Offices & Resources

Mathematics


About

The purpose of a St. Mark’s mathematical education is to develop students’ understanding of and appreciation for the ability to make sense of the world through the lens of mathematics: space and number, logic and pattern. We nurture students’ development by challenging them to be creative, critical and tenacious problem solvers, able to both share and reflect on their process and results with clarity and precision. Throughout, we seek to nurture and develop an appreciation for the joy and beauty of mathematics in all St. Mark’s graduates.

The mathematics department has established the following core values: curiosity, tenacity, and clarity. These values inform and guide all that we do. The program has three principal aims: first, to establish skill and confidence in applying mathematical techniques; second, to convey the analytical power of mathematics in modeling practical applications; third, to develop sound reasoning and communication around the logical structure of the subject. The search for patterns, the recognition of analogies, and the development of various strategies for solution provide the student with insight into and understanding of the problem-solving process.

Throughout our curriculum, emphasis is placed on each student doing mathematics. We believe it is essential students actively write and speak their mathematics in order to learn to develop sound mathematical reasoning and communication skills. Graphing calculators and computers are integrated into our teaching as they have become useful in exploring and illustrating mathematical content. TI-83 or TI-84 calculators are required for most classes.

The mathematics department adheres to the following policies for placement of students in courses and in regards to summer or alternative courses.
  • Every new student completes a placement test. A student is placed in a course based on the score on this test, their SSAT score, the student’s prior course work and the teacher recommendation. There is an additional honors placement test for those students deemed qualified to take an honors level course in Algebra II or Precalculus
  • Placement for returning students is based on their performance in their current courses and the recommendation of their teachers. Students recommended for Honors Algebra II or Honors Precalculus take an honors placement test.
  • While students are allowed, and at times, encouraged to do summer work in math, the St. Mark's math department will only grant credit for a summer or alternative course in Geometry. Credit will be earned, only if the student satisfactorily completes the course and passes the St. Mark’s departmental final exam. It is our belief that when studying specific mathematics topics for the first time, students should not expect that a summer course or alternative course will provide them with the necessary depth of understanding of the material. Students who have performed poorly in a course, who wish to preview a topic, or who want enrichment are encouraged to do summer or alternative courses.

Faculty

Department Chair
email

[Full Profile]

Seth Battis

Jacob Backon


Yue Cao


Allyson Brown


Karen M. Bryant


Scott Dolesh


Eben Healy


Brian Lester


Kinne McBride


Rick Umiker


Courses

Algebra I / Algebra I Enhanced

This course is an introduction to algebra. Extensive attention is given to developing algebraic and graphical problem-solving skills. The coordinate geometry of lines and parabolas, fractions, rational expressions, integral exponentials, and linear and quadratic equations through the quadratic formula are included with an eye toward developing confidence and agility in problem solving. Algebra I Enhanced is designed for students who have already been exposed to some or all of the topics of Algebra I, but who need further reinforcement to support higher levels of study in math. Placement in these courses is determined by the Chair of the Math Department based on a placement test and the background of the student.

Geometry

This course pursues a deepening of the students’ understanding of plane and solid geometrical figures and a building of their abilities to analyze and communicate mathematically. Much attention is given to logical structure and the writing of mathematical arguments as well as to geometrical problem solving.

Algebra II

This course reviews, then extends, the study of algebra with greater attention toward function. Topics include linear and quadratic functions and relations, polynomial functions, rational expressions, exponents, and logarithms and elementary trigonometry. Students are expected to broaden their problem-solving skills and techniques. Increased attention is given to multiple representations with graphs and charts used to illustrate and enhance algebraic manipulation. Honors Algebra II YearThis is an accelerated course in algebra in which topics are pursued in greater depth as well as at greater speed. Students are expected to have strong intuition and motivation for the study of mathematics. Extensive work with trigonometric functions is usually encountered, reaching beyond the topics of Algebra II. (Prerequisites: teacher recommendation, placement test, and Departmental permission)

Statistics, Functions and Trigonometry

This yearlong course is for students who have completed Algebra II. It includes an introduction to the basic concepts of statistics, a review of linear, quadratic, exponential and logarithmic functions and a study of trigonometry. Modeling data using the functions and using statistics to verify the validity of the model is an emphasis. There is extensive use of the software program Fathom. Topics include: descriptive statistics, correlation, quadratic, exponential, logarithmic regression, circular trigonometry, sequences and series. This course will not be sufficient preparation for students to take a Calculus course at St. Mark’s. (Prerequisite: completion of Algebra II. This course is not open to students who have already completed Precalculus.)

Mathematical Modeling in the Life and Environmental Sciences

Fall

We will look at different ways to use mathematics to model phenomena in the natural sciences with special attention to relevant current events. Students will follow the mathematical modeling cycle to develop predictive and analytic tools to better understand complex natural systems, such as predator-prey relationships’ impact on ecosystems or the factors leading to the spread—or containment —of infectious diseases. Students will develop skills in a variety of common and professional modeling tools.
Essential questions
• How can natural phenomena be predicted?
• How can we determine an effective model for a particular phenomenon?
• Why and when do mathematical models fail?
(Prerequisite: completion of Statistics, Functions and Trigonometry or Precalculus. This course may be taken concurrently with Precalculus with departmental permission.)

