Mathematical Surprise

I gave a talk at the Wisconsin state math conference earlier this month and this woman was the best part.

I don’t know her name. I’ll call her Jan. Jan is about to testify to the power of surprise.

I asked the crowd to give me three numbers between 1 and 6, numbers you might get from a roll of the dice. They said 2, 3, and 5. Then I asked all of them to evaluate those numbers in this expression.

Most of the crowd started working on that task, but Jan didn’t. She laughed and said, “I teach second grade,” excusing herself.

I encouraged her to show off whatever she remembered from the last time she worked with expressions like this. She scribbled on the notebook in her lap and we managed to evaluate x = 2 in the time we had, but not 3 or 5.

I asked the crowd to call out the result for 2, 3, and 5. They called out 2, 6, and 20, one after the other.

Then I asked the crowd to evaluate those same three numbers in this expression.

Jan tossed her notepad on the desk, a reaction of “no way, no thank you” to the length of that expression. I decided not to press her at that exact moment, because I had a secret everyone in the crowd would come to understand at different times, Jan last of all and perhaps best of all.

I asked for their result for 2.

“0.”

“Okay, what about 3?”

“0.”

“Okay, that’s weird. What about 5?”

“0.”

I played up my surprise, acting like I didn’t know all of those terms would simplify to 0.

That’s when I noticed Jan. Out of the corner of my eye, Jan straighted up in her chair and then picked up her notebook to sort out what just happened.

I wish I had a sharper vocabulary to describe this transformation, as well as more strategies for provoking it. By showing Jan a situation where order arose from apparent disorder, she felt something in the neighborhood of … cognitive conflict? Intellectual need? “Surprise” feels closest.

I don’t know all the words and I don’t know all the strategies, but I know there are few gifts a teacher can give a student more satisfying than helping her transform from “no way, no thank you” to “okay, let’s sort this out.”

Discuss:

• I don’t think this experience has much to do with Jan’s growth mindset about herself, or mine about her, but I’m willing to be proven wrong. How was this experience distinct (or similar) to a mindset experience?
• Think about the design of this activity, all of its different permutations, and how each one might have affected Jan. What if, for instance, I had given given the class those three numbers instead of soliciting them from the class? What if I had only solicited one number? What if all three numbers didn’t evaluate to the same number? How would these permutations have affected Jan’s interest in picking up her notebook?
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I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. More here.

28 Comments

1. Paul Hartzer

May 18, 2017 - 1:19 pm -
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Fixed this.

At the risk of being pedantic, 0! is unfortunately ambiguous. It took me a second to verify that you mean zero and not one.
2. Paul Hartzer

May 18, 2017 - 1:23 pm -
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To the point, though, I’m curious: Did some of the participants simplify the longer expression before evaluating it? Of those (other than Jan) who evaluated it first, did some of them then simplify it to see what was happening?

My first response would be to simplify, so my sense of wonder wouldn’t have lasted long on this one. :D

• Dan Meyer

May 18, 2017 - 1:26 pm -

Some participants evaluated without first simplifying. Other participants simplified right away.

It seems surprise, like a lot of aspects of learning, depends on what you already know.

• Mr Ruppel

May 19, 2017 - 3:38 am -

I’ve used a pretty similar task to get kids thinking about equivalent functions and polynomials. Many of my kids “simplified” despite that not being in the directions.

I actually think this created much more powerful discussion after the fact…the kids who were done in about 10 seconds were better prophets for my idea about simplifying expressions than I ever could have been.

3. Paul Hartzer

May 18, 2017 - 1:42 pm -
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I just finished reading Jo Boaler’s chapter on gender in “What’s Math Got to Do With It?”, and it occurs to me that this fits right along with that. To summarize (even if you’re familiar with it, your readers may not be): Boaler’s work suggests that females tend to prefer to know WHY and HOW mathematics works, while males tend to prefer to know mechanical processes. This is, Boaler posits, a significant reason for the gender disparity on STEM at the public school level: Traditional mathematics is about learning rules and algorithms, many of which are doled out without explanation.

Today, the new topic in my geometry class was the volume of a sphere. Yesterday, I had given the formula at the end of class, expecting just to do exercises on the volumes of cylinders, cones, and spheres today. I was inspired by that chapter, and while I’m not in the position to radically change curriculum at this point in the school year, I did decide to at least explain where that formula came from, using the classical Greek method (using the area of circular cross-sections). Because I said they weren’t being tested on it, the boys pretty much ignored my explanation while most of the girls watched attentively, and a few even took notes.

