Lecture 0: Welcome to Maths

ISE Block 1.2 Maths · Week 1

Why maths?

When will I ever use this?

• Your maths teacher probably promised you’d use algebra in the real world
• Then you discovered that jobs rarely involve factorising quadratics
• Computer science is different: mathematics sits just beneath the code
• Sometimes it’s the difference between a system that seems to work and one you can show will work

Keep the quadratic joke brief. The point is not that school algebra is useless: much of it turns up again in computing.

An unbreakable cipher?

• By the end of week 1: you’ll build a one-time pad and prove why it offers perfect secrecy
• When its conditions hold, the ciphertext reveals no information about the message
• Then we’ll reuse a key and see what goes wrong
• The maths is sound. The difficulty is meeting its conditions in practice

Do not claim that the one-time pad is cryptanalytically broken. Perfect secrecy holds under its assumptions: a uniformly random independent key, as long as the message, used once and kept secret. The practical conditions are what fail. Keep the promise about week 1 aligned with the lab plan.

The maths underneath the code

• LLMs and neural networks: linear algebra and calculus, including backpropagation
• Algorithms: asymptotic notation to understand how runtime and memory use scale
• Public-key cryptography: number theory behind using a public key and a private key
• Correctness: logic to show what a program does, rather than just test a few examples

Don’t imply that these few topics are all there is to LLMs or cryptography. They’re important foundations. Distinguish material explored here from advanced material students may meet later.

You’ve already met a little of this

• def total(...) -> int: says what kind of result a function should return
• Literal["pending", "paid", "cancelled"] restricts the permitted values
• A type checker can catch violations before you run the code
• Types don’t prove everything: a function can still loop forever or return the wrong integer
• We’ll learn to state and justify stronger claims about programs

Be precise about the limitations of Python’s runtime type annotations: the examples state intended contracts; a static checker can enforce some of them, subject to configuration and assumptions. A return annotation by itself does not ensure termination, correctness, or even runtime enforcement.

The plan

Weeks 1 to 7

• Week 1: knowing you’re right: proof, logic and a perfect cipher
• Week 2: the language of specifications: quantifiers, sets and functions
• Week 3: linear maps and induction
• Week 4: number theory and cryptography
• Week 5: graphs and growth
• Week 6: counting and chance
• Week 7: assessment
• We’ll check the pace after week 1 and adjust if needed

How a week runs

The timetable

Day	09:00–11:00	11:00–12:00	14:00–16:00
Tuesday	Lecture	Lab with lecturer	Lecture
Friday	Lecture	Lab with teaching assistant

Three lectures and two labs each week. Week 7 is revision and assessment.

Lectures

• One topic per lecture, following the textbook’s order (mostly)
• The slides contain the material taught and can be used for revision
• Each deck appears on this site on the day of the lecture
• We will read mathematical expressions aloud and, where helpful, run them in Python

Labs

• Work through problems in a notebook: calculate, test, find counterexamples and write arguments
• Tuesday’s lab uses that morning’s lecture, with the lecturer present
• Friday’s lab revisits earlier material, with the teaching assistant
• Download notebooks from this site on the day of the lab
• Use Google Colab, Jupyter or VS Code; bring a laptop
• Submission instructions, deadlines and marking arrangements are on Brightspace

Starting points

What I assume about you

• You’ve completed seven weeks of programming and can write small Python programs
• You’ve studied Leaving Certificate mathematics, including algebra, functions and calculus
• …it’s okay if you haven’t remembered everything; we’ll refresh the maths as needed
• Formal proof may be new; we’ll work through it from the start

How to work

• Try each problem before looking at a solution
• Write down your reasoning, not just the answer
• If a proof fails, find the step you can’t justify
• Ask when a definition or step isn’t clear
• Read the textbook sections listed with each lecture

