Friday, January 16, 2015

Snippets from the 2015 Joint Mathematics Meetings

I'm back from a trip to San Antonio for the 2015 JMM, and already into the full swing of the spring semester! I've attended the last three JMMs, and each trip is full of many great memories: seeing grad school friends and former teachers, feeling part of a larger community of mathematicians and educators, rubbing shoulders with big names, and attending interesting research and teaching talks. Every year, I take some notes during talks and say to myself, "I'm gonna write something up about my experinces". Inevitably, the notes get lost or my ambition wanes exponentially on the flight home. This time, I'll act, though.

To improve the chances of sharing my experiences, I decided to record at least a brief, one-sentence "nugget of wisdom" from every talk I attended, be it research- or teaching-oriented. Oftentimes, I jotted down other things, too, like references to look up later, or more extensive ideas. But for (pretty much) every talk I went to and paid genuine attention to (which was about 95% of the ones I attended), I wrote down something memorable to take away from that presentation. Gathered below is a curated list of some of those "nuggets". (I've fleshed out some ideas that were chickenscratched and brief, and I've tossed out some that were too similar to others.)
  • Create a "moment" or "story" for every class you teach; in years to come, students might forget the material, but they'll remember a significant moment.
  • We mathematicians have trained our brains to switch from "proof mode" to "example mode"; we need to help students do the same, or even think to do so.
  • In an intro to proofs course (or any course, really), a good activity is to have students convince their classmates that their work is valid and clear.
  • It sometimes seems like students read a proof like a story, so they can get a good overall sense for it, but they will miss little statements that are either incorrect or subtle or otherwise difficult to follow; can/should we teach them to fixate on particular lines of a proof?
  • If you want to write a "fun" exam problem (or even have a theme throughout the exam), be sure to know your audience and make sure it won't be disruptive.
  • There are many comic strips that have mathy content; an activity where students have to explain the joke to someone else can build their conceptual understanding.
  • Beginning class with a joke or short story, even if it takes up a few minutes of time, can get the class more engaged at the beginning and yield more participation/attention throughout the session.
  • First year students may view math more cohesively (as opposed to procedurally) than we give them credit for, and our reaction to that shapes their future learning.
  • Students apparently misunderstand what a function is; but what can we do about that?
  • Baseball is known for being a sport full of statistics, but folks should be aware that the PGA Tour's ShotLink system will willingly share with you, for free, more data than you know what to do with!
  • Have students take conceptual quizzes before class (perhaps online), and shape discussion at the beginning of class based on those results.
  • Mentoring an undergrad research project is both time-consuming and rewarding, in large amounts (of both).
  • Why is it so hard to come up with combinatorial proofs for identities that are trivially true algebraically?!?!
  • When training future teachers, be sure to teach them how to give good criticism; it's not all about praise.
  • Adjacency matrices and their properties can be great topics to use "easy/intro" graph theory to motivate linear algebra.
  • A great speaker can make a topic that I really don't know anything about seem completely fascinating and understandable.
  • If you want to motivate epsilon-delta definitions, use in-class numerical explorations/"lab work" to make the notion of "closer and closer" feel like cheating; then, when the epsilon-delta definitions are introduced, it will feel rewarding.
  • Using primary literature (i.e. journal articles) in undergrad classes improves understanding, but it's very difficult to choose appropriate articles. (We need more people writing such articles!)
  • One way to introduce students to reading scholarly articles is to choose a particularly readable article and have students turned in an annotated copy after reading, making notes and highlighting parts that are relevant to your course.
  • We need to ensure that high school students understand there is so much more to math than just calculus, especially talented students who might be jaded by math by the time they enter college.
  • (re: last point) It would be great to provide activities to AP Calc high school teachers to use in school after the AP exam, but we need to keep the needs and struggles of those teachers in mind (e.g. provide them a guide to the math involved, to save them time/effort).
  • As much as possible, be playful with math; show students that one can unwind with math, that it's not always "stressful".
  • There are many academic journals that embody exposition instead of creative research, but I wonder why these don't have the same level of "prestige" attached to them as other journals.
  • Seeing "3/10" on a proof is demoralizing to a student; seeing something like "progressing" (e.g. on a descriptive scale) is better, but still, students want and need a sense of where they stand. Is there a happy medium?
  • Historically, divergent series were once a contentious and important topic; nowadays, not so much. What happened, when, and why?
  • When we plan courses, we should ask ourselves, "What are the skills we want students to have 10 years from now?" We can contribute to that positively, and we really should.
  • Encourage false starts in class, and be willing to follow through to a dead end; if students complain about their notes, just say, "Put a box around this part and, underneath, explain WHY it's wrong, then move on."
  • When giving online quizzes, make the last question, "Ask one question about this section (or earlier material)" and give credit for this. Making the students formulate a question pays dividends for them and us.
  • I appreciate the teaching talks from large universities where their class sizes are 200+ so they can have lots of data to analyze, but my classes are so dissimilar that it's hard to compare. I also appreciate the teaching talks from small colleges where their class sizes are more like mine (10-20), but their results are almost entirely anecdotal. Is there any way to have large-scale data about small classes?
  • If you're making students present in class (be they homework solutions or final projects or anything), tell them what skills they're supposed to gain from doing so (beyond just sharing with the class).
  • All of those "series convergence tests" we teach in Calc II are fundamentally comparisons (direct comparison, comparison to a geometric series, etc.) and presenting them as such gives that section a narrative students can hang on to.
  • The mainstream does an undeniably terrible job of presenting mathematicians and their work accurately and meaningfully; we need to do better self-promotion by writing articles ourselves.
  • Might it be worthwhile to encourage (or at least support) "guess and check" methods of integration? Integration by parts and u-substitution have an inherent "guess" step anyway...
  • If you have a class of size 6n+3, you can split the class into triplets for 3n+1 sessions such that each student works with every other student exactly once. (Such arrangements are fairly easy to find by hand for small n but the difficulty increases greatly...) [This is related to the "Kirkman Schoolgirl Problem"]
  • Paul Seeburger has a collection of awesome applets for Calc class demonstrations.
  • Consider adding a "How confident are you?" button to online quizzes (or even exams). Gathering such data can be enlightening in the long term, and can guide class discussions in the short term.
  • Consider letting students choose the weighting scheme for your syllabus (within limits you define). Letting them take ownership pays off for everyone.
  • Take an equilateral triangular board, whose "squares" are hexagons, with n hexagons on each side. There exists a Knight's tour on such a board if and only if n is at least 8. One proof of this is a neat "non-standard" induction: show a bunch of base cases and then show KT(n) and KT(n+1) imply KT(2n) and KT(2n+1).
  • Letting students write and turn in exam corrections for credit can increase their engagement and show that we actually care.
  • Fractals can be a great, engaging motivation for studying infinite series.
Here are just a few pages from my stack of notes that I generated ... perhaps more to come?

