Quite a few months ago, I was engaged in a somewhat heated discussion with a visitor about the nature of science, and physics in particular. The debate tended to orbit around the issue of whether or not one could be engaged in learning or doing physics without mathematics. Of course one's answer depends on what one mean by mathematics and what one mean by physics, so there was much to discuss.

If you know me, I tend to espouse a view of physics (and science) in which explanation and argumentation are central to its practice. This visitor espoused a view in which mathematics was central to physics. In the abstract, of course, these two commitments are not contradictory, but it can help to discuss specific examples, because we disagreed on much.

At some point, I introduced an example. I asked him to imagine that two friends are walking out in a field and find that the grass seemed taller on one end then the other. Both friends decide to investigate.

One friend starts by wondering what causes this to happen: Does it have to do with sunlight? Does it have to do with water? Did someone cut it this way? Do animals graze on one side more than the other? So, he walks around the field, feeling the soil for moisture or color difference. He examines the grass to see if it's been bitten or cut. He tries see if there are different species of grass. He tries to imagine how the sun traverses over the sky, and wonder if the trees on the fields edge would shade one side more than the other. He tries as best he can to determine if the field is slanted one way or the other, thinking that water would flow differently. He goes to the edges of the field to look for any creeks or other water sources. He looks for evidence of animal tracks or tracks from a machine.

The other friend wonders if there's a discernible pattern in how the grass height varies: Does it really vary? Does it vary linearly? How quickly is the grass height changing across the field? So he gets out his ruler and starts measuring the heights. He carefully lays out a grid of measurements that spans the field and begins tabulating the data, making sure to get track of his uncertainties. He uses the table to generate various plots showing height of grass vs. various positions. He uses the graphs to draw in trend lines, and then starts modeling the data with various mathematical functions. He goes on to determine values (and bounds) for the parameters in his mathematical models, and even estimates the goodness of his fit.

Now granted, both kinds of activities have value in science: (1) exploring plausible causal mechanisms and looking for qualitative evidence to help define the possible space of explanations; and (2) carefully using measurement tools to quantify aspects of phenomena in order to look for mathematical structure and relations. Let's call the first kind of activity: "the pursuit of causal explanation" Let's call the second kind of activity: "the pursuit of quantitative structure"

Of course, as I mentioned above, these two need not be disjointed activities. Looking at mathematical structure can lead one to ask new and different questions requiring explanation. Seeking evidence to support or refute an explanation may require that one collect evidence that is quantitative in nature. But the visitor saw little value in the first activity. He likened it to the kind of science that took place with aristotle–loosey goosey ideas about how the world works with no mathematical structure. He also saw immense value in this second kind of activity–he likened it to how mathematical structure is used in quantum physics.

He went on to argue that the mathematical model for the grass was an explanation and that it could be a good explanation if it could predict with some degree of reliability the height of grass anywhere in the field. To him, it didn't matter whether the grass was cut by a lawn mower or whether it was a matter of water source. Of course, someone might care, but caring about those questions is not science but of agriculture. To him, science was about laying down the quantitative structure and coordinating that structure with quantitative data in increasingly precise and accurate ways. I felt like all the mathematics wasn't an explanation, but that it might serve the point of helping to better define the space of possible explanations or to point to new questions or inquiries about the phenomenon.

Our conversation went on for quite sometime, and it was never quite resolved. Afterwards, I came to think that the root of our different views of science were actually in much deeper differences in worldview. The visitor was a professor of physics from a developing country, and I am a physics education researcher who grew up in a US middle class home. Based on much more conversation with him, I have come to see that his views are much more rooted in pragmatic needs for technical training and economic development, while my views are much more rooted in some idealistic notions of the enlightenment. We are both a product of our personal, cultural, and nationalistic histories. I have tried for sometime to write up more about this difference, and how our views of science are embedded with deep issues of culture and history. It's something I need to think about more and write up carefully, so I'm going to get to it eventually. But for now I wanted to just write about the back story of our science conversation and foreshadow that later blog post.

I've been thinking a lot about this over the last several years as I've taught science teachers who've come back to school to get their physics license. Some of them have very poor math skills and we've had many classroom, hallway, lunchroom, and email conversations about this very topic.

ReplyDeleteI'll admit that often I'm in the role of your colleague and they are in your role, as you've described them. Some of the reason for that is to motivate them to learn some of the things the state says physics teachers should know. The teachers sometimes want to discuss the qualitative aspects of various phenomena, but if I put the math to it, some of them, sometimes, shut down.

When I reflect on the time I've spent with these teachers, I like best the memories of sitting around discussing things like how we have to redefine momentum if we want to say momentum is conserved at high speeds. Just yesterday I taught that and we quickly went over the algebra that shows that gamma m v is conserved while mv isn't. Some of the teachers nodded their heads (or closed their eyes!) at the not-so-straightforward algegra, but ALL of them were engaged in the discussion about whether to rename momentum.

I'm torn about this a lot. You can hear me saying "math is the language of physics" and "I'm nervous about Physics First because the relatively low math skills of 9th graders." On the other hand, I love talking with people about the ramifications of things, and about how to ask questions to figure things out.

The way you presented the analogy, I liked the first way better. It resonated with me. However, if I wanted to then try to predict a pattern of how other farm fields might respond to similar interactions, I think I'd pull out the math skills you mention in the second version.

Thanks again, Brian, for getting these thoughts going in my head. I'm looking forward to the next blog post! -Andy

I keep thinking that, with these populations, we need to find better ways for the mathematics to serve the conversations we are having about the physics, and not just let the math be the whole of the conversation of the physics. Like you said, it can cause shutdowns cognitively; but more so I see how it becomes associated with shutdowns in curiosity. How we get people to a place where the two become more and more indistinguishable and more and more integrated is not an easy question. I have no doubt that the math-physics problem is strongly associated with "end-of-chapter" like problems, and how they require math in a weird way that has little to do with physics.

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