My 2 developing from a typewritten style to one with a loop at the bottom left may have preceded mathematics. I just like the look. However, it probably would have happened eventually anyway – I cross my Zs for good measure, so they are decidedly not 2s.

I also cross my 7s, and in many cases (though not typically when it is part of a longer number) I give my 1 a top hook and a bottom bar. My lowercase i and t curve up at the bottom right, and my lowercase x has a hook at the upper left.

I’ve been out of mathematics for nearly two and a half years now, but those handwriting changes show no sign of fading: an odd little calling card left by my former vocation.

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Inclusive or is a low hurdle. In plain English, when someone asks “Should we take our vacation to New York or Boston?” the assumption is that the answer will be “New York” or “Boston” (or “I don’t care” or “neither”). The geeky joke – sometimes serious – answer of “yes” is totally unhelpful. However, it’s not too hard to get used to inclusive or, and we do have examples in natural language. One of the best is “Would you like sugar or cream in your coffee?” Of course, even then “yes” isn’t a useful answer, since there are three possible coffee fixings that would lead to it.

Implication is a much higher bar; there’s still a part of me, even, that doesn’t think A implies B means much of anything when A is false. Implications where there is clearly no causal relationship between A and B can be helpful, since they rarely appear in plain English (outside of statistical correlations, I suppose) and thus resist natural language intuition. For teaching, in addition to that, I settled on the approach of “an implication is true unless proven false.” You can only prove that it’s false by having A be true and B be false, so in the A-false situations the implication is therefore true. This is basically the conversion of “A implies B” into the disjunction “B or not-A,” but hopefully in a way that doesn’t just shift the confusion to a different location.

I think the base of that confusion is a disconnect between allowed truth values. In plain English, a sentence can be true, false, or nonsensical. The third option is not permitted in mathematics (except in the sense of ill-formed formulas), and it is confusing that many implications that seem nonsensical or are constructed from false clauses (“if the moon is made of green cheese, then fish swim in the sea;” “if the moon is made of green cheese, then carriages turn into pumpkins at midnight”) are logically true.

In the fifteen-plus years since my bridge class, I have found only one plain English example of an implication considered true but constructed with false clauses, and in general my students were unfamiliar with it: the adage “If wishes were horses, beggars would ride.” I would love another, even though nowadays it would be purely for my own interest.

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• Respect your students: Learn their names. Start and end on time to the best of your ability. Admit your mistakes and when you don’t know an answer. Answer questions without implying they are stupid ones. Set out all expectations in the syllabus and stick to them. Be prompt returning graded work.

   

• On fairness: Strike a balance in class between answering all questions and keeping lecture moving. Enforce the rules you set and do so consistently (to facilitate this, only set rules you really care about). Compassion is good, but keep the phrase “the squeaky wheel gets the grease” in mind when you are tempted to make concessions for a single student without a very good (e.g., dean’s office approved) reason – generosity to one student can be unfair to the rest of the class. Even what seems to be generosity to the entire class may not be; extending a deadline at the last minute does not benefit the students who did the right thing: buckled down and got it done by the original due date.

   

• Find your style in the classroom. There are many ways to engage with your students: tell jokes, ask them questions, ask them for their questions (noting that a sufficiently long pause will likely feel uncomfortably long at first), make a few mistakes on purpose, bring them to the board, walk among them, give them small group work. Figure out what behavior bothers you and what doesn’t. Do you care about attendance? tardiness? whispering? eating? reading the paper? sleeping? My recommendation is to ignore it unless it really bothers you, but do feel entitled to ban actions that get under your skin. I didn’t worry about any of those examples in general, but at the beginning of the semester I told the students that the standard in my classroom was whether an action was disruptive to other students. You can be late, but don’t plow through four students to sit in the middle of a row. You can eat, but make sure the packaging isn’t loud and the food doesn’t have much odor. Etc.

   

• If a student starts crying in your office, pass the tissues and carry on. Gently, of course. In my experience most crying students are embarrassed they couldn’t control themselves, and making a big deal out of it just makes things worse. The other two situations I can conceive of also bear this reaction: the student is glad to be crying because maybe it will make you feel bad, or the student is genuinely upset but is okay with you seeing that because they’re that comfortable in their own skin.

   

• Key elements to students liking you: Be clear, be accessible, care. Clarity is lecture organization, lecture content, and boardwork (don’t erase things you’ve just written, use the boards in a sensible order, erase thoroughly, don’t write across the lines in the board because that makes some people CRAZY). Accessibility is arriving at class early and staying late to answer questions, if possible, and having a variety of office hours to accommodate schedules. Caring is learning names, keeping track of how your students are doing, learning about them as individuals to whatever extent possible, and being responsive – if something really isn’t working in the classroom, change it. Students will forgive a multitude of sins if they see that you genuinely care about their success in your class.

