This is a pitch for a new text that I’m thinking of writing. I want to post it here to solicit opinions from the general community before investing a lot of time into the actual writing.
Summary
There are a lot of students who ask me a question isomorphic to:
How do I learn to write proofs?
I’ve got this on my Q&A. For the contest kiddos out there, it basically amounts to saying “read the official solutions to any competition”.
But I think I can do better.
Requirements
Calling into question the obvious, by insisting that it be “rigorously proved”, is to say to a student, “Your feelings and ideas are suspect. You need to think and speak our way.”
Now there is a place for formal proof in mathematics, no question. But that place is not a student’s first introduction to mathematical argument. At least let people get familiar with some mathematical objects, and learn what to expect from them, before you start formalizing everything.
— Paul Lockhart
There was a while I tried to look around to find an introduction-to-proofs textbook that I liked. I specifically wanted to have the following requirements:
- Pragmaticism: the textbook should not start with foundational issues like logical quantifiers or set theory. I have held a long belief that these are emphatically not the right way to start proofs, because in practice when one really does proofs, one is usually not thinking too much about the axioms of set theory.
- Substantial: the results one proves as practice should feel interesting. They should have meat. For example, the statement that a tree always has one fewer edge than vertex is not obvious at first, so when one sees the proof it gives an idea. I believe this is important because I want to develop a student’s intuition, rather than try to teach them to work against it.
- Intuitive: I reject the approach of some other instructors in which students start by proving basic results from first principles like the well-ordering principle, “all right angles are congruent”, etc. I think this is an experience that is worth having, but it should not be the first experience one has. (This is the same reason people’s first programming language is Python and not assembly.)
- Combinatorial: for competition reasons. My currently recommended combinatorics textbook by Pascal96 is a bit on the difficult side. It would be nice to cover some ground here.
The closest I got was Jospeh Rotman’s Journey Into Mathematics textbook, which satisfies the first three conditions but not the fourth (the book draws from algebra, geometry, and number theory). I adore Rotman’s book and the copy I read at age 12 is tattered from extended use. I’d like to get the combinatorics in, too.
Picking a fight
I should state now this is against common wisdom. Terence Tao for example describes mathematical education in three parts: pre-rigorous, rigorous, post-rigorous. Relevant quotes:
[In the rigorous stage], one is expected to be able to comfortably manipulate abstract mathematical objects without focusing too much on what such objects actually “mean”. … The transition from the first stage to the second is well known to be rather traumatic.
My thesis is that for high school students with an enriched math background, the rigorous and post-rigorous stages should be merged or even inverted. Attending a math circle, going to math camps, or participating in competitions gives you a much better intuition than a typical starting undergraduate would otherwise have access to. I propose that we take advantage of this intuition, rather than ignore or suppress it.
Content
I’m eyeing graph theory as a topic to start off on, if not use wholesale. I think it is an amazing topic for teaching proofs with. Definitions that make sense, proofs that are intuitive but not obvious, lots of pictures that don’t lose rigor, and so on. I imagine I would start there and see where it takes me.
If I go through with it, I think it would take about a year for me to get some initial drafts available to the public.
Pay-what-you-want model
I want to try this out. I think it would look something like:
- You can download the nicely typeset PDF for 20 dollars;
- The entire source code is publicly readable on GitHub, so if you can’t pay or don’t want to pay just download the source and compile it. It might not have some formatting polishes or whatever but all the content is going to be there.
- If you don’t have a computer to compile things on, email me nicely and I’ll send you a copy.
- Pull requests welcome, and if you fix some sufficient number of typos or some major errors I’ll add your name to acknowledgments.
But I’m not sure yet.
Questions for the audience
- Is this something people would want to see?
- Is there any existing text that already satisfies my requirements?
- Is the payment model fair?
- Other comments or suggestions?
Power Overwhelming 
1. Absolutely! I’ve been working with IMO kids from my country (~75th at IMO) for a couple of years and I was shocked to see how bad their proof-writing was. I’ve tried to help some of them improve at that but it’s quite difficult since there doesn’t seem to be a text that has the same purpose (it’s not as easy as “oh you’re bad at geo? Read egmo”). I think such a book is much needed.
2. Not to my knowledge.
3. Sounds reasonable
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As a Fourth Year Undergrad who has no IMO and is relatively considered “weak” as a Math Student. If your book can do what you initially proposed and also provide the student the elusive world of Mathematics without being frightened/scared/overwhelmed, I think it could do very well and be valuable. Personally, I would read over it if it ever comes into actuality.
I might even skim the Journey of Mathematics book due to your affinity with it.
Best of luck Evan!
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Hi Evan,
Just to register enthusiastic support for all aspects of this project – not least because it matches my prejudices in how to instruct about proofs in every way :) I think “Pay what you want” is fine, but you might want to look at some other options as well (e.g., the text itself is free but you commit to say monthly bits of additional content available to subscribers — and eventually to everyone). Still, the commitment to open access is definitely laudable.