At any rate, I wonder if Jan didn’t see the point in evaluating that expression as a mechanical process (I don’t blame her, neither do I). Once it became a matter of “why?”, though, her interest was piqued.

• Chester Draws

May 18, 2017 - 10:53 pm -

Traditional mathematics is about learning rules and algorithms, many of which are doled out without explanation.

I was taught very traditionally. I was never taught without explanation. Picking up old textbooks, you can see that this is a straw man attack on traditional education. They are, if anything, more full of explicit explanation — given that they aren’t big on discovery. My own teaching style is quite traditional, and gives a high premium to why a technique works — I basically never teach a technique without some amount of explanation.

We might argue that traditional mathematics leaves the explanation to the teacher, whereas a progressive might want a student to discover it for themselves. Or we might disagree when in the process of teaching that explanations are important — I personally like to deliver them at the end of a topic rather than the start.

But can we please stop pretending that traditional maths teaching never thought explanations were important. If anything it tended to be rather too focused on esoteric proofs for my preference — giving me a proof of differentiation when I was 16 was too much, too soon.

• Paul Hartzer

May 19, 2017 - 3:14 am -

Chester, here’s a typical proof that the tangent line is perpendicular to a circle in a traditional textbook:

Take line PQ tangent to circle O at P. Assume that PQ is not perpendicular to PO. Then there must be some line RO that is perpendicular to PQ, with R on PQ. There must also be some point S on PQ such that SR = RP. By SAS, since segment RO = RO, SR = RP, and angle ORS = ORP, triangles ORS and ORP are congruent, so OP and OS are congruent. But since OP is a radius of O, so is OS, and PQ touches O twice. This is contradiction, so PQ is perpendicular to PO. QED.

Is this an explanation? in a very dry way, sure. You yourself refer to “esoteric proofs”. It’s a boring explanation, and it’s a traditional explanation.

What I don’t think it is is a “why this works”. It’s actually fairly circular: It ultimately relies on the notion that, if the tangent line isn’t perpendicular, there would be a second point on the same line that’s the same distance as the alleged point of tangency. We could start with chords first, and then show how, as a line moves away from the circle, there is some distance at which it touches exactly once. That’s got to be perpendicular (no proof needed) because that’s a distance, and distances are always measured with a perpendicular (in Euclidean space). But the textbooks I’ve seen, both modern and older, discuss tangents before chords. Is that universal? Probably not. The book closest to me on the shelf right now (Kusch’s Mathematik fuer Schule und Beruf, 1952) defines chords before tangents. But it is fairly common. (I’m not saying proofs aren’t needed, I’m saying that proofs aren’t evidence that “why” is being taught.)

• Dan Meyer

May 19, 2017 - 6:10 am -

A suggestion. Divest yourselves of emotional or intellectual attachment to terms like “traditional” and “progressive.” You’re using the same words but talking about different things anyway. instead, describe the practice. You both seem to agree that we shouldn’t ask students to follow rules and procedures without an explanation. Perhaps start there.

If anything it tended to be rather too focused on esoteric proofs for my preference — giving me a proof of differentiation when I was 16 was too much, too soon.

I find Chester’s comment interesting. He may have received a deductive and mathematically correct explanation for the limit definition of the derivative but it wasn’t right for him. What makes an explanation productive? Interesting? What are good preconditions for explanation? I’m trying to offer one here.

• Chester Draws

May 19, 2017 - 9:34 pm -

I explain why a tangent is at 90 degrees to a radius exactly as Paul describes. That was how it was explained to me, incidentally, 40 years ago. Good teaching is timeless.

Where I part company from many teachers is that, for me, a good explanation is always explained by the teacher. So having played that trick where the expression always equates to zero, and I have done similar in classes, I would explain very explicitly why it worked. I would not leave loose ends, where some students didn’t know why it worked and others thought they knew but were wrong.

Where I also part company from many here, is that I do not insist that they understand my explanations. I want them too, obviously, but I don’t labour it until they insist they get it so we can move on. I feel that understanding comes from a slow process of explanation, practice, more explanation, more practice. My explanations are explicit, but also brief. More than five minutes and they will have all lost the plot anyway.

