MATHEMATICS AS A DEDUCTIVE SCIENCE
City Tutoring Mathematics Academy is devoted to the disciplined study of mathematics.
We regard mathematics not as a collection of techniques for passing examinations, but as a deductive science:
an ordered body of knowledge founded upon definition, logic, proof, and reason.
Instruction emphasizes rigor, abstraction, symbolic fluency, and precise thought. Students are expected to
attend carefully, reason correctly, and express their work with clarity and discipline.
The aim of study is not merely to obtain answers, but to understand the principles from which those answers
necessarily follow. Mastery is achieved through effort, attention, and intellectual discipline.
Assertions without proof, shortcuts without understanding, and habits of intellectual carelessness have no place
in mathematical study. The purpose of instruction is not simply to produce correct answers, but to cultivate
mathematical judgment and genuine understanding.
COURSE CATALOG
[00] BASIC MATHEMATICS
The preparatory course for all subsequent mathematical study.
Students acquire mastery of the fundamental operations of arithmetic while developing the precision and
discipline necessary for higher mathematics. Topics include computation with whole numbers, fractions,
decimals, percentages, ratios, and proportions.
In addition, students are introduced to the symbols, notation, and elementary principles of logic that form the
language of mathematics. Attention is given not merely to obtaining correct results, but to presenting work
clearly, accurately, and according to accepted mathematical conventions.
Successful completion of this course is required before enrollment in Algebra I.
[01] ALGEBRA I
An introduction to algebra as a formal system of mathematical relationships. Students study equations,
inequalities, functions, and symbolic operations while developing fluency in mathematical notation and reasoning.
The course employs an elementary set-theoretic approach, introducing students to the language of sets,
relations, and functions as foundational concepts of modern mathematics. Particular attention is given to
precise definitions, logical argument, and the orderly development of mathematical ideas.
Students are expected not merely to manipulate symbols, but to understand the structures and principles that
give those symbols meaning. The course establishes the intellectual foundations for all subsequent study in
mathematics.
[02] ALGEBRA II
A continuation of Algebra I devoted to the systematic study of algebraic structures and relationships.
Students examine polynomial and rational expressions, exponential and logarithmic functions, complex numbers,
and advanced methods of equation solving. Throughout the course, emphasis is placed upon general principles,
formal reasoning, and the recognition of recurring mathematical patterns.
Particular attention is given to the development of algebra as a coherent deductive discipline and as a
preparation for higher mathematical study.
[03] GEOMETRY
A rigorous study of Euclidean geometry founded upon definition, deduction, and proof.
Students investigate congruence, similarity, circles, geometric constructions, and coordinate methods while
learning to distinguish carefully between assumption, proposition, and demonstration. The course emphasizes the
construction of valid mathematical arguments and the orderly development of geometric knowledge from first
principles.
Geometry is presented not merely as the study of figures, but as a model of logical reasoning and deductive thought.
[04] PRE-CALCULUS
A comprehensive study of the mathematical ideas that prepare the student for the rigorous study of calculus and
analysis.
Topics include functions, trigonometry, analytic geometry, and elementary mathematical structures. Emphasis is
placed upon abstraction, symbolic fluency, and the unification of concepts developed throughout the preceding
algebraic and geometric courses.
The course seeks to cultivate a mature understanding of mathematical relationships and to establish the
intellectual foundations necessary for advanced study.
[05] CALCULUS I
The first systematic study of the differential and integral calculus.
Students examine limits, continuity, differentiation, and integration with particular attention to the concepts
and principles upon which the subject rests. Rather than treating calculus as a collection of computational
techniques, the course presents it as a logical development arising from the study of change, quantity, and
functional relationships.
Emphasis is placed upon careful reasoning, precise notation, and conceptual understanding. Students are
expected to comprehend not only how mathematical results are obtained, but why they follow.
[06] CALCULUS II
Sequences, series, transcendental functions, techniques of integration, and further study
of the definite integral. The course continues the development of rigorous mathematical
thought introduced in Calculus I.
[07] CALCULUS III
Functions of several variables, vector-valued functions, multiple integration, and vector
calculus. Attention is given to the geometric and structural ideas underlying multivariable
analysis.
[08] LINEAR ALGEBRA
Vector spaces, linear transformations, matrices, determinants, eigenvalues, and eigenvectors.
The course emphasizes abstraction and generality, introducing students to one of the central
organizing ideas of modern mathematics.
[09] ABSTRACT ALGEBRA
Groups, rings, fields, homomorphisms, and quotient structures. Students are introduced to the
study of algebraic systems as mathematical objects in their own right and develop familiarity
with proof-based mathematics.