By Steele, Diana F.
Teaching Children Mathematics , Vol. 6, No. 7
Students should value mathematics in their lives and become confident in their abilities to solve problems, learn to reason, and communicate their mathematical ideas (NCTM 1989). Using these mathematical skills gives students school experiences that come close to the experiences of mathematicians. Such experiences help students learn what it means to know mathematics as a discipline as they actively construct their own knowledge and develop mathematical concepts.
To find out how children think like mathematicians, I observed several elementary classrooms as a teacher and researcher (Steele 1995). I became very interested in one particular fourth-grade classroom. The students in this classroom exhibited characteristics of what mathematicians do when they discover new mathematical ideas. These students made conjectures and presented their points of view to convince others of their validity. They saw patterns and relationships emerge as they made sense of mathematical situations. They reflected on their thinking, speculated about mathematical ideas, and drew logical conclusions. The students examined and justified solutions to problems and extended their thinking into higher levels to generalize their solutions to new problems. This article describes how these fourth graders thought like mathematicians and illustrates their thinking through classroom vignettes.
Clarifying Questions and Representing Ideas
The first vignette shows students explaining their mathematical ideas and using visual representations to help others understand their thinking. Hal seeks to clarify the question that the teacher has asked. Interpreting what the problem is asking is an important attribute of the work of a mathematician. Mathematicians clarify questions and restate problems in their own words. Only after understanding what they are looking for can they begin to answer the question.
Teacher. Can more than one circle have the same center? Raise your hand if you think that more than one circle can have the same center. It looks like we have about half and half.
Hal. Can I ask you one thing? If it's like a circle that goes around and has a point in the middle--can every one that goes around have the same middle of the circle? [He uses his hands to show one circle oriented vertically and one oriented horizontally.]
Even though Hal has trouble forming his question, he demonstrates clearly with his hands that he is talking about circles in two planes.
Teacher. That's an excellent question, and I didn't think that anybody was going to ask that. I think what he is basically asking ... is what if he has circles like this ...? [Teacher draws fig. 1].
To make sure that she and others understand Hal's question, the teacher helps him illustrate his thinking; however, she is not satisfied with her sketch. Instead, she uses two rubber bands, looped around fingers on each hand and overlaps the rubber bands.
Teacher. Right, Hal? And the center is in the middle of it. What he is wanting to know is, What if [we] have circles that go like this?
Hal agrees that the teacher is illustrating his idea. Some students initially disagree that this idea, two circles in different planes, could be possible. In the end, the teacher and students accept Hal's idea because the question does not state that the circles must be in the same plane. Through discussing Hal's question, the students move beyond the assumption that their circles must lie in the same plane. Mathematicians know through experience that they must not make unfounded assumptions when solving a problem.
In the following excerpt, we can see that Hal believes that his thinking has value. We can also see that Hal's idea opens up the discussion.
Teacher. Hal has thought of an example of two circles that do have the same center. One circle is headed in this direction in one plane and one, in this direction in a different plane. …