*A child’s insight*

“I know how to find out how many divisors a number has. You factor it into primes….” Alejandro was with a virtual group of four enthusiastic ten year olds, in the midst of exploring a problem. He gave the usual result, using his own somewhat makeshift words. But not too distant really from what I would have said: If $N$ factors as $p_1^{a_1}p_2^{a_2}p_3^{a_2} \dots$, then the number of divisors is $(a_1+1)(a_2+1)(a_3+1)…$. His description was less economical, but still accurate.

His virtual friend Xue said: “That’s great. Let’s look it up on Wikipedia.”

Then, “No. Let’s not look it up. Let’s pretend we don’t know it and see if we can prove it.” It is this insight into his own learning, not any mathematical breakthrough, that I remark on in the subtitle to this section.

Dear Reader: I swear to you, on Galois’ grave, that I am not making this up. Nor the rest of the vignette I will be recounting here.

*The venue*

This spring, in reaction to the COVID crisis, I was part of a team developing an online ‘webinar’. The team was from the Julia Robinson Mathematics Festival (JRMF) program. At the time, I was its Executive Director. In normal times, we run non-competitive after-school mathematics events (“Festivals”) in which students are offered interesting games, puzzles and problems, assisted by a facilitator. Since face-to-face work with students has lately not been possible, we have sought to continue the work virtually.

The program has met with success. The JRMF team works on the presentation of a problem each week, polishing it for a group of about 200 students who `tune in’ to the event internationally. The students are split into groups of fewer than ten, and put in breakout rooms to discuss the problem. An adult facilitator guides the discussion, not to achieve a particular goal, but as a moderator, letting the students’ insights emerge naturally. Facilitators meet for half an hour after webinar, to pool their experiences and offer ideas for refining the program.

Problems are “Low threshold, high ceiling”. That is, very young students can work on them, have fun, and achieve insights that will eventually take them farther. More advanced students can use them to engage in thorny issues or deep mathematical concepts.

For examples of such problems—and an open invitation to participate in these webinars—see www.jrmf.org. We post the problem on Mondays and discuss them on Thursdays so participants have time to explore the question. On Thursday, we ask students two questions: (1) How long have you worked on the problem? (2) What is your age? They are assigned to breakout rooms depending on their answers to the questions.

Attendance has grown steadily. We find that students who come to the webinar tend to return. Thus we have created a virtual community, all around the globe, of students who enjoy mathematics.

*The Zoom Room*

Last week, I was assigned a room with four energetic and highly motivated young students, ages about 10. The facilitators were familiar with these four. When we first started the program, we found them difficult to work with. They had often gone far into the problem: the amount of time they spent on it could not tell us that. They often had bits of mathematical background that other students lacked. And their youthful and overflowing exuberance made it hard to integrate them into a group. They were always a challenge to any facilitator.

So we decided to create a special breakout room for them, the “Zoom Room” where they could race ahead. The success of this effort varied with the mood of the children. At best, they urged each other forward and vied with each other for insight. At worst, they would try to show off to each other what they already knew, without contributing to either the group effort or their own knowledge.

This past week the group clicked. I was delighted to find that the four boys (they were all boys) worked beautifully together as a team, and got further than any one or two of them could have in the short time available. I led them with but a light touch of the reins.

They did not solve the given problem. They didn’t even work on it. They created their own, and the last thing I wanted to do was confine them to what I thought they should be learning.

Here was the problem we had set (briefly): Given a large square with integer sides, how can you tile it with smaller squares. also with integer sides? The problem was presented in a more structured way, to offer `on ramps’ to the mathematics. An interesting problem, combining elements of combinatorics and geometry. And, as is typical of JRMF problems, it can be worked on many different levels. I was eager to find out where the discussion would go with my four young students. It took a turn that I could not have predicted—or prepared for.

They looked at the first problem and immediately answered that for $1 \times 1$ squares, you can tile any $N \times N$ square. The important point here is that they saw this as a special case: it was a sophisticated insight for children that young. They then went on to consider the question of tilings with $2 \times 2$ squares. I asked if you could tile a $7 \times 7$ square with $2 \times 2$ squares. They again saw that they couldn’t, and articulated the reason: 2 does not divide 7.

So I asked, “If $a$ does not divide $B$, then clearly an $a \times a$ square cannot tile a $B \times B$ square. But is that enough?” My point was new to them. It was the difference between a necessary condition and a sufficient condition. Very generally, I find that the core difficulty in learning mathematics—for anyone, at any level—is the logical structure behind the assertions or computations. Even these very experienced students had to take a minute to understand what I was saying.

In fact the condition is sufficient as well as necessary. They seemed to understand this particular example, but I am not so sure that they will understand the distinction between a necessary and a sufficient condition in another context. No matter. They are ten years old.

To guide the discussion a bit, and to get what I could out of their intense engagement in it, I asked how many ways they can tile a $7 \times 7$ square with identical squares. Dan (I am not using the students’ real names here) immediately said, “Only with $1 \times 1$, because 7 is prime”.

“No,” countered Alejandro. “You can tile it with one big fat $7 \times 7$ square. Does that count?”

“Well,” said Titus, “A prime number has only two divisors: one and itself. So we can use the same idea to count these tilings, if we count $7 \times 7$ as a tiling.”

Titus may have wanted simply to show what he already knew. But this seemingly innocent and perhaps boastful remark turned out to be a fertile one. Dan generalized immediately: “For an $8 \times 8$ square, there are four tilings.” (He meant tilings with identical squares, and everyone knew it.) “That’s because 8 has four divisors: 1, 2, 4, and 8.”

And this is where we came in. Alejandro took up Xue’s challenge, and his ten-year old explanation was wonderfully simple. “Say there are two primes, $p$ and $q$. Say the number is $p^2q^3$ You just make a picture.” And he drew this on the shared screen:

In another group, Alejandro’s explanation would have been a mystery. But these four looked at it and understood.

“You need a 1 to count the 1,” said Dan, “and also the singles: $q, \ q^2, \ q^3$.”

“Right,” said Xue. “So if $p$ is squared, you have three columns, not two. That’s why we add one to the number on top.” He meant the exponent.

“But what if it’s like $p^2q^3r^4$?” asked Alejandro… and answered his own question. “Oh. It’s the same thing. You can just list the twelve divisors we have already down the side, and list $r, \ r^2, \ r^3, \ r^4$ on the top.” As facilitator, I squirmed a bit at the error. But in this virtual environment, no one saw it. And knowing these kids, I remained silent.

“No,” said Titus quickly. “You need five columns: $1, \ r, \ r^2, \ r^3, \ r^4$.”

“That’s right,” said Alejandro. My silence had paid off: the point was made better and faster than I could have. The interaction at once exploited the benefits of kids working together and increased the bond between them. Boastfulness and ego were quickly put on the back burner.

I didn’t want to rest there. They could recite the formula. They could prove it. I wanted to make sure they could use it. So I asked them a question that they were unlikely to have seen before: What two-digit number has the most divisors?

Their thought was swift, and collective. They quickly saw that they had to look at prime divisors and balance the number of divisors with the exponent in the formula. All this without writing anything down.

Titus led off: “It probably should have lots of 2’s and 3’s. Because we don’t want the number to get too big.”

Xue: “Well, it can’t have more than six 2’s, because $2^7$ is already 128. And $2^6$ is 64 and has seven divisors.” He had intuited the formula for the case of a single prime. I did not need to call his attention to this special case.

Titus again: “What if we put in a 3? Three times 32 is 96. It has. . . ” He thought a minute. “ It has $6 \times 2 \dots$ twelve divisors.” I didn’t have to ask him to explain.

Indeed, I didn’t have time. Alejandro jumped right in: “It depends on the exponents. The primes don’t matter. They just can’t be too big.”

Xue: “Can we have a 5 as a prime factor? Well, we can’t have two 5s. We can, but that will give us 25, 50, 75, and they don’t have enough divisors.” He was imagining what applying the formula would do, and his intuition told him (correctly) that these numbers would have fewer divisors than the 12 that they already saw for the number 96.

Dan: “And if we have one five, the rest of the number is 20 or less. We would need 6 or 7 divisors for that kind of number. Can we do it? ”

Silence.

Then Dan again: “Seven divisors can’t work. It’s prime. Six divisors? It’s $2 \times 3$, so we need $pq^2$. That’s $2 \times 3^2$ or $2^2 \times 3$. Eighteen or twelve. Five times these give 90 or 60. Each of these also has 12 divisors.”

Alejandro: “I don’t think we can beat 12. We just have to look at 2’s and 3’s. No. We can’t get 13 or 14 divisors. We would need too high a power.” (I didn’t stop him—everyone seemed to understand.) “Can we get 15 divisors? We’d need $2^2 \times 3^4$. That’s too big. Or $2^4 \times 3^2$. What is that? $16 \times 9$. No, still too big.”

Titus: “So only 12 divisors.”

I asked, “Which two-digit numbers have 12 divisors?” The list came tumbling out of them, and they all contributed.

*Generalization*

Unbidden, the group asked the next question: “What three-digit number has the most divisors?” They started working on this, and the ideas flowed. Ramsey Makan, my techical assistant, himself quite young, had been listening. The number 720 came up, and someone remarked that this was $6!$.

Ramsey asked them, “How many divisors does $6!$ have?” They worked it out. Then of course started thinking about factorials in general.

Titus was out of the discussion for a few minutes, then came back. “I wrote a Python program to list the divisors of $n!$.” They all wanted to see, so Titus ran it, for $n = 1$ through $6$.

“Can it do $10!?$” someone asked. Titus ran it for $n=10$. The screen went blank.

“The numbers are pretty big,” he finally said. “So it’s going slow.”

And indeed it was. The program was using brute force. I wanted to keep the momentum of the group up, so I said: “Can you figure it out yourselves? Maybe you can beat the machine.”

And they did. When the number finally popped up on the screen, it matched their result.

With time running out, I wanted to leave them with something to work on. So I said: “Suppose you know the number of divisors of 12!. Suppose some wizard told you how many there were. Would there be more divisors of 13!? Or fewer?”

The group responded easily: “More.” And then Dan said, “Twice as many. Because 13 is prime.” This was met by a chorus of “Oh, yeah.”

“But it wouldn’t work for 14! if you knew 13!,” said Xue. Then, a moment later, “What would work?”

They started thinking. Titus said: “Four times as many…”

Titus’s idea was not quite right. But the time was up. The breakout room was closing. I said goodbye and the webinar came to a close.

*Conclusion*

Teaching online can be tough. You lack certain means of communication: gestures, looks, posture. And if a student is silent, it’s hard to tell if he or she is engaged. My experience with these four students may not generalize easily. But it does give us a picture of what can happen when students encounter each other virtually.

And it gives us another picture. Sometimes it is argued that we must not do anything special for students who need more mathematics. It inflates their egos, makes them think that somehow they are ‘superior’. Well, it can, if done badly. And it can damage a child if we value the gift and not its bearer. The experience I chronicle here shows how important it is that such students meet other such students, that they come to see themselves as no different from a whole group of peers. My experience has shown is that once they are in such a group, they grind off each other’s rough edges. Being a ‘nerd’ or a ‘brain’ is a costly defense, like a suit of armor. And the defense, in such situations, is quickly and eagerly shed.

*Acknowledgments*

I would like to thank Ben Blum-Smith and Yvonne Lai for their invaluable input into the writing of this piece.

]]>For reasons that will not be considered here, I recently learned this dance:

Although I have no background in any style of dance, I can now do the whole thing, start to finish. I am very proud.

My purpose in attaining this objective was unrelated to mathematics or teaching. Nonetheless, the experience put an eloquent fine point on a certain basic dialectic in math education.

I spent a decade working in middle school and high school math classrooms before I trained as a research mathematician. Conversations regarding goals for students in elementary and secondary math education, and math education research, often distinguish between two types of knowledge: procedural and conceptual. These are fraught words, and you have your own ideas about the meanings.^{[1]} Nonetheless, for the sake of clarity (at least internal to this blog post), I will offer some definitions.

Conceptual knowledge: knowledge of what things really are, what they are all about, and how they are connected.

Procedural knowledge: knowledge of how to actually do things.

I hope with these definitions that I have not accidentally tripped any wires. If your background is anything like mine, the mere mention of this dichotomy may have already given you some unpleasant flashbacks. In one of my first teaching jobs, almost every department meeting eventually devolved, in a practically ritual way, into a bitter fight. And one of the perennial sticking points was which of these two knowledge types deserved priority. Those days were a high-intensity period in the Math Wars, and the “procedural vs. conceptual” dichotomy served, in my experience, as a kind of a “Math Wars bat signal”: once it came up in a conversation, powerful ideological fault lines showed up soon after, as though they had been summoned.

The terrain has shifted a bit since then. It eventually became fashionable, uncontroversial—indeed, *obviously true*—to assert that these two types of knowledge are both important, and are mutually reinforcing.^{[2]} Interest has grown in creative ways to serve both masters at once.^{[3]}

Nonetheless, educators still often have a propensity one way or the other at the level of educational values and aesthetics. For some, a calculus student who can differentiate elementary functions flawlessly, but doesn’t know what any of it means, ‘hasn’t actually learned any math.’ To others, ‘at least they can solve the problem!’ For some, it is distressing and concerning when a fourth grader can accurately identify a wide range of contexts modeled by subtraction, but can’t compute except by counting down on their fingers. Others feel this student has already learned the hard and important lesson, and believe that this will make learning a better computational method easy. These differences can persist even among educators who believe passionately in the joint value and mutual complementarity of the two types of knowledge.

