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Mathematics

The Math Department at Columbia Prep believes everyone learns math best through exploration, discovery and problem solving. Our students are asked to be thoughtfully engaged in problem solving at every level and be inspired by mathematics as well as appreciate the process of productive struggle.

The Math Department collaborates to design the problems that shape our curriculum with the belief that every concept is best explored from four different perspectives: numerical, visual, algebraic and verbal. Teaching through these four lenses ensures genuine and comprehensive understanding. Teachers help students recognize patterns and construct knowledge through questions and observations. In the Columbia Prep mathematics program, students begin with an introductory Algebra 1 course followed by Geometry, Algebra 2, Precalculus and Calculus. All of the courses are problem-based and student-centered: students are exposed to problems for which they have not been given specific procedures to follow, allowing them to grow as mathematicians through a productive struggle that helps develop independence, creative thinking and flexible mathematical habits of mind. Each of the classes are taught in a way that reflects the department’s commitment to all eight New York State Standards for Mathematical Practice. Our math classrooms are dynamic places. Students are often at the board working in groups or pairs. Creative assessments and homework assignments stretch students beyond memorization and promote collaborative, thoughtful work.

 

Course Offerings

Algebra I/Honors Algebra I

The Algebra I course, designed by the faculty, has no textbook and the sequence of content is similar to that in The Art of Problem Solving: Introduction to Algebra. Hundreds of problems are curated online (through Google docs) and accessed by students through the School’s Schoology system. In working though these problem sets, students learn to see structure in algebraic expressions, perform arithmetic on polynomial and rational functions, generate and solve equations that describe numerical and geometrical relationships and create and analyze graphs of these equations. Students use both the online graphing calculator Desmos and the mathematical computation program Mathematica to enhance and inform their exploration of these topics.

Geometry/Honors Geometry

In Geometry, students are given a problem-based introduction to a standard treatment of Euclidean plane geometry. There is no textbook, but students work on problem sets that they access through Schoology and the sequence of topics is roughly the same as The Art of Problem Solving: Introduction to Geometry. In working though these problem sets, students learn about congruent and similar polygons, circles and their relationship to trigonometry and the importance of scale factor when dealing with problems of area and volume. In addition to solving problems, students prove theorems about the invariant properties of shapes in two and three dimensions. Classical straightedge and compass construction is introduced along with the dynamic geometry software Geometer’s Sketchpad. An emphasis on deductive reasoning and creative problem-solving is present throughout the course.

Algebra II/Honors Algebra II

The concept of functions and linear equations is explored in Algebra II, along with an emphasis on creating mathematical models for more complex numerical and geometric patterns. Students continue their study of polynomials with higher degree polynomial equations and graphs, solve problems involving arithmetic and geometric sequences and series, and are introduced to the basic concepts of combinatorics and probability. The class uses Discovering Advanced Algebra: An Investigative Approach as well as the online graphing calculator Desmos, the mathematical computation program Mathematica and the TI-89 graphing calculator.

Precalculus/Honors Precalculus

In Precalculus, students explore trigonometry from a functional perspective and are introduced to rational, exponential and logarithmic functions. Emphasis is placed on algebraic invariants over entire classes of functions and the course ends with an introduction to limits. The class uses CME Project’s Precalculus as well as the online graphing calculator Desmos, the mathematical computation program Mathematica and the TI-89 graphing calculator.

Mathematics Electives

About our Electives program: A hallmark of Columbia Prep is the value placed on individual students and their interests. Beginning in the ninth grade, the curriculum offers many electives in art, music, theater, technology and physical education. As the students progress through high school, they are given increasing autonomy to choose their courses and, by the time they reach their junior and senior years, they are creating their entire academic programs in all subjects—English, history, math, science, world language and all other elective courses. Harnessing and building upon their passions within the context of our courses results in interested, successful and independent graduates who are ready for the next educational step in their lives. The elective system allows for flexibility in course offerings, so no two academic years are identical.

Mathematics Electives: Some previous Mathematics electives offered in the Prep School are:

Number Theory and Pattern Recognition
Seminar in Advanced Mathematics
Geometric Applications and Statistical Analysis
Advanced Topics in Geometry
Computer Mathematics

Elective classes are taught in a way that reflects our department’s commitment to teaching mathematics for understanding through a problem-based style of instruction.

AP Statistics

This course is designed to teach high school students the equivalent of a one semester college statistics course. In it, students learn to collect, explore and draw conclusions from data. The main purpose of course projects is for students to gain strong experience in developing statistical studies and making sound connections and judgments between the design and the results of the experiment.

AP Calculus AB/BC

In either Calculus AB or Calculus BC, students are introduced to fundamental notions such as limits, continuity, differentiation and integration. Many students will also sit for either the AP Calculus AB or Calculus BC exam. Each exam deals with challenging applications of the derivative and a more sophisticated approach to infinite series. The Calculus courses use the Calculus: Graphical, Numerical, Algebraic textbook as well as the online graphing calculator Desmos, the mathematical computation program Mathematica and the TI-89 graphing calculator.

Advanced Mathematics Seminar

 This course is designed for students concurrently enrolled in BC Calculus and have a particular interest in continuing their studies of mathematics in college. The two major topics of the course are Linear Algebra (matrices, vector spaces, dot and cross products, eigenvalues and eigenvectors) and Abstract Algebra (symmetry, fields and groups). A variety of other higher-level mathematics topics (probability, game theory and complex analysis) are explored. These are considered common gateways to developing the mathematical sophistication expected at the university level. Proof, mathematical rigor and elegant approaches to problem-solving are emphasized in this course aimed at mature and self-motivated young mathematicians.

New York State Standards for Mathematical Practice
  • Make sense of problems and persevere in solving them
  • Reason abstractly and quantitatively
  • Construct viable arguments and critique the reasoning of others
  • Model with mathematics
  • Use appropriate tools strategically
  • Attend to precision.
  • Look for and make use of structure
  • Look for and express regularity in repeated reasoning