Designing for Mathematical Surprise

“Surprising” probably isn’t in the top ten list of adjectives students would use to describe math class, which is too bad since surprise lends itself to learning.

Surprise occurs when the world reveals itself as more orderly or disorderly than we expected. When we’re surprised, we relax assumptions about the world we previously held tightly. When we’re surprised, we’re interested in resolving the difference between our expectations and reality.

In short, when we’re surprised we’re ready to learn.

We can design for surprise too, increasing the likelihood students experience that readiness for learning. But the Intermediate Value Theorem does not, at first glance, look like a likely site for mathematical surprise. I mean read it:

If a continuous function, f, with an interval, [a, b], as its domain, takes values f(a) and f(b) at each end of the interval, then it also takes any value between f(a) and f(b) at some point within the interval.

[I slam several nails through the door and the floor so you’re stuck here with me for a second.]

Nitsa Movshovits-Hadar argues in a fantastic essay that “every mathematics theorem is surprising.” She continues, “If the claim stated in the theorem were trivial it would be of no interest to establish it.”

What surprised Cauchy so much that he figured he should take a minute to write the Intermediate Value Theorem down? How can we excavate that moment of surprise from the antiseptic language of the theorem? Check out our activity and watch how it takes that formal mathematical language and converts it to a moment of surprise.

We ask students, which of these circles must cross the horizontal axis? Which of them might cross the horizontal axis? Which of them must not cross the horizontal axis?

They formulate and defend their conjectures and then we invite them to inspect the graph.

In the next round, we throw them their first surprise: functions are fickle. Do not trust them.

And then finally we throw them the surprise that led Cauchy to establish the theorem:

But you can’t expect me to spoil it. Check it out, and then let us know in the comments how you’ve integrated surprise into your own classrooms.

Related: Recipes for Surprising Mathematics

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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.


  1. Reply

    I love this activity! Can’t tell you how much I love it!

    I LOVE the way the Desmos team use a straight line, which is continuous, but f(a)=f(b)=c

    I LOVE the last example with discontinuous functions!!! I know something will happen:)

    I like it a lot!! I will also introduce it with y=x when I use it in my class. :)
    I really really like it :DDDD

  2. Reply

    Great activity – thoughtfully sequenced. Thank you for sharing this. I think that the surprise you refer to is often the same reason that we appreciate great visual art. When you look at art, there are often layers that reveal themselves and surprise you. What appeared to be simple at first might actually have a depth and a novelty that we didn’t expect, and it’s this moment of joy at a violation of our expectation that sticks with us and makes us keep thinking about it. As a teacher, choreographing this moment, and observing this joyful lightbulb happen in students is one of the greatest gifts we can give to them. Mathematicians naturally understand this moment because we have had the chance to experience this joy. Most students’ opportunities for this experience are pretty limited by the idea that mathematics is a series of procedures that students need to be told how to do.

  3. Reply

    This is one of my favorite activities with Calculus students… Especially those that think they already understand and get surprised!

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    I think simple surprises in my classroom can be powerful. Graphing two functions in different forms and see they are the same. Graph a derivative and have it match a function we already know. Use any rule and get the same answer from a different method.

    The best surprises are those that leave the students asking “but, why?” at the end.

  4. Reply

    Thank you for sharing Dan!

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    I teach 1st grade and find that a well constructed number string is full of surprises, like, “How did Samaya solve that second problem mentally?!?” And then, “OH! It’s like the same thing as the first one!!”
    • My students are fluent with making 10 with single digit numbers. This week I will show 7+8 and get all of their solutions. Then I will show a visual I created showing starting at 7, adding 2 to make 10, then 5 more to make 15. Then they will talk about how 7+8 could be similar or different to 27+8, and the surprises begin! Kids will be surprised that make 10 doesn’t just work for single digit numbers, that they can already solve 27+8 mentally, that the reason it works is because of place value (which they might not have realized was related to solving add/subtract yet), etc. Here is the visual I’ll be using – if anyone has feedback on how to make this task better I’d appreciate it!

  5. Scott Farrand

    March 31, 2018 - 4:48 pm -

    Great topic!! I love using surprises in my lessons, and not just because it is fun.

    I do a lot of re-teaching of mathematics, both reteaching of material from previous courses, and reteaching of students who are repeating a course.

    Strong argument for WTFBL.

    Students tend to stay in the same rut if I teach the material the same way they saw it before, and nothing seems to move them to a different perspective like a surprise.

