How to teach someone how to prove something
In a couple of my posts (most recently here), I’ve talked about the need for a course early on in undergraduate math classes on proof techniques.
The goals of the class are two-fold: first, teach the students basic skills, and second demystify the concept of proof. The students should come away from the class thinking, no it’s not magic, and I’ve learned how to do this stuff, and there are a few basic techniques which seem to come in handy.
Today I want to go further into what a curriculum for such a course might look like.
And I will, in a moment, but first I want to explain something. It’s actually a really important and dangerous question, how to teach such a course, because it could go wildly wrong, and sometimes does. From my commenter Jordan:
… “Numbers, Equations, and Proofs,” which I started at Princeton in 2002 and which is still going as well. Though here’s an interview with a dude who was an ace math competition dude and found the course so hard as to drive him out of the math major! So maybe it’s no longer as “for everyone” as I designed it to be….
This struck me, how perverted Jordan’s class became. For that matter, Math 55 at Harvard could have started out as a good idea as well, but by the time I got to Harvard as a grad student it was the reason so few math majors ever stuck at Harvard and why there were especially few women.
I remember Noam Elkies taught it while I was there and was famous for asking questions in class and getting students to compete to answer them quickly. It makes sense that he’d run a class like this, because he’s so fast and clever, and he’s naturally wondering, am I the fastest and clevererest of them all? But rather than a place where proof is demystified and people feel safe asking dumb questions, he’d created the polar opposite, a live quiz show of clever competition. Ew!
In order to combat this downfall and decay, I think the class needs to have a clearly stated mission as well as built-in curriculum requirements that works against ostentatious displays of cleverness, which indeed only serve to further the “I got it but you don’t” stereotype of math skills (but which mathematicians themselves are incentivized to further since that magical aura comes in handy).
For example, when I taught it, I let the students hand in homework again and again until they got a score they liked. Of course, this depending on me having an awesome grader (and a relatively small class), which luckily I had.
Also, I asked each student to give a presentation to the class on some proof they particularly enjoyed, and I sat through a preview of their presentation and gave them extensive advice on board work and eye contact, which took a lot of work but really helped them prepare and also boosted their egos while at the same time increased their sympathy with each other and with me.
But of course the most important thing was that I clearly stated at the beginning of each class in the first two weeks that proving things in math was a skill like any other that you get good at through practice. And when I left Barnard Dusa McDuff took over the class and still teaches it, so I know it’s in good hands.
If I hadn’t had Dusa, I’d probably have written a manifesto to be given to each person who would teach the class after me. Of course anyone could have just thrown that away but it’s an idea.
As for content, I taught them really basic proof techniques, so induction, proof by contradiction, the pigeon-hole principle, and some epsilon-delta practice. We covered some basic logic, graph theory, group theory, ordinals, and basic analysis. We constructed the reals two ways and the complex numbers once and talked for a long time about whether “i” is real and what that even means. We used A Transition to Higher Mathematics, which I recommend with a few reservations (please tell me if you’ve found a better text for something like this!).
Everything was done super explicitly and carefully, no rushing. I said things three times in three different ways. I wasn’t expecting people to be fast or clever, because I know intelligence works in different ways and that this stuff was completely new to most of the students. And at least one student in the class, who had been an artist, is now a grad student in math at Berkeley.
Looking over my post I realize I spent way more time talking about the tone of the class than the content, but that’s totally appropriate, since I think of this class as an introduction to the culture of mathematics (or rather the culture I wish we had) just as much as mathematics itself.
After all, there really is no time limit on good ideas, and you do get to do it over if you make a mistake, and going over things slowly gives you more time to ask good questions and find mistakes.
mathbabe 
Here at Stony Brook, we have had such a course since at least 2001. The emphasis of the course is neither to promote competition-style mathematics (we do have a math puzzles course) nor to be an inspiring magic show. The purpose is to help ease the transition to abstract algebra, real analysis, etc., where proofs are essential.
