 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, viewing functions as rules, sets of ordered pairs, tables, graphs, and mappings). They should be taught to avoid nonsensical representations (such as price signs one sees in stores confusing dollars and cents, or misuses of the equals sign in chains of calculations).
1. Number and operations
Students should be proficient in the basic arithmetical calculations on integers and fractions, mentally, with paper and pencil, and with calculator. They should not have to — or want to — use a calculator for problems that can be done mentally. They should understand the different kinds of numbers and their various representations (for example, being comfortable converting among fractions, decimals, and percentages). They should also have "number and operation sense" — knowing intuitively, for example, that for positive numbers, dividing by a larger number produces a smaller quotient. We find that many students are hopelessly confused by such things as division involving 0, the difference between A
divided by B
divided by A
, and even adding fractions. Students should develop a good intuition for how the arithmetic operations relate to concrete situations — for example, to understand how the remainder from an integer division may provide the answer to an applied problem.
The important link between arithmetic and algebra should be emphasized. We find that students who cannot simplify arithmetic expressions or are weak in applying critical thinking to problem solving in arithmetic will show the same deficiencies in algebra.
Technology should be used appropriately, efficiently, and creatively. Students need to understand that they need to think through the solution of a problem and not try to use the calculator to replace analysis.
As stated above, algebra should be viewed as an extension of arithmetic. In arithmetic we do operations on specific numbers we know; in algebra we do the same operations on numbers we do not know (but are thereby able to discover). Students should understand deeply the use of variables to represent unkn