martin’s (2004) paper “Inventing to Prepare for Future Learning,” they give an example of four different pitching machines that aim baseballs at a target. They present positions where each of the four machines landed a number of balls relative to a target (contrasting cases) – one of them very close together away from the target, one more scattered but centred on the target, one with a different number of balls thrown, and so on. Students must invent a way to quantitatively determine which machine is more reliable. An expert solving this would use standard deviation to describe which set of hits are closer together (ie more often hitting the same spot). Inventing a solution, even though most often an incorrect one, before learning about the concept (in this case, standard deviation) has shown large learning gains over just being taught the concept without the invention activity, even when controlling for time on task. I also recommend the short Physics Teacher paper by Day, Nakahara, and Bonn, “Teaching Standard Deviation by Building from Student Invention” (2010).

Back to the point about transfer

We use a number of these types of tasks to teach data analysis skills in our course. The main four for this discussion are on the following formulas:

  1. Least squares fitting, especially the chi-squared formula: \chi^2=\frac{1}{N}\Sigma{(y_i-f(x_i))^2}
  2. Uncertainty in the slope of a best fit line with the intercept fixed at the origin: \delta_m=\frac{1}{N} \frac{\Sigma{(y_i-f(x_i))^2}}{\Sigma{x_i^2}}
  3. Weighted Average: \bar{x_w}=\frac{\Sigma{\frac{x_i}{\delta x_i^2}}}{\Sigma{\frac{1}{\delta x_i^2}}}
  4. Weighted least squares fitting: \chi_w^2=\frac{1}{N}\Sigma{\frac{(y_i-f(x_i))^2}{\delta y_i^2}}

The idea is that the final activity (a screenshot of which is included below) combines what they’ve learned already about weighted averages and least squares fitting to get the weighted least squares fitting equation. In its simplest form, you take the unweighted chi-squared formula and add the weighting concept from the weighted average, this time using the uncertainties in the y-values as the weight.  As someone who has developed all the activities and used the formulas many, many times, it seems an almost too simple transfer task to me.

But it isn’t.

I had a number of conversations with students while they were trying to invent the formulas. They were indeed transferring their knowledge, but there were some very inappropriate examples of it. The main problems came from trying to adapt the wrong equations. One group wanted to take a weighted average of all the points. When I asked what they meant by “the points” they were as confused as I was. I pointed out that there were x and y values, and so again asked which they were going to take an average of. Then I asked them what the goal of the task was and how this average was going to help achieve it. Eventually we broke down this idea and started from scratch. It seems they saw the uncertainties in the data points and wanted to apply the one equation that they knew involved weighting by uncertainties in the data.

Another group really wanted to use the uncertainty in the slope equation, but they weren’t sure how to incorporate the uncertainties in the data points. Instead of the denominator just being the average x^2 values, they turned that into a weighted average of the x^2 values. That is, they made the denominator in equation 2: \frac{\Sigma{\frac{x_i^2}{\delta x_i^2}}}{\Sigma{\frac{1}{\delta x_i^2}}}. I had to work through a rigorous discussion about what the goal of the task was and why they wanted to include the uncertainties at all. Eventually we got to something a bit more meaningful.

By far the most interesting image of transfer, though, was a superficial notion of how students incorporated the uncertainties. A large number of students, even those who invented a close-to-correct solution, wrote the uncertainties in their equation as \delta x, even though the uncertainties were clearly in the y-values. They still used the uncertainties in the y-values in their calculations, but kept the notation of \delta x in their formula.  They transferred a very surface-level notation from the weighted average activity into this new activity.

Why? I’m not really sure, but I think it’s really interesting.

What do you think?

Have you seen weird notions of transfer in your activities? What’s the most surprising example that comes to mind? Is there anything worth studying in these activities with respect to transfer, do you think? We get logs of all student actions during these activities, so there is lots of data to be mined!

Screen shot of the weighted-least squares invention activity.

Screen shot of the weighted-least squares invention activity.