Courses
MATH 486: CONCEPTS BASIC TO SECONDARY SCHOOL MATHEMATICS
Content: Math 486 can be described as the rudiments of analysis and algebra underlying theorems used in secondary mathematics. Math486 examines the principles of analysis and algebra underlying theorems concerning fields  especially the rationals, reals, and complex numbers; and concerning functions  especially polynomials, exponential functions, and logarithmic functions. Mathematical underpinnings of these ideas can serve as intellectual resources for secondary teachers. Major topics covered by Math486W11 vary from year to year, but have included:
 Properties of fields, including the parallels between Q[√p] and C=R[i], for positive integers p such that √p is irrational.
 Properties of the rational numbers, including density.
 A rigorous description of the long division algorithm for integers and for polynomials; invariance of the sequence of remainder terms for a particular integer dividend, assuming an integer divisor.
 The proof that a degree n polynomial must have exactly n complex roots (counting with multiplicity); the proof establishing an equivalence between factors of a polynomial and roots of a polynomial.
 Limits, convergence, and divergence of sequences of real and complex numbers; accumulation points of sets of real and complex numbers.
 Limits, convergence, and divergence of sequences of real functions; uniform convergence, pointwise convergence.
 Properties of functions including injectivity, surjectivity, invertibility, continuity, and periodicity.
 Definition of the exponential function and logarithmic function on a complex domain.
The students are engaged via constructing collective explanations of key mathematical topics. Mathematical practices emphasized by Math486 include:
 Assessing the completeness and soundness of explanations and proofs of mathematical ideas.
 Reading, explaining, and writing conjectures and proofs.
 Alertness to mathematical language and precision; how small differences in phrasing may have significant mathematical implications (e.g., the phrase "the solution" versus "a solution").
 Giving mathematical motivations for different given formulations of equivalent mathematical ideas; using different ways of representing the same mathematical idea, and giving explanations of why they are equivalent.
