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(IV) The Role of Practice in Problem-Solving

"It is important to separate 'outcome goals' – the big stuff – into 'process goals' – the little stuff you can accomplish every day. Larger goals, like winning championship titles, can often seem abstract and distant, but when you break them into things you can do every day, your dreams gain power, and it gives you a sense of purpose and control. By doing it that way, I've also learned to enjoy not just reaching the goal, but the daily pursuit of it."

- Jay Martin, 53, soccer coach,
Wesleyan University, U.S.A., taken from
Where Pride Still Matters,
Men’s Health (Brooks, 2002, pp. 80-83)

There are two important issues here, and the first is not complex.

Tackling large projects can be paralyzing unless the large project is broken down into more immediately attainable objectives. When things are going well, it is easy to feel encouraged and to continue to work hard at something like mathematics. But when you are facing a brick wall, it is easier to retreat than it is to attempt to scale the heights of that wall, especially when you anticipate many such brick walls to come. Often, when a student at the Mathematics Learning Centre is having difficulty with a particular task in his/her program, that student loses focus because of the anxiety generated. In those instances, we frequently suggest that the student back away from the brick wall and regroup with the assistance of a program planning template similar to the one illustrated on p. 51. This enables the student to concentrate on one troublesome topic in mathematics at a time by assigning each topic to specific future days and weeks.

That’s what Jay Martin is talking about on the left-hand page. He calls the immediately attainable objectives process goals and the large project an outcome goal, but it is the same principle being applied to athletics.

The second issue is more complex because it has nothing to do with planning and everything to do with cognitive psychology.

Contemporary curriculum developers often undervalue practice. They focus instead on "pursuing open-ended problems and extended problem-solving projects" (National Council of Teachers of Mathematics, 1989, p. 70). Yet, there have been arguments made in the literature of cognitive psychology that support the importance of practice, indeed, declare it to be of paramount importance in the effective pursuit of open-ended problems and extended problem-solving projects. We cite some of this literature here.

Campbell and Xue (2001) have demonstrated that differential amounts of practice over more than a decade facilitates arithmetic skills. These scholars presented 72 students registered in undergraduate and graduate programs 360 arithmetic questions. Participants were required to add three 1- or 2-digit numbers, divide 2- or 3-digit numbers by single-digit numbers, subtract 2-digit numbers from 2-digit numbers, and multiply 2-digit numbers by 1-digit numbers. The number of these problems participants successfully completed in 15 minutes was predicted by reported calculator use before entering university and mean reaction time to answer simple arithmetic (e.g., 4 x 3) questions. Presumably early practice in single-digit arithmetic led to faster access to basic facts which transferred to more complex arithmetic while access to calculators reduced the number of opportunities to practice more complex arithmetic.

In addition, Haverty (1999) has demonstrated that concentrated practice facilitates complex problem solving. Haverty trained seventh graders to generate either the 17 or 19 multiplication tables, but not both the 17 and 19 tables. Following their mastery of these facts, the children were presented inductive reasoning problems at a variety of difficulty levels by providing them with tables, each of which contained six x, y number pairs, and requiring them to describe the mathematical relationship between the variables represented by the numbers in each table. The prior learning of the relevant multiplication facts facilitated solving the most difficult problems [e.g., y = 17 (x + 1) ]. Haverty’s finding is consistent with the assertion that mathematical discovery is dependent on prior skill level, and consistent with our contention that skill development usually is a consequence of practice over extended time periods that results in both the availability of and speedy access to facts and procedures.

This research in cognitive psychology makes a mockery of the "move away from a tight focus on manipulative facility to include a greater emphasis on conceptual understanding, on algebra as a means of representation, and on algebraic methods as a problem-solving tool" (National Council of Teachers of Mathematics, 1989, p. 150). One can no more learn to do mathematics well by immediately immersing oneself in problem solving than one can learn to play soccer well by immediately concentrating on playing games. As Jay Martin puts it, it’s important to separate outcome goals into process goals. In so doing, "your dreams gain power, and it gives you a sense of purpose and control" (Brooks, 2002, p. 78).

- taken from The Math Plague, 2007, pp. 78-81.

For more detail, please see May, Rabinowitz and Mantyka
, 2002, pp. 20-23.