Algebra Course Plan — Structural / Galois Trajectory

Algebra for Teaching — Structural / Galois Trajectory (Spring 2026 • Meets Mondays)

Algebra Course Plan

Purpose & Structural Trajectory

This course develops a structural view of algebra aimed at future secondary mathematics teachers.
We travel from number systems and their axiomatic foundations, through functions and polynomial
equations, to symmetry, fields, and a first encounter with Galois theory. At every step, we return
to the question: what does this perspective change about the way we teach algebra in high school?

Course arc.
Unit I: Number Systems & the Birth of Structure →
Unit II: Functions as Algebraic Objects →
Unit III: Equations, Symmetry & Groups →
Unit IV: Fields, Extensions & Galois Theory →
Unit V: Synthesis & Teaching Algebra Today.

Meeting alignment.
This plan is keyed to your once-per-week Monday meetings.
Although the semester begins Wed Jan 21, your first actual class meeting is Mon Jan 26, 2026.
Weeks with no Monday meeting are marked explicitly.

A Motivating Number: Comparing \(\alpha\) and \(\pi\)

Consider the following real number:

\alpha = \left( \left(\sqrt[4]{\tfrac{2}{3}} + 1\right)^{3/7} + \sqrt[16]{\tfrac{5}{3}} \right)^{2/3}.

At first glance, \(\alpha\) may look like an arbitrary expression built from roots and rational numbers,
but it raises natural questions: is it algebraic (like \(\sqrt{2}\)) or transcendental (like \(\pi\))?
The course builds the structural toolkit needed to answer such questions responsibly.

Structural lens.
We begin with axioms (\(\mathbb{N}\) via Peano), then construct \(\mathbb{Z}\) and \(\mathbb{Q}\) to repair limitations, and finally reach \(\mathbb{R}\)
as a completion of \(\mathbb{Q}\). Only after that do we treat polynomials and symmetry (groups) as the correct language
for “why equations behave the way they do.”

Capstone Project: Final Reflection Journal

Your capstone project is a Final Reflection Journal built incrementally.
Each meeting corresponds to a chapter prompt (mathematical content + computational/CAS exploration + pedagogy).

Guiding theme.
How are number systems developed (\(\mathbb{N}\to\mathbb{Z}\to\mathbb{Q}\to\mathbb{R}\) and beyond),
and how does this structural reconceptualization reshape how you teach algebra?

End-of-semester synthesis.
In the final meetings you will revise and compile chapters into one coherent document.
Mathematical accuracy, quality of computation, and pedagogical reflection are all evaluated.

Course Meeting Plan — Mondays (Spring 2026)

The table is aligned to your once-per-week schedule. The first meeting is Mon Jan 26, 2026.
Holiday adjustments and “no meeting” weeks are marked explicitly.

