en specific problem-solving strategies as well (a la Polya's four-step process), such as looking at simpler cases and discerning patterns, drawing pictures, or working backwards; they should never be allowed to say "I don't know how to start." They should be encouraged to reflect on a problem after they have solved it, asking whether the answer obtained makes sense and thinking of ways to go beyond the specifics to generalize their ideas.
Another aspect of problem-solving is to think about mathematics algorithmically, and the study of computer programming is an excellent choice for high school students preparing for college. The computer is an excellent tool when used effectively, and we would encourage high school teachers to spend some time applying such software as spreadsheets, computer algebra systems, statistical software, or interactive geometry programs to nontrivial mathematics problems.
2. Reasoning and Proof
The most important word in learning mathematics is "why", not "how" or "what". Students should, for example, understand why the quadratic formula gives the roots of a quadratic polynomial. They should expect to justify their solutions to problems, not merely to provide answers. They should understand some simple logic (for example, what it means for an "if...then" statement to be false). They should understand the important role of definitions in mathematics and distinguish between definitions, axioms, and theorems. They should understand the role of counterexamples, and the idea that in mathematics we require a convincing argument before accepting something as true, not just some numerical evidence or the word of an authority. Mathematics should be presented to some extent as an experimental science, in which the student or mathematician investigates situations and makes conjectures, but then tries to understand why these conjectures are or are not true.
Currently students come to college thinking that there is no need to communicate ideas in a mathematics class — that what they write down is just for themselves and need not communicate anything to anyone, because all that matters is the answer. Nothing could be further from the truth. Students should be taught to organize their mathematical thoughts and communicate them clearly, orally and in writing. (For example, they should never introduce a variable for solving a problem using formal algebra without making it clear what the variable represents.) They should also be required to spell mathematical words correctly.
Of course, before writing come reading. One might argue that the development of reading skills is not the responsibility of the mathematics teacher, but such division of labor is counter-productive. Students should learn how, for example, to read a page of text, out of which they will extract the relevant information for the analysis and solution of a problem. Moreover, they should be expected to read critically assertions related to numbers, whether presented in the text or in tables or figures.
Although abstract mathematical reasoning and skills are important, we want students to think of mathematics as having application to the real world as well. We hope that teachers will try to do a lot of mathematical instruction within realistic contexts, to motivate the mathematics and give students a reason to want to know the answers. When teaching applied problems, it is important to pay attention to units and to precision of reported answers.
The crux of mathematics is abstracting — translating unclear, informal, incomplete, or badly worded problems into precise problems using the languages of algebra, geometry, and other formalisms of mathematics. This is the mathematical modeling process. In order to do this well, students must value reading in mathematics classes as much as they value it in language arts, science, or humanities classes.
We want students to use mathematical notation correctly and confidently. They should understand equivalent ways of representing the same concepts and easily move among multiple representations (for example, vie