Two Examples

I wanted to zero in on two examples from the UPI article cited below that indicate exactly what is wrong with some of the instructional techniques of NCTM math. I liked these two examples because I think they illustrate what is wrong with NCTM math on both ends of the ability spectrum.

Cheney used an example from what she called an NCTM-inspired program called Mathland. Students were asked to solve this problem: "I just checked out a library book that is 1,344 pages long. The book is due in three weeks. How many pages will I need to read a day to finish the book in time?" The traditional method would be to divide the number of days (21) into the number of pages, getting 64. But, Cheney said, students today are often not taught long division. She held up a huge poster board covered with numbers, displaying the work of the student that Mathland featured as exemplary. "This particular student added up 21s until reaching 1,344," she said. Later in the program, NCTM President Lee Stiff said that he would never recommend such a method with numbers that large. But Stiff, a professor of mathematics education at North Carolina State University in Raleigh, said the technique is useful in teaching math concepts with much smaller numbers.

This is an excellent example of the problem at the low end of the spectrum. Yes, this will give you the right answer, but (a) understanding of what’s actually happen is nil and (b) if the assigned reading were 10541 pages and you had 83 days to get it done, the “technique” the student has learned is useless and he hasn’t learned an alternative technique that works in all cases. Prof. Stiff’s last comment in my mind completely misses the point – you want to teach the correct idea (division!) with small numbers, so that the student can confirm his understanding of what’s going on by doing something like this repetitive addition with small numbers. Then when presented the more complicated problem, he can just apply division (and UNDERSTAND why he's doing it). I don’t think this intellectual construct is beyond the ability of even below-average students, yet they are denied this true learning in lieu of mechanical meaningless manipulation. No wonder so many kids are growing up math-phobic.

[Edited to add: The mathematician in me couldn't let this go. It's worth pointing out that even conceptually adding 21 a bunch of times is meaningless. What's physically meaningful here is chunks of 64 (pages per day). Adding in increments of 21, while arriving at the correct answer, doesn't actually mean anything.]

In the 7th grade, he and his classmates were asked to find the area of a circle. Four weeks were devoted to the task. Traditionally, children were given the formula, but apparently these junior Archimedes were supposed to rediscover the uses of pi.

This is an excellent example of the problem at the high end of the spectrum. How many students among even our highest ability students are capable of deriving A = πr² on their own, even given all the time in the world? I took a course in topology and was able to understand and eventually replicate proofs of such famous theorems as the Urysohn Lemma, the Tychonoff Theorem and the Jordan Curve Theorem. According to NCTM math, I would be expected to construct this understanding on my own without a teacher lecturing to me. Not being a mathematician of the caliber of Urysohn, Tychonoff or Jordan, I would of course have failed miserably. Yet this is the same flawed idea that is used to justify asking 7th graders to derive A = πr². Absurd!