The language of mathematics, 2022 Semester 1

The aim of this course is to develop written and oral communication skills in mathematics. Through weekly readings, discussions, and presentations we will learn

Structure of the course

We will meet twice a week, on Tuesday from 12 to 1:30 and Thursday from 2 to 3:30. One meeting will be devoted to writing and one to oral presentation. In the meeting about writing, we will review the week’s reading, do in-class exercises, and discuss the writing assignment. In the meeting about presentation, one or two students will give short talks, and we will discuss various aspects of presentation raised by the talks.

  1. Tuesdays 12 to 1:30, MSI Board Room (HN 2.48) - presentation
  2. Thursdays 2 to 3:30, MSI Board Room (HN 2.48) - writing

Assessment

The mark will be based on class participation (10%), writing assignments (60%), and in-class presentations (30%).

Resources

  • Francesco Vivaldi, Mathematical Writing for Undergraduate Students (Required textbook)
  • Primer, University of Michigan, Demonstration, Proof beyond the possibility of doubt (Online textbook)

Rough plan for the writing component

  1. Weeks 1-3 : Fundamentals
  2. Weeks 4-6 : Writing proofs
  3. Weeks 7-9 : Micro writing (small-scale issues)
  4. Weeks 10-12 : Macro writing (large-scale issues)

Collaboration and late policy

You may collaborate with others, but everything you submit must be your own work. That is, you must write your own submissions, and not copy it from anywhere else. If you get any outside help, you must acknowledge it by writing (for example) “I worked with … and … for this homework.” Or “I took help from … website for this homework”. Failing to do this is a form of academic dishonesty.

Late homework will be penalised at 10\% of the grade per day, for a maximum of 3 days, after which it will receive a score of 0. The exception is if you ask me for an extension in writing (email is fine) at least 24 hours before the deadline.

Detailed weekly plan

The writing assignments are due on Mondays by 5pm. I expect you to be familiar with the reading by the class on Thursday.

Week 12

Reading and discussion

Writing

  • Submit a revised draft of your final paper.
  • Submit the final draft of your final paper (at the end of the exam period).

Presentations

  • Continued fractions
  • The wave equation

Week 11

Reading and discussion

Writing

  • Submit a first draft of your final paper.

Presentations

  • Central limit theorem
  • RSA crypotgraphy
  • Primality testing

Week 10

Reading and discussion

Writing

  • Sumit a tentative title and outline of your final paper.

Presentations

  • The RSA cryptosystem
  • Ordinary differential equations

Week 9

Reading and discussion

Writing

  • Submit the final draft of your first paper.

  • The final writing assignment.

    Your goal is to write a 5-10 page paper about the topic of your second talk. It should have a title, an abstract, a section called “Introduction”, other sections, and a bibliography. There must be more than one mathematical results with proofs. The mark will only be based on the final submission, but there are the intermediate submissions before the final one.

    1. In week 10, a tentative title and an outline of the paper.
    2. In week 11, a first draft
    3. In week 12, a revised draft
    4. In the exam period, the final draft.

Presentation

  • Taylor Series
  • Fermat’s Little Theorem
  • The binomial distribution
  • The Poisson distribution

Week 8

Reading and discussion

Writing

  • A first draft of your first paper.

Presentation

  • No presentations this week

Week 7

Reading and discussion

  1. Keith Conrad, Advice on mathematical writing

Writing

This is a longer writing assignment. The first draft is due in Week 8, and a revised draft is due in Week 9. Your mark will be based only on the revised draft.

Write a 1 to 2 page paper about the content of your first talk. Include a title and one or two short introductory paragraphs. Your write-up must contain at least one definition, one example, and one result with proof. See “6. Types of Mathematical Results” and “7. Definitions” in Keith Conrad’s document. For this assignment, you do not have to include references, and it is OK to not divide your document into sections.

Presentations

  • Explain the intermediate value theorem.
  • Discuss the determinant of a matrix.
  • Explain the fundamental theorem of algebra.