For example, I fall on the conceptual side. Not intellectually: I believe strongly that mathematical knowledge comes in both types, that they’re both crucial, and that they’re mutually supportive. Every time I reflect on my own learning with this question in mind, it’s obvious how much my procedural knowledge has done for me. *That said*, I’m simply more passionate about teaching concepts than procedures. I am lit up by the challenge of getting students to perceive an unexpected connection or to understand the purpose of an important definition. I can also get excited about the how-to-do-this stuff when I know it will make my students feel powerful, or put them in a position to think about a particular interesting question or concept, but even in these cases it’s a means to an end. Meanwhile, my heart sinks a little when I read student work that evidences thoughtless application of a formula, even if the answer is correct.

These differences in taste can shape our curriculum design and our teaching choices even if we believe at the intellectual level in the importance of both types of knowledge. For example, my gut orientation toward conceptual knowledge means that when a student presents as stuck or lost and asks me what to *do*, my first instinct is always to pull their attention away from that question, down to the level of “what is this all about, and how is it connected to other things you know?”

I don’t think there’s anything wrong with these tendencies toward the conceptual or the procedural, and in any case, we have them whether we like them or not. But because they shape our teaching practice, I do think it’s useful to recognize them. Sometimes, the thing a student needs is conceptual; other times it’s procedural. I think both types of bias have their strengths, but each can also lead to teaching blunders caused by failing to recognize the needs of our students.

For example, my strong habit of assuming that the obstacle facing a student is conceptual, can make it hard for me to recognize when a student has a procedural need that’s not being met. I, and I think many conceptually-oriented educators, have a tendency to see the procedural knowledge—*what to actually do*—as a consequence or corollary of conceptual knowledge. So if a student presents with a difficulty *doing* something, I (we) take aim at the concept of which the desired action is (to us) a consequence.

This does actually work a lot of the time! *And*, there are plenty of times when it doesn’t, because it isn’t always reasonable or fair to assume that the student can get from “I know what’s going on” to “I know what to do” on their own.

Much to my surprise, learning a K-pop dance routine provided me with an incisive opportunity to reflect on both of these possibilities—from the student side.

When I set out to master the dance from BLACKPINK’s 2018 hit song “DDU-DU DDU-DU”, it was kind of like learning to walk. My lack of any kind of dance training, combined with my gender socialization, meant that half the stuff Jennie, Lisa, Jisoo and Rosé do in the dance practice video was missing entirely from my movement vocabulary. But I was up for a challenge.

I started with the chorus. I got as far as the first “Hit you with that DDU-DU DDU-DU,” but that little 4-beat bouncy lean thing that immediately follows it—

[The video is cued up at the exact point I’m talking about, but you lose the cueing once you play it. To rewatch, reload this webpage.]

I mean, I was lost. Right shoulder down, right hip up, lean back, left shoulder down, left hip up… while the hands are moving? How do you do the weight transfer smoothly while you’re bouncing? How do you bounce and lean at the same time? *Where do I put my head this whole time??* Trying to assemble this strange little movement felt like trying to hold too many things in my hands at once: something was going to fall. If I got my hips in the right place, I’d forget about my shoulders. Get the shoulders? Mess up the bounce. The idea of doing all of it at once felt overwhelming. The idea of ever making it look cute felt *way* out of reach. I needed help.

My wife has an actual background in a highly relevant field, namely hip-hop dance. Also, as it turns out, she is a completely conceptually-oriented dance teacher. Her first move was to tell me to stop thinking about what to do with each body part. Instead, she said, focus on the attitude. She illustrated it with other, more familiar movements that differed in their details but shared the attitude. “It’s like, ‘Eyyyyyyyyyy!'” she said, demonstrating.

The parallel to how I respond to analogous situations as a math instructor was extremely apparent. There was a main idea here. My wife was pulling my attention away from the impossible-feeling task of assembling the whole out of a bunch of disconnected details, and toward a single main idea from which all those details would flow. She was elucidating that main idea through its connections to more familiar knowledge. The main idea was what was important. The details would work themselves out.

*It worked!* By focusing on the attitude, everything came together. The bounce was nothing more than feeling the music. The whole thing with the shoulders, the hips, and the lean, turned out to be nothing but a right-to-left weight shift shaped by the appropriate attitude. The hands were, like, I mean obviously, I just hit you with that DDU-DU DDU-DU—now I have to put the “guns” away, and where else would they go? The entire motion felt logical and coherent, and I could do it without even thinking too hard.

*Score one for the conceptually-oriented lesson!*

I kept going. Exactly 7 seconds deeper into the chorus, there is a second “Hit you with that DDU-DU DDU-DU,” and again the four beats that follow it threw me completely:

It’s just a turn. No fancy roll/lean/bounce stuff this time, just rotate 360 degrees over four beats, stepping on alternating feet, and end up in that same little shoulder-shimmy as before.

But I wasn’t getting it! I felt off-kilter, gangly, uncoordinated. I felt I had to keep lurching, yanking my weight in different directions—this did not feel cute at all. I kept being late to finish the turn and set up for the shoulder-shimmy. Furthermore, I didn’t understand how it was possible not to be late. I repeatedly watched my wife and all four members of BLACKPINK pull it off, but this seemed like magic.

Fresh off our previous success, my wife again took a conceptual approach. To her, the main idea of the turn is to feel the beat in the alternation of your steps. She had me practice those 4 beats without turning, just stepping right-left-right-left in place.

This was easy for me—but this time, it didn’t actually help. My problem wasn’t, as it turned out, a failure to feel the beat in my steps. I realized I had a more fundamental question: *where should I put my feet?*

When my wife responded with, “It doesn’t *matter,*” I had a little moment of acute empathy for every student I’ve ever driven up the wall by insisting they focus on an underlying concept when they want me to tell them what to do. In that moment, I was the student who needed some concrete steps to follow (pun intended), and I wasn’t getting them.

On the one hand, in saying “it doesn’t matter,” my wife was obviously telling me the truth. The four members of BLACKPINK are at that point in the song rotating their whole formation. They’re all turning, it’s all synchronized, but they’re not putting their feet in the same places at all. My wife’s own rendition involved turning in place, so that was different too. All five of them—Jennie, Jisoo, Lisa, Rosé, and my wife—were evidently successfully executing the same fundamental dance idea, while putting their feet in different places. It follows that this particular dance idea is not determined by the locations of the feet.

On the other hand, I *understood* the underlying concept, at least as my wife was presenting it to me, but this understanding was not clarifying for me *how to actually do the turn*. She saw the procedural knowledge as an immediate corollary of the conceptual knowledge, but to me it was apparent that she was using some additional, not-entirely-conscious prior knowledge to translate this underlying concept into actual steps to take, and this was knowledge that *I didn’t have*.

This elucidated a mistake I’ve made countless times in teaching. The student is stuck and asks me how to proceed. I assume it’s a conceptual problem and take aim at the underlying concept. The student seems to understand the concept and is frustrated I won’t just tell them what action to take. Because the appropriate action, AKA procedure, feels *to me* like an immediate corollary of the concept, I assume that there’s a subtler, undiscovered conceptual problem still lurking. Because, furthermore, I fear that I’ll short-circuit the student’s opportunity to address this underlying conceptual issue by revealing the appropriate action prematurely, I hesitate to answer the question about what to actually do.^{[4]}

*But sometimes, that’s what the student needs!* The piece the student is missing may not actually lie in the concept, but instead in the way the concept entails the appropriate action—this is a kind of knowledge often not even visible to me, as focused as I am on the concept. In this situation, the student may need direct information about what to do. Seeing a complete solution demystifies this missing link, providing an opportunity to coordinate the underlying concept with the appropriate action.

This is what was happening to me with the turn. My only way forward was to *directly mimic a correct example*. I played the video back several times, focusing on Jennie—she’s the one in front at the beginning of the turn. Right foot steps out; turn 180 degrees on the right foot while swinging the left foot around the front; shift the weight; turn the other 180 degrees on the left foot, this time with the right foot moving backward; shift weight again; left foot behind right; step out with the right. The body is moving in the same absolute direction the whole time. Lemme try that…

*It worked!* Directly mimicking Jennie’s footwork gave me a structure to follow that solved the problem of how to turn around in exactly 4 beats without awkward direction changes. The abstract concept of feeling the rhythm in my feet could now inhabit the concrete set of motions I was following.

*Score one for the procedurally-oriented lesson!*

All of this is to say—we are hopefully on our way out of the false dilemma of procedural vs. conceptual knowledge, and toward a consensus that they are both critical, and are mutually reinforcing. Nonetheless, this wisdom can function as a bit of a platitude—preached, without always being lived. So I think it’s a worthwhile exercise to look, both in the classroom and outside of it, for opportunities to go beyond knowing it, to *feeling* it. And—who knew?—but learning a K-pop dance routine gave me the opportunity to feel it in my bones. Literally.

[1] Indeed, the lack of consensus about the meanings even extends to the possibility that by calling them knowledge *types*, I’m not being entirely faithful to the full range of their uses. See J. R. Star and G. J. Stylianides, Procedural and Conceptual Knowledge: Exploring the Gap Between Knowledge Type and Knowledge Quality, *Canadian Journal of Science, Mathematics, and Technology Education* Vol. 13, No. 2 (2013), pp. 169–181 (link), which argues that while the terms refer to knowledge types among psychology researchers, they are better seen as referring to knowledge *quality* among math education researchers.

[2] An illustration: In 2015, in the *Oxford Handbook of Numerical Cognition*, Bethany Rittle-Johnson and Michael Schneider wrote, “Although there is some variability in how these constructs are defined and measured, there is general consensus that the relations between conceptual and procedural knowledge are often bi-directional and iterative.” B. Rittle-Johnson and M. Schneider, Developing conceptual and procedural knowledge of mathematics, *Oxford Handbook of Numerical Cognition* (2015), pp. 1118–1134 (link).

[3] An example: M. Schumacher, Developing Conceptual Understanding and Procedural Fluency, on the Illustrative Mathematics Blog (link).

[4] While this and the next paragraph are focused on the situation in which I am wrong to withhold the “what to do” information, I hasten to add that this is, in general, a reasonable fear. If a student is in fact missing a conceptual piece of the puzzle, premature information about what to do may allow them to walk away from instruction with the belief that they have fully learned the concept when they actually did not. The student who applies a procedure in inappropriate contexts probably mis-learned it in this way. Judgement is required to determine what the student needs.

]]>CUNY Brooklyn

The forced conversion to distance learning in Spring 2020 caught most of us off-guard. One of the biggest problems we face is the existence of free or freemium online calculators that show all steps required to produce a textbook perfect solution. A student can simply type in “Solve ” or “Find the derivative of ” or “Evaluate ” or “Solve ,” and the site will produce a step-by-step solution indistinguishable from one we’d show in class. With Fall 2020 rapidly approaching, and no sign that distance learning will be abandoned, we must confront a painful reality: Every question that can be answered by following a sequence of steps is now meaningless as a way to measure student learning.

So how can we evaluate student learning? Those of us fortunate enough to teach courses with small enrollments have a multitude of options: oral exams; semester-long projects; student interviews. But for the rest of us, our best option is to ask “internet resistant” questions. Here are three strategies for writing such questions:

● Require inefficiency.

● Limit the information.

● Move the lines

**Require Inefficiency**

One of the goals of mathematics education is developing adaptive expertise: the ability to identify which of the many possible algorithms is the best to use on a particular problem.

For example, consider a quadratic equation. We have at least two ways of solving quadratic equations: by factoring; or by the quadratic formula. Which do we use? Since the quadratic formula always works, there’s no obvious reason why we would ever want to use anything else. But sometimes using the quadratic formula is like using a chainsaw to cut a dinner roll: we wouldn’t use it on “Solve $(3x-7)(2x+5) = 0$ ,” and we probably wouldn’t use it on “Solve $x^2-9 = 0$,” though we’d almost certainly use it on “Solve $6x^2 – 19x – 36 = 0$.” The boundary between the problems we’d attempt to solve by factoring and the problems we’d solve using the quadratic formula can’t be taught: every student has to find it for themselves through firsthand experience.

It should be clear that requiring inefficiency is a possibility every time there is more than one way to solve a problem. This approach works even better when one method is clearly (to us) less efficient. Indeed, the least efficient method is one that doesn’t work, and in some ways, requiring inefficiency in such cases may give us more insight into student learning than their ability to solve a problem.

For example, consider the problem:

If possible, solve by factoring: $x^2 – 3x – 12 = 0$. If not possible, show why; then solve using the quadratic formula.

Since the quadratic expression is irreducible over the integers, no online calculator will produce a factorization. Thus, a student can’t simply look up the answer. More importantly, in order to provide an answer, they must check every possible pair of factors (and show that none of them work).

There’s an added bonus. On the same exam, we might ask students to factor various quadratic expressions. We argue that a student’s attempt to factor $x^2-3x-12$ will actually reveal more about whether a student understands factoring than the successful factorization of an expression like $6x^2 + 19x – 36$. Thus, we can omit straight factorization questions (which, in any case, can be “solved” by an online calculator).

**Limit the Information**

Another way to thwart the use of internet calculators is to provide incomplete data. For example, Wolfram Alpha can find the derivative of any function—provided you give it the function. Thus we might ask students to solve problems without giving them equations.

This might sound hard to do, but it’s actually pretty easy. Since the 1990s, state and national mathematics standards have called for increased use of graphical and tabular representations, so source material is plentiful. Even the most traditional texts include problems based on interpreting graphical and tabular data. For example:

Suppose you know $f(3) = 5$ and $f'(3) = -4$ . Let $h(x) = ln f(x)$ . Find $h'(3)$.

