What Does It Mean to Do Mathematics?

What Is Basic in Mathematics?

In a climate where "basics" are once again a matter of public discussion and there is an unrelenting pressure on teachers to raise test scores, it is useful to ask, "What is basic in mathematics?" The position of this text is as follows:

The most basic idea in mathematics is that mathematics makes sense!

• Every day students must experience that mathematics makes sense.
• Students must come to believe that they are capable of making sense of mathematics.
• Teachers must stop teaching by telling and start letting students make sense of the mathematics they are learning.
• To this end, teachers must believe in their students--all of them!

Every idea introduced in the mathematics classroom can and should be completely understood by every child. There are no exceptions! There is absolutely no excuse for children learning any aspect of mathematics without completely understanding it. All children are capable of learning all of the mathematics we want them to learn, and they can learn it in a meaningful manner in a way that makes sense to them.

An Environment for Doing Mathematics

Look again at the verbs of doing mathematics. They are action verbs. They require reaching out, taking risks, placing ideas out where others can see. Contrast these with the verbs that might reflect the traditional mathematics classroom: listen, copy, memorize, drill. These are passive activities. They involve no risks and little initiative. Doing mathematics takes effort and initiative.

Though thinking, reasoning, and sense making can be fun, it can nevertheless be a bit frightening to stick out your neck when no one tells you exactly what to do. The classroom must be an environment where doing mathematics is not threatening and where every student is respected for his or her ideas. Students should feel comfortable taking risks, knowing that they will not be ridiculed if they are wrong.

The teacher's role is to create this spirit of inquiry, trust, and expectation. Within that environment, students are invited to do mathematics. Problems are posed; students wrestle toward solutions. The focus is on students actively figuring things out, testing ideas and making conjectures, developing reasons and offering explanations. Students work in groups, in pairs, or individually, but they are always sharing and discussing. Reasoning is celebrated as students defend their methods and justify their solutions.