Ad Fontes Upper School Math Program

At Ad Fontes, our classical Christian educational approach uniquely prepares students to become excellent mathematicians, scientists, engineers, and programmers…

by equipping them with both essential math knowledge and the capability to think and engage real mathematical challenges critically and creatively as individuals and in small groups.

The teaching philosophy of modern math changes about every decade between a focus on memorization and algorithms and a focus on problem solving and soft skills. Classical education incorporates both into a coherent instructional methodology that equips students to be contributors to mathematical community. This means students work at hard and rewarding problems and then use that experience and understanding to creatively develop their own solutions.

Good, hard problems have table stakes:

The mathematical grammar that makes the problem accessible. So, we also spend time drilling the fundamentals. We emphasize internalizing vocabulary, algorithms, and new techniques. Not for their own sake, but to allow students to work on problems that point them at beauty and truth.

Ad Fontes students are not intimidated by new or difficult challenges and are exceptionally prepared to engage with mathematics in a changing world that requires curiosity and initiative. AFA graduates have been exceedingly well prepared for success in technical college and career fields.

Upper School Courses

Algebra I

The foundation for our upper level courses is laid in the routine of the Algebra I course. We build on the arithmetic and fundamentals of the arithmetic and Pre-Algebra courses, transitioning from the grammar to the dialectic stage. Much of the year is spent firmly in grammar mode, so that students can solidify basic concepts. We emphasize solving and graphing equations, with a later focus on factoring polynomials, to prepare students for the rigor of the future classes. Students reinforce and strengthen skills through small group engagement.

Geometry

Students will enter the Great Conversation as it pertains to mathematics, learning the foundational concepts of Geometry from the source. They will study the definitions, postulates, and common notions at the basis of Geometry as well as learn to follow and replicate deductive proofs, with attention to learning to justify their mathematical work, in this class and beyond. They will study plane and space figures, with special emphasis on congruence, similarity, and transformations. Using Euclid’s Elements as our text creates opportunity for discussion in class, as students study his proofs, come to class with questions, and discuss the concepts. In the first semester, students make monthly presentations of proofs, explaining these foundational concepts. By second semester, students engage in small group presentations to explain and support their reasoning for their answers.

Algebra II

Building on the formative work students have done in Euclidean Geometry, this course cements the idea that explanation and argumentation are central to mathematical work. We approach algebra with Diophantus, discussing what constitutes an algebraic proof and tracing Diophantus’s arguments. After Diophantus, we derive the quadratic formula from first principles, and dig into conic sections. The year culminates with an extended unit on trigonometric definitions and arguments, as students prove familiar theorems (like the perpendicular line theorem) and explore new theorems about triangles and circles.

Students learn to use programs – both calculator and computer – to help with the complex calculations their arguments require. By the end of the year, students will come to enjoy the creative play of constructing their own mathematical explanations about numbers, equations, and figures.

Pre-Calculus

As the saying goes, there are many mansions in the house of mathematics. Pre-Calculus introduces students to the breadth and variety of mathematical thought. The year is organized as a series of short courses on number theory, data analysis, cryptography, analytic trigonometry, game theory, linear algebra, and graph theory. One aim of the course is to show students that there is something to love in mathematics for everyone. Throughout the course, we focus on functions as a tool to think about mathematical objects, and examine the different styles of argument and proof present in the different branches of mathematics. We read original sources, including some Euclid, Euler, and Diophantus. We also read modern mathematical writing, including texts by Conway, Rivest, Shamir, and Adleman, Calkin, and Wilf.

The course offers students many opportunities to construct mathematical arguments of their own. Our school gives outside the box awards to students who develop ideas new to the teachers, and this course generates most of the math awards because students have many opportunities to construct creative answers to beautiful questions. The course culminates in a thesis of sorts where students choose a hard problem from one of the areas they’ve studied and spend several weeks tackling it as best as they can. They then present the results to their classmates!

AP Calculus I (AB)

Students finishing our Pre-Calculus class enter the Calculus class excited to do more mathematics because they have learned to enjoy the subject by enjoying the pursuit of solutions to the various problems it presents. Much of what we do in Calculus brings the rigor of the study of mathematics back to the students who have learned to enjoy the subject.

Calculus is a beautiful way to look at the created world. We see rates of change and how those rates are determined and interrelated. We see continuity of functions, and work with the limits and discontinuities there. This course is firmly planted in the Rhetoric stage, and yet we are continually adding basic pieces to our arsenal of mathematical tools. We use the language of functions developed in Algebra II and Pre-Calculus, using polynomial, logarithmic, exponential, and trigonometric functions in our studies of rates of change, tangents, area, volume, and other problems. Students are prepared for the AP exam.

AP Calculus II (BC)

In this course we explore various topics for a fuller understanding of power of calculus. Students learn to represent functions in the cartesian plane or in polar form. We use this to springboard into discussion of parametric equations of all kinds, and then dive into a representation of functions as series, to allow us to use the calculus with polynomials in lieu of transcendental functions. Students are prepared for the AP exam.

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