Yesterday’s post recounted a conversation with Tabitha (5) in which she asked for a “math class” problem.
I focused the initial discussion on where she learned what constitutes a “math class” problem.
But there’s lots more in there that’s interesting.
It wasn’t just her affective response (rejecting the driving in a car context, asking for a naked number problem) that matters here. Notice the way she engaged with the two problem types.
When I relented and posed the question, What is two plus three? she guessed. I know that she guessed because she (a) took no time to process, (b) asked rather than told me her answer, and (c) it was wrong despite being within her grasp.
When I posed the exact same problem in a situation she could imagine (she and Griffy had been to the arcade just that day), she engaged quite differently. Her body position changed. She paused. Her fingers moved. Each of these is an indication that she was thinking. And when she had an answer, she stated it; she did not ask.
The central tenet of CGI and an important belief underlying the IMAP work is that children can use contexts to solve problems that they cannot solve abstractly. Here it is in action. 2+3 is meaningless to Tabitha right now. But her 2 tickets combined with Griffin’s 3 tickets? That’s got meaning.
The conclusion here is obvious, right? We start with contexts kids understand and can reason about (here, combining tickets). We move to the abstract mathematical representations (2+3). We don’t save arcade tickets for after the kid understands addition. We don’t wait for symbolic mastery before doing some applications.
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.