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Thanks for discussing this. I have been teaching a similar course but for entering freshman math majors. Actually what I did was hijack a course that was meant to be a soft friendly introduction to math (meaning essays and projects about fractals and all that) and try to introduce the students to what I consider to be the fundamental skills in math, namely logical deduction and abstraction. I don’t even like to use the word “proof”. It’s a question of being given certain assumptions, facts, theorems and figuring out what new facts you can deduce from them. Or, given a hypothesis, trying to figure out whether it can be deduced from what you already know.
Luckily, since my course is not a prerequisite for anything else, I don’t have to worry about covering some set syllabus or covering enough material. So what I do is run it like a workshop. I think expecting anyone but the best students to learn how to write proofs alone at home is maybe the least efficient way to do it. Instead, I lecture for maybe half an hour and then hand out a worksheet for the students to do during class, where they can struggle together and I can keep an eye on them to help them when they get stuck. This way they get to struggle, and I can provide the minimum amount of help needed.
But here’s the thing: to me math *is* the use of abstraction and logical deduction to derive new knowledge from old. So if we don’t teach this to all students (and not just math majors), then I don’t believe we’re teaching math honestly. We’re teaching only a caricature of what math really is.
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While it is written for a somewhat different audience than Harvard/Barnard students, I can’t help but chime in and recommend my colleague Bela Bajnok’s book “An Invitation to (Abstract) Mathematics”, forthcoming in the Springer UTM series, as a book that has been enormously successful at helping students here at Gettysburg College make a transition to proof-based math.
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As for the book, I never considered the one you mention due to its high price. Too many of our students are short of money, so I refused to consider any book that cost more than $60. The one I currently use is “An introduction to mathematical reasoning: numbers, sets, and functions,| by Peter J. Eccles, but I am open to other suggestions.
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OMG I didn’t notice how *expensive* this book is until you just mentioned it. OUTRAGEOUS!!!
Someone really should just post online notes from a really thoughtful version of the course.
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Sam Vandervelde’s book (http://myslu.stlawu.edu/~svanderv/bridge.html) is a lot of fun and definitely takes away a lot of the “math is not for me” feeling that you get from many of these courses.
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Two comments.
1. It should be said that Noam is a great guy and surely was not intending to alienate any segment of the class. Every one of us has the tendency to aim the course squarely at the students who remind us of ourselves, and it requires a lot of vigilance to resist.
2. “Numbers, Equations, and Proofs” was centered on number theory. We used Niven-Zuckerman-Montgomery but there really isn’t a book at the right level. I meant to write one but forgot to get around to it.
3. Revisions and regrading of homework is the best thing ever. I used to do it this way at Princeton. You need an extremely good grader and you kind of need the resources to pay them extra.
4. Here’s a former English major I converted!
http://www.cs.utexas.edu/~alewko/
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Yes, I know those are actually four comments.
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Oh good, so I don’t have to make a joke about how mathematicians can’t count.
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“I meant to write one but forgot to get around to it.” There’s still time to do it! Please?
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I created a freshman seminar of this sort at Duke. One idea I tried that I really liked was to have the students turn in homework as a group, with each student responsible for writing up a fraction of the problems, and each student officially responsible for checking and editing the work of another student. The checking/editing never really happened to the extent that I had hoped, but I think it would be great with the right system in place. Ideally, there would also be a revision round, but I didn’t have a grader and I’m no masochist.
In defense of Math 55, the whole point of it is to give the hot shots something interesting to do their freshman year. (In particular, it has nothing to do with teaching proofs. The Harvard analogue of what you are talking about is Math 101, which I think has/had a decent reputation.) The alternative is to just mix them in with everyone else, which would arguably discourage even more people from being math majors. And in defense of Elkies, he probably taught the most sane and sensible version of this course there ever was (a very difficult course on linear algebra and real analysis), as opposed to other professors who seem to splatter graduate-level topics across the homework for no real rhyme or reason.
