In the semester that just ended, we started and ended with US Census data. The first assignment was to make a “claim” about some data you explored. The final project was to write a medium-sized paper using Census data to illustrate a phenomenon in US History. (More on that later.)

In between, we did a number of things that, if I wanted to look really smart and together, I’d say were designed to take the interest that the first activity kindled, and help students develop the skills they were able to apply in the final project.

The reality is not so organized and purposeful, but I did intentionally use *Sonatas for Data and Brain* as preparation for the final project. Since you’ve never heard of them, I will explain.

In the book *Data in Depth*, now out of print (by me, Key Curriculum Press, 2001; it was replaced by some other books, but I don’t know if the sonatas are in them), I invented this genre of assignment to give students small-scale investigations. If you’re a music type, you know that a piece in “sonata form” has three distinct sections: the *exposition*, in which themes are introduced; the *development*; in which they get all mixed up and transformed; and the *recapitulation*, in which the themes re-appear in close to their original forms.

Taking off from this, a sonata for data and brain has three sections: *prediction*, *measurement* (or *analysis*), and *comparison*. These sections are fairly self-explanatory except that in prediction, the idea is to describe as accurately as possible what you think the data will look like when the investigation is done. Then you do the data thing (measurement, analysis) and finally compare what you got to your prediction.

### Example

One of the sonatas in the book is called “What Are π?” It explains that you’re going to

- cut strings of different lengths
- measure and record the lengths of the strings
- tie each string into a loop
- make the loops as circular as possible
- measure and record the diameters of the circles

The question is, what will be the relationship bewteen the lengths and the diameters?

Note that you measure the lengths of the strings *before* you make the loops.

### About that knot

Some experienced students and teachers ask things such as, “should we make the knots as small as possible?” I assume they’re thinking ahead (too much). They know that so they want the data to be clean.

That’s laudable in its way, but I’m more interested in how we can use data analysis to make sense out of data that we have not worked so hard to sanitize. The whole knot thing adds variability and, more importantly, it gives a meaning to the intercept that appears in the graph: it’s the amount of string used in the knot. So if you want to make clean data, the most important thing is that the knots are *consistent*, not *small*. Then, with simple data analysis, you can completely account for the knots.

### What students did

There were a number of issues that came up in class with this particular sonata, such as whether the intercept was “real.” A second was what goes on which axis. In this case, it’s tricky: the length is really the independent variable, isn’t it? But if you put length on the horizontal axis, the slope is . Not nearly as pretty as the other way around. It’s easy enough to switch; we discussed it and put the diameter on the horizontal axis for the purposes of further analysis.

Another continuing difficulty is in learning to describe the meanings of parameters in the context of the problem. In this case, when they agreed that there was an intercept, I insisted (as usual) that they tell me what it meant. And (as usual) I first got a bunch of answers such as “it’s the length of the string when the diameter is zero.” Yes, but…?

It’s hard to get across the idea that “it’s the length of the knot” is more meaningful to someone than “the length when the diameter is zero.” It also shows a fuller understanding of the problem. Students have similar problems with slope; “the rate the length increases for each change in string length” has fragments of meaning embedded in it, but doesn’t show real understanding. We’re continuing to work on this aspect of communication.

### Other Sonatas

In case you’re interested, here are some other sonata ideas from the book. You can, of course, make up your own. They could start as questions (as in research questions, if you go that way) or as claims:

- Show people a book, and have them draw a rectangle the same shape as the book cover. What’s the relationship bewteen the length and the width of the rectangles people draw?
- What is the relationship between the number of prisoners in a state and the state’s population?
- Get data on buildings in your state. What’s the relationship between the height of tall buildings and the number of stories?
- What’s the relationship bewteen the amount of toilet paper you have pulled from a roll and its diameter?
- What’s the relationship between the “fret number” and the position of a fret on the neck of an electric bass?
- What’s the relationship between the number of puffs you puff into a balloon and its circumference?

There are, of course, more “stats”-oriented sonatas in the book; these are a few of the mathier ones.

### The Prediction Issue

The hardest thing (at which I have yet to succeed) is getting students to predict. And I mean real, on-paper, explicit predictions. My vision is that with practice, you will get better and better at it, and predictions will get more thoughtful *and quantitative*. But it’s inherently risky; people don’t want to do it.

For example, in the “Prisoner sonata” described above, your prediction could be in the form of a graph. Then, when you look at a graph of the data, you can see how good your prediction was. To get a really good graph, you will of course need axes. And points on the graph. That means you have to actually estimate the populations and the numbers of prisoners. And maybe draw a line of pretty-good fit, and know its slope. And identify any outliers. Do you think some states incarcerate a larger proportion of their inhabitants? If they do, where must they lie on the graph? Which states?

Why predict explicitly? Because at the end of the sonata, you have to compare. And if you were explicit, you have a better chance of being surprised, and seeing what preconceptions you fell prey to.

This is great. I’ve done a number of pi labs with my physics students, but never this variation with string. I’ll have to try it out. Too bad Data in Depth is out of print, I’ll have to see if my math department has a copy in its library.

This is great. Thanks for sharing.

Hi, I am still wrapping my mind around this. You mention: “Then, with simple data analysis, you can completely account for the knots.” I can picture the graph, but I am not sure how to lead a discussion of how we account for the knots in the C = pie x D ratio. If you happen to have a document showing this worked out I would love to have it.

I meant to also say thanks, this is a great learning exercise for Stats and a good general one for Pie week.

Ah! I see the confusion, I think.

The students discover that they get a line

with an intercept. So we first wonder, is the intercept “real” or should model really be direct proportion,y=kx? They look at the data, and there’s no way to make a zero-intercept line fit. So what can account for the intercept? The knot.A formal analysis might look like this:

/* where is the amount of string used to make the

Knot *//* where is the total

Length of the string */If you plotted

Lengthon the vertical axis, you’re done. The slope is and the intercept is theknotlength. If you plotdiameter(y-axis) against stringlength(x-axis), however — which is the “normal” way since diameter depends on string length — it gets more complicated. You have to solve the above for D.Is that what you were wondering?

Tim, thanks for the extra details. I was a little slow on the draw, but I get it now. Thanks!

I like this sonata idea. It’s actually what scientists do – predict, experiment, conclude. Maybe instead of prison data & state population we will collect data on prison population & per pupil expenditure for public schooling. Thanks for this idea!

Wow that was odd. I just wrote an really long comment but after I

clicked submit my comment didn’t show up. Grrrr… well I’m not writing all that

over again. Anyway, just wanted to say superb blog!

I hate it when that happens! But thanks for the props 🙂

What exactly genuinely influenced you to publish

“Sonatas for Data and Brain A Best-Case Scenario”?

I personallycertainly appreciated the blog post!

I appreciate it -Jenifer

Originally it came from the problem of helping kids do more open-ended investigative work. The idea is that this structure is a step along the way towards more independence.