I’m suspicious of “Aha!” moments. I’ve seen students have them, only to not be able to recall what they supposedly understood so well the previous week. I’ve seen students have them, only to have actually learned the wrong thing. Real understanding takes time to seep in — even if it seems to come in a flash, it needs repeated exposure to actually stick.

• Paul Hartzer

May 20, 2017 - 5:09 am -

Chester, I sometimes explicitly say something like, “I don’t expect you to understand this particular explanation right now. It’s based on calculus, which you’ll be formally meeting in a later class. But I’m offering it so that you know that it does work, that there’s reasoning behind these mechanisms.”

I think you and I are probably closer to being on the same page then we’re acting. I had some excellent teachers in my day that explained mechanisms, and I had some teachers who just taught the mechanisms.

4. John Golden

May 18, 2017 - 1:52 pm -
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It’s a nice bit of teacher misdirection to make it less likely to simplify first the way Paul would have. (Plus you didn’t say simplify, you said evaluate, which everyone knows means plug in.)

Without the audience offering the numbers or the same result there would definitely be less room for curiosity. And maybe it needed to be zero. Would anything else have that feeling of significance? Those two things piqued my interest even when just skimming.

Jan was at a math conference and chose a Dan Meyer talk, so maybe she’s prone to engage already, but I’m wondering if she would have the 2nd time if you hadn’t solicited the first try.

5. Alison Childers

May 18, 2017 - 7:19 pm -
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Wow this feels like it comes directly out of the #elemmathchat I fell into on twitter where they were taking graphs with blank axes, labels, etc. and revealing information one-by-one. The math/numbers were a relief at the end because you had built them up so much. You wanted to analyze because info had been withheld and it was physically painful not to know. How can we do more of this?!? #goals

6. danny

May 18, 2017 - 9:49 pm -
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As much as gender plays a role, it only plays a role to the extant that we all have individual differences.

This experience was very ripe for a mindset experience. Empathy and human concern was expressed (caring adult noticed her and offered encouragement). Control was given (Jan threw away her notebook), in other words, she had a choice to continue or not. Individual thinking was allowed (Dan allowed the curtain to be pulled back slowly). The self concept of everyone in the room was maintained, and perhaps grew. Courtesy of @loveandlogic

The last two years I discovered the secret to learning is not in pedagogy (though it helps – color of the white house?), but rather psychology. Though, it is a humbling discovery because one realizes how insignificant is the work of my life. :)

• Dan Meyer

May 19, 2017 - 6:15 am -

Thanks for the comment, Danny. I’m sure both aspects – pedagogy and psychology – played a part. But I can’t undervalue the former.

“I believe you can learn this,” I say, endowed of a growth mindset about my pupils.

“I believe I can learn this too,” she says, endowed of the same mindset about herself. “I just don’t want to. There’s zero upside here.”

That’s where, in my view, the psychology has to yield to the pedagogy and, in particular, the curriculum design.

• danny

May 20, 2017 - 12:02 pm -

You’re right.

We can’t under-value either, because they are not mutually exclusive. But let us assume for argument’s sake that they are. Would a teacher be better off devoid of sound pedagogy and yet solid psychology, or vice versa?

I could take a group to the moon without pedagogy, but would sink without psychology.

7. Meredith Thompson

May 19, 2017 - 7:55 am -
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Sounds like you didn’t give Jan the right headache, Dan! ;)
This story brings up a common theme in thinking about our role as educators – whether it’s math or science or history or English. Teaching is a huge challenge, and there are tons of stakeholders who each cast a shadow on what and how things are taught: standards, administration, parents, the norms established by colleagues, ed tech companies, curriculum designers and suppliers, and so on. I

In the midst of all of this, the critical link between motivation and learning sometimes gets lost. Even with the best of intentions, sometimes what we think will motivate students works for some but not all – sometimes students are motivated by understanding the outcome (as Jan was) rather than at the outset. Whenever and however we are able to do it, finding ways to motivate learners is a critical part of being a teacher.

8. Sue Thuma

May 19, 2017 - 8:23 am -
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I love this example. Each student (and teacher for that matter) comes to mathematics from his or her own perspective.

Sharp analysis!