Saturday, September 7, 2013

What Is Math? — Responses to an informal survey of young college students

For the past couple of years, I've been focused on teaching a particular course (whose title is the eponym for this blog, in fact). That course is meant to introduce students to mathematical proofs, transitioning them from computation-heavy, rote, follow-this-example style math into abstraction, problem-solving, and the communication of mathematical ideas. It's notoriously difficult, and rightly so. In particular, it's one of the first occasions where students have to face the truth that mathematics is even harder than they ever thought it was. As mathematicians, we constantly face our own lack of knowledge and the mysteries of the universe, and rarely is there "the right answer" that we just need to get to before circling it at the bottom of our paper. This course forces students to face these ideas head on, and it can be quite a shock. Now, this is true even though the course and its exercises are often structured so that there kind of is a "right answer" or some "correct method" to be used. But that just goes to show how even tossing a little bit of this uncertainty into the mix can be surprising and challenging for students.

On a larger scale, this course is a place where students' fundamental assumptions about what math is are challenged, questioned, reworked, and never really settled. Last summer, I had the idea to "pre-test" students on what they thought about math. Mostly, I was just curious, for myself. I thought that seeing their answers to the question "What Is Mathematics?" might reveal something about their backgrounds in math and their desires for the course; at least, the answers might confirm my suspicions that the course will be as shocking and difficult as it always is. It turns out that they were somewhat what I expected, and the course was, indeed, difficult for everyone (including me). But that was the point :-) (I'll have to go back and find these responses and analyze them as I do below…)

I'm now at another institution in a full-time teaching job. I'm teaching three different courses, none of which are this "transition to proofs" course that I've focused on for the last few years. In fact, two of the three courses I'm teaching now are completely populated with non-math-majors, students who are fulfilling distribution requirements/pre-requisites or getting extra preparation for their next requirement. Even more than ever, I felt the need to understand my students' backgrounds in math, their goals for learning, and what preconceptions about math they might bring with them. The results are both exactly what I expected and completely shocking at the same time.

