I was curious to see what was on the web and what links I could collect for that stats rewrite, when I stumbled upon this series of pages. I wondered why it seemed rather juvenile, until I saw this and followed the link:
This is where the lesson fits in the NCTM Standards.
My first thought was, “When did the NCTM expand beyond primary and secondary into college math ed?” They haven’t. Follow the link, and this lesson is suggested for grades 9-12.
I’ll leave aside the issue of whether statistics belongs in the high school curriculum. I will also leave aside any criticism of the problem or presentation of this lesson (I have several) until later. But I do have several questions, and I’ll try not to rant.
In the statistical scheme of things, chi-square is fairly complex, both mathematically and conceptually (particularly compared to other tests, such as ANOVA, or linear regression). God knows I have nothing against complexity, but given that the students have probably done nothing but macaroni art projects and guess-and-check, and have no understanding even of basic operations (pulling out a calculator is not understanding), how, exactly, does the NCTM think students will be able to grasp chi-square? How, exactly, does one explain goodness of fit, degrees of freedom, or the chi-distribution to someone who can’t calculate a 15% tip or make change?
If students were actually learning math, I’d probably jump on the “Let’s teach stats in high school!” bandwagon, but that’s rather like saying if I were a billionaire, I’d take a Caribbean cruise every winter. I’m not a billionaire, and students aren’t learning math — or I should say that in most cases, students who do learn math do so in spite of the school system and pedagogy.
So what is the NCTM thinking?
But let’s assume for the moment that students are learning math in school. There is exactly one reason to teach statistics in high school, given that the only students who will ever actively use statistics will go into college: So they can’t be fooled by sensationalist crap from math-illiterate journalists. Given this reason, why teach chi-square? When was the last time you saw a reference, good or bad, to chi-square in the newspaper or on the news? Never? Me neither.
So if we must teach statistics in high school, why not focus exclusively on things that will actually be useful to students, like means and medians, standard deviations, polling and surveys, sample and population, and error (margin of error, confidence intervals)? If we must go beyond those and teach tests, why not stick with correlations, regressions, ANOVA, and t-tests, all of which are less complex than chi-square?
The only reason I can think of is that the NCTM folks are education faculty, and ed faculty — on those rare occasions when they do actual statistical tests — seem to love chi-square more than anything else. That may sound cynical, but I really can’t come up with another justification for this nonsense.
As for the lesson, the explanations aren’t bad, but the examples are a bit odd. Okay, so I was being nice. The explanations are vague and sloppy, but I’ll get to that after I deal with the examples. How many students outside, say, Pittsburgh know what a foundry is? And why, exactly, would high school students have an opinion on whether accidents are more likely to happen at one time of the day than another? How many students know what the draft is? And why make vague statements such as:
If a D [difference between the outcome and the expected value] couldn’t happen very often by chance then it is called statistically significant.
The question you should ask yourself is How large a value of D is needed before you can say that, for a fair die, such a value will be obtained very seldom by chance?
How often is “very often”? How seldom is “very seldom”? Why use such sloppy thinking? Isn’t a part of math ed — indeed, ed in general — supposed to be precise thinking?
Then there’s the spreadsheet they’re supposed to download and use. I’m mystified. Read the instructions (it’s an html file) and try to figure out exactly what students are supposed to learn from doing nothing but putting a 1 in a cell and watching rand() function outputs change. What’s the point?
Then, there’s this:
If your D value is on the far right side of the graph then your die is probably not fair. If it is the middle or to the left then it is probably fair and the numbers are a random occurrence.
And why is this true? (Understanding the question and answer, by the way, depends on understanding the chi-distribution.) There is no answer here. It’s presented as a statement of fact.
Then, there’s another spreadsheet with rand() functions, to generate die-roll trials. Here’s a question: If we’re going to use Excel in classes, why not teach students, you know, how to use Excel? Why provide a spreadsheet with =rand() in a bunch of cells? Why not — and I don’t want to shock anybody here — teach students to use the rand() function? It’s not like it’s difficult. A groundhog could type =rand() into a cell if he had fingers.
And again:
Again, to read the table you must recall the table from the D statistic. The first number in the table is the number of c2’s that fall between 0 and 1, the second number in the table is the number of c2’s that fall between 1 and 2, 1 inclusively. Look at the following graph. If your c2 is in the ranges with the highest ten values of the table then it is statistically significant and your die is probably unfair, otherwise it is good.
Why is this true? Why does the author of this lesson never address this? Any student who isn’t comatose will wonder. Why the right side? Huh? Why not the left side? Why not the center? What are you talking about?
I have worked with people who produce crap like this, and I’m glad to say that it never gets through the review process. We mark it up, send it back, and say, “Rewrite this garbage.” Apparently, either this author has no colleagues with standards higher than his own, or this was produced without external review. And that leads me to wonder: How many teachers are using crap like this in the classroom?
To the author’s credit, he does give them two more problems, but by the time the students get to the end of this lesson, they still have no idea what chi-square is, or what is so special about the right-hand side of the graph. The “understanding” in this lesson is superficial, at best. And perhaps that’s appropriate for high school, but the question remains: Why teach chi-square at all in high school — and if you must teach something, why not teach it in depth?
Having known several people who did graduate programs in ed, this may be an assignment for an ed class. But had I produced this for an ed class, I would have taken it offline the minute grades were reported.