One more thought: Most mathematicians did not learn to read and write proofs from an “intro to math” course, so where did they learn it? I know when/where/how *I* did, but I often wonder how others did. It seems like math camp is one common answer. But it also seems like some (smart) people just mysteriously figure it out from studying math.
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“The alternative is to just mix them in with everyone else, which would arguably discourage even more people from being math majors.”
I dunno — this is the way it was done when I was an undergrad (there was no Math 55, just Math 25, which all math majors, the Olympiad royalty and everyone else, took) and I don’t think there was any particular difference in the number or quality of math majors. What was true of that period was that math majors, including Olympiad royalty, took awesome undergrad courses like Dick Gross’s semester of representation theory + Coxeter groups and Siu’s semester of calculus of variations. When Math 55 is in force, you see a lot of hot shot undergraduates skipping over this stuff in order to get to Gromov-Witten theory and derived categories faster, which I believe to be a mistake for most of them.
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I think this sort of class should be taught at a high school level (perhaps, as well). I know too many kids that are turned off by math because all they know is the crap they teach in high schools. I also give guest lectures at high schools, and so I know high schoolers can handle reasoning, proofs, and even constructing useful definitions. Moreover, they enjoy it the same way that I see the great math students enjoy math competitions and such. And at least at the lower tier schools, many of the students who go on to major in math are the ones who actually liked what they did in high school, and by the time they get to proofs and linear algebra, they realize how much they actually dislike reasoning (as opposed to following directions), and they resign to just finish their degree so they can get their credential and teach high school the same cruddy way it was taught to them.
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Sorry to resurrect such an old thread, but this post really caught my attention. I DO teach a course of this type in a high school. (www.jhsct.org) There are several difficulties to doing so. 1) Most high school math teachers are not very good at math, nor do they have much of a math education. It’s really only an option in private schools, where we’re free to hire people with math credentials (like me) instead of math-education credentials. 2) Students haven’t had calculus yet. At best, they’re taking it concurrently with calculus. 3) Because it does not have an “AP” designation, it’s hard to convince parents that colleges want to see this class on a transcript. 4) It’s difficult to get the funding for books and faculty to have an “extra” math class. Most school money gets prioritized for students who are struggling, and there’s not a lot left over for kids who are already doing well. (ps: feel free to donate!)
There are a lot of benefits of college over high school: I have 5-10 kids in the class, most of whom have at least a full year of experience working with each other and me. All the students in the class are there by choice, and they’re very good students; talented, eager, and hard-working.
Here’s how I make it work. 1) It’s an elective class. Everyone in the class is taking it in addition to a “calculus sequence” class (which I teach with a modified “Moore method”). Because it’s an elective, I can grade a little less rigorously. The bulk of their grade is based on collaborative problem solving. 2) I use this book: http://www.amazon.com/Transition-Advanced-Mathematics-Survey-Course, but that’s mostly because the Dumas book is too expensive. 3) I skip the analysis chapters, and focus on logic, set theory, number theory, algebra, and graph theory. If I have time, I also do some point-set topology (my favorite!) 3) It’s a “seminar” style class. Every other day, students read a section and answer the “reading questions” as homework. They email me any questions. On the first class-day, I answer any questions that came up, and provide some general context and background. Students go home and solve problems. The next day, they present their solutions at the board. This is a collaborative process; they frequently get stuck or need help (mostly from each other, but sometimes from me) tightening up their proofs.
One thing I find really important and helpful is to take a whole class at the beginning to explain to them that the goal of this class is to teach them to read and write mathematics, and that its a new skill that takes practice to learn. It’s not like the math they’ve done before, where just being smart basically assured that they’d just “get it” right away.
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Been thinking about this lately and am giving a talk about this very subject soon at NCTM in Dallas. I teach a Transitions course at my college (a small, public open-enrollment college in the Midwest)
Comments:
“1) Most high school math teachers are not very good at math, nor do they have much of a math education. It’s really only an option in private schools, where we’re free to hire people with math credentials (like me) instead of math-education credentials.”