I asked several questions of my students on the first day of class this week, and had them write their answers on an index card. Three of those questions were:
  • Why are you taking this course?
  • What do you hope to learn from this course?
  • What is math?
Again, I was hoping the answers to these questions might help me shape the course, the choice of material and examples and problems, my behavior in class, and so on. Unfortunately, I now feel rather daunted by these responses, but I'm hoping I can turn that into some powerful motivation (for both me and the students).

Let's talk about the third question: "What Is Math?". I knew I'd get a lot of strange responses here, and was expecting a general trend emphasizing computation and numbers, but the overwhelmingness of this trend is still surprising. I fed the 91 responses to Wordle to generate a word cloud:

Over half of the answers used the word "numbers". About a quarter used "problem(s)", and almost all of these were mentions of "problem solving" in some form. One sixth mentioned "equations". About one eighth mentioned "letters" or "symbols", and a handful put forth the idea that math is a "language".
I expected this. Most of my students are college freshman, and high school math emphasizes the idea that math is a system of useful rules for solving equations and problems. That's fine. I didn't necessarily expect these phrases to be so widespread and uniform, though. So many students seem to think that math is completely characterized by algebraic manipulations of letters, numbers, and symbols. I would like to dispel this rumor, but I really don't think I'll be able to in a course that is explicitly meant to teach these students algebraic techniques!

I find it interesting that many students pointed to the "usefulness" or "application" of math to the real world. I'm pleased that at least 20% of the responses indicated some positive usefulness of mathematics, even though many of the phrasings intimate that it's the quantitative/numerical aspect of math that makes it useful. Here are some example phrasings of this kind:
  • Math is how the world runs
  • Math is the concrete explanation to everything
  • Math is an extremely useful tool that can get very complicated
  • Math is difficult! It's the use of numbers to help explain different phenomena we see in our lives
  • Math is a problem solving tool involving numbers and symbols to form equations for real life applications
  • Math is a language which we use to describe and talk about our world
  • Math is the way numbers/measurements/data/etc is used and applied
  • Math is numbers, calculations, things that help justify why things are
  • Math is everything. Math is numbers, letters, shapes, etc. We need/use math daily.

Another aspect I expected is the idea that there's always a "right answer" in math. Admittedly, this is what drew me to math when I was young, and it took me far too many years (more than I care to admit) to realize that this is far from true. Luckily enough, I found that I also loved the difficulties of math no longer cared about the sureties of having a "right" and a "wrong". I'm not so sure that every student would respond favorably to this epiphany, though. And even still, I can't really disagree with a claim like "Math is a way of finding an answer a certain way with certain rules" when, in all likelihood, that is exactly what this student has been taught for years. I can only hope to relax and alter their view a little bit but, again, this will be difficult to do in a course designed to perpetuate those certainties. When their homework is completed online and the system immediately gives them a big green check mark or a big red "X" … what else can I really say/do to convince them otherwise?
All of that said, I marvel at the comfort inherent in the following two statements:

"Math is what you make of it. It is found everywhere and is one of the only truths in this world."
"Math is simple and always has a right or wrong answer."

Maybe I don't want to step in and shatter their worldview. Then again, maybe this is exactly what good teachers are supposed to do!
That last example also brings up the issue of difficulty. I would never have thought that the definition of a subject could have anything to say about how simple/complex it is; that is, shouldn't such a definition just say what the subject is, without making a claim about its difficulty? Several students incorporated the notion that "math is hard" into their definitions, though! Indeed, "Math makes numbers into difficult things, ha" was surely written in jest, so can't read too much into it. But, to receive an answer like "Math is annoying" on this kind of friendly, day-one survey is both incredibly disheartening and motivating. If I can make this one student reverse that thought over the course of this semester, I'll consider my entire efforts worthwhile.
While most of the students had some idea and shared it with conviction, some other students realized that either there isn't a good answer to this question or they didn't have one, themselves. One even told me, "That's an insane question" before going on to actually answer it, saying "Math is a universal language". Another hinted at the dependence on personal viewpoints, and answered "To me? Just numbers". Yet another was probably being cheeky, resorting to "2+2=4" as their complete answer, but I bet Whitehead and Russell would have a field day with that one …