You have three groups there: evaluate without simplification, simplify first then evaluate, and the “this isn’t worth my time” group. Each group brings something to the table and the fact that you hooked the last group is where you want to always be. The second group has a surprise for the first and the third.

Everyone doesn’t need to be surprised every time. Sometimes it is good to let the students do the surprising. Thanks for the example!

9. AL

May 19, 2017 - 12:38 pm -
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I enjoyed reading the dialogues. I think Jan has a subconscious of fixed mindset towards math regardless of her age. She never felt or was exposed to excel in math. From children to adults, we are all human being and display the distinct or similar behavior pattern, reaction to the certain matters. Math is one of these matters.
The Research shows that subconscious thought does affect our behavior. The subconscious mind can be a hiding place for anxiety (Jan’s math anxiety), a source of creativity, and often the reason behind our own mysterious behavior.
Dan encourages Jan to try beyond her comfort zone.
She probably felt more confident to attempt the second half of the question with Dan’s positive contribution to her will and ease her anxiety.

10. Koen Vanhoutte

May 20, 2017 - 12:44 am -
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More of a question, rather than a comment:
Something markedly changed in your description of Jan’s behavior between first contact with the long phrase and the discovery that all of them equal 0, no matter the input.

I find it hard to believe that what changed is the belief in her capacity to solve this problem.

Love this line.

Rather, what seemed to have been a surprise to Jan is that she was – against everything she expected – interested.

It seems she did not necessarily overcome any anxiety or aversion, but rather temporarily disregarded it in pursuit of something worthwhile. This creates a portal, allowing you or her a crack at building a growth mindset, ability, etc.

I think pedagogy is the science of generating and inviting as many such moments as possible.

11. Scott Farrand

May 20, 2017 - 12:58 pm -
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I am not at all sure that a single experience with a surprise can have a significant effect on having a growth mindset. But I do believe that thoughtful repeated use of surprises can change what students believe about what they can do with mathematics.

Perhaps the most inviting reason to use surprises is to get students to pursue the question you want them to consider. Everyone enjoys a puzzle, IF they think that they can figure it out and IF they then are successful. People like to feel clever. And feeling clever builds mathematical confidence.
It isn’t so easy to find problems that create useful surprises, but the more you use them, the more students respond. I want students to gradually build their belief that they CAN figure out whatever weird thing they encounter in my classroom. There’s a crucial unspoken agreement here that I learned from Rick West – I won’t give them a puzzle that they don’t have the means to solve. I want students to progress to the point where they see the surprise and then say something like, “Don’t tell me, I want to figure this out.” You can watch individual students progress in their mathematical confidence in the way that they respond to surprises. I do think that this is related to having a growth mindset.

Jan might be someone whose prior experience with puzzles has provided her with some confidence, some belief that this surprise was worth pursuing, even though it was outside of her area of subject matter confidence. The set-up to the problem might have worried her that the talk had prerequisites that she didn’t meet, but the puzzle drew her in.

One more thought here: I would like to try the same set-up to the problem (perhaps they should choose a whole number between 0 and 4) and go through the same process, except have the students plug their chosen numbers into the polynomial that is the expanded form of
x(x-1)(x-2)(x-3)(x-4).
For students at just the right place in their understanding of polynomials, this could lead nicely to an investigation of the relationship between zeroes and linear factors of polynomials.

12. Vince Hoover

May 22, 2017 - 8:46 am -
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Dan,
I appreciate your story about creating an intellectual need in students. This is desperately needed to create students who have a healthy disposition for math and for learning. Often we kill their curiosity to math by not creating a need first. I experienced this lesson just like Jan did when I saw you at NCTM last year.

Do you have recommendations for sharing this idea of intellectual need with more teachers? I provide professional development for area schools and would love to share this message with teachers who do not attend conferences or don’t read your blog. Do you have examples of teachers creating intellectual need so that I can inspire more teachers?

• Dan Meyer

May 24, 2017 - 10:22 pm -

Hi Vince, thanks for the note. I’d love to help out with your PD efforts. Frankly the only work I’ve ever done that seems to resonate with teachers has involved the same three-step process every time: a) do weird math together, b) debrief the math to decide what made it weird, c) decide what we’d need in order to start being weird in our own practice.

• Pam Harris

May 25, 2017 - 6:50 am -

Yowza. Solid PD outline.