Unfortunately True
“2) Students haven’t had calculus yet. At best, they’re taking it concurrently with calculus.”
I don’t see how that’s a real hindrance. It limits your choice of topics a little, but there’s plenty to investigate
“3) Because it does not have an “AP” designation, it’s hard to convince parents that colleges want to see this class on a transcript. ”
Unfortunately, probably true. Although, from personal experience, I can tell you that it is very useful. (I took a similar course at JHU/CTY the summer after my 9th Grade Year. It was extraordinarily helpful in kickstarting my mathematical career.)
“4) It’s difficult to get the funding for books and faculty to have an “extra” math class. Most school money gets prioritized for students who are struggling, and there’s not a lot left over for kids who are already doing well.”
This one is outside my experience.
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I’m so glad you wrote this post! Some random comments (I won’t try to count them, lest I make a fool of myself a la Jordan):
(*) Cal did not have a class like this when I was an undergrad (which was just a couple of years before you were an undergrad). My first two upper division courses were abstract algebra and linear algebra. I distinctly remember getting back homework after homework with the comment “this is not a proof” and thinking, “well, what the hell is a proof?!” I basically figured it out for myself during that first year, but I’m guessing lots of folks didn’t. (I didn’t have the math camp bona fides that Cathy came in with, though I don’t particularly remember being intimidated by the people who were “getting it.” I just remember thinking that it wasn’t being taught very well if I wasn’t getting it.)
(*) I don’t know when these classes came into vogue, but they seem to be everywhere now. In principle, they are a good idea. But even in my department, the implementation varies drastically based on who’s teaching it. It is classified as a “writing intensive” course where I teach, which means that enrollment is limited to 20, but we do not have graders at all for upper division courses. So I did “infinite resubmission” the first time I taught it, because it is so obviously a good idea. But I dropped any form of resubmission the second time because I needed to be sane.
So what I did instead: Homework was assigned every day but not collected. They were expected to be ready to talk about (either in groups or in front of the class) any of the problems and show their solutions or at least what they tried. They then had to pick one problem from the set to write up carefully. (And we spent a lot of time talking about the difference between presenting a proof in class and writing it up carefully for homework.) The idea was to separate the two steps: figuring out the problem and writing it up carefully and clearly. We talked a lot about how to solve problems / find proofs (and the techniques you mention, though I probably gave epsilon-delta short shrift). We also talke a lot about what makes a good, clear written proof: complete sentences, displayed equations, clearly defined notation, etc. But they never had to write up (for a grade) a problem they didn’t already know the answer to.
This went pretty well for the first half of class, but later on I had fewer and fewer students willing to get the conversation started about the homework. I think they stopped even working on it, hoping that they’d find out the answer in class and then be able to write it up. I don’t know how to fix this, really. There’s this tension between putting someone on the spot who really worked on the problem but couldn’t solve it (so I always allowed students to “pass” on a presentation, but recorded when they did so) and just allowing students to blow off the homework.
(*) The book situation for this class is outrageous. There’s a good book by Paul Sally (http://www.amazon.com/Tools-Trade-Paul-J-Sally/dp/0821846345/ref=sr_1_1?ie=UTF8&qid=1334243838&sr=8-1) if you happen to be teaching at University of Chicago. But it was way too sophisticated for our students. I used “Chapter Zero” the first time through (http://www.amazon.com/Chapter-Zero-Fundamental-Abstract-Mathematics/dp/0201437244/ref=sr_1_1?s=books&ie=UTF8&qid=1334243892&sr=1-1), which is often OK but when it’s bad, it’s really really bad. (I read the chapter on equivalence relations multiple times, and I still have no idea what it says.) Last time, I used this one: http://www.amazon.com/Extending-Frontiers-Mathematics-Augmentation-Curriculum/dp/0470412224/ref=lp_B001IZ0YT8_1_7?ie=UTF8&qid=1334243974&sr=1-7. I really like the idea of the book (they take the “prove, or disprove and salvage” approach to problems). But the upshot is that there are wrong statements in the book, labeled as theorems, and this is an absolute disaster for take-home open-book exams. Oh, and it’s $100 for a skimpy paperback.
