# Advice on teaching abstract algebra and logic to high-school students

NOTE: This question will soon be duplicated, as I didn't make clear that I was a high school sophmore in the beginning. At first I thought it didn't matter, and somewhat arrogant to mention, but in the comments of HFBrowning's answer, we see:

@JohnClever KCd's observation is spot on. The fact that you are currently a student changes the advice everyone would be giving you; I am certain that all the answers you have received were geared around you being older. I would either heavily modify your question, or consider starting a different one, since I'm not sure this one (with all the answers given) can be salvaged in its current state. – HFBrowning 2 hours ago

I'm planning on creating a 'math club' that is, in truth, a semester's course covering set theory, logic, and abstract and linear algebra.

Curriculum:

The bulk of the course is the algebra. The set theory and logic act as preliminaries and are covered in the first pair of months, where abstract and linear algebra are covered in the next two pairs of months respectively. That being said, although I can see myself replacing the algebra with combinatorics, analysis, or number theory, I see the foundations as essential.

Pacing:

We'll do 4-6 problems a day, usually about a third to a half of each section's problems, and working around 2-5 hours each day is expected (a little under an hour for each problem).

This gives about 6 sections a week, assuming you work on Saturdays. This is about a 1.5 chapters a week, and because there are 4.2 weeks in a month, that's 6.3 chapters a month, usually enough to move on to another book. I know this is rather fast, so I half the time to be 6.3 chapters every two months, or one book every two months. This allows us to do 2.75 books by the end of the school year, and assuming that we leave out some stuff, it seems perfectly reasonable.

School starts back up in January, and ends in June, leaving 5.5 months. Since the logic and set theory is rather short compared to either the linear or abstract algebra, we won't need 6 months.

Miscellaneous Things About the Course:

• I have no teaching experience, being a high school sophomore myself.
• The course will be taught from Mendelson, (for logic and set theory) Lang (for abstract Algebra), and Axler (for linear Algebra), the books I, myself, used to learn these subjects.
• It is assumed that anyone who joins the club has a great interest in math, and is sufficiently hardworking.
• Although the previous condition is obvious insofar as it is necessary, I have tried as much as possible to make it true insofar as it is sufficient.
• Thus, I have tried to make the curriculum require as few prerequisites as possible, and may even end up replacing linear algebra with something else to that end, considering the amount of calculus-based examples it often uses.

Questions:

Has anyone else already done this? I would presume yes, but finding a specific example seems difficult.

How do I recruit students? I suppose I could leverage the concept of a 'super-hard math class covering concepts you've never heard of,' but it doesn't really seem true.

Does the curriculum make sense? In my experience, enumerative combinatorics would be the standard 'higher-math' example, but combinatorics is also the bane of my existence, and I hate it with every fiber of my being. If I had to replace the algebra, I could probably have abstract algebra be some point-set topology and a bit of measure theory, which I could use later on to teach fractal geometry, so that they finally understand what's on the posters to every math class in America.

How do I keep the students interested? I would assume it would be easier to do than a required curriculum course, especially considering how much more interesting I would suppose pure math would be than the unmotivated drudgery that is, for example, pre-calculus, but I also think that the sort of people who love math in high school would be the kind of people that sign up, and they might be expecting something more computational compared to the abstract stuff we're doing.

Clarifications: I now realize that people thought I was referring to Lang's graduate textbook, which I am not. I'm "ambitious", but not evil. I am instead referring to his undergraduate textbook.

At our school, I know that that basic computational linear algebra is taught in precalculus, which is why I thought Axler was reasonable, perhaps supplemented by some 3blue1brown for good measure.

Mendelson is required reading for many philosophy majors as an introduction to formal logic, so I hardly think it too harsh for a math enthusiast.

• This plan strikes me as extremely ambitious. I don't think the content of those three texts could be absorbed in a single semester even by motivated and talented students who are not taking any other courses. Perhaps you could clarify (a) the expected number of hours per week in and out of class, and (b) the total number of weeks. Dec 9, 2020 at 21:12
• I'm not sure about the other books but Axler is meant as a text for a second course in Linear Algebra. It's a terrible idea to try to teach high schoolers using it and you're severely overestimating their abilities. Dec 9, 2020 at 22:05
• The clarification helps but it sounds like much, much more than what I'd consider a "math club". More like a serious math camp that would attract students from an entire state or even country. If this is just something you're planning to offer the kids at your local high school I don't think it will work as planned. Dec 10, 2020 at 0:21
• The proposed set of topics is too much, and with no teaching experience on your end I suspect this will turn out very badly. Do something more basic and gentle first to check that there is an audience for your teaching and that you can teach in a way that interests them and accounts for their background. I suggest introducing and using modular arithmetic before going into abstract algebra.
– KCd
Dec 10, 2020 at 12:23
• Quote from a close personal friend of Lang's: "We all know he wrote terrible books, but we all loved him anyway". That's someone with a PhD in math. Trying to cover 3 books like that for high-schoolers in 6 months is an absolute no-go. Dec 10, 2020 at 15:56

