On January 22, 1997, the Old Adobe and Waugh School Districts in Petaluma, California sponsored a presentation by education consultant Ruth Parker. The flyer sent home to parents described her as a “dynamic, highly acclaimed educator, researcher and author” who will discuss “research on how children learn as well as current effective and ineffective teaching and curricular practices.”
Ruth Parker stated in her Petaluma presentation: “We can no longer afford to teach standard U.S. algorithms.” Instead, she proposed we teach critical thinking skills that allow students to “explore, conjecture, and reason logically.”
Unfortunately, what passes for “mathematical power”, “critical thinking”, “problem-solving”, and the other high-sounding objectives is sometimes nothing more than fun with numbers – a ruse which allows problems to be solved diagrammatically. The trick is impressive to parents and teachers who aren’t watching closely or would never suspect such trickery from a professional. It’s also intellectually dishonest to propose these tricks as examples of superior math education.
In her Petaluma appearance, Ruth Parker presented her Turkey Problem.
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? She first described the “traditional” solution to the problem, which is to set up a ratio and solve for x as shown in Figure 1. 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.” 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 first described the “traditional” solution to the problem, which is to set up a ratio and solve for x as shown in Figure 1. 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.” 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.”
| Figure 1: | Figure 2: |
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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.
Ruth Parker and our schools should concentrate on teaching mastery and understanding of the powerful, time-tested methods of mathematics instead of ad hoc parlor tricks.
Is it any wonder why Ruth Parker and other math reformers are opposed to standardized testing?