I add to/tweak Dan’s three steps in my PD a) do intriguing math together,
b) congress about the intriguing math toward a mathematical end to grow teachers’ content
c) ask teachers to predict what students would do with the task (They can’t do this! We just struggled/had more prior knowledge/like math and we struggled!),
d) watch students do the same task with an experience teacher mentoring student mathematicians,
e) debrief the experience,
f) ask participants to predict what the video teacher should/could do NEXT to continue building the student mathematicians (ie. one surprise does not change everything nor teach all the math)
g) then offer a possible next move,
h) debrief – how to sequence multiple entry-multiple exit tasks so that all students continue to learn, are stretched, are intrigued (at least enough).
Rinse and repeat.

For me – it’s about sequencing tasks to mentor student mathematicians (it’s not about discovering, it’s about building mental relationships they can use to solve problems) and it’s about sequencing PD tasks to mentor teachers.
Dan’s sequence in this example is worth studying.

13. rene

May 29, 2017 - 1:26 pm -
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Couple of thoughts:
1) Yes, we don’t always know the level of student understanding…of course you couldn’t know Jan’s, but in our classrooms I am afraid sometimes I might have under or overestimated individual students’ ability.
2) Developing curiosity is another word I’d use here, adding to yours. It’s really combo.
3) Both of the above fit within Dynamic Skill Theory and Perceptual Control Theory.
4) Teachers can’t leave out the rest of the room to focus on the Jans, and they can’t leave out the Jans to focus on the rest of the room. Letting her “not” engage for that very brief bit of time was helpful. In PK-12, teachers monitor that BRIEF disengagement to ensure students are “still there” – students don’t HAVE to be “visibly engaged” to actually BE engaged. LOTS of times I have sat in classes and thought, “Yeah, I don’t get any of this!” but I am still interested and still trying to hook onto something. I doubt very seriously that you would have left her in that disengaged state. Sometimes we don’t actually know what to do in any given moment, that’s where wait time comes in. (And checking back in later!)
5.Growth Mindset? Well, I can’t read your mind, but I think you have a growth mindset. You believe people are capable of succeeding in math, given the right environment, scaffolding (yes, I used that word), and headache. It’s different for teachers who just dismiss the Jans and figure, yeah, she’s out of there, I’ll just focus on the others. As for Jan…we can’t read her mind, but her behavior tells me she has a growth mindset because she continued to listen (in silence) even when her physical behaviors showed something different. If she had started texting, got up and left…gave up…that tells you something. It MAY tell you she doesn’t have a growth mindset, it might also tell you that she had a really bad night’s sleep, didn’t get her coffee, her dog chewed her favorite boots and she is more concerned about that than this little math exercise and so her interest AT THAT MOMENT isn’t enough to try to play along. Though, given her comments about 2nd grade, I’d say she has a lack of confidence. Lack of confidence is not the same thing as not having a growth mindset. GM isn’t that simple (as psychology isn’t that simple); not binary, and it’s highly contextual. Each small success builds on another. Growth Mindset is really tackling cognitive distortions, and being realistic. I can say and even act upon the belief that I have the ability to do “x” – but if I don’t have the skills necessary, I’m going to need those first! AND, I have to want to do it. I love to ski. I “own” the bunny hill and greens on several ski areas in NM and CO. I COULD ski blues and blacks IF I WANTED TO. I’d have to exercise to build leg strength, I’d have to get over my fear of heights, I’d have to feel unsatisfied with my current ability…I’m not willing or interested in doing any of that. I am happy where I am. But, I don’t face negative life outcomes because I give in. Understanding math has implications for life outcomes. It’s our job to monitor and support the growth mindset of our students. That’s WAY more than a cute poster on the wall or verbal encouragement. It’s giving them the chance to experience growth so they know what that feels like! The dopamine release will provide a great environment to seek out more like experiences! (And, for the record, I’d say you just experienced a bit of your own medicine…now your interested in what happened to Jan. You did weird math, debriefed, and now your thinking about what you need to do weird math together (in different PD environments) next time. Pretty much, did you ever have a day in teaching where you did’t learn something? I didn’t. So, that last one…you aren’t ever going to know for sure because you won’t know your audience. Adult learners are just like kid learners, just older. What if Jan hadn’t been there that day, how would you have evaluated that particular PD? What would you have done similar or different next time?