Discipline Elective II – Advanced Mathematics for Teacher‑Education
(B.Ed. – Electives & Projects)
Target audience – B.Ed. students who have already completed a first‑year mathematics elective (basic calculus, linear algebra, statistics).
Goal – Equip future teachers with deep content knowledge and rigorous pedagogical strategies for teaching advanced mathematics at the secondary or early‑college level.
1. Course Overview
| Item | Detail |
|---|---|
| Duration | 14 weeks (28 × 1 h lecture + 7 × 2 h seminar) |
| Prerequisites | Basic calculus, linear algebra, discrete mathematics |
| Learning outcomes | 1. Master advanced mathematics concepts from first principles. 2. Analyze the epistemology of mathematics. 3. Design curriculum units and assessment tasks grounded in research. 4. Produce an original pedagogical project (research paper or curriculum module). |
| Assessment | Weekly problem sets (20 %), research‑based seminar (30 %), final project (30 %), reflective portfolio (20 %) |
2. Content Map & First‑Principles Approach
| Week | Theme | Core Concepts | Representative Problem |
|---|---|---|---|
| 1 | Foundations of Set Theory & Logic | Axioms (ZF(C)), first‑order logic, proof techniques | Prove that the set of all finite subsets of ℕ is countable. |
| 2 | Algebraic Structures | Groups, rings, fields, modules | Show that the ring ℤ/12ℤ has exactly 4 idempotent elements. |
| 3 | Order Theory & Lattices | Partial orders, lattices, Galois connections | Construct the lattice of subgroups of ℤ/12ℤ. |
| 4 | Metric Spaces & Topology | Open/closed sets, continuity, compactness | Prove that any continuous map from a compact space to ℝ is bounded. |
| 5 | Linear Algebra Revisited | Vector spaces over arbitrary fields, dual spaces | Determine the dual of ℝⁿ and its dimension. |
| 6 | Category Theory (Intro) | Objects, morphisms, functors, natural transformations | Define the category Grp and prove that it is not cartesian closed. |
| 7 | Advanced Calculus | Lebesgue integration, measure theory | Show that the indicator function of the Cantor set is Lebesgue‑integrable. |
| 8 | Differential Geometry | Tangent spaces, manifolds, differential forms | Compute the Euler characteristic |