Interestingly enough, one student seemed to answer the question from the point of a teacher, seeking to explain why we teach math and not just what it is: "Math is a subject taught in schools to students. It helps to benefit their knowledge with numbers and critical thinking." I don't disagree with this, and I hope this student keeps this in mind as he/she learns.

In the past, students have approached me looking for my own answer to this titular question. I haven't come up with a good, consistent answer, and I make sure to tell them as much. The real point in asking the question originally is to open the debate, not to indicate that there is a perfect answer that the students just don't have yet. Usually, though, any attempt I make at an answer mentions the search for and analysis of "patterns". I know other people use similar phrases, while others disagree with this characterization, for whatever reason. Regardless, not a single student in this informal survey mentioned anything like this. I wonder why, and I wonder whether I should try to share this view. Would that just be imposing my own beliefs about mathematics on them? Would that then make me a bad teacher, a selfish one? Or would it just be the way that teaching is done?

I'll conclude with my favorite answer, one that I might use myself in the future, and one that I hope will resonate with people who have devoted their entire lives to pursuing mathematics and people who have chosen to run far away from mathematics, alike:
"Math is one big puzzle"

(Next time, I'll talk about responses to the other survey questions and how they're potentially related to these responses.)

Thursday, July 4, 2013

Learning from Common Mistakes: Generality vs. Specificity and Evoking an Emotional Response to Math Problems

It's been too long since I've written, but I'm feeling the writing bug again, especially after discussing a problem recently. It came up in a discussion of widespread student mistakes that make the teacher learn something. This particular problem was assigned to a class I taught last summer, and the results were surprising and made me think about the problem's wording and how to grade it, at the time. In the long run, it has made me carefully consider (a) how students learn, (b) how to properly respond to students' written work, (c) the difficulties inherent to learning abstract mathematics for the first time, and (d) cracking through the widespread misconception that mathematics is a "cold, hard, logical science".

First, the course context: The course "Concepts of Mathematics" (whence this blog's title!) is a transition from computation-based mathematics to proof-writing and abstract problem-solving. During the academic year, this is taught to university freshman and sophomores (mostly) in one semester. We cover sets, logic, and proof techniques (direct, contradiction, induction), and then apply these techniques to chosen topics: relations, functions, cardinality, number theory, combinatorics, and probability (depending on the instructor's choices and time). During the summer, this is also taught to high-school students (simultaneously with university students) in an advanced placement program.

Next, the problem's context: This was an assigned, written homework problem during the unit on relations. We had just spend many days studying logical proof techniques and were beginning to apply them to newly-learned topics. We had just seen relations in lecture, with a definition, several examples, and a discussion and formal definitions of the standard properties: reflexivity, symmetry, and transitivity. This homework assignment was their first opportunity writing formal proofs about these properties. Here is the original problem statement:
1. Define the relation \(T\) on \(\mathbb{R}\) by saying, for any \(x,y\in\mathbb{R}\), \[ (x,y)\in T\;\Leftrightarrow\; \left(\;\frac{y}{x}\in\mathbb{R}\;\wedge\; \frac{y}{x}\geq 0\right) \] (a) Find \(x\in\mathbb{R}\) such that \((x,x)\notin T\). Does this mean \(T\) is not reflexive? Why or why not?
(b) \(\ldots\)
There were three other parts to the question, but part (a) is the one I'd like to discuss. Think about the problem for a minute, if you wish, or check out my solution below.
Consider \(x=0\). Notice that \((0,0)\notin T\) because \(\frac{0}{0}\) is undefined, so \(\frac{0}{0}\notin\mathbb{R}\).
Since we have exhibited a counterexample to the property of reflexivity (that is, we proved that \(\exists x\in\mathbb{R}{\large.}\;(x,x)\notin T\)), we conclude that \(T\) is not reflexive.
This makes sense, right? The property of being a reflexive relation is one of universal quantification: either for all elements \(x\) in the set \(T\), something happens; or else, that something fails somewhere. What could be more clear-cut? This is a "Yes/No" question. Of course, this requires some thinking and "playing around" to come up with an answer, but I predicted this would be a straightforward homework problem, the "easing in" to the assignment's material, as it were. Indeed, it was the first problem on the assignment (for what that's worth).