## 7 Answers

Here's my advice.

I have no teaching experience.

Remedy that first before you lay out plans for a 6-month course of study. Find some way where you can teach just for a single day in some way at the high school that you're targeting. After that, find some way where you can teach for a week (i.e., say, 4 interlinked days). I'd suggest that you restart your math-club planning after that.

I think most of us find that the very first day of teaching forces a radical shift in perspective. I had ideas about teaching and student capacities that were shattered into a million pieces based on that first day. You should get on the other side of that experience first.

Frankly, the outline as given in the OP is galactically infeasible for high school students, and you'll find yourself very frustrated if you try to pursue anything like it.

Regarding "How do I recruit students:"

You should start here -- you have started with this cool thing you want to do, and are wondering how to do it. But you should instead try to find some of these students first. What do they want? Where are they?

If you have a core of 1-2 excited parents and 2-3 excited students, then you can start planning.

I have run events with almost no students before. It is embarrassing and depressing. Don't make the same mistake.

I’d like to give you some ideas, based on my own musings about starting a math club, of what to teach instead of Lang and Axler. You say you want to attract motivated, hardworking students, but that’s not necessarily equivalent to “mathematically mature” students. They will have gaps in their knowledge, and as Daniel Collins said--and I can attest to--you may be surprised at how big and numerous these gaps can be! Things that are old hat to you will be new and exciting to them. Seemingly simple things such as drawing a diagram to illustrate completing the square can be very helpful. Or drawing dots for even/odd numbers and seeing how the rules for combining them work out by pairing up leftover dots or seeing how a dot is still left over. Or going over complex numbers in a more geometric way, perhaps starting from the fact that multiplication by $$-1$$ rotates by $$180^{\circ}$$ and then seeking a rotation by $$90^{\circ}$$ to get negative squares. Don’t assume that they’ve seen any of this stuff. (I’m stressing visualization a lot in this answer partly because I’m a visual learner but mostly because I think it’s neglected in the usual high school curriculum.)

As for number theory, how about starting with “clock arithmetic” (no need for fancy, intimidating terminology yet) and literally drawing a clock and counting steps, just as they were introduced to arithmetic on a number line? You can then “skip count” and see the patterns that emerge, maybe even creating nice shapes in the process by connecting the numbers you hit with lines. When does skip counting make a shape with a vertex at every number? What’s the pattern between the number of “hours” on the clock, the number you skip count with, and the number of vertices? This already getting into pretty serious math.

How about drawing Pascal’s (Khayyam’s? Yang Hui’s?) Triangle with just the dots, no numbers. Ask for the number of paths to a lower point starting from the top and going down exactly one level at each step and only one step left or right. Gets pretty messy. But the only way to get to a point is to first get to one of the two points directly above it, so there’s our recursion formula! Now how about the number of teams we can pick from a given number of players? Well, let’s put one player at each level of the triangle and go left if they’re on the team, right if not. Now there’s a one-to-one correspondence between teams and paths. If they can also solve this by basic counting, then there’s a proof that the numbers in the rows add up to powers of $$2$$. And if they don’t know counting, then you can still get the result by arguing inductively (yet still informally), noting that the zeroth row sums to $$2^0$$.

Some final suggestions: do something like the sum of the first $$n$$ naturals (not by induction at first, but perhaps come back to it later if you teach induction) then ask how you can use this to easily get the sum of the first $$n$$ even numbers. From these two results, can you get the sum of the first $$n$$ odd numbers? Now, start from scratch and find the sum of the first $$n$$ odd numbers by adding L shapes to squares. Can you now use this to get the sum of the first $$n$$ numbers? What about the formula for the sum of the first $$n$$ squares? If they know the formula for the volume of a pyramid, can they get it by taking a pyramid and then cutting off parts of it to get a step pattern?