Assessment and academic integrity

Assessment: what to expect

• Continuous assessment on Brightspace, weekly in weeks 1–6 (20%): short retrieval quizzes, lab checks and written proof tasks
• Two in-class tests, in weeks 3 and 5 (25%)
• A capstone project, due at the end of week 6 (15%)
• A final exam in week 7 (40%)
• In-class tests are closed-book
• The final exam is closed-book
• End-of-block assessment takes place in week 7, 14–18 December, during teaching weeks; there is no separate exam week
• Assessment weights, test format and the final exam’s precise arrangements will be published on Brightspace
• Brightspace is the source of truth for assessment and submissions

Academic integrity

• Submit your own work
• Do not use generative AI in assessed submissions unless the assessment explicitly permits it
• Where AI use is allowed, follow the assessment’s conditions and declaration requirements
• Discussion and collaboration are encouraged for learning, but submitted work must comply with the rules for that assessment

Keep policy wording aligned with the definitive Brightspace instructions. Avoid implying that permitted AI study support gives permission to use AI in submissions.

AI and learning

AI can help… but it can also hide gaps!

• A model can explain a concept, check code or suggest a proof
• It can also give you a convincing answer that is wrong
• Completing a problem with help doesn’t show you can solve it unaided
• Research on AI-assisted study has found both gains and setbacks, depending on the tool and how it is used (Kestin et al. 2025; Bastani et al. 2025)

These are findings from particular studies, not a general verdict on AI tutoring.

Use it as a critic, not a substitute

• Try the problem yourself first
• Ask for a hint, or for a flaw in your proof
• Check the answer against definitions, calculations and the textbook
• Then try a similar problem without assistance
• Studying with AI does not give you permission to use it in assessed work

The text

The textbook

• Lehman, Leighton and Meyer, Mathematics for Computer Science, 2015 revision
• Written for MIT’s mathematics for computer science course
• Free from MIT OpenCourseWare; link on the course website
• Lecture decks identify the relevant sections; we may select or reorder material to suit the teaching sequence

When books disagree

• Mathematical notation is not always uniform
• Example: some texts include 0 in the natural numbers, and others start at 1
• Here, \(\Nat = \{0,1,2,\ldots\}\)
• Check which convention a source uses, and state yours where it matters

The slides

• Use the slides for revision alongside the textbook
• Find the latest published version at maths.eoin.ai
• For assessment rules, use Brightspace

The lecturer

The boring intro slide

• Eoin O’Brien, Assoc. Prof. (Practice) in Software Engineering, University of Limerick
• Sat on the Industry Advisory Board for ISE when the course was being set up
• Founder and CTO of Stokka, building software for manufacturing, stock management and production planning.
• Previously co-founded Tracworx while an undergraduate at UL. Developed hospital patient tracking, Bluetooth-based contact tracing (COVID-19) and supply-chain tracking systems.
• Lecturing at UL: software testing, cryptography, data structures and algorithms, AI/ML
• Research: part of the ABCD (AI for software engineering) and STEER (entrepreneurship) groups
• Irish 30u30, InterTradeIreland Seedcorn, UL Gold Medal etc.

References

Bastani, Hamsa, Osbert Bastani, Alp Sungu, Haosen Ge, Özge Kabakcı, and Rei Mariman. 2025. “Generative AI Without Guardrails Can Harm Learning: Evidence from High School Mathematics.” Proceedings of the National Academy of Sciences 122 (26). https://doi.org/10.1073/pnas.2422633122.

Kestin, Greg, Kelly Miller, Anna Klales, Timothy Milbourne, and Gregorio Ponti. 2025. “AI Tutoring Outperforms in-Class Active Learning: An RCT Introducing a Novel Research-Based Design in an Authentic Educational Setting.” Scientific Reports 15 (1). https://doi.org/10.1038/s41598-025-97652-6.

Questions?

eoin@eoin.ai · eoin.p.o.brien@ul.ie

Leave contact details visible at the end of the brief in-class introduction.