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• Bad Proof: Suppose the hypothesis. We can substitute a times something for b in the equation b times something equals c. The two somethings together multiply a to c so we’re done.[way too vague; might be logically sound but difficult even to assess that]
• Deceptive Proof: Suppose a, b, c are natural numbers such that a divides b and b divides c. All of a’s prime factors must also be prime factors of b, and likewise for b and c. Therefore any prime factor of a must be a prime factor of c, so a divides c.
  [seems convincing, but the prime factor business is without its own proof here, so preliminary work is missing. Even with that, would need repeated prime factors to occur sufficiently many times, so the argument would need cleaning up.]
• Good Proof: Recall a divides b if and only if there is an integer k such that ak = b. Suppose, then, that ak_1 = b and bk_2 = c for k_1, k_2 natural numbers. By substitution, ak_1k_2 = c. Since the product k_1k_2 must be some natural number k, ak=c and a divides c.

Multiple proof techniques for one statement:

Quick definition: integer n is even if, for some integer k, n=2k, and odd if, for some integer k, n = 2k+1.

Claim: If n is even, n+1 is odd.

• Direct proof: Let n be even and k be such that n=2k. Then n+1 = 2k+1, so by definition n+1 is odd.
• Contrapositive proof: Suppose n+1 is not odd. Then it must be even, so for some k n+1 = 2k. That means n = 2k-1 = 2k-2+1 = 2(k-1)+1. Since k is an integer, so is k-1, so n is odd by definition. Therefore, if n is even, n+1 must be odd.
• Contradiction proof: Suppose n is even but n+1 is not odd. Then for some integers k, l, n=2k and n+1=2l. However, then n+1 = 2k+1 = 2l, so l=k+1/2. Since k and l are both integers, this is impossible, so whenever n is even n+1 must be odd.

Proof using multiple methods together:

Theorem: The polynomial p(x) = x^3 + x – 1 has exactly one real root.

Proof: First we show that p(x) has at least one real root. Note that p(0) = -1 and p(1) = 1. By the Intermediate Value Theorem, since 0 lies between -1 and 1, there must be a real number c between 0 and 1 such that p(c) = 0. [Direct]

Now suppose for a contradiction that for c_1 ≠ c_2, p(c_1) = p(c_2) = 0. Then by the Mean Value Theorem, there must be some b between c_1 and c_2 such that However, p'(x) = 3x^2 + 1, which is always strictly greater than zero. Therefore there cannot be such c_1 ≠ c_2, and p has only one real root.

Case proof:

Claim: If n is an integer, then n^2+n is even.

Proof: Case 1: n is even. Then n^2 and n are both even, and the sum of two even numbers is even.

Case 2: n is odd. Then n^2 and n are both odd, and the sum of two odd numbers is even.

Proof of equivalence using two implications:

Theorem: Let a be an integer. Then a is nondivisible by 3 if and only if a^2-1 is divisible by 3.

Proof: (->) Suppose a is nondivisible by 3. Then a = 3k + r for some integer k, and r equal to 1 or 2. Substituting, a^2-1 = (3k+r)^2-1 = 9k^2+6kr+r^2-1. Clearly this is divisible by 3 exactly when r^2-1 is. Since r=1 gives r^2-1 = 0 and r=2 gives r^2-1 = 3, both of which are divisible by 3, a^2-1 is divisible by 3 for any a nondivisible by 3.

(<-) Suppose a^2-1 is divisible by 3. Since a^2-1 = (a-1)(a+1) and 3 is prime, 3 must divide at least one of a-1 or a+1. However, in either case 3 cannot also divide a itself.

The key to a (one-part) “if and only if” proof is in the -> half:

Proof: Any integer a may be written as 3k + r for some integer k with r equal to 0, 1, or 2; a is divisible by 3 if in this form its r is 0. By substitution, a^2-1 = (3k+r)^2-1 = 9k^2+6kr+r^2-1, which is divisible by 3 exactly when r^2-1 is. We may plug in each of our values of r: r=0 gives r^2-1 = -1, r=1 gives r^2-1 = 0, and r=2 gives r^2-1 = 3. The latter two, corresponding to a nondivisible by 3, are divisible by 3, and the first one, corresponding to a divisible by 3, is not divisible by 3, giving the desired correspondence.

Depending on context, these might qualify as “bad” proofs since they take for granted the division algorithm and the fact that if a prime number divides m it must divide at least one of the terms in any factorization of m.

