Children's thinking has often been described as something like a staircase, in which children first use one approach to solve problems, then adopt a more advanced approach, and later adopt a yet more advanced approach. For example, students of children's basic arithmetic (e.g., Ashcraft, 1987) have proposed that when children start school, they add by counting from one; sometime during first grade, they switch to adding by counting from the larger addend; and by third or fourth grade, they add by retrieving the answers to problems. More recent studies, however, have shown that children's thinking is far more variable than such staircase models suggest. Rather than adding by using the same strategy all of the time, children use a variety of strategies from early in learning, and continue to use both less and more advanced approaches for periods of many years. Thus, even early in first grade, the same child, given the same problem, will sometimes count from one, sometimes, count from the larger addend, and sometimes retrieve the answer. Even when children master strategies that are both faster and more accurate, they continue to use older strategies that are slower
and less accurate as well. This is true not just with young children, but with pre-adolescents, adolescents, and even adults (Kuhn, Garcia-Mila, Zohar, & Anderson, 1995; Schauble, 1996).
This cognitive variability is a spontaneous feature of children's thinking. Efforts to change it do not usually meet with much success. For example, in one study, first-to-third-grade teachers were interviewed regarding their beliefs about their students' arithmetic strategies and their evaluation of whether the students' use of multiple strategies was a good thing (Siegler, 1984). All of the teachers recognized that the children used multiple strategies, though most viewed this as a bad thing. One teacher said that she was constantly discouraging students from using strategies such as counting on their fingers. When asked how often she had done this with the pupil in her class who did it the most often, she asked, "How many days have there been in the school year so far?" This teacher and others recognized that even when they explicitly told students not to use their fingers, they did anyway, even if they had to do it by putting their fingers in their lap, under their leg, or behind their back. There is a certain logic that supports the teacher’s view. Older students and those who are better at math don't use their fingers, whereas younger and less-apt students do. One goal of education is to make younger and less apt students more like older and more apt ones. Therefore, children who use their fingers should be discouraged from doing so.
However, children actually learn better when they are allowed to choose the strategy that they wish to use. Immature strategies generally drop out naturally when students have enough knowledge to answer accurately without them. Even basic strategies such as counting fingers allow students to generate correct answers when forbidding use of the strategies would lead to many errors. Further, students who use a greater variety of different strategies for solving problems also tend to learn better subsequently (Alibali & Goldin-Meadow, 1993; Chi et al., 1994; Siegler, 1995). This is in part because the greater variety leads the students to be able to cope with whatever kinds of problems they encounter, rather than just being able to cope with a narrow range. Allowing children to use the varied strategies that they generate, and helping them understand why superficially-different strategies converge on the right answer and why superficially reasonable strategies are incorrect seems likely to build deeper understanding (Siegler, forthcoming).
Children’s use of diverse strategies makes it essential that they choose appropriately among the strategies. To choose appropriately, they must adjust both to situational variables and to differences among problems. Situational variables include time limits, instructions, and the importance of the task. For example, in a magic-minute exercise, it is adaptive for children to state answers quickly, even if they aren't absolutely sure of them. Similarly, if it's very important to be correct in the particular situation, then checking the correctness of
answers becomes more worthwhile. At least from second grade onward, children shift their choices appropriately to adapt to such situational variations.
Adaptive choice also involves adjusting strategy use to the characteristics of particular problems. When children are faced with a simple problem, it often is ideal for them to use a strategy that can be executed quickly, because it will be sufficient to solve the problem. In contrast, when faced with a more difficult problem, they may need to adopt a more time consuming and effortful strategy to generate the correct answer. Adaptive choice involves using quick and easy strategies when they are sufficient, and using increasingly effortful ones when they are necessary to be correct. Research on strategy choice has also revealed some surprising similarities in the performance of children from different socio-economic groups. Children from low-income backgrounds, particularly low-income African-American backgrounds, are often depicted as choosing strategies unwisely. Suggestions have been made that instruction focus on improving their metacognition and their strategy selection. However, selection of appropriate strategies does not seem to be their main problem, at least in the context of arithmetic. Their strategy choices are just as systematic and just as sensitive to problem characteristics as those of children from middle income backgrounds (Kerkman & Siegler, 1993).
Instead, their problem seems to be that they do not possess adequate factual knowledge. This in turn seems to be due to less practice in solving problems, and to less good execution of strategies, rather than to any high level deficiency in their thinking. The findings indicate that greater practice and instruction in how to execute strategies may be the most useful approach to improving their arithmetic skills.
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