"You ask me a question": Part 1 of N. Abstract thought and occupational mathematics.

Last year, I started conducting regular quizzes for my courses online, through our course management system. The regularity, structure, and significance vary from course to course, but I now do this in all my courses. The ease of this (at least compared to quizzes on paper) has allowed me to cultivate another practice: I now (frequently) add an extra question at the end of quizzes that says something like this:

You ask me a question. This can be about recent course material or something that is related to this course.

This has been enlightening and entertaining well beyond what I expected and I wholeheartedly recommend it!

The online submission of quizzes allows students to type their thoughts thoroughly, and I believe the fact that they're taking the quiz in a setting outside of class encourages them to ask more honest questions. From my perspective, this also allows me to respond to questions individually, directly, and thoroughly.

I end up sharing a select subset of questions and my answers in a document online (but I anonymize the questions to prevent potential embarrassment). This can show the students that they have a question similar to lots of others and should not be afraid to ask anything. This process also helps me "check the pulse" of the course. (For instance, I might observe a common theme among several questions and use this to guide my planning.) And, honestly, I think it's great fun! Overall, I believe that students subsequently feel more engaged with both me and the course, and it surely forces me to reflect on my teaching and consider improvements.

I'd like to use this blog to share some questions I've been asked and the answers I've provided. Since I don't know how often I will do this, I declare that this is the first of some arbitrary number, N, of entries.

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Context: Real Analysis course (for advanced math majors).

Question: In what way does a course like Real Analysis, or any that aims at developing deep abstract thought in terms of mathematics, apply to the occupational field of mathematics? 

My Answer: Interesting question! In some sense, I don't have to say much here, because your question boils down to, "How does studying mathematics prepare one for a career in mathematics?" In that case, all I have to say is, "Well ... what ELSE should we do instead? Not study math?!"

But I think your question belies an attitude that deep, abstract thought is somehow not useful, or not applicable. And I have to flat out disagree with that sentiment. Personally speaking, I feel that I have become a better and better mathematician as I've aged, and this has come from both a combination of applying my knowledge to solve problems and studying deeper and deeper into the theory of math. Without one or the other, I would be doomed as a mathematician, and I feel like this is a universal phenomenon. Even the most tried-and-true "applied mathematician" has to study its theoretical aspects for many years, and this isn't just to be difficult or to "weed out" poor students or anything like that. It's because it's genuinely useful and important. It further develops one's understanding of what one is doing and puts things in perspective.

In a broader, sense, as well, a course like this is teaching you several important general skills:

1. An appreciation for the history and philosophy of mathematics and its people. In our course, we have mentioned several famous mathematicians that have contributed to the development of humanity's collective knowledge. We are drinking from the fountain of that knowledge, and I think it behooves us to marvel at its beauty. You may not find any one particular theorem we study all that fascinating, but taken as a collection, its a remarkable body of knowledge that human beings have put together.
2. Problem-solving. This idea is trotted out in just about every mathematics course, but I think that's for a good reason. Think about the homework exercises and quiz problems you've done in this course. Haven't they helped you become a better logical thinker? You've had to grapple with ideas you've never even heard of before, putting them into logical order and making sense of them, and then writing down your ideas about them to explain them to others. Surely, this has helped you become a better thinker and expositor, right?
3. General content knowledge to be used in other areas. You may be surprised by how widespread many of the key ideas and results from this course are. Sequences? All over math! Functions and continuity? You bet! Derivatives? Ummm, calculus amirite? Just because we're looking at these ideas from an abstract viewpoint doesn't render them meaningless. If anything, it gives them even more meaning and, I'd argue, a profound beauty.

All of these skills are important for occupational mathematics.