Nonetheless, many of my latest naysayers have been high school teachers who simply can't believe that such research, which was done mainly in elementary schools, applies to older students and "higher order" skills.
As luck would have it, the DI people helped developed math and science videodisc programs for use with older students for teaching more advanced skills. The discs were effective instructional tools.
One of the programs was an algebra program and it had an interesting history. The algebra program consisted of 15 initial lessons. The program was based on what the DI people assumed the kids knew by the time they reached algebra. Unfortunately, their estimates were seriously mistaken:
As part of the pre-test for the tryout group, we presented a series of problems that involved simple addition, subtraction, and multiplication of fractions.
When we tabulated the results of the pre-test, we knew that we might be in deep trouble. Of the 32 junior-high kids in the tryout group, one could add fractions with unlike denominators:One kid in the group could multiply fractions:
--- + --- =
3 4Unfortunately, the kid who could add was not the kid who could multiply.
--- x --- =
They went ahead with the program field test anyway, but soon had to face the grim reality that things weren't going to work out.
[I]t became apparent that the only thing to do with these kids was to bring them back to frame one and teach them about the properties of fractions and basic fraction operations. Their misconceptions were amazing. Some of them could tell whether a fraction was more than one or less than one; however, they didn't seem to understand that the 1 referred to is the same 1 you say when you count: one, two, three ...
We looked at some other "pre-algebra" groups and observed the same problems we saw in our group. So we scrapped the algebra program and started over about ten rungs lower on the academic ladder, with fractions, what they are, and how they work.
So, the program was rewritten before it was released, which is all but unheard of in education. A study was then conducted with the program. The study involved two teachers:
One was a devotee of manipulatives and the NCTM approach. This teacher spent lots of time teaching math -- 1 1/2 hours a day. She gave her students big time homework assignments. She worked until eight every evening preparing for the next day. The other teacher did not believe in homework (yea for her). She spent far less time on teaching math.
The study was great because the kids (6th graders) were matched in performance, and pairs were randomly split for distribution to the two classrooms. The resulting classrooms were greatly heterogeneous. In the end, the "NCTM" group was slightly ahead (but not significantly) above the video group.
After both groups worked on fractions, decimals, and percents for a semester, they received a three-part test: the first part on tasks and problems that were unique to the video program; the second on the tasks and problems unique to the NCTM program; the third on tasks and problems common to both programs.
The lower half of the videodisc program outperformed the upper half of the NCTM group on everything. On the items presented only in the NCTM program, the lower half of the group averaged 65 percent correct; the upper half of the NCTM group averaged 51 percent correct. On those items common to both groups, the lower half of the video group averaged 65 percent correct; the NCTM group averaged 35 percent correct. The upper half of the video group averaged 90 percent and 97 percent on these two parts of the test.
Clearly the problem of providing effective math instruction has far less to do with kids then with the delivery of instruction.
I'd like to think I've made my point by now about the importance of effective instruction, but our educators won't let a little thing like facts get in the way of their pet biases.
I do enjoy hearing their rationalizations though, so I'm going to stir the pot a little over the course of the next week or so and run through all we learned with these videodisc programs.