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• explicit/specific (non-vague)
• logically sound, including complete
• lacking irrelevant statements
• understandable to the reader
• self-contained (may assume basic things; anything else needs explicit reference to previous work or must be written out in the proof)

Proof kinds

• Direct proof: Assume hypothesis and march to conclusion.
• Contradiction: If proving a single clause, assume its negation and prove something that contradicts laws of mathematics. If proving A->B, assume not B and (implicitly or explicitly) A, and prove something that contradicts one of those two assumptions or general laws of mathematics.
• Contraposition: Arguably a special case of contradiction; used only for proving implications, A->B. Assume not B and prove not A.
• Induction: Theorem asserts something about an infinite collection, such as all natural numbers. Prove it for the lowest value (starting point) and then show that if you know it for n you can prove it for n+1. Conclude it holds for all values.

Proof tips

• Bi-implication is often easiest to prove as two separate implications, possibly using different methods.
• Existence: Either show the statement holds for a specific value or kind of value (e.g. all even numbers), or draw a contradiction from the assumption that it holds for no value.
• Universal: Leave the value as a variable and use only properties common to all possible instances of the variable.
• “For all x there exists y” (AE): Universally-quantified value must be represented as a variable, existentially-quantified value may (often must) be a function of the universal one.
• “There exists x such that for all y” (EA): Universally-quantified value must still be a variable. Existentially-quantified value can be specified, but must not be a function of the universal one because the same value (not just the same function) has to work for all possible values of the variable.
• Disjunction: “A or B” is often proved as “not A -> B” or “not B -> A”, whichever is more straightforward.
• Cases: Sometimes your universally-quantified values fall into a few neat categories (odd/even, positive/negative/zero) and it is easier to use different proofs for each category.
• If it’s not coming together, two things to check: Can you work backwards? Are there any definitions you haven’t fully unpacked?

Proof warnings

• Existence of an item satisfying the given criteria may be shown by presenting an example (constructive proofs) but need not be (nonconstructive proofs). For example, the Intermediate Value Theorem says if f is continuous and f(a) ≠ f(b), for any r between f(a) and f(b) there is c between a and b such that f(c) = r. Its proof gives no clue as to where c might be found, not even in terms of a, b, and r.
• If you are trying to prove something holds of all items in a certain group you may assume only traits that hold of every item in the group. For example, if you are proving something about all rational numbers you may use the fact that they may each be written as p/q where p, q are integers, q is nonzero, and gcd(p,q)=1. Be careful not to assume rational numbers are nonintegers, integers are positive, and so forth.
• If you are asked to prove an implication, it is likely it is not actually an equivalence, and if you “prove” an equivalence it will be incorrect. This trap catches more students than you might expect.

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A beginner’s guide to quadric surfaces (pdf), by me, for my calculus students and intended to supplement the textbook rather than be used as a replacement.

I found the first paragraph of this article on seat belt use to be a great example of awkward statistics writing.

Chapter 23 of The Art of Public Speaking by Carnegie and Esenwein is a thorough exploration of logic in debate.

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There are many questions one can ask about counting: how many ways are there to fill a bag of chocolates (i.e., how many distinct lists of varieties might a bag contain)? How many ways to fill a bag include orange-filled chocolates? How many ways include both chocolate cream and mocha-filled chocolates? How many ways include neither vanilla cream nor raspberry-filled chocolates?

The question I’m more interested in discussing, however, is this: Suppose something is going wrong in the assembly line and over half the bags are coming out overfilled. Before you break open the chocolates and look at their insides, form a hypothesis as to what has gone wrong. What machine or machines are most likely to be malfunctioning? Why?

Your answer consists of your assumptions and the arguments based on them, with enough context restated from the problem that the answer reads smoothly.

There are a few conflicting but appropriate assumptions you could make here. The first is that either the malfunction is consistent, affecting all bags that come through that machine, or it’s inconsistent, affecting at most all of the bags, but possibly fewer.

The second choice of assumptions is whether the description “over half” implies “not all.” If every bag were coming out overfilled, you’d expect that to be specified… unless you’re a logician teaching a math class, in which case “over half” definitely includes the possibility of “all.”

But the important thing for writing up an answer is to be clear on what you’re assuming above and beyond the information given in the problem.

There is a third assumption that I expected everyone to make: that the most likely explanation involves malfunction of the fewest machines. It may not always be the case, but we’re assuming the candy factory is run well enough that machines are more likely to be working properly than improperly.

So what’s your answer?

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You can turn this lack of clarity to your advantage if you are an evil marketer, pitching either in favor of or against the product. Here’s an exercise:

Suppose a medication for condition X has the side effect of raising a person’s risk for disease Y. On average, the risk increases from a 1% chance to a 3% chance. Represent this data in a way that
a) makes it seem like a terrible side effect,
b) makes it look like a completely dismissible side effect, and
c) gives full disclosure on the risk.

