own 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 terms of measurable parameters like area) and how these notions can be understood through transformations. They should understand the ideas of symmetry.
Geometry is important because of the level of mathematical and critical thinking required, its connection to algebra via analytic geometry (which should be given more emphasis at the high school level), its relationship to trigonometry, and the visual aspect of mathematics embedded in it. This is an area of mathematics where early introduction to the importance of definitions, proofs, logical reasoning, and making connections will be highly productive.
Students should understand the idea behind measuring attributes of objects, the limitations inherent in measurements (notions of accuracy and precision), and the arbitrariness of most units of measurement. They should understand how to work with units the same way one works with factors and fractions. They should understand the effect on measurements when sizes change (for example, that area changes as the square of linear measure, and volume changes as the cube). They should know how to compute some basic derived measurements (such as the area of a circle in terms of its radius).
5. Data Analysis and Probability
Although we would not assume any specific knowledge in probability or statistics in our courses, every informed citizen (i.e., every high school graduate) should understand some basic notions in these areas, so as to be able to interpret the results of political polls or medical studies, know how unlikely they are to have success when gambling, and know what is meant by such terms as mean, median, and mode. Beyond that, developing critical thinking skills by studying these subjects would be a bonus. It would certainly be better for students to take a good probability and statistics course in high school than to rush to study calculus. A course involving probability also provides an opportunity for discussing combinatorics and other aspects of discrete mathematics, which are increasingly important in light o