As the nature of mathematical work evolves rapidly, many of us are asking: how should we prepare our students for successful careers? Which skills will be crucial for mathematicians ten years from now — or one year from now, or even one month from now? We would like to share a few ideas.
As AI becomes increasingly capable of producing mathematical arguments, surveying the literature, and drawing connections between different areas of mathematics, the following skills, among others, are becoming paramount:
- The ability to read and analyze mathematical arguments quickly and critically; to absorb new ideas; to distinguish genuine insights from hallucinations; and to spot errors, gaps, and tautologies.
- The ability to ask interesting questions, formulate conjectures, judge which directions are worth developing, and identify the tools that might make such developments possible.
- The ability to communicate mathematical ideas clearly, both in writing and verbally.
As instructors and mentors, we should place greater emphasis on fostering these skills. Here are several kinds of classroom assignments that seem particularly relevant, especially in graduate and upper undergraduate classes:
- Proof-auditing assignments. Students could be given a theorem together with several proposed proofs. They would be asked to decide which proofs are correct, identify the precise errors in the incorrect ones, and repair those arguments whenever possible.
- Assignments that cultivate mathematical taste. Students could be introduced to a new mathematical object and asked:
- What would you like understand about this object? Formulate one or two interesting problems involving it.
- What known objects, results, or open problems might it be related to?
- Which tools would you try to use in studying it, and why?
3. Assignments in mathematical communication. Students could be given AI-assisted mathematical documents and asked to rewrite them as clear, concise, human-readable expositions. They could also serve as peer reviewers, reading one another’s work and offering constructive feedback.
4. In-class presentations. Regular short presentations could help students practice explaining mathematical ideas clearly and responding to questions in real time.
These assignments need not be isolated from one another. After proposing ideas in an in-class quiz of type (2), students could pursue them in a homework assignment, potentially with the help of AI. They would then need to produce a human-readable report as in (3), and present and defend their conclusions in class as in (4).
In short, we should be thinking ten steps ahead about how we teach students, train junior researchers, and conduct research ourselves. We hope these ideas will be useful, and we would be very interested to hear what others are trying in their own classrooms.
Received 14 August 2026.
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