How do the best and the brightest boys do it? Boston's Roxbury Latin School is the oldest continuing operating school in the US and places the highest proportion of it's students in elite colleges. Here is an excerpt from their catalogue.

Mathematics

We live in a fascinating time in which to teach mathematics. New voices have urged the use of calculators and computers, the adding of "relevant" applications for motivation, the encouragement of students to express themselves in writing, and the importance of conceptual learning.

Roxbury Latin has embraced these innovations, but it has also resisted a rush to change for change's sake. We still use problem solving as a motivation for and a natural outcome of the study of each new topic. We expect our students to learn algorithms and pencil-and-paper skills for calculations. Unifying themes and concepts, such as functions, graphing, and factoring, serve to integrate and connect various topics and are repeated in most courses, but at increasingly higher levels of sophistication. Real-world applications also connect topics to a variety of disciplines and expose students to the power and usefulness of mathematics.

The graphics calculator has changed how we teach. We use the calculator in Algebra 2, Pre-Calculus, Calculus, and Statistics. Many of our better students were always good at graphing, but now all of our students, regardless of their ability, can "see the picture." Students who are visual learners and those prone to calculation errors now enjoy greater success. Students who, in the pre-calculator era, might have been uninterested in mathematics, now become excited by their newfound ability to explore and solve problems that arise from real life situations. While it is impossible to predict where this new technology will lead us, we believe that calculators help students complete our courses with a far richer understanding of mathematics than their predecessors had in years past.

We realize, of course, that we must teach students fundamental classroom skills: paying attention, organizing their thoughts, listening to and working with others, translating between real world problems and mathematical models, and communicating their insights effectively. But the goal of our mathematics program is to enable students to analyze carefully, to reason logically, and to think conceptually so that they may not only solve the problems of everyday life but see the "big picture" and sense something of the grandeur and order of the universe.

After Class VI, students at each level are divided into two or three groups according to their background and ability.

Class VI (ie 7th grade) mathematics is designed to give all students a strong background in basic mathematical concepts and skills. Fundamental properties, notation, and relationships are introduced in a more rigorous way than students might have been exposed to before, and mastery is gained through drill work and extensive real-life applications. Time is spent investigating the collection of data (sampling problems, relative error and significant digits in measurements) as well as its presentation and use (tables, graphing, scale drawings, statistical measures), in order to prepare students for Class V science. Helpful calculation short cuts are taught. Topics for the first two-thirds of the year include: different number bases, set theory, divisibility rules and factoring, operations with and conversion among fractions, decimals, and percentages, business applications of percentage, ratio and proportion, rates, distance problems, and English and Metric system units and conversions. The final third of the course introduces students to fundamental geometric definitions, concepts, and constructions. The perimeter, area, and volume of plane and solid figures are studied in detail. Finally, the course introduces students to algebra: operations with signed numbers, simplifying and evaluating expressions, and writing and solving elementary equations. The text is Dolciani, Pre-Algcbra, An Accelerated Course.

Class V (ie 8th grade) mathematics is a full first-year course in algebra using Dolciani's Algebra: Structure and Method. We feel that this is our most important course. It introduces the basic skills that students will use throughout their study of mathematics. The most important of these are factoring, breaking polynomials into smaller, and often linear, parts, and graphing, giving students the ability to see the picture represented by the given algebraic expression. Specific topics covered include: evaluating and simplifying polynomial expressions, factoring, solving first and second-degree equations and inequalities, operations with algebraic fractions, systems of equations, inverse and direct variation, graphing linear and quadratic functions, and solving parts of right triangles by using trigonometry. Structured approaches to solving an extensive assortment of word problems are taught.

In Class IV students continue their study of algebra using Dolciani's Algebra and Trigonometry: Structure and Method, Book 2. Algebraic skills are reviewed and refined during the study of sets of real numbers, inequalities and absolute value, linear equations including systems of equations in both two and three variables, factorization, rational expressions, irrationals, and quadratic equations. The course then goes on to cover more advanced topics including complex numbers, functions and graphing, exponents and logarithms, introductory trigonometry, conic sections, polynomials, and sequence and series. Extensive table work including interpolation accompanies the study of both logarithms and trigonometry, but having demonstrated competence in these skills, students are encouraged to obtain values from their calculators. Word problems are assigned throughout as applicable. Simple motion problems that the students will see in Physics are solved by algebraic techniques such as completing the square.

Class III mathematics provides an introduction to Euclidean geometry. For perhaps the first time, students develop a self-contained system of theorems and results, all built upon a few fundamental axioms and postulates. The concept of proof and the art of mathematical writing are also presented more than in previous courses. Geometric topics include results on lines, triangles, circles, perpendicularity, congruence, and similarity. Algebra and arithmetic are reinforced in this course by work in inequalities, areas and volumes, and proportions. Advanced sections also cover trigonometry. The text for the course is Geometry by Moise/Downs.
 
 

Though two years of algebra and one year of geometry fulfill the School's minimum requirement for a diploma, Roxbury Latin students rarely drop mathematics at this point.
 
 

The Class 11 mathematics course returns again to more algebraic topics and it prepares boys for the Math 2C SAT 11. The text for the course is Dolciani's Introductory Analysis. A comprehensive coverage of topics provides solid preparation for college-level ' coursework in Calculus in Class 1. These topics include analytic geometry, sequence and series, functions, polynomials, trigonometry, and exponents and logarithms. In addition we lay the groundwork for Calculus by covering limits and derivatives of all algebraic and some transcendental functions, as well as introducing antiderivatives and their applications.
 
 

In Class I students cover - according to their ability and background - either the AB or BC Advanced Placement Calculus syllabus. This is our culminating course and its foundation consists of all the algebraic, trigonometric, and geometric skills that students have learned in their previous courses. Both groups study derivatives and integrals of algebraic and transcendental functions. Derivative applications include tangents and normals, curve sketching, and related rate and max/min word problems, while area under and between curves and volumes of revolution are studied as applications of integrals. The BC group studies advanced integration techniques, differential equations, and sequence and series as well.
 
 

We also offer Advanced Placement Statistics as an alternative to Calculus. In this course, students explore data and interpret patterns (and departures from patterns) by observing graphical displays of distributions. They plan a study, decide what and how to measure, use random sampling, and determine the error inherent in their survey. They also anticipate patterns and produce models using probability and simulation, and then use statistical inference to confirm the models that they produce.
 
 

In addition to course work, students compete in the Continental Mathematics League and the New England Mathematics League. We also participate annually in the Massachusetts Mathematics Olympiad.