erms 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 of computer applications.
In summary, we want students to come to college thinking that mathematics makes sense, that they can solve nontrivial problems (both in pure mathematics and within real-life contexts) by reasoning and using powerful mathematical tools, and that they are expected to communicate their solutions effectively. They need to be on top of arithmetic and algebraic skills. But perhaps most important of all, students need to develop some degree of love and enthusiasm for mathematics. We would like to see a typical high school student think like a mathematician, develop his or her critical thinking, apply the critical thinking skills in solving problems, and be able to see the interconnections among different areas of mathematics and between mathematics and the real world.
To accomplish this, students should be encouraged to take as much mathematics as possible (at least learning well the content of two years of algebra and a year of geometry, so that we don't have to offer remedial courses in college). A fourth or fifth year of college-preparatory mathematics is to be strongly encouraged, such as some advanced concepts of algebra (e.g., including mathematical induction, set theory, formal logic, the binomial theorem, functional transformations, advanced work with exponential and logarithmic functions, matrices, and lots of applications to real-life situations), trigonometry, discrete mathematics, probability, and statistics. (Students should not be rushed to take calculus in high school, and studying calculus outside the AP program makes no sense at all.) It may well be that not doing a mathematics course in their senior year is likely the single most damaging choice students can make in high school with regards to their success in college.