Semester 1: Mathematical & CS Foundations
The required expansion strand, evidence contract, assessment integration, AI-use declaration, and selective reading route are defined in World-Class Strand.
Year 1 — Fundamentals | Phase 1
Weeks 9–20 are historical roadmap coordinates, not a validated 12-week duration. The upgraded core has the following author-estimated focused workload:
| Component | Hours |
|---|---|
| Proofs and discrete structures | 70–100 |
| Combinatorics and graph theory | 90–130 |
| Probability and statistics | 70–100 |
| Linear algebra for CS | 100–145 |
| Problem-solving strategies | 65–95 |
| Additional semester integration | 30–45 |
| Integers and modular arithmetic | 25–35 |
| Calculus foundations | 70–100 |
| Total | 520–750 |
Module totals already include their lessons, practice, labs, assessments and projects. Integration adds project 18–26, review 4–6, exam 3–4, checkpoint/defense 2–4, and repair 3–5 hours. E12 adds 15–25 minutes already counted in the calculus module allowance. Reuse module artifacts; do not count their work twice. At 10 hours/week this is 52–75 study weeks; at 20 hours/week, 26–37.5. Compressing into 12 weeks would require roughly 44–63 focused hours/week before additional prerequisite repair, breaks and optional full-book reading. Re-estimate using actual learner times; retention intervals may add calendar time.
All seven modules have authored teaching and assessment material. The upgraded path has executable calibration checks; independent subject review and learner pilots remain pending. No percentile or professional-rank outcome is guaranteed.
Goal
Build mathematical maturity for later algorithms, systems, and rigorous engineering reasoning.
Prerequisites
Complete Semester 0, including the checkpoint gate, cumulative review, and a stable study system.
Semester Structure
The public learner-facing material for Semester 1 covers the full mathematical foundations block:
| # | Module | Status | Focus |
|---|---|---|---|
| 1 | Proof Techniques & Discrete Structures | Authored; validation limits noted | Logic, sets, functions, relations, induction, and proof discipline |
| 6 (after proofs) | Integers and Modular Arithmetic | Authored; review and pilot pending | Divisibility, Euclid, primes, inverses, powers and complete congruence solutions |
| 2 | Combinatorics & Graph Theory | Authored; validation limits noted | Counting principles, graph fundamentals, trees |
| 7 (before continuous probability) | Calculus Foundations | Authored; review and pilot pending | Limits, derivatives, integrals, tails and numerical error |
| 3 | Probability & Statistics | Authored; validation limits noted | Probability models, Bayes, random variables, expectation, variance, and statistical reasoning |
| 4 | Linear Algebra for CS | Authored; validation limits noted | Systems, spaces, orthogonality, spectra, decompositions, and numerical judgment |
| 5 | Problem-Solving Strategies | Authored; validation limits noted | Polya framework, heuristics, debugging |
Why This Semester Matters
Proof-writing and discrete reasoning are where the curriculum stops being slogan-driven and starts demanding rigor. If this layer is weak, later algorithms and systems work become shallow very quickly.
Core Resources
| Book | Role |
|---|---|
| Mathematics for Computer Science | Primary reference for proofs and discrete structures |
| Discrete Mathematics and Its Applications | Secondary source for extra examples |
| Linear Algebra and Its Applications | Primary reference for Module 4 linear algebra |
| How to Solve It by Computer (Dromey), with Pólya's framework | Problem-solving, specification and verification |
Use Rule
Complete the relevant proof, discrete-math and programming gates before elementary discrete algorithms. Calculus is required before the continuous-probability lessons, not before every discrete algorithm. Follow each later module’s stated prerequisites. All seven modules and the semester evidence gates remain required for full semester completion.
Capstone Throughline
Every semester must leave behind evidence that can survive into the final capstone defense.
- Artifact carried forward: proof notebook and math reasoning log.
- What to preserve: Preserve representative proofs, failed attempts, and repaired reasoning so later design and correctness claims can cite concrete mathematical discipline.
- Module threads: Module 1: Proof Techniques & Discrete Structures, Module 2: Combinatorics & Graph Theory, Module 3: Probability & Statistics, Module 4: Linear Algebra for CS, and Module 5: Problem-Solving Strategies.
- Defense prompt: In Semester 10, explain how this semester's artifact changed a capstone decision, reduced a risk, or made the final system easier to defend.
Model Artifact Calibration
Before submitting proof-heavy work, compare your notebook to the proof notebook model artifact.
Enrichment Pages
Portfolio Artifact | Common Failure Modes | Bridge Review
Entry readiness check
Before starting proof exercises, complete the Semester 00 notation bridge and all four Launchpad algebra reassessment tasks. If either is weak, repair that dependency first. These short bridges prepare notation and algebra; they do not replace this semester’s proof, combinatorics, probability or linear-algebra instruction.
Integer arithmetic integration
Study Integers and Modular Arithmetic after the proof and invariant prerequisites. Reuse its cycle audit at the semester project and E11 exam gate; the additional 25–35 hours already include that evidence. No calculus prerequisite is introduced. Existing divisibility examples in the proofs module provide context, while the new module supplies algorithms and complete-solution reasoning.
Calculus integration
Study Calculus Foundations before continuous probability. Pass CAL1–CAL6 for that transition and CAL7 for the executable strand. Reuse the approximation audit for the semester project and E12; its 70–100 hours include this evidence. Optional precalculus repair adds time beyond the total. The module uses free OpenStax books and optional MIT explanations; no purchase is required.
Probability inference integration
The inference extension adds 25–35 hours within the revised probability total. Its PS9–PS12 gates and memo appendix are required and reused at semester review. The calculations do not replace the original probability, sampling and confidence gates.
Cross-module connections
Complete the model, units and error check during the existing integration review. Its four subchecks must each reach level 2 under the existing probability, numerical-judgment and calculus categories; no extra semester hours or replacement of module gates is implied.