    It also helps me when I am planning lessons to focus my thinking on identifying for myself what is surprising in the math in the lesson, and planning the lesson to dramatize it so as to get as much surprise as I can. If I don’t think about the right things when I am planning, my lessons suffer. There are some big surprises to be found, but I also enjoy the search for the small ones. Surprises are good for me, as a teacher, because of how they makes me think about my lessons.

    Love this surprise.

    Here’s a big surprise, about exponential functions, that I learned from a colleague many years ago. I tell my class to imagine that 200 years ago, a distant grandparent of theirs had the foresight to put $1 into an account for them, earning 1% interest. I ask them to guess how much that account would now have in it. Go ahead, guess! They are quite surprised when I tell them that there is $7.32 in that account – they all had guessed much more than that. When the dust settles, I tell them that another distant grandparent of theirs also put $1 into an account for them, 200 years ago, but this one has earned 10% interest. Go ahead, guess! Most of my students over the years have guessed $73.20, which is great because I am trying to highlight a non-linear relationship. So when I tell them that the second account has about $190 million in it, they are totally surprised. I’ve done this with hundreds of students and never had a student underestimate with their first guess or overestimate with their second. This works like a charm. And their appetites for understanding exponential behavior are whetted.
    • That’s a peach, Scott. Love how the design anticipates students who are anticipating a surprise and subverts their expectations.

  6. William Carey

    March 31, 2018 - 5:05 pm -

    It’s interesting to me that you specify that the function is continuous for the first three, but not for the last function. The surprise is something of a magic trick here, a misdirection driven by an expectation you’ve created in the first three that the function will be continuous. I wonder whether we could think about two kinds of surprise: those arising from assumptions we’ve incorrectly conditioned into the universe of the problem, and those arising from genuinely counterintuitive bits of mathematics.

    For example, Calkin and Wilf’s lovely paper on recounting the rationals (or even Cantor’s diagonal method) is, I think, the former. Students make assumptions about what it means to count infinitely many numbers, and those assumptions are wrong in a surprising way. Perhaps the definition of the zeroth power is the latter?

    Certainly that sort of mathematical magic is powerful in the classroom – one of my favorites is to introduce logarithms by putting up a few values on the board and then make use of the arithmetic properties to compute more in my head. Not know how I’m doing that makes the students batty.

  7. Reply

    Really like Carey’s categorization of surprises and love the Cantor diagonal method example – that STILL surprises me.
    Examples for middle school for me include:
    *Add 10% to 100 and get 110, but subtract 10% from 110 and you do not get 100
    *Negative exponents mean divide
    *Getting rid of radicals in denominators simply involves multiplying by a special form of 1
    *Finding out how different values for a and b affect the graph of a line in ax + b = y.
    *Finding out how different values for a, b, and c affect the graph of ax^2+bx+c=y
    *Finding out how different values of a, h, and k affect the graphs of a|x-h| + k = y and a(x-h)^2 + k = y and y = a*b^(x-h) + k for positive values of base b.

  8. Scott smartt

    April 4, 2018 - 7:21 pm -

    I had just come up with the observation that “most of the concepts of geometry are obvious to students (once they see them) and that most of the challenge comes from synthesizing multiple concepts within a problem. But now I have to seriously consider how I can surprise students in geometry. Thanks, Dan, you always throw a change up. I will be back with surprises.

  9. Reply

    One last example – that some infinite sums converge. I’ve explored this idea with Middle Schoolers using images related to proofs without words. 1/2 + 1/4 + 1/8 + …. converging to 1 can be shown dividing a box in half successively. And there is a great graphic on the web showing 1/4 + 1/16 + 1/64 + … converges to 1/3 using an equilateral triangle. Thanks again for such a great prompt.

  10. Reply

    (a) This is awesome. (b) I generated surprise and suspense around the IVT in a completely different way today, with a different goal. The context was a Real Analysis course so I wanted to motivate not the statement of the IVT but its proof. I had students work with a function defined only on the rationals, by f(x) = -1 when x^2 > 2 and f(x) = 1 when x^2 < 2. We applied the definition of continuity to this function and found that it was actually continuous at every point in its domain. We thus had a function with a "jump" that managed to be continuous anyway. How to account for this?

  11. Reply

    I was really surprised by that last graph. This sequence was so interesting, and such a weird and fun way to bring that theorem to life. I was sure of what you were going for – sure that as a math teacher I understood the pattern and rhythm of the sequence and lesson.

    But then the reveal!

    You preyed on my overconfidence. I can imagine this working very very very very well in a classroom. Thanks to the team.

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