Meeting (Date)	Unit / Theme	Main Topic	Key Mathematical Ideas (Structural / Galois Trajectory)	HS / Pedagogical Focus	History / CAS & Activities
1 (Mon Jan 26)	Unit I – Number Systems & the Birth of Structure Reading	Natural Numbers & Induction	• Peano axioms as an axiomatic description of \(\mathbb{N}\). • Recursive definitions of addition and multiplication. • Mathematical induction as the structural principle for \(\mathbb{N}\). Deep Idea Arithmetic rules come from an underlying axiomatic structure.	• Where induction appears implicitly (patterns, sequences, identities). • Explaining “why” arithmetic algorithms work.	• CAS: generate patterns → conjecture → prove by induction. • Journal chapter focus: induction as a structural proof principle.
2 (Mon Feb 2)	Unit I – Number Systems & the Birth of Structure Reading Integers, Rationals, and Cauchy Sequences — Reading Notes	Finish Induction → Begin \(\mathbb{Z}\) and \(\mathbb{Q}\)	• Close remaining induction foundations (as needed). • Construct \(\mathbb{Z}\) as equivalence classes of pairs \((a,b)\) for \(a-b\). • Construct \(\mathbb{Q}\) as classes of \((a,b)\), \(b\neq 0\), for \(a/b\). • Group/ring/field viewpoint (early exposure). Deep Idea New number systems arise by repairing limitations of old ones.	• Negative numbers and fractions as constructed objects. • Address typical HS misconceptions (sign, zero, division).	• CAS: rational approximations; density of \(\mathbb{Q}\) on the line. • Journal chapter focus: “construction” vs “rule memorization.”
3 (Mon Feb 9)	Unit I – Number Systems & the Birth of Structure Reading Cauchy’s Construction of \(\mathbb{R}\) from \(\mathbb{Q}\) — Reading Notes	Real Numbers & Completeness	• Irrationals (\(\sqrt{2}\notin\mathbb{Q}\)), “gaps” in \(\mathbb{Q}\). • Cauchy sequences as constructions of \(\mathbb{R}\) (idea-level). • Completeness and least upper bound property (idea-level). • Algebraic vs transcendental (stated, not proved). Deep Idea \(\mathbb{R}\) is the completion of \(\mathbb{Q}\); “continuum” is structural.	• The number line as continuum; \(0.999\ldots = 1\) as a structural statement. • Language for limits appropriate to HS.	• CAS: sequences converging to \(\sqrt{2}\), \(\pi\); visualize convergence. • Journal chapter focus: what “completeness” changes pedagogically.
4 (Wed Feb 18)	Unit I → Unit II Transition Reading Homework Schedule note: Mon Feb 16 is Presidents’ Day (no classes). Wed Feb 18 follows a Monday schedule.	From Numbers to Abstract Structures	• Definitions: semigroup, monoid, group, ring, field. • Why these axioms capture “reusable algebra.” • Examples across \(\mathbb{Z}, \mathbb{Q}, \mathbb{R}, \mathbb{C}\) (and mod \(n\)). Deep Idea Number systems are instances of general algebraic structures.	• Turning “properties” into explicit objects in instruction: identity, inverse, distributivity. • Connecting HS rules to structural axioms.	• CAS: verify identities across multiple structures (mod \(n\), polynomials, matrices). • Journal chapter focus: how structure clarifies “why the rules work.”
5 (Mon Feb 23)	Unit II – Functions as Algebraic Objects Reading	Polynomials & Polynomial Rings	• \(F[x]\) as a ring: formal vs functional viewpoint. • Degree; operations; division algorithm; factorization. • Roots and multiplicity as structural constraints. Deep Idea Solving equations is factorization in the polynomial ring.	• Factoring as structure (not just technique). • Degree bounds number of roots; conceptual basis for quadratic formula.	• CAS: factor, solve, graph; multiplicity and graph behavior. • Journal chapter focus: “polynomials as objects,” not just expressions.
6 (Mon Mar 2)	Unit II – Functions as Algebraic Objects Reading	Rational Functions + Brief Complex Lens; Midterm Checkpoint	• Rational functions as elements of \(F(x)\). • Domains; asymptotes; removable singularities; partial fractions (intro). Deep Idea Fields of fractions extend rings in a controlled way (\(\mathbb{Z}\to\mathbb{Q}\) analogy).	• Domain restrictions as a core HS skill; linking algebraic simplification to graphical meaning.	• CAS: asymptotes, holes; symbolic simplification and graph comparison. • Midterm checkpoint: short written synthesis emphasizing structure → pedagogy.
— (Mon Mar 9)	Spring Break	No meeting	—	—	Calendar note: Spring Break begins Mon Mar 9 and ends Fri Mar 13. Optional: journal drafting / reading.
7 (Mon Mar 16)	Unit III – Equations, Symmetry & Groups Reading Graduate Notes (SIUE) Complex Analysis, Part 1 — notes I took as a graduate student at SIUE; included here because students need some exposure to complex numbers in order to complete the story of solving cubics.	Solving Low-Degree Polynomials	• Quadratic formula from completing the square. • Structural overview of cubic and quartic formulas (story-level). • Symmetric functions of roots as the “hidden organizer.” Deep Idea Solution formulas hint at hidden symmetries of roots.	• Choosing among HS quadratic methods (graphing, factoring, completing square, formula). • Enrichment: why cubic and quartic are historically important.	• CAS: inspect symbolic outputs; discuss expression structure. Calendar note: Tue Mar 17 third quarter ends; Wed Mar 18 fourth quarter begins.
