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.