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1. Yes, as a teacher I think such a text, which does not focus too much on proving basic results from first principles would be really appreciated by students and faculty alike.
2. You might want to check out “Proof and the art of mathematics” by Joel David Hamkins
3. Yes, it is. You could also consider a crowdfunding campaign (I, for one, would gladly contribute!)
I hope you go forward with the text. Best of luck!
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u spelled Terence Tao’s name wrong
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f
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Yeah, that part of the essay is, like, the wrongest thing Terence Tao ever wrote :) A smart student can do a lot of math and computer science without ever knowing the formalities.
About 1. – I suspect most contestants who have issues with writing 7-point proofs actually have more basic issues with mathematics — they can’t distinguish between a correct and a false proof. However, the book may be useful to them too.
About 3. – I would ask an economics grad student, I guess? You can easily model your readers in the USA. Only two options for payment — it sounds low.
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“My thesis is that for high school students with an enriched math background, the rigorous and post-rigorous stages should be merged or even inverted.”
I actually don’t really agree with you (or Lockhart) at all. This may have to do with my focus on analysis, which might draw me more into Tao’s camp, but I remember feeling this way before I knew any substantial math. (Btw, I’m happy to talk more about this over messenger)
I went to a math circle for a bit as a kid and learned a little math here and there, but I always felt a bit “adrift;” like I wasn’t ever quite sure if I was saying something brilliant or stupid, and had no firm footing to stand on. It wasn’t until 9th grade geometry, and moreso 11th grade calculus, that I felt I could really feel confident enough in myself to produce mathematical proofs and not be worried about some subtle flaw somewhere. (For background: I went to an unusual high school, and the advanced math track did everything fully rigorously from the start. 11th grade calculus was taught out of Spivak.)
“This is the same reason people’s first programming language is Python and not assembly.”
I don’t think this analogy is relevant for the following reason: programming, unlike writing proofs, has a quick certificate of correctness; we can just run the program and see that it outputs the right answer. Imagine we lived in a world where computing power was a rare and valuable resource. In this world, you’d have to learn programming entirely without a computer. I would argue that in this world I’d prefer to learn assembly first, at least enough to understand what was going on in python behind the scenes.
What’s more, I have heard sometimes that new mathematicians from that learning fully rigorous proofs can be intimidating at first. On the contrary, I’ve found that all the times I’ve most “intimidated” by math (whether in a class, textbook, or online forum) were when people were using “post-rigorous” language. I could understand the beauty of the solutions, sure, but I felt very hesitant to try to offer my own, as I didn’t know what was right and what was wrong.
I think such a “post-rigorous from the start” paradigm would be a good way to help people _appreciate_ math; indeed that was the function of some such lectures at Canada/USA Mathcamp, for example, and could be good for the very first lecture of a discrete math course. But for _learning_ math, I feel the traditional method made me most comfortable.
“in practice when one really does proofs, one is usually not thinking too much about the axioms of set theory.”
I also don’t really draw the same conclusions from this that you do. In practice when one plays basketball, one does not usually attempt uncontested shots, or dribble without any defender nearby. When playing violin one does not usually think about individual bow strokes. To appreciate those activities, it may be worth watching Lebron Highlights or listening to Mozart. But the reason we drill the basics so much is _so that_ when more complicated situations arise, we have the _confidence to not worry about the basics._ At least that’s how I see it anyway.
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I think we actually agree more than you are suggesting and I was just not clear in my original post what I was going for.
(1) I think I wasn’t clear what “post-rigorous” meant for me. I don’t mean post-rigorous as not-rigorous-but-intuitive, like in math circles. I mean it as rigorous-AND-intuitive.
That’s why graph theory is the big example topic I have in mind. It’s specifically a topic for which a formal proof and the intuitive explanation are nearly exactly the same. Euclidean geometry is also pretty good, although I still give the evil eye to teachers who start by proving all right angles are congruent. Analysis is near pessimal and I have absolutely no intention to cover analysis in this proposal.
(2) In particular, I have had the same criticism of math circles, and even the napkin for that matter. (I feel bad saying this because I also love math circles, and I understand that they’re making a conscious trade-off between accessibility and completeness.)
(3) In particular, I agree that *if* I was teaching analysis to a class without proof experience I would probably be more pedantic. But instead of concluding that this means to follow Tao’s approach, I instead concluded analysis is a terrible way to start introducing people to proofs. (I’m looking at you, MIT.)
(4) Re assembly: I think even in the world with limited computing power, I would still fundamentally teach Python first. Or to be more precise, I think there’s a difference between understanding how a program *could be* written in assembly, and *actually writing* the program in assembly. The former seems fine, the latter seems unreasonable.
Pushing the analogy, I would teach pseudocode over assembly any day. I don’t care that the pseudocode doesn’t actually run.
(5) I feel like the last paragraph is the one for which I actually don’t agree with you. When you play the violin, you don’t _think_ about the bow strokes, but you still _do_ them, which is why they’re important to practice. But when doing proofs, ZFC is so far removed from the discussion that it seems nonsensical to me to assert ZFC is important. It would be like starting violin instruction by explaining how the digestive system converts food into calories into signals to your brain into motion in your muscles.