So, Jordan, I’ll co-author with you if you want. I actually have a lot of curriculum development experience, and I suspect we’re largely on the same page as to what this should be. We’ll make the book available free on our website. You in?
Wow… that was long. Sorry.
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“I let the students hand in homework again and again until they got a score they liked” — someday, there will be an easy-to-use interactive proof checker that you can use to let the students write fully rigorous proofs and debug them until they work. (There are many automated proof checkers already today, but they work at too low a level of detail to be fun for anyone besides a logician.)
I teach a class where the students have to write simple “programs” (like “design a Turing machine that adds two binary numbers”), and I made an online submission system to check everything automatically. A lot of extra effort the first year, but wonderful for me and my grader since then. And better for the students too — instant feedback, and you naturally want to keep submitting until it works. Some students will submit a single problem hundreds of times as they work on it. Even the best grader would never do this. A psychology major once commented on what a strange style of thinking was required for the homeworks, which I figured was a good sign that learning was happening.
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We have a transition course that is (at least should be) like this, and now I’m feeling inspired to teach it. It is a big effort on the part of the instructor and grader, but it can have such amazing outcomes.
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Woohoo!!
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Proving things is an undervalued life skill.
If I recall correctly, the first time I learned proofs in a math course was in 7th grade geometry. I regarded Mr. Cannon as a great teacher and still found the fourth quarter very difficult. From my years as an undergrad, my courses with Serge Lang in real analysis, and with Shelly Kagan in ethics stand out as exceptional experiences. In all cases, the courses that stand out were both challenging and rewarding.
Professor Lang emphasized speed and memory, often telling us, “You should know this three days after you’re dead.” Professor Kagan demanded similar rigor in prose, with a more measured approach; I recall writing one paper three times before my grader was satisfied with my argument, and before I was satisfied with my grade.
Your measured approach to teach proving as a life skill through math is refreshing. Of course, speed still matters. In a course setting, time is bounded by the end of the semester. In life, it’s bounded by death. All the more important to learn life skills early. I see no reason why the opportunity of learning with your measured approach should wait until college. I hope we see K-12 schools teaching proofs using your approach even earlier.
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Having been through an Eastern-Block (when it still existed) education, I find this fascinating.
I was introduced to proofs when I was about, what, 10-12 year old, as part of maths competitions. High school maths had some basic proofs, but induction and contradition was part of basic curricula for everyone IIRC. At the uni (I did maths for a year), doing proofs WAS the exam, I remember having to prove L’Hopital rule in my first semester of calculus (mostly because it was about a densly written A4 lenght – I didn’t know about Cauchy mean value,which makes it a bit shorter).
I say this mostly because when recently I attended some lectures on basic stats, I was bored to death by people using the approach you suggest – going through everything three times in a great detail, over and over.
I understand why you want to do it, but this is as good a way of turning someone from maths as the “magic shootout” one.
It’s the eternal dillema of whether you go at the slowest speed and lose the fast, at the fastest speed and lose the slow, or medium (and lose everybody) – I never could get a right speed.
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Japheth just posted this – I haven’t read all the way through, but it seems like it has some potential!
Click to access proofguide.pdf
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Students in transition to proof-based mathematics may find my interactive, self-study tutorial useful (available free at my website http://www.dcproof.com). Using the simplest possible examples from formal logic, set theory and elementary number theory, the student, working at her/ his own pace, is introduced to the following methods of proof:
– direct proof
– proof by contradiction
– proof by contrapositive
– proof by cases
– proof of biconditionals
– manipulating quantifiers
– proof by induction
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Just and FYI, Keith Devlin has recommended students to read this blog post for his Mathematical Thinking course on Coursera. 🙂
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As an alternative (win-win) course set up consider: In the Johns Hopkins programs, the Center for Talented Youth , the Independently-Paced Math course makes the student take each test until they get at least a 90 before they can go on to the next chapter. If they can’t get a 90, there will be an accumulated deficit of knowledge which hampers their complete understanding. In this way the students ALL do well, and many accomplish in three weeks math courses for a whole year of school.
