Early one morning in the halls of a typical mathematics department, Katie, a graduate student in the field of higher category theory, walks into her final exam for grad algebra 1. She had enough sleep the previous night, and feels confident about her abilities. The first question is a routine application of Nakayama’s lemma, and the next an exercise in computing a $ \mathrm{Tor}$ group. After half an hour of deftly dealing out solutions, she comes to the last item:

**Explain the importance of module theory in ring theory using a few examples.**

*What kind of exam is this?* Katie thinks. *The question is not true, false, a computation, a proof, or undecidable in ZFC + V=L! Madness!*

### The Role that Essays Could Have in Math

I made this story up entirely. However, believe incorporating a small amount of such questions would be useful in emphasising intuition and the aesthetic side of mathematics, and this is something that could be used in upper undergraduate and all graduate courses.

Don’t get me wrong; I think conventional mathematical problems are one of the most important parts of learning mathematics. However, there could be other aspects to a solid mathematical education besides problems and reading theorems. Personally I love to read solid, austere, and elegant math, but I also enjoy the informal atmosphere of attempting to explain intuition.

Thinking about the *meaning* or intuition behind concepts is vitally important. The true meaning of things cannot be ascribed a rigorous mathematical definition, but mathematicians are constantly using their own mental “picture” or intuition about their research to advance their field.

Having such vague questions in assignments would give students a chance to really think about the line of thought emphasised through intuitive sentiment. Moreover, students would be encouraged to express their intuition and aesthetic appreciation, and might find that although the final product of mathematics is and should be proof, it is also worthwhile to examine one’s subconscious driving force behind creative output. Moreover, vague questions do not have to have vague answers; a good answer I think would be a combination of vague intuitive statements with precise examples and statements of theorems that would motivate the topic of discussion.

I also feel that mathematics is a very beautiful subject, filled with vast scenes of the elegant and sometimes bizarre objects that come out of the deceptively simple axioms and ideas; this aesthetic feeling is something that should be encouraged. “What parts of this theory do you find most elegant?,” might be a fairly unusual question to ask, but given to a sufficiently advanced class might yield interesting answers.

Are such questions practical? Perhaps on a final with tough questions, alternative questions might not be appropriate. However, assignments are certainly a good place to ask these essay-type questions.

Would these questions be difficult to mark? There is certainly more ambiguity in marking essays that attempt to explain intuition compared to the correctness of a proof, but on the other hand, reading an essay would often be easier and far less mind numbing than marking thirty attempts at Urysohn’s lemma.

Examination and assignment questions are already good at testing technical ability and proof-writing technique as well as a basic knowledge of theory, but are somewhat poorer in testing the level to which a student has integrated new knowledge with existing knowledge. Furthermore, typical assignment questions rarely test creativity, as most of them have very few (if more than one at all) path to the solution. By allowing students to discover connections through mathematical essay writing, we would be giving them the opportunity to be creative in their exposition and discover how they think of the subject, and what it means to other parts of mathematics.

I totally agree with the post. And I’d love to see an essay on the importance of module theory in ring theory! (Even better: many essays)

Thanks for the comment! I feel like the connection from modules back to rings is something people often wonder about when encountering rings and then modules, and one of these days I’ll try to write about it because I really think a lot of cool stuff like homological dimension theory, which is actually fairly elementary, could motivate modules better and also be easily included in most discussions on rings and modules (in a course for instance).