Comments on: Designing for Mathematical Surprise

By: Sean

Sean — Tue, 10 Apr 2018 11:09:48 +0000

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.

By: Ben Blum-Smith

Ben Blum-Smith — Sat, 07 Apr 2018 05:50:47 +0000

(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?

By: Dianne

Dianne — Fri, 06 Apr 2018 13:04:48 +0000

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.

By: Scott smartt

Scott smartt — Thu, 05 Apr 2018 03:21:41 +0000

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.

By: Dan Meyer

Dan Meyer — Tue, 03 Apr 2018 02:59:23 +0000

In reply to Scott Farrand. That's a peach, Scott. Love how the design anticipates students who are anticipating a surprise and subverts their expectations.

By: Sharing Diigo Links and Resources (weekly) | Another EducatorAl Blog

Sharing Diigo Links and Resources (weekly) | Another EducatorAl Blog — Mon, 02 Apr 2018 03:34:29 +0000

[…] Designing for Mathematical Surprise — dy/dan […]

By: Dianne

Dianne — Sun, 01 Apr 2018 17:57:49 +0000

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.

By: Berkeley Everett

Berkeley Everett — Sun, 01 Apr 2018 15:40:13 +0000

In reply to Berkeley Everett.

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! https://twitter.com/BerkeleyEverett/status/980467114954895360

By: William Carey

William Carey — Sun, 01 Apr 2018 01:05:23 +0000

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.

By: Scott Farrand

Scott Farrand — Sun, 01 Apr 2018 00:48:11 +0000

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.