I was wrong.

I wasn't vastly wrong, but I was decidedly incorrect in my assumptions that (a) students would formalize their "playing around" in the correct way, and that (b) they would recognize this as a "Yes/No" question. I was, and still am, far more surprised that they shattered assumption (b).

Several students got this question correct, with no issues. (FYI, there were 45 students in this class. About 30 were from the advanced high-school program, and about 15 were from the university.) And a small number of students were way off (misinterpreting the problem, failing to turn it in, etc.). But the majority of students made the same mistake that simultaneously showed (a) they understood the problem statement and thought about it carefully (and correctly, even!), but (b) failed to present those ideas formally and, more importantly, correctly.

It wasn't really the mistakes that caught my attention, but their universality. I expected some students to forget the \(\frac{y}{x}\in\mathbb{R}\) condition, perhaps, or to neglect to factor in negative real numbers. Neither of these occurred. Instead, many, many students said something like this:
Only when \(x=0\), the fraction \(\frac{x}{x}=\frac{0}{0}\) is undefined. But elsewhere, \(\frac{x}{x}=1\). Thus, \(T\) is reflexive everywhere except the origin.
What? Did you expect this? I certainly did not. Clearly, all the ideas are correct. Indeed, \((0,0)\notin T\) but \(\forall x\in\mathbb{R}-\{0\}{\large .}\; (x,x)\in T\). But the point is this: a single counterexample to the "for all" claim means that \(T\) is not reflexive.

This idea had been drilled over and over. I saw that the students had trouble with quantifiers when we studied them, so I chose to spend extra class time then (even during the already time-crunched summer term). I assumed that issue with the idea of "generality vs. specificity" had been learned. But it reared its head again here. The students all saw that some specific example occurred, but they didn't think that meant anything in general.

At the time, I was baffled. The first few homework assignments I saw this on, I wrote some comments about how the negation of "\(\forall\)" is "\(\exists\)". Eventually, I wrote shorter and shorter comments, and decided to make a blanket statement in lecture the next day. And I did. I pointed out that negation again. I told them that a relation is either reflexive or it isn't. It can't be "kinda" reflexive. It's not a sliding scale. This is the Law of the Excluded Middle all over again. Reread the definition. Etc etc etc. I asked them how they collectively and independently (I assume!) came up with this notion of relative reflexivity. And I got … nothing in response. They shrugged their shoulders and I reiterated my prior ideas more concisely and moved on. Thereafter, I was careful to indicate the "Yes/No" nature of such questions, and I believe the ideas sunk in over time. But for that assignment, I was concerned.

But the oddest thing is that, in retrospect, I feel like them. Or at least, I appreciate their response and believe that they were expressing some other aspects of learning mathematics. For one, they were indicating how difficult it is to learn abstract mathematics. They can't "see" a binary relation, let alone visualize its properties. This goes against most things students learn in pre-calc and calc. (Keep in mind, 2/3 of my students were high-school juniors/seniors.) There, you graph a function and assess it pictorially (for the most part). The "vertical line test" is a visual thing, not a formal, quantified statement that you negate. And so, when you see a definition like \[ \text{A relation }R\text{ defined on a set }S\text{ is reflexive iff }\forall x\in S{\large .}\; (x,x)\in R\] it's difficult to "see" what it means; in particular, it's difficult to "see" what its opposite (read: logical negation) means!

For that reason, I somewhat understand why the students mostly failed to properly write up (formally) that "this relation is not reflexive". Mind you, they lost points because, yes, their answers were incorrect. It represents a lack of full understanding that the negation of generality is specificity. But in another sense, I am retrospectively proud of these students. I see that they realized the relation \(T\) came "oh so close" to being reflexive. "There's only this one counterexample, man … Only 0 doesn't work!" In that sense, "kinda reflexive" is an appropriate assessment of this relation. (Thankfully, no one used that much of an informal phrasing!) Accordingly, I like to think that their investment in figuring out the problem and presenting their work in written homework elicited an emotional response from them. In assessing \(T\)'s reflexivity, they saw it came close and presented their ideas as such; they wrote up their thoughts, not just "the right answer".

