Learning and Mathematics

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Strategy Acquisition & Application - Siegler and Jenkins (1989)

Robert S. Siegler and Eric Jenkins of Carnegie-Mellon University discuss how children acquire and apply strategies by looking closely at a small group of students over a long period of time. Strategies differ from algorithms in that they are generated by the student and are a nonobligatory, goal-directed procedure. Anything that does not accomplish a goal or accomplishes an unintended goal is not a strategy.
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Chapter:

Siegler, R., and Jenkins, E. "Strategy Discovery and Strategy Generalization," in How Children Discover New Strategies. Lawrence Erlbaum Associates, Hillsdale, NJ, 1989, pp. 1-20.

Overview:

There are two distinct phases associated with successful strategy use: strategy discovery and strategy generalization. The first refers to initial use of the strategy, the second to use in the full range of situations in which a strategy might be applicable.

New strategies appear within the context of existing knowledge and strategies, building new strategies by combining old ones, and must be assimilated into that context. Aside from the general difficulty of articulating a goal, part of the difficulty in strategy construction lies in the students' abilities to select appropriate segments of old strategies to address new goals-- goals the new strategy is to achieve.

Students develop the same strategy at different times and in different contexts. Thus, providing students with opportunities both to talk about what they are trying to do and to work with this information across different kinds of problem types should give them the kinds of tools necessary to become more strategic.

Even after a strategy has been discovered, the student may not use it consistently. Students tend to use a given strategy most often when it yields the greatest advantage, such as speed and efficiency, relative to other available strategies. For example, a student may use retrieval (producing the answer from memory) to generate the solution to 5x20, but successive addition to solve a problem whose answer may not be readily available, such as 5x75.

In addition, students who use a strategy in one context may not use it in a different problem that features the same concepts. For example, students who use fractional arithmetic to find the amount remaining given 8 ounces of soda, 1/4 of which has been consumed, may, given 2 pizzas with 8 pieces each, determine by counting how much is left if 1/8 of the total amount of pizza has been eaten. Students learn to choose the most efficient strategy through experimenting with the use of different strategies in different situations, thereby discovering the relative speed and accuracy of a given strategy in a given situation.

Strategies may be developed for many different reasons. The most obvious is a failure of existing approaches, such as may occur when a student who adds by counting is suddenly confronted with the problem 378+274. However, elegance, novelty, and efficiency, such as retrieval over counting for simple arithmetic, also motivate the discovery of new strategies.

Direct Quotes (and some comments)

"Our everyday prototype of strategy discovery seems to be Archimedes' insight in the bathtub concerning how to find out whether the king's crown was made from pure gold. Within this prototype, strategy discovery involves a sudden burst of understanding and is accompanied by a conscious 'aha' (or 'Eureka') experience; the discoverer not only uses the strategy for the first time but immediately understands why it works and what types of problems it can solve. The appeal of this view lies not only in its drama but also in its simplicity. Discovery is seen as involving a sudden, discontinuous change from not knowing to knowing." (p. 15)

"[E]ven after children first used the most efficient experimental strategy available, they continued to use a variety of unsystematic strategies as well....[O]nly a few children's first use of a strategy was accomplished through a sudden conscious burst of insight. For the others, the discovery process was less conscious and often less explicit....[E]ven children who had earlier seemed to abandon the less systematic strategies returned to them when confronted with slightly more difficult problems." (p. 10)

"[W]hat seem in retrospect to be dramatic qualitative breakthroughs are in fact the culmination of a long series of smaller realizations concerning how existing ideas can be recombined." (p. 17)

-- Summarized by Andrea Hall

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