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First, addressing slides or boardwork:

• Don’t put too much on a slide.
• However, don’t put so little on each slide that you’ve moved on to the next before the audience has time to read it.
• Feel free to only say some things. In fact, if literally everything is written out the audience may wonder why they’re listening to you when it would be much faster to read the slides. You need the key points, the technical definitions, and anything else the audience may want to re-read, as well as diagrams or pictures. Think of your slides as the highlighted or boxed portions of a textbook.
• It is helpful to repeat important information (especially definitions) on later slides, or at least verbally.
• Turn down the contrast and brightness on your computer screen (both separately and together) to check readability if you’re using anything other than black and white; projectors often wash things out a bit.
• In a chalkboard talk, write top to bottom, left to right, and respect the seams of the board as edges of paper – it is difficult to write across them neatly. When going back for a second round, erase your previous writing thoroughly.
• Make sure the type (or your handwriting) is large enough.
• Use whitespace generously – avoid large blocks of solid text as much as possible.
• Landscape orientation allows higher magnification since it matches the usual projection screen dimensions better than portrait.
• Stand to the side of your slides or what you’ve just written. It is continually surprising to me the number of people who stand in front of their writing until they have moved far enough past it in the talk that reading it would be a distraction to the audience instead of a help.
• Proofread! You might not catch everything, but you’ll catch the most egregious errors. Putting the slides aside for a day will help your observational abilities – when you look at something repeatedly you often start seeing your memory of it rather than the actuality.

Secondly, your speaking:

• Practice! When you have given many talks you will need less practice, but when you are starting out, multiple sessions are desirable.
• I have never attended a talk that was read aloud from a prepared paper that did not bore me to tears. Use notes, but don’t read word for word.
• Speak loudly but do not shout. While whispering and mumbling are clearly not desirable, neither is yelling. Note that you can yell without raising your voice, as well. The audience will not understand more if you are strenuously emphatic; they will simply feel accosted.
• Note that while practicing your talk by yourself is good, it is inexact for timing. Err on the side of “too short” (within reason, of course – prepare more than thirty minutes of material for an hour-long talk); between extra things you say and questions from the audience the length will most likely increase, and if not, well, no one minds ending early.

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The key to giving a good talk is to remember that talks are about the audience, not you. If the audience comes away understanding your result and its significance, you win. If the audience comes away confused, annoyed, or otherwise feeling that you’ve wasted their time, you lose.

I always tell students that talks are an exercise in letting go. You must accept that you won’t be able to say everything you know. There’s a time limit and a speed limit, and both need to be respected to keep the audience’s respect. Pick a goal for your talk (understanding the statement, motivation, and significance of one specific result would be a typical one) and ruthlessly tear out anything that does not assist you in getting to that goal. Omit proofs if you can and give high-level summaries or “proof by example” if you cannot; give only special cases of definitions and results if that’s all you need (but always label them as such).¹ Less is more.

Your audience will be more forgiving of seeing several things they already know than of an unfamiliar idea being flashed before them with inadequate preparation. Take into account the audience you are going to address, of course: a seminar in which everyone is at least a graduate student in your field requires less background than a colloquium talk wherein, although everyone is a mathematician, many will have seen your topic most recently in a class they took a decade or more past. Within reason, it’s all right to be imprecise, as well. I have seen many talks where the speaker said something along the lines of, “I’m lying to you now, but it’s morally correct.”² Examples are helpful and getting bogged down in details is not; it will require more than a talk worth of information to truly understand the details of your work no matter what you do.

Give context for your work. As with background, how much you have to say will depend on your audience. The history of your work can be helpful in understanding how you came up with your proof, if it is not a standard technique (e.g., “we tried this and it didn’t work, so we looked into why…”). Clarify your contribution but don’t obsess over it; while you certainly don’t want to appear to take credit for someone else’s major idea, you don’t have to itemize who worked out each specific detail.

Think carefully about organization. The best order to explain things in a talk may be different from the best order for the full research paper. Remind your audience of your goal periodically and repeat key definitions or lemmas if they are being used any time other than immediately after their original statement. Don’t insult the intelligence of your audience, of course, but remember that most or all of them haven’t been working on this topic recently, unlike you, and can’t refer back to earlier material like they could in a paper.

The 3 Cs are to be clear, concise, and charming. I have no advice for the last one, but I hope this post has been useful for the first two.

¹ Of course in a topic-specific seminar, you may have been asked to explain a proof, in which case this rule is clearly changed.

² Any mathematician will understand this phrase, but non-mathematicians might not. “Lying” is used typically to mean “being somewhat imprecise,” and “morally correct” means “literally false, at least in some aspects, but gives good intuition.”

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