Week 6

Reading and discussion

  1. Primer, Proof techniques: Proof by induction
  2. Vivaldi, Chapter 8
  3. In class exercise 6

Writing

  1. Revise and resubmit one proof from assignment 4. Schedule a meeting to talk about the revision.

Presentations

  • Convince us that \ (\sqrt 2\) is irrational.
  • Explain the binomial coefficients \(n \choose r\).

Week 5

Reading and discussion

  1. Primer, Proof techniques: Contraposition, Contradiction
  2. Vivaldi, Chapter 7
  3. In class exercise 5

Writing

  1. Vivaldi, Chapter 7, Exercise 7.2
  2. Vivaldi, Chapter 7, Exercise 7.3
  3. Vivaldi, Chapter 7, Exercise 7.4 (5 more)

Presentations

  • Explain eigenvalues and eigenvectors.
  • Explain unique factorisation of integers.

Week 4

Reading and discussion

  1. Primer, Proof Techniques: Uniqueness, Casework, Either/Or Max/Min
  2. Vivaldi, Chapter 6
  3. Vivaldi, Sections 7.7, 7.8
  4. In class exercise 4

Writing

  1. Vivaldi, Chapter 7, Exercise 7.1 (any 2)
  2. Vivaldi, Chapter 7, Exercise 7.4 (any 5)

Presentation

  • Define the limit of a sequence.
  • Explain the quadratic formula.

Week 3

Reading and discussion

  1. Primer, Functions (part two), negating universal quantifiers, negating existential quantifiers, negating nested quantifiers
  2. Vivaldi, Chapter 4
  3. Vivaldi, Chapter 5
  4. In class exercise 3

Writing

  1. Vivaldi, Chapter 4, Exercise 4.3 (any 10)
  2. Vivaldi, Chapter 4, Exercise 4.4 (Only state the negations for any 5)
  3. Vivaldi, Chapter 5, Exercise 5.1

Presentation

  • Define a polynomial function and give some examples.
  • Define similar triangles. Define equilateral triangles. Prove that any two equilateral triangles are similar.

Week 2

Reading and discussion

  1. Vivaldi, Chapter 1
  2. Primer, Fundamentals: Existential quantifiers, universal quantifiers, combining quantifiers
  3. In class exercise 2

Writing

Vivaldi, Chapter 1, Exercise 1.2.

Presentation

  • Define an even number and prove that 0 is even.
  • Define a prime number and prove that 2 is the only even prime number.

Week 1

Reading and discussion

  1. Primer, Fundamentals: Set Theory, Functions (part one)
  2. Vivaldi, Chapter 2
  3. Tips on giving talks, Jordan Ellenberg
  4. In class exercise 1

Writing

  1. Convert a page of handwritten mathematics into LaTeX.

Resources

Comments on presentations (updating list)

Many of these comments also apply to clear writing.

General

  1. Make eye contact with the audience as much as possible. Avoid speaking to the board.
  2. Speak so that everyone can hear clearly (not too softly or too fast).
  3. Begin informally, before going into technical details.
  4. Motivate your topic. Get the readers interested in what you are saying.
  5. Keep the talk focused. Do not try to do everything. Some strategies to keep focus: making simplifying assumptions, doing special cases, explaining by an example.

Board/slides

  1. Write the title.
  2. Write legibly, clearly, in sufficiently big font.
  3. Develop the habit of speaking when you writing, avoiding long silences.
  4. Use the board to write the important points, not as a place for scratch work.

Content

  1. Illustrate definitions with examples and non-examples. Comment on edge cases.
  2. Give precise definitions. If that is not possible, explicitly indicate what is left vague or undefined.
  3. Mention each non-trivial mathematical result you use. Indicate if it is easy or difficult. Give proper attribution.
  4. Take every opportunity to draw a picture.
  5. Take every (reasonable) opportunity to tell a story.

