(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."
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- Jay Martin, 53, soccer coach, |
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.
