Part A
In a typical mathematics curriculum/subject, students learn a wide about variety of mathematical ideas (e.g., definitions, theorems, techniques, etc…). However, in most cases they have very few explicit opportunities to engage with ideas around culture, society and communication in mathematics. This webpage presents some examples of how I have incorporated some of these ideas to support student learning and to expose students at all levels and all motiviations some key ideas about the practice and culture of mathematics, and, as a corollary, how to proceed as a student studying mathematical ideas. I call these exercises “Part A”.
Part A tasks are short written components embedded within technical mathematics assignments. They are designed to foreground aspects of mathematical practice that are essential to undertanding the nature of modern mathematics, but which are often implicit or invisible to students: communication, justification, judgement, revision, and reflection on meaning and purpose. See here for an example of how this looks in practice.
Across subjects and year levels, these tasks invite students to engage with mathematics as something done by people, for people, within a community. These tasks sit alongside traditional problem-solving, not as replacements, but as complements that make the norms and values of the discipline more explicit.
Browse by task type
- Mathematical Communication & Writing
- Proof, Justification, and Legitimacy
- Language, Definitions, and Precision
- Mathematical Beauty, Elegance, and Taste
- Mathematics as a Human and Social Endeavour
- Learning from Error, Revision, and Growth
- Big-Picture Synthesis & Mathematical Identity
Mathematical Communication & Writing
Read Writing Mathematics Well (Guidelines for Good Mathematical Writing) and write a short typed response (approximately 150–250 words) that considers:
- Was there anything about the reading that you found surprising?
- What other experiences, if any, do you have in mathematics classes where you were expected to write in full sentences as part of your solutions? Do you find it difficult to write in mathematics courses?
- How will your approach to crafting your submission for this assignment (or for assignments this semester) change, if at all, as a result of having done this reading?
In the place you downloaded this file you will find a number of sample solutions for a problem on the previous assignment.
- Write a short (100 words) “peer review” of each of the sample solutions. Comment on correctness and clarity.
- Prepare a new version of your solution, and
- Write a short reflection (approximately 100 words) commenting on how your revised solution differs from your initial solution.
Your reflection should address, as appropriate:
- What changes did you make between your first and second solution, and why?
- What did you decide to keep the same, and why?
- What new knowledge did you take away from the peer-review activity?
- Did the activity help you understand the mathematics better, or help you communicate your thinking more clearly to the reader?
Proof, Justification, and Legitimacy
Read the article
Titans of Mathematics Clash Over Epic Proof of the ABC Conjecture (Quanta Magazine)
Write a short response (no more than 250 words) addressing the following questions:
- How does the use of the word proof in this article intersect with the meaning of the word as you understand it?
- The ABC conjecture has no obvious scientific or industrial applications. Do you think there is value in researchers undertaking these sorts of research questions?
Proof as a social compact
Read the article
Why Mathematical Proof Is a Social Compact (Quanta Magazine)
Write a short typed response (approximately 150 words) that considers:
- How has your understanding of the word proof changed as a result of your work so far in this subject?
- A key part of a proof is shared understanding. Where is this idea of shared understanding relevant in other subjects you are taking?
- The article contains a brief discussion of generative AI and proofs. How does the idea that an AI system can construct a proof challenge the idea of proof as a social compact?
Computer-assisted proofs and Preprints
Read Computer Proof ‘Blows Up’ Centuries-Old Fluid Equations (Quanta Magazine)
Write a short response (approximately 150 words) addressing questions such as:
- The article discusses the idea of a computer assisted proof. Given what you have learned about the nature of proof in mathematics this semester, why should the idea of a computer assisted proof give you pause? The first line in the article gives a preprint for the paper that is discussed in this article. – Do an internet search to figure out what a preprint is. Why do you think these authors would publish a preprint of their work? Access the preprint. Does it look the way you would expect a mathematical research paper to look? Explain why or why not. What audience do you think the preprint is written for?
Language, Definitions, and Precision
Mathematical vs everyday language
Read the article
Why Isn’t 1 a Prime Number? (Scientific American)
Write a short response (no more than 250 words) that considers:
- Was there anything about this reading that you found particularly surprising or interesting?
- Do you think most non-mathematicians would be surprised to learn that 1 is not prime?
- Do you think it is a problem that mathematicians and non-mathematicians sometimes use the same word to mean slightly different things?
Historical development of definitions
Read the article
The Jagged, Monstrous Function That Broke Calculus (Quanta Magazine)
Write a short response (approximately 150 words) that considers:
- The article describes early definitions of continuity and differentiability as relying on vague language and inconsistent notation. How does this align with or challenge your impression of where mathematical ideas come from?
- Even if the function discussed had never found practical applications, do you think it still has value?
Mathematical Beauty, Elegance, and Taste
Mathematical beauty
Read the article
Two Forms of Mathematical Beauty (Quanta Magazine)
Write a short response (no more than 250 words) addressing questions such as:
- Do you identify with the idea of mathematical beauty?
- Are there particular mathematical ideas or arguments you find beautiful?
- Has anything in this subject surprised you in a positive way?
Evaluating elegance in an argument
Read the article
The Simple Math Behind the Mighty Roots of Unity (Quanta Magazine)
Write a short typed response (approximately 150 words) that considers:
- Mathematics is often described as purely logical and objective. Does it surprise you that mathematicians describe arguments as elegant or beautiful?
- The article presents an elegant algebraic argument showing that the sum of the (n)th roots of unity is zero. Do you agree that the argument is elegant?
- Are there any arguments or ideas you have encountered in this subject that you would describe as elegant or beautiful?
Mathematics as a Human and Social Endeavour
Critiques of mathematics education
Read Lockhart’s Lament
Write a short response (no more than 250 words) discussing:
- How your own experience of learning mathematics aligns with or differs from the critique presented in the reading.
- Points of agreement or disagreement you have with the author.
Value of abstract mathematics
Read Why the Sum of Three Cubes Is a Hard Math Problem (Quanta Magazine)
Write a short response (no more than 250 words) addressing:
- Why the problem described is mathematically difficult.
- Why mathematicians care about problems like this.
- Whether you think this kind of work has value for humanity, even without immediate applications.
Learning from Error, Revision, and Growth
Mid-semester reflection
Reflect on your performance in the subject so far. Your response should address:
- The grade you currently expect to achieve and how satisfied you would be with that outcome.
- Whether your performance so far aligns with the effort you have put in.
- What resources or strategies are available to help you improve, if improvement is desired.
Your response should be concise (approximately 200–250 words) and, where appropriate, include specific and realistic goals.
Learning from mistakes
Choose a question from a previous assignment or test that you did not receive full marks on (excluding multiple-choice questions).
- Rewrite a full-mark solution to the question, in your own words.
- Answer the following questions:
- Was the original error primarily due to misunderstanding of content, or to issues of communication?
- What do you now understand that you did not understand at the time?
- How can you avoid making similar errors in future assessments?
Big-Picture Synthesis
Explaining mathematics to a peer
Imagine you have a peer who missed the first two weeks of the subject.
Using your own words, write a brief explanation of the main mathematical ideas from the opening lectures.
Include informal definitions and examples where appropriate. Assume your reader has access to the course text but no greater mathematical experience than you.
You may also comment on anything you found particularly interesting or challenging. Your response should be approximately 300 pages.
End-of-subject reflection
Write a short reflective response (approximately 150–250 words) addressing:
- An idea you learned in this subject that you found particularly interesting.
- Why you found it interesting.
- What this idea tells you about doing or creating mathematics and how doing mathematics is a separate act from computation.