Engaging in the science of pattern and order--in doing mathematics--is effortful and often takes time. There are lots of ideas to learn. Often these ideas show up on lists of "basic skills." For example, children should be able to count accurately, know their basic facts for addition and multiplication, have efficient methods of computing whole numbers, fractions, and decimals, know measurement facts such as the number of inches in a foot or quarts in a gallon, know the names of geometric shapes, and so on. But to master these bits and pieces is no more doing mathematics than playing scales on the piano is making music.

The Principles and Standards document makes it very clear that there is a time and a place for drill but that drill should never come before understanding. Repetitive drill of the bits and pieces is not doing mathematics and will never result in understanding. Drill may produce short-term results on traditional tests, but the long-term effects have produced a nation of citizens happy to admit they can't do mathematics.

### The Verbs of Doing Mathematics

Envision for a moment an elementary mathematics class where students are doing mathematics. What verbs would you use to describe the activity in this classroom? Stop for a moment and make a short list before reading further.

Children in traditional mathematics classes often describe mathematics as "work" or "getting answers." They talk about "plussing" and "doing times" (multiplication). In contrast, the following collection of verbs can be found in most of the literature describing the reform in mathematics education, and all are used in Principles and Standards:

explore | represent | explain |

investigate | formulate | predict |

conjecture | discover | develop |

solve | construct | describe |

justify | verify | use |

These are science verbs indicating the process of "making sense" and "figuring out." When children are engaged in the kinds of activities suggested by this list, it is virtually impossible for them to be passive observers. They will necessarily be actively thinking about the mathematical ideas that are involved.

In classrooms where doing mathematics this way is a daily occurrence, the students are getting an empowering message: "You are capable of making sense of this--you are capable of doing 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.

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