While this is an algorithmic question that can be easily answered by invoking the chain rule, doing so relies on correctly interpreting the written statements about the function and derivative values. As such, it is currently beyond the capability of online calculators.

We can also present data graphically:

The graphs of y = f(x) (solid) and y = g(x) (dashed) are shown:

Find the sign of $(fg)’)(0)$.

Again, this is an algorithmic question that can be answered by invoking the product rule. However, it relies on being able to extract information from a graph, then make a quantitative argument based on the signs of the functions and their derivatives.

**Moving the Lines**

Requiring inefficiency and limiting information should be viewed as stopgap measures at best. Thus, when calculators were first introduced, math teachers insisted on “exact answers,” since the student who returned the answer “1.4142135” instead of $\sqrt{2}$ was clearly using a calculator. But now, even a \$10 calculator can return “exact answers” like $\frac {3+\sqrt {5}} 2$ , so this distinction is no longer useful as a way of distinguishing between students who used a calculator and students who didn’t. Similarly, while I’m not aware of any app that allows for the user to select a solution method, or that can read graphical or tabular data, there’s no a priori reason why there couldn’t be one. This means we need a more powerful method of creating internet resistant questions that can adapt to advances in technology. This leads to a strategy I call “moving the lines.”

To begin with, it’s important to understand that the problem “Solve $x^2 – 3x – 12 = 0$” does not exist outside of a mathematics classroom. So we should ask two questions:

● Where did this problem come from? This moves the “starting line,” where the problem begins.

● Why do we want the solution? This moves the “finish line,” where the problem ends.

Our long-term goal as mathematics educators should be to shift the lines and turn a sprint into a marathon.

Let’s consider this problem. What leads to “Solve $x^2 – 3x – 12 = 0$?” For that, we might consider some of the basic steps in solving any quadratic equation. One of those steps is to get the equation into standard form. So our problem “Solve $x^2-3x-12 = 0$” might have come from “Solve $x^2 – 3x = 12$.” In fact, you’ve probably asked this question before, specifically to identify students who failed to understand the necessity of getting the equation into standard form.

Now where might we have gotten a problem like that? We might have gotten it from “Solve $x(x-3) = 12$.” In fact, you’ve probably asked this type of question as well, to identify the students who failed to understand the zero product property.

Note that we still have an equation that can be dropped into an online calculator, so the next step is important: What type of question leads to a product equal to a number? There are many times we multiply two numbers to get a quantity of interest; for example, the product of a rectangle’s length and width gives us the area. This takes us to the problem:

*A rectangle has an area of twelve square feet, and its width is three feet less than its length. Find the length of the rectangle.*

In order to answer this question, a student would have to translate the given information into a mathematical form. This is beyond the capability of online calculators (especially if, as in this case, the numbers are also spelled out). If you enter the question into Google, you’ll get examples of similar problems, but no solution, effectively reducing you to your class notes and textbook. If you’re clever enough to switch numbers for the words, you’ll get an answer—which is incorrect (4 feet).

We can further improve the problem by changing the finish line. Remember that once a student translates this problem into the equation $x(x-3) = 12$, an online calculator can produce the algebraic solution, showing all the steps. One way to further blunt the ability of the online calculator to answer all questions is to require another step beyond the mathematical solution. Thus we should ask why we’d want the answer.

Let’s consider: we obtain the length (and width, since we know it’s three feet less than the length). So why would we want the length and width of a rectangle? There are three obvious possibilities: to find the rectangle’s area; to find the rectangle’s perimeter; and to find the rectangle’s diagonal. Since we already know the area, we might want either the perimeter or the diagonal. So we could ask:

*A rectangle has an area of twelve square feet, and its width is three feet less than its length. Find the perimeter of the rectangle.*

Even better:

*A homeowner wants to fence a garden in the shape of a rectangle. The garden must have an area of twelve square feet, where the width is three feet less than its length. The fence will cost two dollars per foot. How much will it cost to enclose the garden?*

The best part about this approach is that as technology advances, we can shift the lines in response. Perhaps some day we’ll be able to enter the above problem into a search engine and get the correct answer. So the next step will be to shift the lines again: move the starting point further back by imagining where the problem might come from; and move the finish line further forward by considering why we’d want to know the cost.

**The Road Ahead**

Notice that we end with something that might be called a “real world” problem. But a homeowner rarely has to build a garden with a specific area and relationship between the sides: it would be a stretch to call the problem above a real world example of how to use mathematics.

What’s more important is that real world problems don’t come with instructions on how to solve them, so they must be solved inefficiently, by trying different approaches until we find one that works. Real world problems don’t come with formulas attached to them, so they must be solved without complete information. And real world problems often change, so we must expect that the starting and finishing lines will change on us.

What this means is that regardless of when or if we can resume traditional resource-restricted exams, we should consider requiring inefficiency, limiting information, and shifting the lines on all our assessments. Sooner or later, our students will leave our classroom. If what they learned can be replaced by someone using a free internet app, then they can be replaced by a free internet app. So it’s not just about making our questions internet resistant: it’s also about making our students internet resistant.

**Addendum**

We’re stronger together. Readers interested in sharing their “internet resistant” questions should email them to me at jsuzuki@brooklyn.cuny.edu, and I’ll put up a selection of these in a later post.

]]>In 2012, 100 years after Henri Poincare’s death, the magazine for the members of the Dutch Royal Mathematical Society published an “interview” with Poincare for which he “wrote” both the questions and the answers (Verhulst, 2012). When responding to a question about elegance in mathematics, Poincare makes the famous enigmatic remark attributed to him: “Mathematics is the art of giving the same names to different things” (p. 157).

In this blog post, we consider the perspectives of learners of mathematics by looking at how students may see two uses of the word tangent—with circles and in the context of derivative—as “giving the same name to different things,” but, as a negative, as impeding their understanding. We also consider the artfulness that Poincare points to and ask about artfulness in mathematics teaching; perhaps one aspect of artful teaching involves helping learners appreciate why mathematicians make the choices that they do.

Our efforts have been in the context of a technology that asks students to give examples of a mathematical object that has certain characteristics or to use examples they create to support or reject a claim about such objects.^{1} The teacher can then collect those multiple examples and use them to achieve their goals.

Kayla: Algebra 2 students often get a super minimalized and overbroad definition of an asymptote. Many leave Algebra 2 saying something like “a horizontal asymptote is a line the graph gets close to but doesn’t touch.” In calculus, they get a limit definition for asymptotes. As a teacher, I’m prepared for students to enter calculus with the Algebra 2 definition—it’s acceptable knowledge for Algebra 2—but if a student left calculus with the impression that a horizontal asymptote is a line we get close to but don’t touch, I would be mortified.

Willy: I think the purpose of learning about asymptotes changes too, right? In Algebra 2, students are getting an overview of a lot of functions and their general behavior. At that point, it seems fine to have such a loose definition. Calculus introduces limits to explain function behavior at various parts of the domain. That includes wrestling with infinity.

Kayla: Yes, yes, but what I hadn’t noticed until recently was that students’ understanding even of tangent in calculus might be influenced by what they retained from geometry.

Willy: Right! The terms shift meaning a bit. When I took calculus and geometry as a student, I don’t recall any emphasis or discussion of a shift in the definition of tangent. In geometry, the only use of tangent that I remember was with circles: the tangent is perpendicular to the radius. That’s not at all how we talk about tangents in calculus.

Dan: And that’s Poincaré’s “giving the same name to different things.” David Tall (2002) argues that evolutions in definitions of mathematical concepts are natural in a curriculum—he calls the phenomenon “curricular discontinuities”—because you can’t unfold the complete complexity of a concept all at once. In different contexts, you think about particular dimensions of concepts. So it’s natural that when we’re just talking about circles, tangent is a special case of a broader concept. It’s one that you meet first. Lines whose slopes describe the instantaneous rate of change in graphs of functions are mathematically different, but it can make sense to give them that same name in order to capture some way in which they’re the same. Kayla, it sounds like you hadn’t thought as much about how differently the word tangent was used in calculus and geometry. What in particular, now strikes you as different?

Kayla: I believe most calculus students learn the new definition—how to derive a tangent, what it looks like, what it tells us about a curve—but I worry they may leave calculus still expecting tangent to mean “touching only at one point” as it did in geometry. I also worry that the geometric idea that the tangent line must lie on just one side of the circle causes some students to trip up and struggle in calculus when they encounter a tangent line that crosses the graph either at a point of inflection, or just at some other point. I also have students who think it is not possible to have a vertical tangent; they conflate the derivative being undefined with the tangent line not existing.

Willy: I wonder if that could be a result of trying to make sense of the idea that there is no linear function of x that will give a vertical line.

Dan: Kayla, it sounds like you’re saying that, on the one hand, there are things that are called tangents in calculus that wouldn’t have been called tangents in geometry and also the reverse, that there were tangents in geometry that calculus students would not think are tangents.

Kayla: Yes.

Dan: That’s really helpful, because it identifies a challenge beyond the curricular discontinuity of changing definitions. When definitions change, people might recognize and remember the changes—a changed concept definition—but the things that come readily to their minds might not change, what Tall and Vinner (1981) call a “concept image.” So really, Kayla, what you were saying is that only some of the things that come to students’ minds as tangent lines from a geometry perspective remain useful when they’re thinking in a calculus sense. A tangent sharing more than one point with a curve is acceptable in calculus but didn’t make sense in geometry; a vertical tangent made sense in geometry but worries the calculus student. The tricky thing is that students might notice that while their concept definition has evolved, their concept images might not have.

Kayla: Yeah. A couple years ago, when we had students sketch a graph with a vertical tangent, a lot of what we got was graphs like x = abs(y), a 90° clockwise rotation of the absolute value graphs students have seen, which doesn’t define a function of x at all. And, they treated the y-axis as the “tangent.” I just wonder if, to students, the picture just seems really similar to a circle despite its shape.

Dan: Right. One point of contact with the vertex of the “v” curve, the curve all on one side of the “tangent,” just like the tangent to a circle. From a geometry perspective, a student could think, well, that’s a reasonable example of a tangent. But, from a calculus perspective, it’s not. In calculus, we want the derivative to be well-defined, determining one specific slope for the tangent at a point.

Willy: If there is an art to the way mathematics names different things with the same name, then students should be able to understand why mathematicians over time decided to use the same name. It seems like the teacher has to help students appreciate the benefit of having the derivative as a well-defined function, with either one unique tangent line or none at all.

Kayla: I agree, but I don’t feel like I have a great answer to a student who asks why it is important that there not be multiple tangents to a point on the graph of a function. I would probably say something like: “At the vertex of the graph of abs(x), the slope to the left of the vertex and the slope to the right of the vertex are really different (one positive and one negative) creating a drastic change in slope where the two lines meet. And unlike a parabola where the slopes change from positive to negative across, those slopes are both approaching zero—just one from the negative direction and one from the positive direction. So, when looking at the vertex of the graph of abs(x), when you go to draw the tangent line what slope would you choose? The two drastically different slopes is why the derivative does not exist at that point—the slope from the right and left are different and the derivative function cannot take on two values for one x.

Willy: This is one of the reasons that asking students to produce examples of concepts has been really thought provoking when I think about teaching. Asking students to sketch a function that has a vertical tangent has the possibility of having students stumble upon things that might challenge their conceptions of how mathematics operates across contexts.

Dan: Those sorts of tasks can also give teachers information about what definitions their students are using, and what kind of concept images they have. But then, Kayla, it seems you’ve also been saying that such tasks give you a way to influence students’ concept definitions and concept images. Is that true?

Kayla: Yes, tasks like these help surface students’ concept image for me to work on with them. With some tasks, students all basically submit the same thing, showing how limited their image is. And, this applies not just to tangents. I especially like asking students to submit multiple examples. When we were doing rational function tasks, we asked them to submit multiple functions that would have a seemingly identical graph to a linear function and students could not think of multiple ways to do so. And from these sorts of tasks, I can also learn about how students think about related concepts: Do students think that points of tangency are different from points of intersection or just special ones? Or, do students think that a horizontal asymptote is a tangent?

Dan: So, your comments are about not just the match between the concept image and the concept definition, but also the richness and variety of the concept image space and connections to nearby concepts. Having surfaced all of those examples from students, in what way do you feel that those are a resource for your teaching separate from their role in assessing students?

Kayla: For the past couple years, students’ submissions have ended up being used in future discussions. When you have this bank of submissions that students actually submitted, you can develop a whole lesson based on what a couple students have submitted. I think the ability to see all those submissions easily, pick ones that are interesting, and use those, is great. Sometimes just seeing someone else’s submission can shift your concept image or support the new definition you are learning in a way that you weren’t able to without that extra nudge. I think that part is key. It can be super powerful just for students to see each other’s work.

Willy: I agree! And in the context of teacher preparation I also think about how difficult and time consuming it is for teachers to make up a variety of examples. So using student generated work helps! The work is already done for you, and then you can select the most appropriate examples for your purpose and have more time for other things.

Kayla: And I think often we make fake student work to use as teachers, we are saying these are the common submissions we know to expect. But now that we’re presenting this task to students, it has been interesting to see examples year after year that I hadn’t expected the first time around.

Dan: What’s an example of that?

Kayla: Year after year, students seem to think that there is a horizontal tangent on an exponential function where the horizontal asymptote is; they think the same line is both an asymptote and a tangent.