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Since this blog may now be inundated by “non-initiates” (because it is linked to in Keith Devlin’s course), it may be fair to provide the following link: http://en.wikipedia.org/wiki/Math_55
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This is similar to how our Discrete class is taught at Berkeley. Before taking the class, I hadn’t written many proofs before — only inductive proofs. Even though I was fairly good at math, I still felt that there was some magic in proof writing. If you’re familiar with the ARML contest, during the power round I wouldn’t participate nearly as much as I would in the team round — I wasn’t sure of my proof writing skills.
I think that an introduction to proof writing should be taught to high school students. Many people take AP BC Calc in their junior year of high school and then are left with no math class to take senior year. That math class should be introduction to discrete math.
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I can’t resist mentioning that I help start that class, along with Sara Robinson and Karen Edwards, when I was an undergrad at UC Berkeley in 1992.
Cathy
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Oh wow, what a small world.
Well, it’s been my favorite class by far over here; by helping start the class, did you help set the “curriculum”? It seems to be exactly what you say in your post.
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With many of these “MOOC”s, one could put one of these classes together and have a large pool of users to see what works best. Have you thought of doing this at like Coursera or Udacity ( or some such place )?
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I’m a new reader, and I’m catching up on old posts. This one caught my eye, because proofs were the reason that I didn’t become a math major way back when. I was in a good (and in many ways very supportive) department that was actively trying to make math more accessible, esp. to women, than it been, but I think I came along just prior to their realizing that some transitional course to proofs was necessary for the non-geniuses they had decided that they wanted to invite along for the ride. Of course, the really bright kids figured it out for themselves, and could see the path from A to B, but I could have used some help – more than the 10 minutes I got in office hours at least! How I wish you’d been there to teach this course! Although I’ve found my way to a pretty quantitative gig anyway, I’d sure like to have the stronger conceptual background I would if I had gotten a few more courses under my belt.
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How To Read and Do Proofs, by Daniel Solow. It’s short, sweet, and nearly perfect.
The first edition was required reading when I took abstract algebra. A miraculously simple way to learn proof techniques. It really works! Don’t know about the recent editions, but the early edition of this work did for me exactly what the post above calls for, and brilliantly.
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What are the reservations you have to using A Transition to Higher Mathematics? I ask this as I am always interested in refining my knowledge base, and though I have a firm basis in logic (and happen to be an attorney aka 1st against the wall when the time comes) I like taking a different approach at times to keep my mind limber.
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The problem with it for a high school class is that calculus is “snuck in” in almost every chapter. I mean, I get that me teaching graph theory to kids who dont’ know logs yet is a little weird, but it’s working out very well. I just gave up and taught epsilon-delta limits to them and now we mostly do some of the analysis parts too. Next year, I might require Algebra 2 (polynomials and beginner transcendental functions) as a prerequisite.
So far this year, we’ve done Logic, Set Theory, Number Theory, Abstract Algebra, Topology, and Graph Theory. By a significant margin, the logic and set theory were the kid’s favorites (although they might just have inherited that from me) We also staged a (super geeky) play based on Lakatos’s “Proof and Refutation”.
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Wow, I wish I were a student in your class!
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You should visit via skype sometime!
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I have always had trouble going from Step A to Step B etc. I intuitively went from A to D or get stuck. Didn’t help me much in math and I’ll never be something like a programmer where that step by step is important.
I would have liked some of this education even in high school. Maybe I can learn how I make those leaps now I have some experience behind me. I’ve found that a bit of life before schooling is a very valuable asset.
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