Notation

  1. Use consistent notation, and alert the audience if notation changes.

Handouts

  1. Keep consistent with the talk.
  2. Comment on everything that is in the handout.

General comments on the first paper

Introduction

The introduction to your paper must answer the following two questions

  1. What is the paper about?
  2. Why should the readers care?

Number (2) is harder to achieve and takes practice. Number (1) is easier. To achieve (1), your introduction should explain the main topic and the main results. After reading the introduction, the reader should not be surprised by anything that comes later.

Body

Use the appropriate LaTeX environments for definitions, theorems etc. For example, put the theorems in \begin{theorem} … \end{theorem} and the proofs in \begin{proof}…\end{proof}.

The body must be more than just a sequence of definitions and theorems. You must provide guides to the reader, by periodically reminding them what has been done and what comes ahead. You can do this by including sentences like:

  • Having defined eigenvectors and eigenvalues, we now see how to compute them.
  • We are now ready to state the main theorem.
  • Before we state the main theorem, we need some notation.
  • We have finished proving the lemma; now we turn to the proof of the theorem.

All major definitions and theorems should come with an introductory sentence.

For example, before you state a theorem in \begin{theorem}…\end{theorem}, include a sentence like:

  • The following is the main theorem of the paper.
  • Having established the notation, we state the main theorem.
  • The following theorem states that solutions exist and are unique.

Before you give a major definition in \begin{definition}…\end{definition}, include a sentence like:

  • We now give the precise definition.
  • The following is the key definition of the paper.

Before an example, include a sentence that explains why you are including the example. For example:

  • The following example shows that the solution is not always unique.
  • We now give an example of a continuous function that is not differentiable.
  • Not all matrices admit an eigenvector, as the following example shows.

An introductory sentence is also helpful for minor definitions and theorems, but in those cases, it is more acceptable to skip.

Topics for second talk and paper

  1. Fermat’s little theorem
  2. The chromatic polynomial of a graph
  3. The four colour theorem
  4. Modelling radioactive decay
  5. The mathematics of contagious diseases
  6. Linear differential equations
  7. Euler’s method for numerically solving an ODE
  8. The mathematics of carbon dating
  9. The heat equation
  10. The wave equation
  11. Harmonic functions and the maximum principle
  12. Primality testing
  13. The RSA cryptosystem
  14. The conjugate gradient method
  15. The Poisson distribution
  16. The binomial distribution
  17. The central limit theorem
  18. The maximum likelihood method
  19. Primitive roots in modular arithmetic
  20. Planar graphs
  21. Recurrence relations
  22. Exact differential equations
  23. The prime number theorem
  24. Taylor Series
  25. Continued fractions
  26. Any other topic (with my permission)

Judgement criteria for the papers

  • The title
    • is descriptive.
    • is concise.
  • The abstract
    • adequately conveys the main topic.
    • is concise.
  • The introduction
    • motivates the topic.
    • summarises all the key results.
    • gives an outline of the rest of the paper.
    • is not overly technical or overly vague.
  • The sections
    • are titled informatively.
    • are arranged in a logical order.
    • are of appropriate length.
  • Theorems and definitions
    • are in the correct environment.
    • are stated precisely.
    • are stated concisely.
    • are motivated.
  • Proofs
    • are in the correct environment.
    • are correct.
    • are readable (not overly long or short).
  • Surrounding discussion
    • guides the reader by periodic recaping and fore-shadowing.
  • Language and grammar
    • Everything is a part of a complete sentence.
    • Spelling, grammar, and punctuation are correct.
    • Sentences are not overly long.
    • Writing is organised in logical paragraphs.
  • Mathematical matters
    • Sentences do not start with a mathematical symbol.
    • Mathematical expressions are adequately separated.
    • Notation is not too cumbersome.
    • Notation is consistent.
    • Mathematical terms are used precisely.
  • References
    • Proper citations are given.
    • A list of references is included in a consistent format.
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