Mathematical Modeling for Finance and Economics

Spring

We will examine how mathematical models can be used to answer key questions in finance and economics. What is the time value of money? How do we understand and consider models of risk and return? Can we model the fluctuation of share prices to our advantage? We will use simulations to revisit scenarios in the world of finance, such as the recent over-application of the Black-Scholes equation. Students will use the tools of linear algebra to model problems in economics and will develop skills in a variety of common and professional modeling tools.
Essential questions

  • How do we measure and manage risk?
  • How do we predict the seemingly unpredictable?
  • Why and when do mathematical models fail?
  • How does our manipulation of a system change the system?
  • (Prerequisite: completion of Statistics, Functions and Trigonometry or Precalculus. This course may be taken concurrently with Precalculus with departmental permission.)

    Precalculus

    Extensive discussion of polynomial, rational, exponential, logarithmic, and trigonometric functions and their properties and applications is encountered. Sequences, series, and limits are also introduced. A graphic scientific calculator is required at this level for its aid in visualization and calculation. Students gain skill in analyzing functions and drawing connections between symbolic, graphic, and numerical representations. This course is primarily designed as a final preparation for the study of calculus. (Prerequisite: completion of Algebra II and departmental permission)

    Honors Precalculus

    This is an accelerated course in the elementary functions that includes an introduction to the study of calculus during the second half of the year. Students are expected to have exceptional intuition and motivation for the study of mathematics. (Prerequisites: one full term of analytical trigonometry from Honors Algebra II, a placement test and departmental permission)

    Calculus

    This is an introduction to calculus designed for students who want to study calculus before college, but who do not yet feel prepared for the pace or depth of an Advanced course. Students who have not already done so are encouraged to consider math electives as an alternative unless they expect calculus to be a required part of their future studies. (Prerequisites: Precalculus and departmental permission)Advanced Statistics YearThis course introduces the basic concepts of statistics. Every student will become familiar with the use of the statistical functions found in the more useful graphing calculators. Computer statistical applications are studied using one which is commonly available and well recognized. By completing the course, a student will also be prepared to use statistics in the various pursuits which draw on this mathematical process to study and analyze data. (Prerequisites: Precalculus or an Honors grade in Statistics, Functions, and Trigonometry and departmental permission)

    Advanced Statistics

    This course introduces the basic concepts of statistics. Every student will become familiar with the use of the statistical functions found in the more useful graphing calculators. Computer statistical applications are studied using one which is commonly available and well recognized. By completing the course, a student will also be prepared to use statistics in the various pursuits which draw on this mathematical process to study and analyze data. (Prerequisites: Precalculus or an Honors grade in Statistics, Functions, and Trigonometry and departmental permission)

    Advanced AB Calculus

    This course develops an understanding of the major concepts of calculus. Applications and techniques of differential and integral calculus are illuminated. The concepts of calculus are studied with problems and results expressed graphically, numerically, analytically and verbally. Graphing calculators are used to help the students visualize critical concepts. Successful completion of this course is roughly equivalent to one-half year of college calculus. (Prerequisites: completion of Precalculus and departmental permission)

    Advanced BC Calculus

    This course is an extension and enhancement of Advanced AB Calculus. It is a more rigorous course, with a more extensive syllabus, including an introduction to power series expansions, vector calculus, and the calculus of polar curves. Successful completion of this course is roughly equivalent to a full year of college calculus. (Prerequisites: completion of Honors Precalculus and departmental permission)

    Advanced Topics in Mathematics

    Semester: The following courses are offered on a rotating basis as dictated by the instructor and the interest and background of the potential students. (Prerequisite: completion of Advanced AB or BC Calculus and departmental permission).

    Abstract Algebra: This course is an introduction to the principles and concepts of modern abstract algebra. Topics will include groups, rings, and fields, with applications to number theory, the theory of equations, and geometry.

    Linear Algebra (Fall 2015–2016): This course presents the main concepts and terminology of Linear Algebra. Topics include: matrices, linear equations, determinants, eigenvalues and eigenvectors, vector spaces and linear transformations, and matrix diagonalization. Applications typically include polynomial interpolation, electrical networks, cryptography, computer graphics, Markov chains, and linear programming.

    Multivariable Calculus: This course re-examines the differentiation and integration processes by looking at it from the perspective of more than one variable. Topics will typically include: partial derivatives, level curves and gradients, double and triple integrals, Lagrange multipliers and optimization in several variables.

    Differential Equations: This course offers an introduction to the theory, solution techniques, and applications of ordinary differential equations. Models illustrating applications in the physical and social sciences are investigated. Linear differential equations and linear systems of differential equations are explored in depth. Some numerical techniques to solve differential equations will be also introduced as well as using power series and the Laplace transform.

    Number Theory (Spring 2015-2015): In this course, students will investigate topics in number theory, which will include: the elementary notions of primes and divisibility, factorization, congruences; quadratic residues and reciprocity; sums of squares; continued fractions and approximations; and selected Diophantine equations.

    Complex Analysis: This course covers the basic concepts and applications of complex analysis by extending the concepts of calculus to the complex plane. There will be an emphasis on computation and applications. Topics will include: Complex differentiability, Cauchy-Riemann differential equations, contour integration, residue theorem, harmonic functions, and geometric properties of complex mappings.

    Mathematical Research

    Fall

    This course is for students who have successfully completed one year of Advanced Topics. The course includes independent or collaborative research on research problems chosen by the students. The students either write a report or make presentations to each other, the mathematics department and interested students at least once each marking period which will serve as progress reports. A goal of the course is try to get their research published in the Math Horizons or Focus publications of the Mathematical Association of America. (Prerequisite: completion of one year of Advanced Topics and departmental permission)