The New New Math, Applied
by Nilanjan Banerjee, The Dartmouth Review, January 13, 1999

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Editor's Note: The following are excerpted from two published examples of traditional mathematics problems and how Fuzzy Mathematics proposes to solve them.

The Sandbox Geometry Problem

The following is excerpted from an article in The New York Review of Books by Martin Gardner. It describes two teachers, Sandra and Valerie, who are trained in fuzzy math techniques and eschew traditional (algorithmic) methods of teaching math.

"Two of the teachers, called Sandra and Valerie, are enthusiastic users of new new math techniques... Sandra was very good at getting students to cooperate in groups. However, in one exercise she told students that one could obtain the perimeter of a rectangular field by multiplying its length by its width! In another project she calculated the volume of a sandbox by multiplying together its length and width in yards, then multiplying the product by the box's height in feet!

"In an interview Sandra said that while working on the sandbox problem her pupils asked what a cubic foot was. `You know, the thing is that I couldn't really answer that question. Then I thought and thought, then I remembered how to measure a cube.' Neither Sandra nor her students were ever aware of her two huge mistakes. In spite of these errors, the author of the article about her said she was an `exemplary teacher.' Sandra is praised for getting her students to enjoy their cooperative efforts to solve problems `in the context of real world situations.' Finding a correct answer was less important than having fun working on the problem."

"Valerie made an equally astonishing blunder. The task was to determine the average number of times her thirty students had eaten ice cream over a period of eight days. This was `solved,' by dividing 30 by 8, to get 3.75, which Valerie rounded up to 4!

"As with Sandra, neither Valerie nor her students ever became aware that they obtained a totally wrong answer. Nevertheless, the author of the paper about her forgives her mistake on the grounds that she had succeeded so well in getting her students to work on a problem in the context of their experience. Moreover,the work had impressed on the students the `usefulness and relevance of averages.' No matter that they completely failed to find an average."

The Turkey Problem

Education consultant Ruth Parker, an ardent proponent of `Fuzzy Math,' routinely presents this problem as an example of how New New Math is a better teaching tool than traditional Mathematical approaches. This critique of the Turkey Problem is excerpted from an essay by Charles Beavers of Accuracy in Academia.

"The Turkey Problem: Ruth's diet allowed her to eat 1/4 pound of turkey or chicken breast, fresh fruit, and fresh vegetables. She ordered 1/4 pound of turkey breast from the delicatessen. The sales person sliced three uniform slices, weighed the slices, and said, "This is a third of a pound." How many of the turkey slices could Ruth eat and stay on her diet?

"[Ruth Parker] first described the "traditional" solution to the problem, which is to set up a ratio and solve for x. Three slices is to 1/3 of a pound as x is to 1/4 of a pound. Instead of simply solving for x, she chose to cross-multiply and highlight the point that most people don't know why cross-multiplying works. When she divided by a fraction she was able to derisively repeat the old saw, "ours is not to wonder why, just invert and multiply." In her paper, people who solved the problem this way are described as not understanding "why their procedures worked" and "confused about the information they were dealing with."

She then described how "mathematically powerful" students solved the problem as shown in Figure 2. First she drew 3 circles to represent the turkey slices. Then she added 6 more circles to represent the 9 slices in a pound. Then she divided the 9 slices into quarters to represent the 1/4 pound she could eat and then counted the whole and partial slices to reach the answer of 2-1/4. No formulas are used. Heck, written numbers aren't even used. At first blush, this solution seems simple and creative.

But is it?

Remember that the purpose of this problem is to demonstrate why "standard U.S. algorithms" should not be taught. They should be replaced by methods that give students "the ability to explore, conjecture, and reason logically; to solve non-routine problems; to communicate about and through mathematics; and to connect ideas within mathematics and between mathematics and other intellectual activity."

Try changing the numbers in the problem from 1/4 and 1/3 to 1/5 and 1/4 respectively. The ratio method works fine – it always works. Now try diagramming the problem as before. First you draw 3 circles to represent the 3 slices. Next draw 9 more circles to represent the 12 slices in a pound. Now all you have to do is divide the 12 slices into 5 equal parts. Well, there's a bit of a problem there. Unlike the original problem, you can't graphically divide the 12 slices by 5 and get any useful information. Of course, at this point (without dividing the diagram) you could determine that the answer was 12/5 or 2-2/5.

Let's try another possibility. Suppose the sales person sliced four uniform slices and said, "this is 3/8 of a pound." The ratio method works fine – it always works. Now try diagramming the problem as before. First you draw 4 circles to represent the 4 slices. Then what? You can't evenly draw a pound. What do you do?

Try again. What if the numbers 1/4 and 1/3 were changed to 1-1/4 and 1-1/3 respectively? What would you do? What would you draw? What would you sketch? What "mathematically powerful" solution would you use? Of course, you could use the ratio method – it always works.

But Ruth Parker doesn't want students to learn the ratio method. Why not? Well, I thought, maybe I was missing something. So I decided to ask Ruth Parker herself. I e-mailed her and asked how she would solve the problem if the numbers where changed as described above. She replied this way:

`I don't have much time for a reply at this point. I'd suggest you give the problems you're curious about to children and see what they do to solve them. That's what I did. It was actually a real problem that happened to me and I was curious to see what they would do with it. I constantly find myself surprised by children's thinking.'

When I pointed out to her that the "mathematically powerful" strategies almost never work, she declined to reply. She is not teaching a general and potentially powerful strategy for understanding problems. Instead, she is showing a strategy that works, graphically, for one carefully chosen problem. Generally, this strategy is useless.

This is the conclusion I reach from the Turkey Problem:

The ratio strategy Ruth Parker derides ALWAYS works no matter what numbers are used in the problem, whereas the diagrammatic strategies she hails as "mathematically powerful" ALMOST NEVER work.

The problem with the ratio method, and standard methods in general, is not that they don't work. Indeed, they have immense mathematical power. The problem is that too few people, including many elementary teachers who explain them to our children, understand the simple mathematical manipulations behind the methods. Textbooks should include this type of information.