wing 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 unknown or arbitrary quantities and how they can change, the use of the equals sign as expressing a relationship and not as a command to perform a calculation, and the concept of a function in mathematics (which is central to all advanced work).
They should have technical skills that are usually taught in courses with titles like Algebra I and Algebra II: solving equations and inequalities, isolating variables in an equation, solving systems of equations, and manipulating algebraic expressions of all kinds (fractions, exponents, logarithms), using properties like the distributive law — and avoiding incorrect symbol-pushing, such as squaring a binomial by squaring each term.
They should completely understand the way graphs in the coordinate plane represent relationships between variables. They should be able to work with ratios and proportions to solve problems (and know when proportional reasoning does not apply). Students who plan to pursue advanced work (for example, engineering majors) should have a solid understanding of trigonometry, as well.
Most students should not be encouraged to study calculus in high school; rather they should be given a solid background in precalculus areas, which will better prepare those who need calculus to study it in college. However, those students who have completed this preparation before their senior year can take AP Calculus with the intent of earning college credit by passing the national exam and going on to subsequent courses in college.
Students should have a good intuitive grasp of geometry facts (for example, that opposite sides of parallelograms are congruent, and the Pythagorean theorem) and knowledge of the vocabulary needed to express them. They should understand the idea that mathematical objects can be defined, axioms assumed, and theorems proved; and geometry provides a good vehicle for doing these things. They should confront the idea of what it means for two geometric objects to be the same or different in various ways (ideas of congruence and similarity, and in t