Wednesday, November 21, 2007

Teaching: understanding graphs

Do you remember when you first learned to read a graph?

I don't. And that might be a problem for teaching.

I'm not talking about the various weird graphs that geologists use. (Stereonets, Ar release spectra, U-Pb concordia plots, ternary phase diagrams... ) I'm talking about a nice, straightforward, x-y plot. Two dimensions, commonplace variables like temperature and time, no logarithmic scales or ratios between variables or anything like that.

(Image source: Global Warming Art)
I started thinking about this while explaining a weather journal assignment to my Earth Systems Science class. I cover weather and climate at the end of the semester, in the slot that always gets shorted when I run out of time. But I figure that weather is something that the students ought to be able to observe, at least qualitatively, and I want to give them a reason to read the book and think about it, even when I don't get to all the material in class. So I have them keep a weather journal: every day they have to record temperatures, cloudiness, precipitation, wind direction, and anything they find interesting. The goal isn't to teach them how to professionally record weather observations - the goal is to get them to observe the world around them and relate what they see to the explanations from their book.

This year, I lost nearly an entire class because the students asked one good question: when, and where, and how, should they make their observations?

It's always a bit dangerous when students ask a good question that I don't have a canned answer for. They get to see me think on my feet - not always a pretty sight. And this time, to try to explain my thinking (and perhaps inspired by reading graphical arguments on Tamino’s blog), I started drawing a graph.

It wasn't a very good graph - it was qualitative, illustrative of an idea rather than a compilation of data. But it showed (maybe) what I was trying to illustrate: that temperatures vary throughout the day, so perhaps they should describe their observations qualitatively, and then look at the National Weather Service official maximum and minimum daily temperatures for comparison from one day to another.

I don't know whether the graph made sense to the students or not - I don't have a good technique for quickly assessing whether students understand what I'm saying. I’ve recently learned that graphs don’t make immediate, intuitive sense to a lot of people, and I didn’t take the time to slowly explain what I was showing them. If the graph made sense, then the discussion might have been effective – I asked the class to tell me how the temperature should change throughout the day, and built the graph pretty slowly, so I hope they thought through the problem with me. But if they didn’t understand the graph... well, I might as well have been lecturing in Klingon.

So I’ve been thinking about what I could do to improve their graph comprehension (and to assess whether I’m doing it effectively). The weather journal assignment might be a good place to start – it should be relatively easy for them to graph the daily minimum and maximum temperatures over these three weeks. I wonder if they could use the graph to look at other things – maybe qualitatively annotate the graph with wind direction, wind speed, precipitation, types of clouds, so they could look for patterns? (It seems like that is one of the uses of graphs: they show patterns.)

But when I start thinking about what to make them graph, maybe I’m getting away from the point. What kinds of activities help them really understand what they’re doing?

- Does it help to graph points by hand, or is that simply tedious and frustrating?

- Does it help to graph data they’ve collected, or data related to things they’ve experienced, so the numbers are associated with what they measure?

- How much data is enough, and how much is too much? It seems as though there might be a balance between having enough data to see a pattern, and having so much data that the exercise becomes tedious. (If the students graph data from a spreadsheet, the tedium disappears... but the kinesthetic experience of plotting the dots, and the spatial sense that comes with it, might diminish.)

- If I ask questions about the graph, how do I avoid mistakenly teaching them that correlation = causation? Or is it ok to first see that correlation can help one to figure out causation, and then explain the fallacy?

- Are they going to be too burned out after Thanksgiving to handle an addition to their assignment?

(Side note: there’s a project in the works at SERC to create online modules to give math practice to students in introductory courses. Graph-reading is one concept that the developers may add. If you’ve got math concepts that you need geoscience students to understand, take the survey for the project.)


Thermochronic said...

I think this is a very serious issue for the geosciences. Graph reading develops the skills that go into understanding more complicated visual documents, like geologic maps. Many of the same skills that you need to make good graphs go into making good maps, and the same can be said for reading both. How to collect data, should I stop collecting, am I optimizing my strategy, etc. From personal experience I think that it is important to build graphs with data the students collect, and then ask them to predict something based on their graph. I did this with my M&M simulation of radioactive decay and thought most of the students got a great deal from it. I was then able to move to less intuitive examples, but because they felt confident with the use of the graph, they still "got it" (at leas based on problem sets and exam responses).