Dan: And, they aren’t thinking about a point at infinity!

Kayla: This comes usually in response to a prompt like “Enter a symbolic expression for a function whose graph is a line parallel to the x-axis. Then write a function, or sketch its graph, such that the line is tangent to the graph of the function at two or more points.”

Willy: To help us learn how students think about a concept, we can design assessment tasks that reveal students’ concept images or the definitions they’re operating from. Students can produce examples that do not fulfill all or any of the requirements of the task but still reveal possible gaps in understanding or overly broad or narrow concept images. For example, the “sideways absolute value” graph is not a function and does not have a tangent at the vertex. We can also design tasks that push students in a particular direction to further their learning—to encounter a concept in a certain way so that there is no prescribed solution or method and responses will vary. Such tasks could be used to shift student thinking for the purpose of, say, evolving their definition of tangent lines from a geometry sort of definition to one more appropriate for calculus. Interestingly, when I spoke with calculus teachers from my old school, one of the teachers thought it was weird that we would care whether a tangent line intersected the graph somewhere else because the curriculum focuses on tangents locally, not more globally. I wonder how extending the tangent line in calculus is helpful.

Dan: I was asking myself that question with a focus on the mathematics. I don’t have anything conclusive, but I have an observation to offer. On the interval between a point of tangency and a point of intersection farther down the line, even if that point of intersection is not another point of tangency, I think the average value of the derivative function is equal to the derivative at the point of tangency or the slope of the tangent line. For example, consider Red(x) = (x-1)(x-2)(x-3), and Green(x) = 2(x-1). The point of tangency is (1,0) and Red'(1) = 2. The point of intersection is (4,6).

Think about the interval [1, 4]. This interval reminds me of Algebra One where we often work with average rates of change and linear functions, rather than more complex curves. As long as we know the values at two points, in order to interpolate or extrapolate, we imagine a hypothetical situation where the change is distributed evenly, rather than the messy reality of change that is not evenly distributed. This observation about the interval between the point of tangency and intersection seems like it might suggest a mathematical value for considering when the continuation of a tangent line intersects with a function.

Kayla: I see the mathematical promise in that direction but wonder how many teachers would see that as standard calculus material. I wonder what it might take to have my colleagues consider using these tasks. I know I am a bit of an outlier. At the beginning of the year, I generally move through content with my BC Calculus class at a slower pace than other teachers in my district. From what I’ve heard from other teachers, many either skip the limits unit (assuming students understand the content from precalculus) or simply do a quick review (a week or so of class time). Similarly, with tangent lines, the concept of tangent line is pretty much skimmed over (pun intended!). The introduction to derivatives usually begins with defining derivative and then a quick transition into derivative rules, the relationship between functions and their derivative graphs, and applications of derivatives (related rates, optimization, linearization, etc.). Our district’s curriculum materials frequently ask questions about calculating derivatives and writing the equation of tangent lines at specific point, but there’s little digging into what the definition of a tangent line is and how it might have changed from geometry. Personally, I think it’s important to spend time on the issues about tangents that we’ve been discussing, but I worry many teachers may find these tasks a distraction that would take time away from other topics and skills in the curriculum that they see as more important/relevant to the AP exam.

Willy: Does that influence what you are going to do next year?

Kayla: No, not really. Using these tasks over the last few years has surfaced important areas of student confusion, even beyond the ones we’ve talked about here. I want students to think hard about definition and how definitions change. These “give-an-example” tasks help. They engage students with something interesting and challenging, and help them to pay careful attention to mathematical definitions and to be precise in using them.

**Endnote**

**References**

Verhulst, F. (2012). Mathematics is the art of giving the same name to different things: An interview with Henri Poincaré. Nieuw Archief Voor Wiskunde. Serie 5, 13(3), 154–158.

Olsher, S., Yerushalmy, M., & Chazan, D. (2016). How might the use of technology in formative assessment support changes in mathematics teaching? For the Learning of Mathematics, 36(3), 11–18. https://www.jstor.org/stable/44382716

Yerushalmy, M., Nagari-Haddif, G., & Olsher, S. (2017). Design of tasks for online assessment that supports understanding of students’ conceptions. ZDM, 49(5), 701–716. https://doi.org/10.1007/s11858-017-0871-7

Nagari-Haddif, G., & Yerushalmy, M. (2018). Supporting Online E-Assessment of Problem Solving: Resources and Constraints. In D. R. Thompson, M. Burton, A. Cusi, & D. Wright (Eds.), Classroom Assessment in Mathematics: Perspectives from Around the Globe (pp. 93–105). Springer International Publishing. https://doi.org/10.1007/978-3-319-73748-5_7

Tall, D. (2002). Continuities and discontinuities in long-term learning schemas. In David Tall & M. Thomas (Eds.), Intelligence, learning and understanding—A tribute to Richard Skemp (pp. 151–177). PostPressed. http://homepages.warwick.ac.uk/staff/David.Tall/pdfs/dot2002c-long-term-learning.pdf

Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics, 12(2), 151–169. https://doi.org/10.1007/BF00305619

]]>by Jeff Suzuki

Unless you’ve been living under a rock for the past decade, you know that one of the buzzwords in education is *active learning*: Be the guide on the side, not the sage on the stage. One of the more common approaches to active learning is the so-called flipped or inverted classroom. In a flipped classroom, students watch lectures at home, then come to class to do problems. This is actually a 21st century implementation of a very traditional approach to pedagogy, namely reading the textbook before coming to class. Many of us embraced this idea, and shifted our approach to teaching.

Then came the era of social distancing and forced conversion to distance learning. It might seem that those who switched to the flipped classroom model had an advantage: Our lectures are *already* online. And that’s true. But the second part of the flipped classroom involves working problems *in class.* This is now impossible, and those of us who had embraced the flipped classroom model have spent the past few months in existential agony. The “sage on the stage” can still give lectures through Zoom, but the “guide on the side” can’t guide.

**The New Normal?**

And yet…it’s now more important than ever to be the guide on the side.

We don’t know how long the current phase of social distancing will last, but even after it ends, we can expect that distance learning will be the new normal: it’s a trend that began long before the pandemic. And this forces us to deal with a new problem: It is impossible to monitor student activity remotely.

We accept this when we assign homework, and expect students will do the work with their books open, their notes in front of them, and a half-dozen math help sites open in different browser tabs. Before the pandemic, we told ourselves it didn’t matter, since they’d have to do the exam without all these study aids. But in the post-COVID world, there is now no difference between the resources available to students on homework assignments and on exams.

Don’t believe the hype about lockdown browsers (which work fine for the students who don’t have smartphones). Live webcam monitoring can be defeated by taping cheat sheets on the wall behind the computer. And if a student turns in a textbook perfect answer, it’s possible they listened to us when we explained how the answers should be written.

Will students cheat on exams? We’ve found copies of our exam questions posted to Chegg (with answers). This shocked me: Who would pay for a Chegg subscription, when there are so many *free *sites that show all steps to solving a problem and, unlike Chegg, leave no evidence behind?

The bad news is that *every* exam question that can be answered by following an algorithm is now obsolete, because such questions can no longer distinguish between the student who understands the material and the student who knows how to use Google.

Here is where the flipped classroom can be our salvation. A key component of the flipped classroom is letting your students figure things out for themselves, and *not* giving them a step-by-step algorithm for solving a problem.

For example, let’s consider a basic problem in calculus: Finding the derivative of a function. In the internet era, *any* function that can be described algebraically can have its derivative found, with steps, by a free online problem solver. So we have to ask questions that can’t be resolved by typing the problem into www.findthederivativewithstepsfree.com (not, so far as I know, a real website, but a thirty-second Google search will give you a plethora of possibilities).

**Transcending the Machine**

The good news is that computers are good at exactly one type of problem: problems that have algorithmic solutions. If you can *describe* the exact sequence of steps needed to solve a problem, then a computer can implement those steps faster, more accurately, and more cheaply than any human being. The real moral of the story about John Henry is *don’t compete with the machine in the machine’s areas of strength. *Instead, find the things the machine is *bad* at. In this case, the easiest way to neutralize these problem solving sites is to make every problem a word problem.

Of course, “students can’t do word problems.” This is a meaningless objection: at the start of calculus, students can’t integrate, but we still ask them integration questions on the final exam! Our job is to teach these students how to do these things. Here’s where the flipped classroom becomes a key part of the solution. *Don’t* spend class time lecturing: students can view lectures on their own time. Instead, class time should be spent working problems, especially those that can’t be solved by following a sequence of steps.

It’s helpful in the discussion that follows to think of problems as falling into one of two categories:

- Routine problems, where the mathematical question and the relevant information are explicitly given: “Find the derivative of
*y*= tan(3*x*).” - Non-routine problems, where this information is not given explicitly. Roughly speaking, every word problem is non-routine, and such problems form the bulk of the questions in “reform-oriented” textbooks.

**Flipping Your Class, Social Distancing Edition**

Here’s one possible structure for such a class (where “class” means any time you’re working with students in realtime). All of the following takes place before class:

- Students watch assigned lectures on the topic.
- Students complete routine homework problems, using some online homework management (OHM) system. If you’re using a commercial text, there is an OHM associated with your text. If you’re using your own, there are free products (MyOpenMath is my go-to) that can be used.
- Students are also assigned a set of non-routine problems to consider. These don’t have to be separate from the OHM: again, almost every word problem should be considered non-routine.

How should you run class itself? Class time is the most valuable resource available to students; using it efficiently and effectively can be challenging. Here’s a few things that may help.

At the start of class (online or in person), take down a list of student questions. One risk is that the more outspoken students tend to dominate the discussion; taking down a list of all questions at the start of class is a way to make sure that every student has a chance of getting their question answered, and to ensure that a sufficient variety of problems are presented.

Establish from the start that the routine problems have lowest priority: these are problems that should be solvable by students who followed the assigned lecture. It is vitally important that you keep to this rule: The biggest challenge to running a flipped classroom is students who don’t watch the lectures beforehand. Depending on how you’ve set things up and the system you’re using, it might even be possible to determine whether a student has watched the assigned lecture (though trying to do this realtime requires a bit of practice); another option, which I use, is to assign simple 1-point problems that students answer after they’ve watched the lecture. Remember: *Class time is the single most valuable resource available to the students; it should not be spent on things that can be done out of class time.*

One way to efficiently use class time is to focus on the setup. For example, let’s consider the following problem, which probably appears in every calculus text ever written:

*A 25-foot long ladder rests against a wall. The base of the ladder begins sliding away from the wall at 2 ft/second, while the top of the ladder maintains contact with the wall. How rapidly is the top of the ladder falling when the base is 10 feet away from the wall?*

The “sage on the stage” would identify the relevant parameters and write down the mathematical problem to be solved. The “guide on the side” would lead students to the mathematical problem. For this, it’s important to ask leading questions and not give outright answers. For example:

- What’s going to answer the question “How rapidly?” (Students should identify that this is an instantaneous rate of change, so it’s a derivative)
- What other things are changing? (Students should recognize that the distance of the base of the ladder from the wall is also changing, but the length of the ladder is not)
- Is there a relationship we can write between the quantities?
- Which derivative do we want? (Students should identify that they want $\frac{dy}{dt}$; it’s also worth making them explain why $\frac{dy}{dx}$ is not relevant).

In the end, we have the mathematical problem, “Find $\frac{dy}{dt}$ when $x^2 + y^2 = 25$ and $\frac{dx}{dt} = 2.$” At this point, it becomes a routine problem—and if you’ve established that minimal class time will be spent on routine problems, you can leave the problem at this point, perhaps with a directive of “Finish the problem after class.”

It’s worth noting that, at this point, the problem can be handed off to an online calculator, which can then solve the problem. You might even go so far as to point students to the online calculator, lest they develop a mistaken belief that you’re unaware of the existence of such things. This epitomizes the idea that humans should do what humans are good at, namely extracting the mathematical problem to be solved; while machines should do what machines are good at, namely applying an algorithm.

As the preceding example suggests, it’s possible to teach a flipped class with very little change in how you’re already teaching. The main difference is establishing the expectation that students watch lectures before class.

Let’s see how we might take a larger step, using a standard topic: finding the extreme values of a function. A traditional approach might be to have students find derivatives, then critical values, then apply some test to decide whether a critical value corresponds to a maximum or minimum.

In a flipped classroom, students wouldn’t be given this algorithm. Instead, they’d create their own approach, typically through some guided exploration of a question. Coming up with good questions is challenging; fortunately, thirty years of reform calculus have provided us with an abundance of material, and many of these questions have been incorporated into every standard calculus text, so you needn’t write your own.

For example, I like to give students the following question:

*An accelerograph records the acceleration of a train (assumed to be moving in a straight line); some of the data values are shown below. Assuming the velocity of the train at t = 0 was 0 m/s, estimate when the train was moving the fastest; defend your conclusion.*

t (seconds) | 0 | 1 | 2 | 3 | 4 |

a(t) (m/s^{2}) |
3 | 2 | 1 | -1 | -2 |

* *

A sequence of leading questions can guide students to creating their own approach:

- What does the acceleration have to do with the velocity? (The student should identify that it’s the rate of change of the velocity)
- So
*a*(0) = 3 and*a*(4) = -2. What does that tell you? (The student should identify that the velocity is increasing at t= 0, and decreasing at t = 4) - So when is the velocity increasing, and when is it decreasing? (The student should identify the velocity’s increasing at t = 0, 1, 2, and decreasing at t = 3, 4. I usually find I have to ask “Is the velocity increasing or decreasing at
*t*= 1?*t*= 2? ) - So where is the velocity going to be the greatest? (The student will
*probably*say*t*= 2, at which point remind them that they just told you velocity is increasing at t = 2)

and so on, leading to an answer like: *The train’s velocity appears to be increasing until at least t = 2, and is decreasing from t = 3, so there’s a maximum velocity between t = 2 and t = 3.*

What’s worth noticing here is that *none* of these questions can be answered by appeal to a formula or an algorithm. Consequently, any attempt to use an online calculator on this type of question will result in, at best, a nonsense answer. The closest thing to an “algorithm” is recognizing that the change from increasing to decreasing is where the local maximum value will occur, but even then, since that change occurs “offscreen”, students must consider how they know that the change has occurred.