Of course, you can introduce plenty of basic linear algebra as well, which I didn’t even attempt to outline here. Even if you do none of this, hopefully it gives you an idea of what (I think) may appeal to bright high schoolers. Good luck!

The idea as you've outlined just doesn't sound like a math club. I would start by first asking yourself what it is that you want to achieve. Just some possibilities, are you looking to:

1. Encourage more local high school students to pursue math/see the beauty in it?
2. Give kids who are already into math a rigorous outlet?
3. Promote logical thinking in general?
4. Or is it that you want the experience of teaching, and these are the topics you are interested in? (By the way this answer is really not a great one)

The answer to this question will probably be illuminating.

If the answer lies closer to #1, that could be a good fit for a math club, if you are willing to change your approach to be (1) simpler/gentler, and (2) more focused on exploration and letting the students' interests guide the topics. You could end up with some students who think fractals are pretty, and want to know how they "work". Or pre-engineering students who would like to explore math because of an interest in robotics. But at the age you're targeting, I really doubt any student (unless you're surrounded by prodigies) is going to have an interest in abstract algebra.

• I apologize for deleting my previous comments, I wish I could undelete them like you can with a question. It's a habit I've been trying to break. If someone with more reputation can read them, it would be nice if he would tell me what it is that I wrote. By "give kids who are already into math a rigorous outlet", I thought you meant the subset of people who were interested in math such that they also like the kind of rigor I am talking about, and want an "outlet", a place to practice and hone it that they can't find elsewhere. Dec 11, 2020 at 5:28
• @JohnClever Something your comments here made me realize, which you never explained in your post: you are currently a high school student. This information really should be part of the original question, to give clearer context about what is going on. Now I understand completely your comment about having no teaching experience.
– KCd
Dec 11, 2020 at 8:15
• @JohnClever KCd's observation is spot on. The fact that you are currently a student changes the advice everyone would be giving you; I am certain that all the answers you have received were geared around you being older. I would either heavily modify your question, or consider starting a different one, since I'm not sure this one (with all the answers given) can be salvaged in its current state. Dec 11, 2020 at 15:30
• @JohnClever For what it's worth I don't feel you've wasted my time. Asking well thought out, good faith questions on SE grants you plenty of clemency for making a little goof. And besides, there wouldn't be an edit button if we didn't all occasionally forget something or change our minds :) Dec 11, 2020 at 18:56
• @JohnClever HFBrowning is right, you certainly have not wasted anyone's time. And if you want to ask a similar question including the new information that you're still in HS, I'm sure people will provide answers; it's what this site is for, after all! However, even as is, this question and the answers are valuable. Dec 12, 2020 at 19:59

I'm sorry, but like many commenters I do think your plan is far far too ambitious. Going over that material (I only know about Lang and Axler personally) will not give students any motivation for the extremely abstract concepts, and I think motivation is crucial, especially at the high school level. At best, you will teach them this abstract math and they will follow what you tell them, but have no idea why you are considering the questions you are considering/why you are using the definitions you are using/what the concepts mean on an intuitive level.

This summer, I started and taught an abstract algebra class for high schoolers (also my first), but I used the book "Abstract Algebra An Introduction, 3rd Edition by Thomas W. Hungerford" instead of say Lang or Dummit&Foote. This introductory text by Hungerford starts with super concrete examples like $$\mathbb Z$$ and modular arithmetic, allowing me to introduce fields and then rings. Then, we moved onto polynomial rings $$F[x]$$ and began talking about quotients. Then we moved onto more abstract but basic ring theory (isomorphism theorems, ideals, basic facts like these), and then onto field theory. Unfortunately the summer ended before I could get to basic group and Galois theory (which I really wanted to get to because it's really cool and deep -- and so makes students feel good about themselves; and it's good advertising to potential students -- i.e. "wanna know why there's no quintic formula? Come find out with some awesome college-level math!"), but that's when I would have introduced groups -- after rings and fields.

I think that introducing groups first is too abstract for first-time students, and only after gaining intuition about more general abstract algebraic structures like rings/fields, can I more easily generalize to groups (by saying "rings are designed to be 'like' $$\mathbb Z$$ or $$F[x]$$ where $$+$$ and $$\cdot$$ are defined, and in fact because of this we can make rings out of functions $$f: X\to \mathbb Z$$, but beyond addition and multiplication functions have a very important operation: composition. Thus we define groups based off of this idea of function composition" (thus shedding light for students on why associativity but not commutativity is included in the axioms).