8 (Mon Mar 23)	Unit III – Equations, Symmetry & Groups Reading	Permutations & Symmetry Groups	• Permutations; cycle notation; symmetric group \(S_n\). • Subgroups; dihedral groups as symmetries of polygons. Deep Idea Symmetries of equations can be encoded as permutation groups.	• Connecting geometric symmetries to algebraic transformations. • Classroom tasks for composition and inverses.	• CAS/code: compose permutations; cycle decompositions; small Cayley tables.
9 (Mon Mar 30)	Unit III – Equations, Symmetry & Groups Reading	Groups as Abstractions of Symmetry & Arithmetic	• Group axioms; cyclic groups; order of an element. • Homomorphisms (intro) as structure-preserving maps. Deep Idea Groups unify arithmetic and symmetry in one language.	• Modular arithmetic as an HS bridge to groups. • Exponent rules as group laws (repeated structure).	• CAS: arithmetic mod \(n\); patterns; small-group visualizations.
10 (Mon Apr 6)	Unit IV – Fields, Extensions & Galois Theory Reading Automorphisms of Field Extensions and Introductory Galois Groups	• Field axioms; \(\mathbb{Q}, \mathbb{R}, \mathbb{C}\), finite fields \(\mathbb{F}_p\). • Adjoining elements: \(\mathbb{Q}(\sqrt{2})\), \(\mathbb{Q}(i)\), \(\mathbb{Q}(\zeta_n)\). • Algebraic vs transcendental (classification-level). Deep Idea We build larger number systems by adjoining algebraic elements in towers.	• Explaining why complex numbers “complete” solving polynomials (HS language). • Teaching algebraic vs transcendental responsibly without proof.	• CAS: factor over \(\mathbb{Q}\) vs \(\mathbb{F}_p\); compare root behaviors.
11 (Mon Apr 13)	Unit IV – Fields, Extensions & Galois Theory	Galois Groups — Simple Cases	• Field automorphisms fixing the base field. • Examples: \(x^2-2\) (≈ \(C_2\)); discussion of cubic symmetries. • Informal form of the subgroup ↔ intermediate field correspondence. Deep Idea Subgroups of the Galois group correspond to intermediate fields.	• Reading solution methods as “symmetry stories.” • What parts are appropriate for HS enrichment.	• CAS: approximate roots; plot in complex plane; discuss permutations conceptually.
— (Mon Apr 20)	Holiday	No meeting	—	—	Calendar note: Mon Apr 20 – Patriots’ Day (no classes). Optional: Abel–Ruffini reading prompt / journal drafting.
12 (Mon Apr 27)	Unit IV – Fields, Extensions & Galois Theory Reading Chapter Abel–Ruffini and Solvability by Radicals — Reading Chapter	Abel–Ruffini & Solvability by Radicals	• Statement: general quintic not solvable by radicals. • Solvable vs non-solvable groups (intuition); why \(S_5\) is not solvable. • Towers of extensions ↔ radical formulas (conceptual correspondence). Deep Idea Non-solvable Galois group implies no general radical formula.	• Framing “no formula exists” as meaningful mathematics, not failure. • How to present this as enrichment rather than routine assessment.	• CAS: quartic radicals vs quintic numerical methods; compare outputs.
13 (Mon May 4)	Unit V – Synthesis & Teaching Algebra Today Calendar note: Mon May 4 – Last Day of Instruction (day classes).	Integration, Presentations, Reflection	• Student teaching-module presentations (number systems / structure / symmetry). • Global synthesis: number → structure → symmetry → Galois narrative. • Final journal assembly expectations and submission standards. Deep Idea Structural algebra can unify and deepen classroom practice.	• Concrete plan: “what changes in my teaching tomorrow?” • Turning course ideas into HS tasks, explanations, and assessments.	• Peer review of journal drafts; finalize compilation checklist.
Reading / Finals (May 5–12)	Reading Day & Final Examination Window Final examinations must be administered as scheduled.	Final Assessments	—	—	Calendar: Tue May 5 – Reading Day (day classes only). Wed May 6 – Day class final examinations begin. Tue May 12 – Day class final examinations end. Final grades due Fri May 15.

Spring 2026 Calendar — Quick Reference

Jan 21 (Wed): First Day of Classes (your section meets Mondays, so first meeting is Jan 26).
Feb 16 (Mon): Presidents’ Day — No classes.
Feb 18 (Wed): Monday schedule of classes (Wednesday classes will not meet on 2/18).
Mar 9 (Mon)–Mar 13 (Fri): Spring Break (no classes).
Mar 17 (Tue): Third quarter ends.
Mar 18 (Wed): Fourth quarter begins.
Apr 20 (Mon): Patriots’ Day — No classes.
May 4 (Mon): Last Day of Instruction (day classes).
May 5 (Tue): Reading Day (day classes only).
May 6 (Wed): Day class final examinations begin.
May 12 (Tue): Day class final examinations end.
May 15 (Fri): Final grades due; Undergraduate Commencement.

Algebra for Teaching

Algebra Course Plan

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