**A Return to Normalcy**

Suppose, against all predictions and the entire trend of human history, we go back to how things were at the beginning of 2020: traditional in-person classes, no social distancing, exams where we could control the resources used by students.

*None* of the preceding needs to change. In fact, *all* of the preceding alterations in our pedagogy and our assessment are worth doing regardless of how we will give exams. The hard truth is that sooner or later, our students will leave the classroom. If what they’ve learned from our classes can be done by a free internet application, then their education is worth a free internet application.

We owe it to our students to give them something more.

]]>

Fostering an understanding and appreciation of the deep, beautiful threads that unite seemingly disparate areas of mathematics is among the most valuable outcomes of teaching. Two such areas that are ripe for bridge building—functions and geometric transformations—are the focus of our NSF project, Forging Connections Through the Geometry of Functions. In this post, we describe the pedagogical benefits of introducing students to functions through the lens of geometric transformations.**Geometric Transformations as Functions**

The most common representations of functions are symbolic and numeric in nature. This emphasis on number limits students’ images of the variety of mathematical relationships that can be represented as functions. As such, it contributes to common student misconceptions. Students may conclude that:

- every function turns an input number into an output number;
- every function can be expressed as an algebraic formula;
- a formula is the primary representation of a function, and all other representations derive from it; and
- the ultimate test of a function requires graphing it in rectangular coordinates and applying the vertical line test.

Although students investigate reflections, translations, rotations, dilations, and glide reflections in a geometry course, they typically do not regard them as functions; the functions they encounter in algebra always have numbers as input and output. We can expand students’ horizons and deepen their concept of function by treating geometric transformations as functions that take a Euclidean point as input and produce another point as output. Coxford and Usiskin pioneered this approach a half century ago in their ground-breaking *Geometry: A Transformation Approach*, but very few of today’s geometry students encounter it.

In Figure 1, a student has used three Web Sketchpad tools to construct the independent variable *x*, the mirror *m*, and the reflected dependent variable *r _{m}*(

*Function notation is meaningful.*The use of function notation gives students language to describe the specific elements that constitute the function: independent variable x, function rule*r*(“reflect in_{m }*m*”), and dependent variable*r*(_{m}*x*) (“the reflection in*m*of*x*”).*Functions need not be algebraic formulas with numeric inputs and outputs.*Point*r*(_{m}*x*) depends on*x*: Students can drag*x*in order to make point*r*(_{m}*x*) move, but cannot drag*r*(_{m}*x*) by itself.*Variables really vary.*As students drag independent variable*x*, red and blue traces memorialize the kinesthetic experience of varying the variable. The traces form a pictorial record of the dynamic interaction and help students analyze the covariation.*Relative rate of change can be observed and described.*By dragging*x*, students observe that*x*and*r*(_{m}*x*) always move at the same speed, but not always in the same direction, and they can investigate how to drag*x*so the variables move in the same direction or in opposite directions.

**Constructing a Dynagraph**

Having explored reflection and other geometric transformations in two-dimensional Flatland, students then restrict the domain of these transformations into the Lineland (one-dimensional) environment of a number line (Abbott, 1886). They focus in particular on connecting the geometric behavior of dilation and translation to the observed numeric values of their variables on a number line.

In Figure 2, students use the Number Line, Point, and Dilate tools to create a point restricted to the number line and dilate it about the origin to obtain a point labeled *D*_{0,s}(*x*). While this notation may at first seem daunting, it actually may be less mysterious than the traditional *f*(*x*) language. Rather than write out or speak all the words “the **D**ilation (of ** x**) about center point

Students measure the coordinates of *x* and *D*_{0,s}(*x*) and drag *x* to compare the values. When asked to describe how *D*_{0,s}(*x*) moves when *x* is dragged, a student might respond, “As I drag *x*, *D*_{0,s}(*x*) moves faster. It seems to move twice as fast, and I notice its value is always twice the value of *x*. I wonder if its speed is related to the scale factor *s*.” By experimenting with different scale factors, the student concludes that *s* represents the relative speed of *D*_{0,s}(*x*) with respect to *x*, and that the coordinates produced by this dilation satisfy *D*_{0,s}(*x*) = *x*·*s*. Students can then experiment with a translation restricted to the number line and conclude that this new function, translation by a vector parallel to the number line and of directed length *v*, causes the two variables to move at the same speed, and satisfies the equation *T _{v}*(

Students are now ready for a new task: What happens when you dilate *x* and then translate the dilated image; in other words, how does the composite function *T _{v}*(

Students who construct *T _{v}*(

- As x is dragged,
*T*(_{v}*D*_{0,s}(*x*)) moves in the same direction as*x*, but 3 times as fast as*x*. - As x is dragged,
*T*(_{v}*D*_{0,s}(*x*)) is stuck at 2. - As x is dragged,
*T*(_{v}*D*_{0,s}(*x*)) moves at the same speed as*x*, but in the opposite direction. - As x is dragged,
*T*(_{v}*D*_{0,s}(*x*)) moves in the opposite direction as*x*, but twice as fast. - As x is dragged,
*T*(_{v}*D*_{0,s}(*x*)) moves 2 times as fast as*x*, and is always 4 behind*D*_{0,s}(*x*).

**The Cartesian Connection**

To conclude, students create the Cartesian graph of a linear function using geometric transformations. As Figure 4 illustrates, students start with the same initial tools that they used to create a dynagraph, but this activity’s Transfer tool rotates a variable by 90°, transferring it to a vertical number line perpendicular to the original, horizontal number line. After using this tool to rotate *D*_{0,s}(*x*) to a vertical axis and translating by vector *v*, students use the Perpendicular tool to construct lines that keep track of the horizontal location of *x* and the vertical location of *T _{v}*(

**Conclusion**

By using web-based dynamic mathematics software and tools tailored to carefully structured tasks, students can enact geometric transformations as functions, create them, manipulate them, and experiment with them. In the course of their explorations they can develop a solid understanding of geometric transformations, explore connections between geometry and algebra, and construct and shed light on linear functions by using a dynagraph representation.

By beginning with R^{2}→R^{2} functions (transformations in the Euclidean plane) and connecting them to R→R functions in algebra, these activities can help prepare students for later study of complex (C→C) functions, and functions with three-dimensional domains and ranges (R^{3}→R^{3}). A further benefit is the gentle visual introduction of the concept and notation of function composition.

Pedagogically, the constructive nature of activities such as these has the potential to engage students, to give them opportunities to assess their own work, to encourage mathematical discussions, and to help students bridge the gap between the concrete, physical world and the profound elegance of abstract mathematical insights.

**Acknowledgments**

This post is based in part upon work supported by the National Science Foundation under NCSU IUSE award 1712280 (July 2017 through June 2019) and KCP Technologies DRK-12 award ID 0918733 (September 2009 through August 2013). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

We are deeply grateful to McGraw-Hill Education for making Web Sketchpad available for the activities described in this post.

**References**

Abbott, E. A. (1884). *Flatland: A romance of many dimensions*. San Antonio, TX: Eldritch Press.

Coxford, A. F., & Usiskin, Z. (1971). *Geometry: A transformation approach*. Laidlaw Brothers Publishers.

Goldenberg, P., Lewis, P., & O’Keefe, J. (1992). Dynamic representation and the development of a process understanding of function. In E. Dubinsky & G. Harel (Eds.), *The concept of function: Aspects of epistemology and pedagogy* (pp. 235-260). Washington, DC: Mathematical Association of America.

Recently as a graduate student I taught a week-long boot camp for incoming mathematics graduate students at Oregon State University. It was my first foray into teaching under the active learning model and it was a completely transformative experience for me as an instructor. The change in my own attitudes towards teaching and pedagogy were so abrupt, so all-encompassing, that I felt compelled to immediately record my thoughts on the experience.

The purpose of the free, optional boot camp is to ensure that incoming students have a common base of content knowledge and proof techniques. The camp is run by an advanced graduate student, with every aspect at the discretion of the instructor. In designing the boot camp I chose to focus on concepts from real analysis and linear algebra (the subjects of our PhD qualifying exams). Each day I would choose a major idea and orient readings and problems around it. In the mornings we explored the historical development of the idea using guided primary source readings and exercises from the excellent TRIUMPHS projects[1]. The afternoons ran as problem sessions where I posed more sophisticated problems with modern definitions and theorems.

The model of active learning that I chose meant that I did almost no lecturing at all. I assigned readings and exercises to be completed before each session. This meant that the students never came at a definition or theorem completely cold. The readings, exercises, and later problems were discussed first in small groups, and then as a whole class with a student presenting on the board. I believe that the boot camp had a strong positive impact on the students. However, the impact on me was even greater. I am a total convert… an evangelical, born-again, active learning enthusiast. This is my testimonial.

My Own Transformation

I hope never to lecture again. This experience of teaching with active learning was eye opening for me. So many things jumped out at me as reasons to prefer active learning to lecturing. First of all, I noticed that students were much more engaged when their classmates presented on the board than when I presented on the board. The students seemed to feel freer to ask “dumb questions” to their classmates than to me. If they didn’t understand a line in a proof that the student had written they would just ask them to explain more. However, the few times that I wrote on the board I could feel the room tense up and the students’ eyes glaze over (and I pride myself on being an engaging instructor!). The problem is that they felt like what I had to say was gospel, and so they didn’t engage as much or question as much. They just accepted what I wrote, copied it down, and then waited for the next piece of information.

The next thing I noticed is that students often came up with solutions very different than what I had in mind. When there were multiple solutions to the same problem I had students write them all up. We then looked them over, compared them, and discussed the virtues and drawbacks of each attempt. This would never have happened if I had simply written my own proof on the board. Also, it sometimes happened that there would be a subtle error in a proof that made it to the board. These errors prompted great discussions every time. Usually one of the students would notice the error but maybe not know how to fix it. Often some students would still need convincing that there really was an error in the first place. As a class we would discuss the various nuances, and I pretty much never had to swoop in and resolve the mystery. The students would solve it themselves. At the end of these debates I would spend a minute summarizing the debate and emphasizing the main takeaways. When appropriate I would also place the debate in a broader context or relate it to something else we had done that week. These moments allowed me to share my expertise in a way that would not have come out if I had been lecturing.

This led me to the realization that lecturing is a waste of an instructor’s expertise. What is the benefit of having me write down the standard proof of a famous theorem on the board when this is something that the students could find in any textbook? I could perhaps explain the reasoning behind certain moves. But then, if the students have read the proof beforehand, I could still give that explanation, and without wasting time writing out symbols on the board that the students already have in front of them. Even better, I could ask questions or assign exercises that allow the students to discover the subtleties on their own. That is where my experience and knowledge becomes useful. It is in designing the readings, exercises, and discussion questions that best facilitate understanding. It is in answering questions on the fly to help students get beyond some mental block. It is in facilitating thoughtful discussions that bring out the nuances in subtle reasoning. Writing something on the board that the students all have in front of them already is not only a waste of my own expertise, it is a colossal waste of everyone’s time.

Teaching with active learning was way more fun than I had anticipated. I have always enjoyed lecturing, but teaching with active learning was more fun, more engaging, less stressful, and ultimately less work than lecturing. The fun was being able to interact with the students on a more personal level. It was also a fun challenge to meet the students where they were and figure out how to get them on board. Writing proofs on the board can be stressful. Subtle errors easily creep in, and when you are writing on the board you don’t always see the mistakes that you made (even if you would notice them immediately if someone else was writing on the board). With the model of making students write their answers on the board, the stress of presenting is broken down into small pieces and spread out amongst the whole class. Students also gain experience and confidence presenting their work (something that academics are expected to do frequently). And students often enjoy showing off a little bit when they have proved something on their own and they are sometimes quite eager to share.

By the end of the week of boot camp, I trusted the students enough that I could relax a bit on my own preparation. At the beginning of the week I had my own answers to every problem or exercise that I assigned written out in painstaking detail. This was useful as the students were getting accustomed to the format of the class and needed more precise prodding. However, by the end I was no longer worried that no one would be willing to write on the board or that a subtlety would be overlooked if I didn’t micromanage. As a result, by the end I was assigning problems that I knew were doable and that I had an idea of how to do, but that I hadn’t worked out fully on my own. This made class even more fun for me because I got to think along with the class. It was also good for the students because they got to see more into how a relative expert tackles certain problems and why I can do them faster even though I’m not any smarter than them. I would say things like “I’m not sure how that goes, but when I see something like this it always makes me think of…” So my being slightly less prepared actually allowed students to gain new insights into how to problem solve at the graduate level. (Not that I am advocating being under-prepared for class! I am only saying that after a while it was useful to hand the reins over to the class a little more, and that my relaxation of control had its own benefits.)