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I actually think that math circles (at least the more advanced ones I’m aware of) do this pretty well. They give students an approach to mathematics and mathematical techniques through more intuitive areas like discrete math or graph theory. I think teaching a student how to formalize the pigeonhole principle from their intuitive understanding of it or learning proofs by induction or contradiction come more naturally from combinatorial problems that are a lot more tangible and students can come up with many examples and get their hands dirty figuring out the framework, while maintaining the same kind intuition they might start with. And then later you can translate that into more intensive subjects like analysis. In terms of the way teaching proofs has been done traditionally, I’d say recreational math topics are more accessible than analysis or even the way euclidean geometry is usually taught.
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I can not tell you the number of times I (and other people on AoPS) have wished for combo equiv. of EGMO or an easier intro to Pranav’s book. Right now, the usual way to learn combo is just spam random ISL’s and learn enough GT till you are good enough to do Pranav’s (Pascal 96) book, but for a lot of people (including me), the initial part is really tough. Even though I would have most probably grown past the introductory stage by next year, still if you were to write an intro to proof writing using combo (and GT), I am sure a lotttt of people will really love it, and It will help lots of kids also.
Also, Thanks for writing EGMO and all of your handouts!
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Great idea!
s/pragmaticism/pragmatism
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This would be awesome. I will definitely recommend it when it comes out. I can say doing high level proofs is a weakness of mine. This sounds like an exciting undertaking.
2. I don’t know because complicated proofs is not my specialty. I just have always felt for the students who are getting to a USAMO/JMO level that there are already far more qualified and competent teachers than me for doing proofs. So I focused on what I was good at and accepted my shortcomings. It’s not that I don’t think I could learn to think at that level, it is simply a supply and demand equation for me. There is not a great number of student seeking such learning, and so I have not undertaken the journey. But a book like this, if well written could be a game changer. I always dream of when I generate high enough residual income to be able to take a few weeks off, learning such content would be one of my goals.
3. I really like the payment model. It is more than fair. I like that you understand that some families or students truly can’t afford it, and you take that into consideration, and rather than deny a young mind access to the resource, you accept some people will not be able to pay and embrace that reality. I will gladly buy 2 copies, and that way you can give one away. I would even be willing to purchase a copy every few months for the same purpose.
4. I have nothing to add. Just keep doing what you do. It’s making a difference. And that’s what matters the most.
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Grad student doing algebraic geometry here:
1) I don’t actually think you are in as much disagreement with Terence Tao as it may seem. He writes:
“The point of rigour is not to destroy all intuition; instead, it should be used to destroy bad intuition while clarifying and elevating good intuition. It is only with a combination of both rigorous formalism and good intuition that one can tackle complex mathematical problems; one needs the former to correctly deal with the fine details, and the latter to correctly deal with the big picture.”
He also mentions that the fact that your intuition is discarded during the rigorous stage is an unintended consequence. Also, the description of mathematical education in three stages seems to be a statement of how things are (at least for the vast majority of students) rather than a statement of how things should be.
2) I would definitely be in favor of such a text as you propose, and I definitely don’t think it would only be useful for those with an enriched mathematical background. At least, I attended a summer program in high school before I had such an “enriched mathematical background” where the instruction satisfied all four of your requirements and learned quite well how to write proofs. On the other hand, I was once (as an undergrad) a TA for an intro-to-proofs class that started with logical quantifiers and the entire course just looked like something that would dissuade people from ever doing more proof-based math.
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I am curious about how could the target audience, “high school students with an enriched math background”, not have seen and worked out enough cases and styles of proof, e.g., in Euclidean geometry, basic number theory, real analysis or CS type discrete combinatorics/algorithms, to already absorb the general idea before they pick up such a book.
Even in my antiquated generation, where you had to be in the know or in one of the nuclei of American meritocracy to be aware of summer programs (which were few and literally far between geographically), people super interested in math and/or olympiads would have also had experience in working through books or (again, if in the know) college courses that covered this stuff. Not necessarily in an explicit, in your face manner with formal logic, set theory, statement/reason sequences or the Moore method, but one way or the other enough to get the point.
Are things more narrowly competition focused these days? The tests are harder and there is more to learn, but also many more opportunities and access points to learn the theory and to start the whole process earlier.
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With all respect, I am not sure that writing on proofs a book warrants the spending of your valuable time. There many great books on the cubject. So if anyone, good or even better at math, wants to develep, they can.
I am more than happy to give you an idea about other subject matters that I see there is an absence of a good reference. [preferably, I would send this privately]
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[apologies, I pushed the button accidentally, wihtout proof-reading]
With all respect, I am not sure that writing on proofs a book warrants the spending of your valuable time. There are many great books on the subject. So if anyone, good or even better at math, wants to develep, they can.
I am more than happy to give you an idea about other subject matters that I see there is an absence of a good reference. [preferably, I would send this privately]
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