Should you show students how to solve problems?
For many people, the answer is “obviously, yes.” But when and how? Research on young children’s mathematical thinking has shown that children can invent strategies to solve problems that are posed within their zone of understanding. Asking children to invent strategies aids the growth of understanding; how a child solves a problems can tell you a lot about what the child understands. Yet certain tools for representing and supporting understanding, such as numerals or number lines, must be shown to children, because they are conventions or because inventing them from scratch is inefficient. But there is a large gray area here where situational variables and teachers’ beliefs and experiences come into play and decisions are not clear cut.
How does a teacher decide when to show and when to not show students how to solve problems? I asked two veteran teachers when, if ever, they show young children how to solve problems. They have been involved in Cognitively Guided Instruction (CGI) for over 15 years each. Both now work as math specialists for the district. Before that, they were classroom teachers. Carrie Valentine taught upper elementary and Mazie Jenkins (sometimes known as Ms. J in CGI writings) taught primary. Both have worked a lot with kids from all kinds of backgrounds, including low income and low achievers in math. It’s certainly a question worth thinking hard about.
Here’s what they had to say:
MJ: The famous question always gets asked. The answer and how you go about getting students to exhibit different strategies is very complex.
Teachers have to learn how to get students to reflect on the strategies that are shared. For example, how is Susan’s strategy like Megan’s strategy? How are they alike and how are they different? Who else solved their problem like Susan’s? Who solved their problem like Megan? Who has a completely different strategy? How is it different than Susan’s? Ask specific questions for the strategy, how are tens used in this strategy? Was this a good way? How do you know? Have we seen this strategy before? When? What kind of problem did we see this strategy used? Did anyone use numbers to solve this problem?
I usually do not show a strategy. I might talk about how I have seen another student solve a similiar problem. (Deborah Ball demonstrates this on a videotape).
I listen and observe to find students who are solving a problem in a different way and build upon that. I ask students to solve problems in at least two different ways (direct modelers show two ways of direct modeling – but they do not know this)
Teachers need to have a good understanding of the development of strategies to know what is developmentally appropriate.
CV: I think it’s possible but requires a great deal of expertise. Skill in problem posing and questioning is essential. I wonder what is meant by strategies? I think of them from a cognitive perspective. Is there something going on internally different in the math understanding? But, I think most people think about the representations that their students use. In that case I definitely see utility in showing ways to represent to hasten the learning. Kids love ‘tools’ such as the empty numberline, arrow language, and ratio tables when they are ready. I would introduce them after some level of understanding emerges. The empty numberline after counting on emerges, arrow language after decomposition and facts of ten are known, and ratio table after kids can double using base ten. There are other ‘tools’ but these seem to capture kids’ attention and are efficient ways to record their thoughts and later to help them solve problems.
You get the gist. It’s actually a complicated question and deserves a nuanced response.
SE: What about upper grades?
CV: Same way. First what is meant by a strategy vs. a tool if there is a difference. Then talk about ways to record thinking.
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