And then during Galois theory ideas like series, solvability, and simplicity fall out more naturally. If I had even more time in my summer class, after Galois theory, I would go back to develop more theory about rings (PID, domains, whatever) and/or groups (orbits/stabilizers, class formula, Sylow, more series, free groups, whatever).

To summarize, my advice for you is the following:

1. Find easier textbooks, i.e. ones with more concrete examples/intuition/motivation. I've already recommended an algebra book that I like for these high school classes; I'm sure there are many linear algebra ones out there.
2. Do not set expectations so high, for yourself or students. First times are always challenging, so be patient and keep and open mind. If you do this more years, it will only get better as you get more experience.
3. If after going through these more concrete books you still have time, go ahead and use Lang or Axler or whatever. But I doubt that you will have time. But again, to reemphasize: DO NOT undervalue good motivation. I promise you if you take the time to guide your students to really get a tangible feel for the subjects through lots of examples/pictures/analogies, they will internalize it much better then if you talk about the snake lemma on week 2.

As I do have a bit of experience on teaching an abstract algebra class for high schoolers, I'd be happy to give you more details (like what my lesson plans were, or how I ran the class) if you reach out by email (which you can find by digging around my profile). I do think my class turned out decently; although the students weren't many, those that stayed around were just normal high schoolers from my old high school, and they grasped the material decently well, though I did get the impression that even my "slower" curriculum was a bit too fast for them. Best of luck to whatever plan of action you will take!

This sounds like a super interesting course - but I'm inclined to agree with the other answers here, it's a lot. As Daniel Collins and Thierry pointed out, there are some things about how students think that are hard to anticipate before you've spent some time teaching. I've had some similar difficult experiences myself - the first time I tried to teach a proper logic class was awful! I had this idea that we were going to try to build arithmetic up from foundations over the course of a five-week intensive class. That plan went out the window on day three.

Here's some observations that I think are relevant to your plan as it currently stands.

• Abstraction is hard. I mean, really hard. Think about how students usually make the transition from arithmetic to basic algebra - it takes a year! And that's peanuts compared to the transition from ordinary algebra to abstract algebra. For that reason, it would actually make more sense to focus on linear algebra first - there's more opportunity to be concrete.
• In the same vein: Just because one topic is philosophically prior to another doesn't mean that it should be taught first. Just like we don't try to teach Peano Arithmetic (a formalized system laying out the fundamental definitions of addition, multiplication, and so on) before teaching arithmetic, and no one introduces Dedekind cuts as a first presentation of irrational numbers, set theory and logic don't need to be covered in order to adequately prepare a student for abstract algebra. For a formal treatment, you need some basics; but it's fully possible to teach finite group theory with no understanding of set theory at all. I worked with a teacher once who successfully taught the basics of finite group theory to fifth-graders!
• It sounds like your planned class isn't a prerequisite for anything and doesn't have a standardized curriculum. That could be an opportunity - try an open-ended class. Introduce some fundamental topics, and then choose what to do next based on the abilities and interests of the students. That'll help you adapt your expectations, too. I've only rarely had the opportunity to do this, but it can be a lot of fun, for you and the students. If you choose to do this, aim to teach two- or three-week "modules", introducing a new topic or delving further into a topic you've previously covered.
• Students are slow. Obviously I don't mean that in an insulting way - but every single time I've taught a new topic, I've been surprised at how long it takes. And the younger the students (no matter how clever and motivated they are!) the longer it takes. For perspective: A typical high-school calculus class covers in about a year the amount of content covered in a quarter at a university (in my part of the US, at least). Combining this with the fact that a typical undergraduate abstract algebra class covers part of one textbook, 3 in a year is asking high-school students to pack about six years of learning into one!
• Don't focus so much on textbooks. If the textbook was the most important thing, these highly motivated students you're planning to teach would just get the textbook and learn it themselves. Focus on what you can offer - what can you explain well? What are your favorite examples of key ideas? What can be made with the ideas you want to teach?

I agree with many others here that your plan in its current form is overly ambitious. I will not repeat the excellent advice given so far, but I will add that one way to attract interest and bring things down to earth would be to use or even focus on concrete applications of abstract algebra. Students may enjoy learning about secret sharing, secret codes and the like. See this question for some ideas. (Full disclosure: I asked the question and posted one of the answers.)