Conclusion

In the year since the boot camp I have occasionally found myself lecturing (usually by accident), and each time I have deeply regretted it. I can always tell that I’ve started accidentally lecturing by the looks in my students’ faces. They may be following and focused, but they are falling behind. My unease over lecturing is so complete that I now have trouble giving presentations without inserting a significant amount of active learning. This has also worked successfully for me with presentations related to pedagogy as well as with academic presentations to REU students and public outreach. However, I have yet to try this approach in a research talk. My friend and mentor in all things active learning, David Pengelley, has been a great encouragement (bad influence?) in this respect. He is working on incorporating active learning into his research talks. Once he figures out how to do so successfully, perhaps none of us ever need lecture or be lectured to again.

[1] 1https://digitalcommons.ursinus.edu/triumphs/

]]>“The difficulty… is to manage to think in a completely astonished and disconcerted way about things you thought you had always understood.” ― Pierre Bourdieu,

Language and Symbolic Power, p. 207

Proof is the central epistemological method of pure mathematics, and the practice most unique to it among the disciplines. Reading and writing proofs are essential skills (*the* essential skills?) for many working mathematicians.

That said, students learning these skills, especially for the first time, find them *extremely hard*.^{[1]}

Why? What’s in the way? And what are the processes by which students effectively gain these skills?

These questions have been discussed extensively by researchers and teachers alike,^{[2]} and they have personally fascinated me for most of my twenty years in mathematics education.

In this blog post I’d like to examine one little corner of this jigsaw puzzle.

To frame the inquiry, I posit that there are *imported* and *enculturated* capacities involved in reading and writing proofs. Teachers face corresponding challenges when teaching students about proof.

Capacities that are *imported* into the domain of proof-writing are those that students can access independently of whether they have any mathematics training in school or contact with the mathematical community, let alone specific attention to proof.^{[3]} Capacities that are *enculturated* are those that students do not typically develop without some encounter with the mathematics community, whether through reading, schooling, math circles, or otherwise. Examples of imported capacities are the student’s capacity to reason, and fluency in the language of instruction. Enculturated capacities include, for example, knowledge of specific patterns of reasoning common to mathematics writing but rare outside it, such as the elegant complex of ideas behind the phrase, “without loss of generality, we can suppose….”

For imported capacities involved in proof, the teaching challenge is to create conditions that cause students to actually access those capacities while reading and writing proofs.

For enculturated capacities, the prima facie teaching challenge is to inculcate them, i.e., to cause the capacities to be developed in the first place. But there is also a prior, less obvious challenge: we have to know they’re there. Since many instructors are already very well-enculturated, our culture is not always fully visible to us. If we can’t see what we’re doing, it’s harder to show students how to do it. (This challenge has the same character as that mentioned by Pierre Bourdieu in the epigraph, although he was writing about sociology.)

When my personal obsession with student difficulty with proofs first developed, I focused on imported capacities. I had many experiences in which students whom I knew to be capable of very cogent reasoning produced illogical work on proof assignments. It seemed to me that the instructional context had somehow severed the connection between the students’ reasoning capacities and what I was asking them to do. I became very curious about why this was happening, i.e., what types of instructional design choices led to this severing, and even more curious about what types of choices could reverse it.

My main conclusion, based primarily on experience in my own and others’ classrooms, and substantially catalyzed by reading Paul Lockhart’s celebrated *Lament* and Patricio Herbst’s thought-provoking article on the contradictory demands of proof teaching, was this: It benefits students, when first learning proof, to have some legitimate uncertainty and suspense regarding what to believe, and to keep the processes of reading and writing proofs as closely tied as possible to the process of deciding what to believe.^{[4]}

I stand by this conclusion, and more broadly, by the view that the core of teaching proof is about empowering students to harness their imported capacities (in the above sense) to the task, rather than learning something wholly new. That said, in the last few years I’ve become equally fascinated by the challenges of enculturation that are part of teaching proof reading and writing. If I’m honest, my zealotry regarding the importance of imported capacities blinded me to importance of the enculturated ones.

What I want to do in the remainder of this blog post is to propose that a particular feature of proof writing is an enculturated capacity. It’s a feature I didn’t even fully notice until fairly recently, because it’s such a second-nature part of mathematical communication. I offer this proposal in the spirit of the quote by sociologist/anthropologist Pierre Bourdieu in the epigraph: to think in a completely astonished and disconcerted way about something we thought we already understood. Naming it as enculturated has the ultimate goal of supporting an inquiry into how students can be explicitly taught how to do it, though this goes beyond my present scope.

I recently encountered an article by Kristen Lew and Juan Pablo Mejía-Ramos, in which they compared undergraduate students’ and mathematicians’ judgements regarding unconventional language used by students in written proofs.^{[5]} One of their findings was that, in their words, “… students did not fully understand the nuances involved in how mathematicians introduce objects in proofs.” (2019, p. 121)

The hypothesis I would like to advance in this post is offered as an explanation for this finding, as well as for a host of student difficulties I’ve witnessed over the years:

*The way we conceptualize the objects in proofs is an enculturated capacity.*

These objects are *weird*. In particular, the sense in which they exist, what they *are*, is weird. They have a different ontology than other kinds of objects, even the objects in other kinds of mathematical work. An aspect of learning how to read and write proofs is getting accustomed to working with objects possessing this alternative ontology.^{[6]} If this is true, then it makes sense that undergraduates don’t quite have their heads wrapped around the way that mathematicians summon these things into being.

The place where this is easiest to see is in proofs by contradiction. When you read a proof by contradiction, you are spending time with objects that you expect will eventually be revealed *never to have existed*, and you expect this revelation to furthermore tell you that *it was impossible that they had ever existed*. That’s bizarro science fiction on its face.

But it’s also true, more subtly perhaps, of objects appearing in pretty much any other type of proof. To illustrate: suppose a proof begins,

Let be a lattice in the real vector space , and let be a nonzero vector of minimal (Euclidean) length in …

Question. *What kind of a thing is ?*

[The camera pans back to reveal this question has been asked by a short babyfaced man wearing a baseball cap, by the name of Lou Costello. His interlocutor is a taller, debonair fellow with a blazer and pocket square, answering to Bud Abbott.]

Abbott: It’s a vector in .

Costello: Which vector?

Abbott: Well, it’s not any *particular* vector. It depends on .

Costello: You just said it was a particular vector and now it’s not a particular vector?

Abbott: No, well, yes, it’s some vector, so in that sense it’s a particular vector, but I can’t tell you which one, so in that sense it’s no particular vector.

Costello: You can’t tell me which one?

Abbott: No.

Costello: Why not?

Abbott: Because it depends on . It’s one of the vectors that’s minimal in length among nonzero vectors in .

Costello: *Which vector?*

Abbott: No *particular* vector.

Costello: But is it some vector?

Abbott: Naturally!

Costello: You said it depends on . What’s ?

Abbott: A lattice in $\mathbb{R}^n$.

Costello: Which lattice?

Abbott: Any lattice.

Costello: Why won’t you say *which* lattice?

Abbott: Because I’m trying to prove something about *all lattices*.

Costello: You mean to say is *every lattice???*

Abbott: No, it’s just one lattice.

Costello: *Which one?!*

For any readers unfamiliar with the allusion here, it is to *“Who’s on First?”*, legendary comedy duo Abbott & Costello’s signature routine.^{[7]} What’s relevant to the present discussion is that the skit is based on Costello asking Abbott a sequence of questions about a situation to which he is an outsider and Abbott is an insider. Costello becomes increasingly frustrated by Abbott’s answers, which make perfect sense from inside the situation, but seem singularly unhelpful from the outside. Abbott for his part maintains patience but is so internal to his situation—enculturated, as it were—that he doesn’t address, or even seem to perceive, the ways he could be misunderstood by an outsider.^{[8]}

My goal with this little literary exercise has been to dramatize the strangeness of the “arbitrary, but fixed” nature of the objects in proofs. Most things we name, outside of proof-writing, don’t have this character. Either they’re singular or plural; one or many; specific or general; not both. Every so often, we speak of a singular that represents a collective (“the average household”, “a typical spring day”), or that is constituted from a collective (“the nation”), but these are still ultimately singular. They are not under the same burden as mathematical proof objects, to be able to stand in for any member of a class. Proof objects aren’t representative members of classes but *universal* members. This makes them fundamentally unspecified, even while we imagine and write about them as concrete things.

There’s an additional strangeness: proof objects, and the classes of which they are the universal members, are themselves often constituted in relation to other proof objects. We get chains, often very long, where each link adds a new layer of remove from true specificity, but we still treat each link in the chain, including the final one, as something concrete. I was trying to hint at this by posing the question “what is it?” about , rather than , in the example above. As consternated as Costello is by , is doubtless more confounding.

I think there are at least two distinct aspects of this that students new to proof do not usually do on their own without some kind of enculturation process. In the first place, the initial move of dealing with everything in a class of objects simultaneously by postulating a “single universal representative” of that class just isn’t automatic. This is a tool mathematical culture has developed. Students need to be trained, or to otherwise catch on, that a good approach to proving a statement of the form “For all real numbers…” might begin, “Let be a real number.”^{[9]}

But secondly, when we work with these objects, their “arbitrary, but fixed” character forces us to hold them in a different way, mentally, than we hold the objects of our daily lives, or even the mathematical objects of concrete calculations. When you read, “Let be a smooth function ,” what do you imagine? A graph? Some symbols? How does your mental apparatus store and track the critical piece of information that can be *any* smooth function on ? Reflecting on my own process, I think what I do in this case is to imagine a vague visual image of a smooth graph, but it is “decorated”—in a semantic, not a visual, way—by information about which features are constitutive and which could easily have been different. The local maxima and minima I happen to be imagining are stored as unimportant features while the smoothness is essential. Likewise, when I wrote, “Let be a lattice in the real vector space ,” what did you imagine? Was there a visual? If so, what did you see? I imagined a triclinic lattice in 3-space. But again, it was somehow semantically “decorated” by information about which features were constitutive vs. contingent. That I was in 3 dimensions was contingent, but the periodicity of the pattern of points I imagined was constitutive. I’m positing that students new to proof do not usually already know how to mentally “decorate” objects in this way.^{[10]}

Here are some specific instances of student struggle that seem to me to be illuminated by the ideas above.

- In the paper of Lew and Mejía-Ramos mentioned above, eight mathematicians and fifteen undergraduates (all having taken at least one proof-oriented mathematics course) were asked to assess student-produced proofs for unconventional linguistic usages. The sample proofs were taken from student work on exams in an introduction to proof class. One of these sample proofs began, “Let .” Seven of the eight mathematicians identified the “Let …” as unconventional without prompting, and the eighth did as well when asked about it. Of the fifteen undergraduate students, on the other hand, only four identified this sentence as unconventional without prompting, while even after being asked directly about it, six of the students maintained that it was not unconventional. I would like to understand better what these six students thought that the sentence “Let ” meant.
- Previously on this blog, I described the struggle of a student to wrap her head around the idea, in the context of – proofs, that you are supposed to write about as though it’s a particular number, when she knew full well that she was trying to prove something for all at once.
- A year and a half ago, I was working with students in a game theory course. They were developing a proof that a Nash equilibrium in a two-player zero-sum game involves maximin moves for both players. It was agreed that the proof would begin by postulating a Nash equilibrium in which some player, say , was playing a move that was not a maximin move. By the definition of a maximin move, this implies that has some other move such that the minimum possible payout for if she plays move is higher than the minimum possible payout if she plays . The students recognized the need to work with this “other move” but had trouble carrying this out. In particular, it was hard for them to keep track of its constitutive attribute, i.e., that its minimum possible payout for is higher than ‘s. They were as drawn to chains of reasoning that circled back to this property of
*as a conclusion*, as they were to chains of reasoning that proceeded forward*from*it. - In the same setting as the previous example, there was a student who, in order to get her mind around what was going on, very sensibly constructed some simple two-player games to look at. I don’t remember the examples, but I remember this: I kept expecting that when she looked at the fully specified games, “what was” would click for her, but it didn’t. Instead, I found
*myself*struggling to be articulate in calling her attention to , precisely because its constitutive attribute was now only one of the many things going on in front of us; nothing was “singling it out.” I found myself working to draw her attention*away*from the details of the examples she’d just constructed in order to focus on the constitutive attribute of . My reflection on this student’s experience was what first pointed me toward the ideas in this blog post:*I mean really, what***is**, anyway, that recedes from view exactly when the situation it’s part of becomes visible in detail? - This semester I taught a course on symmetry for non-math majors. It involved some elementary group theory. An important exercise was to prove that in a group, implies . One student produced an argument that was essentially completely general, but carried out the logic in a specific group, with a specific choice of , and presented it as an example. Here is a direct quote, edited lightly for grammar and typesetting. “For example [take] ; if we will operate on both sides the inverse of we will get . As we have proven that always , we can change the structure of the equation to , [which] shows that x has to be equal to y.” The symbols and refer to one-quarter and three-quarters rotations in the dihedral group . From my point of view as instructor, the student could have transformed this from an illustrative example to an actual proof just by replacing and with and , respectively, throughout. What was the obstruction to the student doing this?

