A research project at San Diego State University (IMAP) has carefully studied what future and practicing teachers believe about the teaching and learning of mathematics. They have been working on the assumption that there are productive beliefs in this area that will help teachers communicate with their students, and design meaningful learning experiences for them, and that there are destructive beliefs that will constrain the kinds of opportunities children have to learn in these teachers’ classrooms.

These beliefs are just as relevant in the home environment as they are in school. To the extent that parents believe some of the following things, and to the extent that they act on these beliefs, parents will be able to support their children’s mathematics learning in powerful ways:

- Mathematics is a web of interrelated concepts and procedures (school mathematics should be too).
- One’s knowledge of how to apply mathematical procedures does not necessarily go with understanding of the underlying concepts. That is, students or adults may know a procedure they do not understand.
- The ways children think about mathematics are generally different from the ways adults would expect them to think about mathematics. For example, real-world contexts support children’s initial thinking whereas symbols do not.
- Children can solve problems in novel ways before being taught how to solve such problems. Children in primary grades generally understand more mathematics and have more flexible solution strategies than their teachers, or even their parents, expect.

Each of these is a productive belief. The conversations in my *Talking Math with Your Kids* series demonstrate what it might mean to act on each of these beliefs in conversations with children.

Belief number four may be the most important. Throughout these conversations you can see me asking questions that I am not at all sure my children know how to answer. You can see me asking them very genuinely how they found an answer. Of course I have hunches about what they are thinking, but I know enough not to assume I’m right. I know enough to ask and to listen to their answers.

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Some teachers hang onto “real-world contexts support children’s initial thinking whereas symbols do not” much too long. Many “gifted” kids are capable of symbolic thought long before teachers are willing to let them do it. Forcing kids to play with blocks for years past the point that they need them does not do them any service. (And “real-world” contexts are generally only useful when you simplify the real world down to the point where the simple models that the kids can understand apply.)

Two responses:

(1) Blocks are not real-world contexts. They are manipulatives; a way of representing mathematical ideas just as symbols are. There is certainly a place for them, and I think there is even a place for insisting that kids use them to represent their ideas at times just as there is a place for insisting that they express their ideas in words and in symbols. But I don’t we ought to be insisting on this regularly. Instead, manipulatives ought to among the tools we offer to kids to explore and to express ideas.

(2) The belief is about

initial thinking, not about continued and sustained development. If your claim,gasstationis that US teachers are by and large adhering strongly to this belief, I’m going to have to object pretty strongly. In fact, the standard paradigm in much of K-12 instruction (and beyond, for sure) is that we learn the symbols first, then do applications. The idea that kids’ everyday experiences in the world can inform the development of their abstract mathematical thought is absolutely not the standard one underlying US math instruction.CGI is working on the problem, but it hasn’t been licked yet.

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