In my brief stint as a biology major in college, I took a biometrics course (stats for biologists). One of the exercise we had was this: we were give a sheet of points with x,y coordinates. We were told to graph them, and that they were from some experiment where x and y were correlated. By this time we had plenty of experience with log and semi-log plots, we'd plotted things as x versus 1/y, more advanced than an intro student. But we all had to do this individually. We all turned in the assignment, and the prof put it together as overheads. Everyone had done it differently, some more complex than others, all trying to highlight different things. The rub of course was that the numbers were random, but because we has a preconceived idea that they were correlated, we messed with things until we could fit a line or curve to it.

John Fleck said...

Two things.

1) I really worry about this exact problem when we make a graph to put in the newspaper. I've no idea of what is intuitive to me in reading and understanding it is intuitive in the same way for newspaper readers.

2) I'd love to hear more, either in your blog or off line, about the weather journal exercise and how it works. I'm currently writing a book for middle school students on climate in the west, and I'm having the kids do exactly the same thing. Each chapter will have an "activity" for the students, and my idea is to use this to a) get them used to the idea of the way scientists actually collect day-to-day data, and b) get them thinking attentively about the weather.

Does this exercise work for your students?

Kim said...

Thermo - I just read about your M&M simulation - it sounds fantastic. (How long does it take? Can it fit into part of a lecture, or does it take an entire lecture, or is it an exercise that's better done in lab?)

John - I can post more about my weather journal (and other things I do related to weather). The short answer is that it doesn't work as well as I would like it to, and I'm trying to figure out how to push the students a bit more, so they think more about what they're observing. It is late in the semester, and the burnout level is high in all classes. And somehow I always assign the exercise during a time when we've got week after week of high pressure, so there aren't as many changes to observe.

Karen said...

Another variable -- what do graphing calculators contribute to understanding of graphs (or lack thereof)?

I'll admit I've never used one; I did my undergraduate schooling before they were invented, and professionally I use Excel for the kinds of graphs I require. (I did figure out how to make it draw ternary diagrams.)

Do graphing calculators encourage students to bypass the practice of making graphs by hand, so that they don't get the practice of relating data points to graphs and to each other?

Kim said...

Karen - That's a really good point about graphing calculators. I've never used one either - I'm just barely too old. I know that I've never seen a student (even a geology major) pull out a calculator to graph something - even when I derive the equations for a Mohr circle and ask them what shape the equations describe. So it doesn't seem, at least to me, that the graphing capabilities have added anything positive to the students' understanding. Whether it's hurt, by taking away the kinesthetic experience of making a graph... I don't know.

However, I think most of my intro students do know how to plot points on an x-y graph, so presumably they've done it at some point in the past. So maybe the calculators haven't done damage to their intuition. (Though they have made it difficult for me to explain how to use a calculator, because I've got a 1985-vintage one that doesn't handle parentheses and so forth.)

Anonymous said...

"If I ask questions about the graph, how do I avoid mistakenly teaching them that correlation = causation? Or is it ok to first see that correlation can help one to figure out causation, and then explain the fallacy?"

That's a good question. As an undergrad, a good example would be a simple association of different demographics - say a graph between post-secondary education and lower incidence of drug addiction (assuming that is the case, I'm just thinking up a possible example on the spot).

One might introduce the fallacy by noting that just because you enter university doesn't isolate you from the effects of drug use/abuse ~ correlation does not imply causation.

You could then ask students about the correlation/causation idea with respect to temperature and the rotation and revolution of the Earth. "Is this causation, or correlation?" ... "Why?", and use it as a springboard into the idea of being able to falsify a hypothesis, and why that's so valuable.

I don't know if that's worth much, but I think it's an interesting discussion.

Ernie said...

I'm a statistics student, and I'll be honest even as much as I've worked with math equations and graphs, they still kind of scare me. One of my teachers developed this, The Cyber Professor and it has alotta equations and run throughs. We're given alot of problems on there to help prepare us for the course content on tuesdays and thursdays.

These are just some of the ideas that I can share from my side of the country. They work for me and I do utilize the resources he provides online for data collection and all types of different charts and graphs we work with. Good Luck to you though!

Newark, DE

Anonymous said...

Throwing this in here in hopes that it applies.
Anyone out there have to take a year of calculus even though it wasn't there strongest subject? How did you survive?