My claim is that *the mathematician’s skill of mentally capturing classes of things by positing “arbitary, but fixed” universal members of those classes, and then proceeding to work with these universal members as though they are actual objects that exist, is an enculturated capacity.*

[1] I trust that any reader of this blog who has ever taught a course, at any level, that serves as its students’ introduction to proof, has some sense of what I am referring to. Additionally, the research literature is dizzyingly vast and there is no hope to do it any justice in this blog post, let alone this footnote. But here are some places for an interested reader to start: S. Senk, How well do students write geometry proofs?, *The Mathematics Teacher* Vol. 78, No. 6 (1985), pp. 448–456 (link); R. C. Moore, Making the transition to formal proof, *Educational Studies in Mathematics*, Vol. 27 (1994), pp. 249–266 (link); W. G. Martin & G. Harel, Proof frames of preservice elementary teachers, *JRME* Vol. 20, No. 1 (1989), pp. 41–51 (link); K. Weber, Student difficulty in constructing proofs: the need for strategic knowledge, *Educational Studies in Mathematics*, Vol. 48 (2001), pp. 101–119 (link); and K. Weber, Students’ difficulties with proof, *MAA Research Sampler #8* (link).

[2] Again, I cannot hope even to graze the surface of this conversation in a footnote. The previous note gives the reader some places to start on the scholarly conversation. A less formal conversation takes place across blogs and twitter. Here is a recent relevant blog post by a teacher, and here are some recent relevant threads on Twitter.

[3] This and the following sentence should be treated as definitions. I am indulging the mathematician’s prerogative to define terms and expect that the audience will interpret them according to those definitions throughout the work. In particular, while I hope I’ve chosen terms whose connotations align with the definitions given, I’m relying on the reader to go with the definitions rather than the connotations in case they diverge. I invite commentary on these word choices.

[4] This is an argument I have made at length in the past on my personal teaching blog (see here, here, here, here, here), and occasionally in a very long comment on someone else’s blog (here). Related arguments are developed in G. Harel, Three principles of learning and teaching mathematics, in J.-L. Dorier (ed.), *On the teaching of linear algebra*, Dordrecth: Kluwer Academic Publishers, 2000, pp. 177–189 (link; see in particular the “principle of necessity”); and in D. L. Ball and H. Bass, Making believe: The collective construction of public mathematical knowledge in the elementary classroom, in D. Phillips (ed.), Yearbook of the National Society for the Study of Education, *Constructivism in Education*, Chicago: Univ. of Chicago Press, 2000, pp. 193–224.

[5] K. Lew & J. P. Mejía-Ramos, Linguistic conventions of mathematical proof writing at the undergraduate level: mathematicians’ and students’ perspectives, *JRME* Vol. 50, No. 2 (2019), pp. 121–155 (link).

[6] Disclaimer: although I am using the word “ontology” here, I am not trying to do metaphysics. The motivation for this line of inquiry is entirely pedagogical: what are the processes involved in students gaining proof skills?

[7] Here’s a video—it’s a classic.

[8] One of the keys to the humor is that the audience is able to see the big picture all at once: the understandable frustration of Costello, the uninitated one, apparently unable to get a straight answer; the endearing patience of Abbott, the insider, trying so valiantly and steadfastly to make himself understood; and, the key idea that Costello is missing and that Abbott can’t seem to see that Costello is missing. I’m hoping to channel that sense of stereovision into the present context, to encourage us to see the objects in a proof simultaneously with insider and outsider eyes.

[9] Annie Selden and John Selden write about the *behavioral knowledge* involved in proof-writing, and use this move as an illustrative example. A. Selden and J. Selden, Teaching proving by coordinating aspects of proofs with students’ abilities, in *Teaching and Learning Proof Across the Grades: A K-16 Perspective*, New York: Routledge, 2009, p. 343.

[10] The ideas in this paragraph are related to Efraim Fischbein’s notion of “figural concepts”—see E. Fischbein, The theory of figural concepts, *Educational Studies in Mathematics* Vol. 24 (1993), pp. 139–162 (link). Fischbein argues that the mental entities studied in geometry “possess simultaneously conceptual and figural characters” (1993, p. 139). Fischbein’s work in turn draws on J. R. Anderson, Arguments concerning representations for mental imagery, *Psychological Review*, Vol. 85 No. 4 (1978), pp. 249–277 (link), which, in a broader (not specifically mathematical) context, discusses “propositional” vs. “pictorial” qualities of mental images. The resonance with the dichotomy I’ve flagged as “semantic” vs. “visual” is clear. I’m suggesting that the particular interplay between these poles that is involved in conceptualizing proof objects is a mental dance that is new to students who are new to proof. (Actually, it is not entirely clear to me that the dichotomy I want to highlight is “semantic” vs. “visual” as much as “general” vs. “specific”; perhaps it’s just that visuals tend to be specific. However, time does not permit to develop this inquiry further here.)

[11] Because this circle of skills involve taking something strange and abstract and turning it into something the mind can deal with as a concrete and specific object, they strike me as related to some notions well-studied in the education research literature: Anna Sfard’s *reification* and Ed Dubinsky’s *APOS theory*—both ways of describing the interplay between process and object in mathematics learning—and the more general concept of *compression* (see, e.g., D. Tall, *How Humans Learn to Think Mathematically*, New York: Cambridge Univ. Press, 2013, chapter 3).

When Yale-NUS College reviewed the curriculum for its Mathematical, Computational, and Statistical (MCS) Sciences major in the autumn of 2018, I spent several weeks reading about mathematics programs at similar institutions. A common learning objective among many of the programs was a variation of “preparing students to become lifelong learners.” I really like this goal because, among many other reasons, it reminds teachers that students are human beings who have lives beyond their studies, and it reminds students that learning is not confined to the early years of one’s life. As I reflect on my life of learning thus far, I cannot help but notice how significantly the way I learn has changed since I was a student. Some of these differences arose naturally with changes in my circumstances over the years, while others could have been addressed while I was still a student.

In this post, I want to share some observations about how my approach to learning has changed since I started working as a professional mathematician, and how I have changed my approach to teaching with the hope of helping my students develop more effective and relevant learning strategies earlier in their mathematical journeys.

*Reading has become my primary mode of knowledge acquisition.*

When I was an undergraduate student, I rarely read mathematics. It wasn’t from a lack of interest in the subject. I remember being enthusiastic about my courses and the joy I felt from solving problems. I simply didn’t read much mathematics. Not *really *anyway. The closest thing I did to reading was scanning through a textbook for a proposition or theorem that could help me link two concepts that would allow me to solve a homework problem. That jigsaw-puzzle approach to reading mathematics lasted well into graduate school. It is dramatically different from my current situation, where the majority of new mathematics I learn, I get from reading. So what changed? Necessity. As a professor, I can go to seminars and conferences to learn more about certain subjects, but not to a degree that is comparable to taking a course. Instead, I spend a lot of time learning on my own and the available format is almost always written text.

When I was a student, the need to read simply wasn’t there. I was fortunate to study at an undergraduate program with many dedicated teachers, who prepared clear, accessible lectures and class activities, so I could successfully complete my coursework without doing the assigned readings. It didn’t become an issue for me until I was a graduate student when I had to look up details of proofs that didn’t fit into lecture notes and read lots of articles for my dissertation research. It was a difficult transition for me.

*Learning new concepts and techniques becomes much easier when I need them to complete an ongoing project.*

When I was a student, I spent a lot of time learning new techniques, diligently practicing them on problem sets… and then forgetting them almost immediately. I don’t think I was particularly unmotivated or lazy – and I completely trusted my professors when they said certain concepts were important – yet I forgot so much of what I learned shortly after learning it. What was going on? On one hand, it is natural to learn new things in stages, picking up a fraction of the content at the first encounter followed by pieces of new information with each subsequent exposure. At each stage of the process, we internalize a portion and forget the rest of what we observe.

On the other hand, I think my struggles were partially related to context. I remember my professors giving clear explanations for why different techniques were developed and how they were used in practice, but there was a disconnect for me because I didn’t have any personal experience developing mathematical techniques, nor did I have an application of my own in mind. Looking back over my career, my most productive learning experiences have come from working on a project where I didn’t have all of the tools I needed and had to learn them on the fly in order to complete the project. In those cases, I didn’t watch a tutorial or listen to a lecture about standard techniques and then practice them on a variety of examples; I started with the problem I was trying to solve, found a technique in the literature that was used to solve similar problems, and figured out how to apply or adapt the technique to my particular situation.

*All of my best ideas have had humble beginnings.*

* *When I was a student, I had a growth mindset about mathematical knowledge but a fixed mindset about mathematical creativity. I believed everyone could have positive, successful, and meaningful experiences with mathematics by learning new techniques but mathematical creativity was an inherent ability that could not be developed. I don’t know why I felt this way, and I can’t recall anyone ever telling me it was the case, yet I remember that impression weighing on me a lot. Whenever I worked on homework sets with other students and someone would figure out how to solve a problem I was stuck on, I always assumed it was because they had some amazing insight that I would not have been capable of finding on my own. I was so preoccupied with trying to figure out whether or not I had what it took to become a successful mathematician that it never occurred to me to ask them how they came up with their idea. Consequently, I spent a lot of time feeling frustrated, not being particularly productive, and waiting for inspiration to strike because that was where I thought creative solutions originated.

Now that I have more experience – and the confidence that comes with it – I can recognize that all my best ideas started with simple observations. And while there is no clear-cut recipe for creativity and innovation, there are concrete things I can do to cultivate situations that make those important kernels of ideas of possible. Instead of dwelling on what I don’t know how to do, I focus on exploring what I can do that might produce a new insight, such as writing out some examples, constructing a conceptual diagram, or drawing a picture.

*All of my proudest accomplishments were made possible through the generous help of people whose experiences and perspectives are different from my own.*

As a student, I found little satisfaction from working in teams, especially with unfamiliar teammates. Team assignments typically went one of two ways for me: either I was confident in my abilities and did the vast majority of the work or I was insecure about my abilities, didn’t want to look stupid, and held back my ideas thinking it was better to appear ignorant than open my mouth and confirm it. In the former scenario, I didn’t mind doing most of the work because I was confident in my ability to succeed, and it often seemed easier to do most of the work myself rather than try to coordinate my teammates’ efforts.

I didn’t see value in exploring different perspectives because there were never any consequences for taking a narrow approach. Like many who have the same privileges as I do (I am a heterosexual white male from North America), I had a limited understanding of how social identities affect group interactions, and I conflated inclusivity with civility. In the latter scenario, I was aware that teamwork required a lot of effort and collaboration. Even though I was willing to put in the work, my insecurities still got the better of me because I didn’t trust my teammates enough to share my ideas openly.

When I look at how the accomplishments I’m most proud of have come about, and how much I have learned in recent years working at an international college in Singapore, I can’t help but wonder how many opportunities to learn and grow I missed out on because I simply wasn’t looking or I didn’t appreciate how much effort goes in to building enough trust to open up a beneficial exchange of ideas.

Here are a few ways I have changed my approach to teaching in response to these observations.

For starters, I no longer rely on lectures or video tutorials for presenting new ideas. Instead, the lion’s share of content delivery comes in the form of reading assignments. To support my students as they adapt to this model, I use the social annotation platform, Perusall, which allows them to highlight passages and ask questions, contribute or link alternative explanations, and propose solutions to “check your understanding” type exercises. They can also upvote annotations of their peers that they find helpful. In addition to developing technical reading skills, Perusall offers the valuable practice contributing to social media debates and online forums like StackExchange in a safe and controlled environment.

To offer my students an authentic learning environment that emulates the typical “on the job” learning that takes place in many technical professions, I have started to build each of my courses around three or four substantial team projects. Instead of asking students to master content and then apply what they have learned to a bigger project, I design the projects in a way that prompts students to learn the relevant material as they go. Each project is assigned on the first day of its respective segment of the course. The students are typically able to understand what the project prompts are asking but are not aware of any obviously relevant tools to get started.

To facilitate effective teamwork, I have adopted the Team-Based Learning (TBL) model, where each lesson has a reading assignment to be completed before class, individual and team readiness assurance tests at the start of each class, and a substantial problem-solving session that enables students to apply and extend their understanding of the tools they will need to successfully complete the project. Students take the readiness assurance tests and work together on the problem-solving sessions within their project teams throughout the duration of the project in order to develop a productive group dynamic.

To encourage and reinforce good habits for mathematical research and creativity, I have started acknowledging and giving credit to teams when they demonstrate important elements of a productive research process, such as generating examples, identifying patterns, asking questions and making conjectures, testing conjectures with new information, drawing connections between relevant topics in the literature to better understand the problem at hand, and re-evaluating an approach based on preliminary findings. Because many of these elements can be difficult to discern in a final written report, I have started asking each team to submit an activity log that documents their progress throughout the project. My rubric for the activity log was heavily influenced by the Creativity-in-Progress Rubric on Proving.

Finally, in addition to research and creativity, I have started to encourage and reinforce good habits for effective and respectful team interactions by asking each team to prepare a mission statement during the first week of the project where they agree on a team name, tentative work schedule, and initial plan of attack. I also ask each team to prepare a set of guidelines for how they will conduct their meetings and a set of criteria for how they will evaluate each other’s contributions to the project.

The idea for creating guidelines came from my experience facilitating Intergroup Dialogue (IGD) at Yale-NUS College. IGD is a structured conversation between members of different social identity groups that encourages participants to explore singular and intersecting aspects of their identities while critically examining dynamics of power, privilege, diversity and inequity in society. Because the dialogues can be difficult or contentious, a lot of the groundwork for IGD aims at building trust and creating a space in which people can share their ideas freely without judgment. For instance, at the beginning of each dialogue, the participants prepare a list of guidelines. I adapted those guidelines to fit a team-based learning classroom: The IGD guideline *“We all recognize that participation in this dialogue is voluntary. Everyone who is here wants to be here.”* became *“We all recognize that this course is an elective. Everyone who is here wants to be here.”* Most of the guidelines are common sense statements, but articulating them in a mission statement provides avenues for students to speak their discomfort and overcome obstacles in a responsible and respectful manner.