For that, I am proud. One of my major goals in teaching this course has always been to dispel certain myths about what mathematics "is" and what mathematicians "do" and how mathematics is all about "True or False, Right or Wrong". With this homework problem, my students showed me that they could appreciate those feelings, and not just because I told them to. They all surprised me with this development. And I love it. I intend to assign this problem again in the future, just to see what happens …
Now, that said, I'm considering adjusting the problem statement to elicit different responses. For example, I might eliminate the "kinda non-reflexive" answers by asking this:
"… Does this mean \(T\) is reflexive or not reflexive? Explain your answer."
I could be even more explicit by saying: "Explain your answer by appealing to a definition from lecture/the book."

I also wonder what would happen if I doctored the example to be "more non-reflexive". I hope you know what I mean by that by now :-) . For instance, I might define \(T\) as this: \[ (x,y)\in T\;\Leftrightarrow\; \left(\;\frac{y}{(x-2)(x-3)}\in\mathbb{R}\;\wedge\; \frac{y}{(x-2)(x-3)}\geq 0\right) \] Then, students have to think about \(x=2\) and \(x=3\), and even perhaps \(2 < x < 3 \), and even … (Do you see what happens?) There might be more to their answer than just, "It's reflexive everywhere but …". Maybe by making it "less reflexive", they'll shed those intuitions about the property being a sliding scale and realize it's "Yes/No" aspect. Where does that intuition cross to the other side? How "much" does the relation have to be reflexive to get a subjective answer from them?

I'm very much looking forward to experimenting with this question and learning more about student learning!

Thursday, June 14, 2012

Puzzles, motivation, and "expert blindness"

I was sitting in my friend's office yesterday, chatting about math and perusing his officemate's extensive book collection. (I was also sitting at said officemate's desk and pretending to be him for the day, not too successfully, but I'm working on that impression.) We both realized we were incredibly hungry and needed to be back on campus in an hour and a half for commitments, so we walked to Oakland to grab a sandwich at Uncle Sam's. I brought along a book for entertainment: Peter Winkler's Mathematical Puzzles: A Connoisseur's Collection. On the walk, my friend reminded me that he hates puzzles.

As an avid puzzle person myself, I couldn't empathize, so I asked him to elaborate. He used, as an example, the following problem we were talking about that I had randomly flipped to in the book:
Prove that every set of ten distinct numbers between 1 and 100 contains two disjoint nonempty subsets with the same sum.
Winkler, P. Mathematical Puzzles: A Connoisseur's Collection. AK Peters: Natick (2004), p. 22.
If you don't know what this means, here's the idea: if you pick any ten numbers between 1 and 100, like {2, 5, 10, 24, 27, 33, 50, 88, 89, 98}, then you should always be able to find two non-overlapping groups of those ten numbers that add up to the same result, like 50+33+5=88 in this case. Of course, 88+10=98 and even 24+27+50=89+10+2 also work, but that's not the point. The idea is to definitively show that you can always do this in at least one way, no matter what the 10 original numbers were!

Now, try working on this problem yourself for a few minutes and see how you feel about it. Does it seem too easy/hard? In between? Do you get frustrated easily, or do you have a vested interest in figuring it out? Do you feel motivated to ultimately solve it? Do you want to just skip to the answer? Do you want a hint or two? (Highlight the text below to see them.)
Hint 1: Use the Pigeonhole Principle.
Hint 2: Consider the number of nonempty subsets of 10 numbers and the number of possible sums those sets can have.

Here is the point that my friend made: there's no motivation to solve this one. Yes, he could probably figure it out if he sat down and thought about it concertedly (and we did, indeed, talk out the solution, with one hint from the book), but why bother? That time is better spent somewhere else, like his research. I couldn't think of a good response to that at the time, but in retrospect, here's what I have: if you're only worried about the end result, then of course the journey isn't going to be much fun! You don't know ahead of time what a puzzle or exercise might teach you, nor do you know what types of skills you'll get to practice. I thought this was a neat (and somewhat surprising) application of a basic mathematical theorem and was glad I had taken the time to think about it.

