Seeing Math and Seeing Problems

Occasionally it has seemed to me that I see things a little differently than other people.  Several years ago, I was co-authoring a paper on outcomes for married people with mental retardation. My co-author was an extremely intelligent person with far more knowledge of the issue than me. Looking at the first set of figures, which was a 2 x 2 cross-tabulation of had children (yes/no) and living independently without state services (yes/no), I stated,

“It’s obviously a significant relationship.”

Sandi asked in surprise,

How do you know that before you look at the table that shows the chi-square and significance?”

I said,equally surprised,

“Because you can see it. I mean, it’s obvious, isn’t it? Chi-square is the square of observed minus expected divided by the expected frequency. If there was no relationship you’d see those with and without children equally likely to be in the good or bad outcomes categories. That’s what you’d expect. It’s very far from equal and when you square it, the difference will be enormous.”

She said,

All I see are a bunch of numbers. But I can see from the table on the next page that you’re right.”

In the book, Born on a Blue Day, an autobiography by a man with autism who also had a remarkable facility with numbers, Daniel Tammet talks about seeing numbers as different shapes and colors. Well, I certainly don’t see anything like that, but I do wonder if spending lots of time just staring at numbers you come to see things differently. For example, today I was looking at an example of coding a structural equation model using PROC CALIS, a SAS procedure. Reading the equations, it was obvious to me that the first few equations looked like this:


I grabbed someone who was walking by, who happens to be a very smart person and said,

“Look at these equations. What do you see?”

She said,

“Well, I don’t know. Maybe it is sales data?”

I drew out the figure above and demanded,

“Do you see something like this?”

She said,

“No, not at all. If you are teaching or using this, you need to document it more, I think.”

My point, and I do have one, is that if you spend a LOT of time, and I have spent a good bit of the past 27 years, staring at numbers, things may start to look differently to you and that is very useful and helpful. In Daniel Tammet’s case it sounds like it is genetic but for people like me I think a lot of it is environment, specifically experience. I met with someone later in the day and when I was explaining to her how structural equation modeling can be viewed as a combination of confirmatory factor analysis and path analysis she said,

“I see.”

And as we were talking, it was clear to me that, very literally, she did SEE it.

One way that I think this happens was demonstrated when I got home tonight. I was going to take my little Julia to judo practice. In my spare time, of which I have none, I am the president of the United States Judo Association. During my misspent youth I was the first American to win the world championships in judo (it’s true, you can look it up). However, when I got home Julia was on her laptop writing her science project on Index of Refraction, which I decided was more important, so I let her be. Julia turned 12 two weeks ago. Her project involves measuring the angle of refraction in solutions with varying degrees of sugar. Tonight she looked up definitions of density, index of refraction and how to calculate the angle of refraction using Snell’s law and sine functions. Both her father and I gave her definitions of density (I _did_ start out as an engineer). Dennis explained a little bit about refraction, the index of refraction and angles. I showed her a picture of the sine function.

Last year, she did her project on the impact of multi-tasking on performance and did a pre-test and post-test with a cross-over design. She found the information on the Internet and told her parents to mind their own business it was her project and she was going to do it herself, which she did, punctuated only by a bit of crying when she could not figure out how to do the graphs. She eventually figured it out, after the evil mother told her she would spend the rest of her life at that kitchen table working on it if necessary but she WOULD figure it out. Fifteen minutes later it was done.

Lest one misinterpret this as Julia being some type of math/ science prodigy, it should be noted that although she made all A’s last year, she is currently not allowed to do video chat because she got a C+ in science after doing poorly on a test (she gets nervous and she has no concept of test-taking strategies. She will spend all of the time on a question if she doesn’t know the answer, rather than doing the ones she knows and going back to it later). After finishing her research for the evening we had to go to the Third Street Promenade in Santa Monica because her life would apparently have ended did she not get Van’s tennis shoes in two different colors, blue new polish, two new pairs of  “skinny” jeans from Pac Sun and a new jacket from American Apparel. If the economy is not rebounding, it is not Julia’s fault.

AND YET …  I am fairly certain that by the time she is in high school, Julia will, literally, SEE sine functions and a great deal more.

This is good. In a way.

Five days a week, I look down on William Jefferson Clinton Middle School, which is a few hundred feet from the building where I work. It is in a very disadvantaged neighborhood in Los Angeles. On Friday, I attended a presentation at the university where brilliant well-meaning people talked about how they were going to leverage technology to raise the achievement of students like these.


Still, I can’t shake the feeling that three years from now, when they are all in high school, Julia is going to be seeing a sine function when all those student see is a bunch of numbers.

And, as Robert Frost said, that may make all the difference.

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One Comment

  1. I spent years and years taking piano lessons. I could only play exactly what was written in the music. After playing a few different instruments that didn’t require written music, I finally discovered what my poor piano teachers had been trying to drive into my head. Music is just patterns..if you know the chords, you can figure out the pattern and suddenly music isn’t so complicated! I am not a virtuoso, but now I can sit down at the piano and come up with a harmony to go with any melody, and my ability to “sight” read is suddenly improved.

    A similar thing happened with quilt patterns. When I first started to sew, I could only follow the pattern. Now I can look at any quilt and figure out how to make it without a pattern. I love to look at quilt patterns in magazines and devise easier ways to make the blocks.

    I think the “seeing” you have discovered requires experience. Maybe a good teacher could help students “see”…but I suspect we just aren’t ready to “see” until we have that time and experience..whether it is math, music, or quilts.

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