Here is the first project brief from my Discrete Mathematics course, which is typically taken by second-year prospective MCS majors at Yale-NUS College whose primary interest is computer science. The course meets twice per week for 110 minutes at a time. Each lesson consists of a pre-class reading assignment (8-10 pages of text, approximately 2 hours of interactive reading), in-class readiness assurance tests (20 minutes), and an in-class problem-solving session (90 minutes). The project spans six class meetings, including one lesson each on the Pigeonhole Principle, mathematical induction, and basic enumeration, two lessons on combinatorial proofs and bijections with emphases on the Binomial Theorem and Fibonacci numbers, and one class meeting designated as work time so students have a full week free of reading assignments and problem sets to complete their reports.

The project description presents students with eight seemingly unrelated families of mathematical objects and asks them to find a formula for the number of objects in each family. It then asks them to describe how the families are related based on the formulas they find. Over the course of the project, the students discover that the families are all equinumerous. Indeed, they are all manifestations of the Catalan numbers!

While the project initially appears somewhat daunting, the students typically proceed by generating lots of examples of each family. From there, they tend to observe fairly quickly and conjecture that the number of elements in each family appears to be the same. This is a significant discovery for them since it means that, instead of finding the same formula eight different times, they only need to find the formula for one family and then argue why the different sets are in one-to-one correspondence with one another. That prompts them to review the reading assignments on mathematical induction, recursion, combinatorial proof, and bijections. The diversity of the objects themselves also makes the project well suited for teams made up of students with disparate backgrounds since finding all the connections requires a variety of perspectives.

The overall response to these changes has been positive. A number of students acknowledged the stated goals and embraced the project-based approach straight away. For instance one student wrote:

[The project] was actually a very fun and enjoyable experience, while also providing a good amount of challenge and difficulty. When we first received the project brief, we were genuinely stunned by what we had to do – we didn’t really know where to begin, and everything we tried seemed to be useless. But it was really nice to see us slowly progress, picking at the problem bit by bit, sometimes with no results, sometimes with huge chunks of the problem falling off. I really saw the advantage of having very different minds work on the same problem. I believe my teammates and I, having come from different backgrounds in terms of interests and experiences, approached the problems quite differently, and we were able to really complement each other and bounce off each other’s ideas. All of us contributed in big ways, and together we managed to come out with a closed formula pretty early into the project. Eventually, we managed to link the closed formula to one of the combinatorial objects, and quickly pieced bijections together. Even in the final moments of the project, the group shone through as we all picked on different parts of the project, trying to polish it off as well as we could.

Other students struggled at times, but eventually warmed up to the approach. For instance, a student wrote:

Initially, I felt rather excited about tackling the questions. We made some observations that turned out to be insightful and it felt like the project was going in the right direction. When the team and I got stuck at the later stages of this project, I became frustrated and lost motivation. But my teammates continued to encourage me and kept trying to develop new methods of solving the problems. Through this project, I learned that solving problems is not always a smooth path. It is helpful to acknowledge our frustration and to expect difficulties so that we are less anxious when we are stuck.

The most encouraging feedback I received, however, was from the many (more than 1 in 5 across my sections of Discrete Mathematics) students who explicitly described how empowering the experience was for them in their reflections. For instance, one student wrote:

Throughout this project, I have learnt a lot about how mathematical reasoning happens and [I] have changed a lot of my perceptions about how mathematics is done. Being used to the usual individual problem-solving method in high school, where there is only one right answer and a few preset methods that are best for determining this answer, I have come to love the collaborative approach taken in this project and in the whole Discrete Mathematics course in general. It is truly a vibrant environment for learning, and I am very grateful to have the support and knowledge of my team members. I have always felt my mathematical reasoning skills to be inferior to other upperclassmen or people who reason faster, sharper and more elegantly, but I have come to learn that the final polished product is not all that it appears to be – it is the process that is the most important, and there are lots of things I can contribute within the process while I am working on improving the skills I can use to refine the final solution.

Despite these successes, there is still a lot of room for improvement. A common piece of critical feedback I receive from students is that the reading assignments are very difficult, even with the added support from Perusall. There are a lot of factors at play here such as the choice of text, size of the class, and my (in)ability to effectively respond to the Perusall discussions in real time.

Revising my courses to emphasize reading, research, creativity, and teamwork has been a challenging but rewarding process. I am thankful to the many Yale-NUS students who worked diligently on the projects and offered their thoughtful, constructive feedback, and I am curious to see how my approach extends to other topics besides discrete mathematics and abstract algebra, which may, for one reason or another, be particularly well suited for this type of open-ended, collaborative, team project framework. I hope I have convinced you to try out some of these ideas and I look forward to hearing about the outcomes!

]]>Like many of us, I began teaching online this Spring. Unlike many of us, I began doing so at the start of the semester. I am co-teaching a class at Michigan State, and I live in Nebraska. One of the most useful conversations I had in preparation for this assignment happened in 2013, well before the current coronavirus epidemic. The math department at the University of Nebraska-Lincoln had been considering a synchronous online version of a mathematics course, for rural teachers. I chatted with Ari Nieh, then an instructor for Art of Problem Solving, about what it would take to teach online, especially via chat forum technology. (Ari then became a lecturer in Writing, Rhetoric, and Professional Communication at MIT; and now he is a game designer at Wizards of the Coast.) In the end, that course was run asynchronously (and in many ways consistent with the advice given in a previous post). Nonetheless, much of the advice I received 7 years ago aged well. With Ari’s permission, I share snippets of our conversation in this post, edited for readability, and with commentary from present-day me.

First, here is a summary of the key pieces of advice I took from the conversation.

**Key pieces of advice for teaching online using chat technology**

**Tools for making students feel comfortable in class and that their input is being valued**– careful choice of words to maintain a welcoming tone; consistently responding to questions, whether publicly or privately; and using a fair bit of humor at the beginning of class and during transitions.**Opening a class**– it’s kind of like face-to-face, but word choice is perhaps even more important, because you don’t have tone/body language.**Lectures**– don’t really translate. Make sure to have questions for students. This can take the form of closed questions (e.g., multiple-choice questions or ones where there is one right answer) or open-ended questions.**Whole group discussion**– There are some ways that this is easier, because students can see each others’ thoughts for longer. Also, questions can be answered privately as well as publicly. Giving instructions is perhaps easier than in person, because of persistent text: whatever you say hangs in the air and they keep reading it.**Whole group discussion, continued**– In general, the biggest skill that doesn’t translate is improvisation. You can’t improvise spoken words, so you want to develop skill at improvising written words.**Wrapping up discussions/class**– this is a place where it would be good to have some prepared draft/default text.**Diagrams**– should be prepared ahead of time when possible.

Now for our conversation.

**Yvonne (2013):** So, let’s talk online teaching. The context is this. There’s a class we offer that is 2.5 hours long, that alternates between lecture and working time and discussion, and some time to work on homework problems. We want to translate this online. How should or can this work?

**Ari:** I see. What is the format of the online classroom?

**Y2013:** That is a good question. It hasn’t been determined, but there will be an online chat place for teachers and students to interact.

**Ari:** Our online classroom platform [at Art of Problem Solving] has a setup where everything the students type goes to the teacher first, who has the option of showing it to the whole class or not. In any case, there’s probably some form of moderation?

**Y2013:** Let’s assume that for now.

**Present-day Yvonne (2020):** This kind of moderation is available on Zoom, and potentially on other platforms as well.

**Ari:** Right. Working on shorter-length problems in class works fine online.

Lecturing is actually the hard part. The reason it’s difficult is that it’s much less interactive. If the instructor is just typing stuff which gradually appears on the screen, there’s not much incentive for the student to pay attention instead of deciding, “I’ll just read the whole transcript later.” So lectures must be liberally sprinkled with questions to evaluate comprehension or points for discussion by students.

What’s the topic of the class?

**Y2013:** Geometry from a transformation perspective.

Which brings up another question: How do you handle discussions about diagrams?

**Ari:** We have prepared diagrams for geometry classes.

But if the students want to do something that we haven’t prepared in advance, it doesn’t work too well. You want some sort of interactive blackboard thing for that, I would think.

**Y2013:** Hmm … okay. I’m thinking about a question that we often open the course with: *Given two rectangles in the plane, show that there is always a line that bisects both rectangles simultaneously.*

Should I maybe look for a separate program that students can be logged onto at the same time to draw?

**Ari:** Possibly. Prepared diagrams actually work most of the time. If your chat system supports them sending images, that might be good, too.

But it’s also okay for them to express some idea in words, and then you provide the diagrams which demonstrate it. For instance, suppose some student says, “The lines that bisect a rectangle all have to go through the middle.”

You say, “You mean, like this?” (DIAGRAM)

It’s partly a question of whether they’ll be able to make good diagrams on the fly, which students may or may not be able to.

**Y2020:** In 2013, virtual shared drawing spaces didn’t exist the way they do now, and certainly webcam technology wasn’t as prevalent. All that said, there’s still something nice about being able to ask students to articulate, in words, what they are imagining. There’s an entire literature on how connecting diagrams with the logical constraints behind them may be key to learning geometry (e.g., Duval, 2006; Fischbein, 1993; Jones, 1998; Mesquita, 1998; Presmeg, 2007). In the class that I’m co-teaching currently, which uses Zoom, I swap between making diagrams based on what students are saying and asking students to make drawings and shove them up against the webcam. I find both useful, and the latter especially useful for getting a read on the class as a whole.

**Y2013:** How much have you found that your face-to-face teaching skills translate to online teaching? For instance, we’ve been talking about diagrams and discussions; what about building rapport with students on the first day and throughout, or how to give instructions for what would otherwise have been a handout or slide?

**Ari:** Good question!

So, obviously one can’t build rapport via eye contact, body language, tone of voice, etc. Some tools for making students feel comfortable in class and that their input is being valued include: Careful choice of words to maintain a welcoming tone. Consistently responding to questions, whether publicly or privately. And using a fair bit of humor at the beginning of class and during transitions.

On answering questions. When students ask a question, the teacher gets the question ‘privately’. The teacher can then choose to pass the question to the whole class, or answer it privately. This allows the class to proceed without those questions, which might distract. At the same time, it allows student to get answers to individual questions.

Giving instructions is perhaps easier than in person, because of persistent text: whatever you say hangs in the air and they keep reading it.

In general, the biggest skill that doesn’t translate is improvisation. Instead of improvising spoken words, you need to get good at improvising written words very quickly.

Of course, one can always stick to a script, but it’s nice to have the option of exploring tangents, finding teachable moments within alternate solutions or mistakes, etc.

That said, for lecturing, I strongly recommend having the remarks prepared and using some amount of copy-pasting rather than writing them on the fly.

**Y2013:** Wrapping up discussions and summarizing key points – are those also places where generating some draft or default text ahead of time would be a good idea?

**Ari:** Yes, definitely. Teaching online can be pretty tiring because you have to produce cogent text on the fly. Delegating some of it to your past self is a good idea.

**Y2020:** As one of my friends recently commented, “How is it that talking at students for 20 minutes on Zoom is more tiring than jumping around a classroom for 2 hours?!”

I have also found online teaching to be similarly or more tiring than physical teaching because I’m limited to primarily one mode of interaction: words. Even though there is some facial interaction on Zoom, I haven’t found it to work with my intuition in quite the same way.

**Ari:** One other thing I should mention about your class plan: I’m not sure that giving students in-class homework time will work. When you do that in person, you can circulate and watch them and there’s lots of direct social pressure to actually work during that time. But if an instructor did that to me online, I feel like I’d probably get up and eat. This is a gut feeling and not based on any particular experience.

**Y2020:** This is perhaps the hardest thing for any instructor seasoned in discussion-based teaching: that in-person social contracts differ from online ones! Social pressure works in different ways, and circulating also feels different. When I teach in person, I often can make in-the-moment changes to my discussion plans based on what I am seeing from students, and how they respond to my questions just to their group. Although I can do this in breakout rooms, it requires a different kind of concentration than visually scanning the room and making connections between different students’ work. As well, in-person, I often send delegates from one group to meet with another group to compare strategies. This doesn’t translate as easily or well, especially as one function of this technique is to give students a physical break by having them walk instead of sitting.

**Y2013:** Thanks so much. I think these are all the questions I have for now.

**Ari:** You’re welcome.

**Y2020:** Reading this over again now, there are some parts that seem charmingly quaint, such as our implicit question about whether there might even exist accessible platforms where people can simultaneously share images. But there are other pieces that ring true: that carefully chosen words are key to helping students feel welcomed. That teaching online can be tiring in its own way, and one way to mitigate that is to have some prepared text. And also that teaching online, though different from physical teaching, can have its perks. A great affordance of online teaching, that you don’t necessarily get so much in physical teaching, is the capacity for students to ask questions privately, and for you and the student to choose between private and public answering. And personally, I’ve found that having students work collaboratively on google docs is far easier than using a document camera to show multiple solutions.

Because I have a toddler at home, I spend a lot of time looking at her shape sorter, and how she will still sometimes put the square peg in the round hole (and it doesn’t fit). On the other hand, the square peg does fit in the octagonal hole, with appropriate rotation (at least in her set). To some degree, I think my difficulties in online teaching are in part due to fitting a square peg of physical teaching into a round role of online teaching. The times that it’s gone best are when I see online teaching not as a lesser version of physical teaching, but as its own kind of teaching with its own special opportunities.

]]>