Need resources for learning higher level mathematics from an inquiry stand point

Question

I am a high school math teacher looking for resources for students who have completed Calculus and are interested in topics like Set Theory and Abstract Algebra. Ideally, I would love to find some resources that are truly inquiry/discovery based. I see a lot of resources that claim to be such, but they don't really adhere to the progression of specific problems -> generalities -> definitions & theorems. Most resources that I find cannot seem to escape the standard textbook approach and this leaves most high schoolers wanting more. Any help is greatly appreciated!

Answer 1

If memory serves, Transition to Advanced Mathematics by Diedrichs and Lovett was the text used for my undergraduate proofs course. It's what made me want to continue studying mathematics so I recommend it highly.

I'll admit it doesn't exactly follow your intended progression of $$ \text{specific problems -> generalities -> definitions & theorems} $$

but it provides plenty of exercises at the end of each section that can be used to facilitate the process of getting one's hands dirty, and each section comes with multiple examples to build intuition ahead of time.

I hope this helps and thanks for the awesome question! :D

Answer 2

A couple reasons, I suspect, for the dearth of books teaching university-level topics in the "inquiry/discovery" style you describe:

I) As far as I know, that isn't, in general, how mathematical research is done: New fields and new branches of pure (theoretical) mathematics (as opposed to applied mathematics) don't necessarily arise in response to "specific problems" (specific calculation tasks). Newton's development of calculus to describe astronomical orbits is one famous case that does fit that pattern, but many others do not. You might have an applied problem that provides maybe a general theme for a direction worth investigating (like map projections and cartography inspiring work in differential geometry), or maybe you literally have no "specific problem" at all: For over 2,000 years (starting with the ancient Pythagoreans), mathematicians have worked on the number theory of prime numbers (prime factorization, density of primes, etc.), which had no known practical applications of any kind until the mid-20th century (e.g. digital cryptography). Until the last 50 years, this research was motivated purely by a fascination with these special numbers and a desire to explore their properties (like a mountain-climber who climbs a mountain just "because it's there"). I suppose you could try to teach this topic by starting with "specific problems" from cryptography, but that seems awfully contrived (and ahistorical).

You may already be familiar with this, but I think a good accessible resource for getting the flavor of the "pure math" approach to mathematics is the first four books of Euclid's Elements. Note that it doesn't contain any "specific problems" about, say, a farmer measuring a field of such-and-such a shape. Just a steady stream of theorems logically deduced from a small number of axioms.

II) Inquiry/discovery is arguably pedagogically inefficient for "advanced" math: That methodology, as you've described it, sounds like it may well be helpful for primary/secondary school kids, but sounds less well suited to the university level where students are expected to have attained a certain level of mathematical maturity and to already have some idea of why they are trying to learn a given topic (the "motivation"). Personally, I actively dislike the (fortunately few) university-level math books I've encountered that put a lot of "motivating examples" in between me and my goal of learning a new area of mathematical theory (I start grumbling to myself "just get to the point already"). Once I know the theory, then I can use that tool to apply it to specific problems/applications.

An analogy that comes to mind - inquiry/discovery strikes me as a bit like the mathematical equivalent of training wheels: appropriate for children learning how to ride a bicycle, but someone who knows how to ride isn't going to put the training wheels back on when attempting to learn BMX tricks.

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As for recommendations for your students, I'm not sure that abstract algebra and set theory are the best topics as next steps after calculus (especially not set theory, unless you just want a very introductory treatment of the topic, i.e. what Enderton's classic book calls "baby set theory").

• I think some introductory linear algebra could be a good topic. I see it as a fairly manageable "step up" from high school math, and also a good vehicle for teaching rigorous proofs and mathematical reasoning. It also has a wealth of applications (e.g. in computer graphics), though I'm not sure you can get the theory and applications you want all in one book. The classic Linear Algebra and Its Applications by Gilbert Strang is the book I used in college and it seemed pretty good (covers some applications, but overall pretty theoretical-oriented).
• Graph theory might be another good choice (I'm thinking in particular of Introduction to Graph Theory by Richard Trudeau).

Also, here are a couple sites that review a bunch of recommended (or sometimes non-recommended) math books on various topics from high school to PhD level:

• Stumbling Robot
• The "Chicago Bibliography"