What does this mean for teaching? Well, our discussion of the above problem was already tainted by the fact that we were both practicing mathematicians and educators, and there wasn't really a debate over whether the problem was interesting or useful in its own right (my friend's position that he just has "better things to do" notwithstanding). Now, imagine a student who might be unsure about whether he/she likes mathematics (or whatever subject you're teaching), whether the pursuit of that subject is in their future; or, even imagine a student who is just taking a course out of necessity and needs to pass to move on to "better" courses. Replace the above problem with some interesting (to you) topic in the course, or some skill that you see as useful for them to learn. How do you make them "like puzzles", in this sense? Like it or not, we have to address situations like this all the time, on a wide variety of scales: How can I make my students see the usefulness of this example from lecture? How can I make my students care enough about the course content to pay attention in class and spend enough time studying outside of it?

These are tough problems, and I'm afraid that there is no generally helpful advice. This is too situational to address. That said, though, I think it's important to keep this in mind all the time. I remember the phrase "expert blindness", or something like that, coming up in a discussion with some folks from the Eberly Center (the teaching center here at CMU). It's a term to describe how people who are already very committed to their subjects tend to implicitly assume that everyone else already has the same enthusiasm and genuine interest, and will just go along for the ride without any questions or doubts. This can be troublesome, of course! You might neglect to properly motivate your students throughout the course, and their commitment will be lackluster; you might fail to point out the usefulness of a certain branch of your topic, thinking that everyone will just find it intrinsically intriguing in its own right; and you might end up teaching to a select handful of top students while leaving everyone else in the dust. I've surely been guilty of this in the past, to some extent, but it's something that I'm consciously trying to eliminate from my teaching ethos. I make myself think hard about every lecture, every example, every homework problem, every segue (both spoken and written), to ensure that a maximum number of students go along with me. This is not easy, but making a concerted effort to address it can go a long way.

Saturday, June 9, 2012


I had an idea this morning. I was plotting out a calendar for the course I'm teaching this summer (the namesake for this blog, in fact) and also reading an article about teaching creativity—instead of just rote knowledge—that I had clicked on via someone's Facebook post or something long ago and had left open. (No, I wasn't performing these activities simultaneously.) I realized that I frequently have ideas about teaching, and math, and teaching math, that I'd like to remember for myself to use in the future or that I'd like to share with others, or both. Voilà! This blog was born.

I'll use this space as a way to keep track of my teaching of 21-127 Concepts of Mathematics at Carnegie Mellon University during this summer's second session, through July and part of August. I'll post some interesting ideas I hear about or have myself, I'll share how the course is going, I'll pontificate about things that incite passion in me, and so on. Who knows, really, how this will turn out? At the very least, it will make for an interesting "public diary" about my teaching experiences. Maybe I'll keep writing during the fall semester, but I'll make no promises on that at the moment.

For now, here is a link to that article I was reading today. I don't know anything about the experimental schools the author listed, nor do I feel particularly motivated to completely revamp school curricula. I am interested, however, in the idea of training students to be creative and ingenious, to not only answer questions but to ask them. These sorts of skills are always on the objectives list for this course I am teaching—we want our students to not just memorize mathematical "facts" but to understand why they're true and to even gain an appreciation for why we care about them in the first place!—but they seem to be the least stressed during the actual teaching of the course, and the main ones of which the students might not realize the benefits. This might be because these skills are hard to assess accurately, of course, but that's a cop out. All we need is a little ingenuity, ourselves, on the instructors' side.

I've been mulling over several innoventions on the homework/quiz/exam front for this course, and I'm hoping to implement them during this session and assess their effectiveness, so this has already been on my mind. Certainly, though, this article has reminded me that creativity is a skill we can address every day in the classroom, and even just a little emphasis can go a long away. If I find myself describing a difficult proof of an important theorem, I should stop and try to describe how someone came up with such an argument, or why they were thinking about it in a particular way. I think that by simply addressing the creative underpinnings of the daily content, students will benefit, at the end of the course. This doesn't take a lot of effort, just a little bit more talking in class, and maybe a pause for some questions for the students. I will try to keep this in mind every day and see how it effects the students! If I can convey any sense for the beauty and ingenuity that permeates mathematics, I will consider my endeavors a smashing success.