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The Difference Between Math and Modeling with Math in Five Seconds

Jim Pardun sent me a video of a dog named Twinkie popping balloons in the pursuit of a world record. How you train a dog to do this, I don’t know. How there is a world record for this, I don’t know either.

What I know is that this video clearly illustrates the difference between math and modeling with math.

You can’t break math. Some people think they broke math but all they did was break ground on new disciplines in math where, for example, triangles can have more than 180° and parallel lines can meet.

Our mathematical models, by contrast, arrive broken. “All models are wrong,” said George Box, “but some are useful.” And we see that in this video.

Twinkie pops 25 balloons in 5 seconds. How long will it take her to pop all 100 balloons? A purely mathematical answer is 20 seconds. That’s straightforward proportional reasoning.

But mathematical modeling is less than straightforward. It requires the re-interpretation of that answer through the world’s imperfections. The student who can quickly and confidently calculate 20 seconds may even be worse off here than the student who patiently thinks about how the supply of balloons is dwindling, adds time, and arrives at the actual answer of 37 seconds.

Feel free to show your classes that question video, discuss, and then show them the answer video. Or if your class has access to devices, you can assign this Desmos activity, where we’ll invite them to sketch what they think happens over time as well.

The difference between the students who answer “20 seconds” and “37 seconds” is the same difference between the students who draw Sketch 1 and Sketch 2.

You might think you know how your students will sort into those two groups, but I hope you’ll be surprised.

That difference is the patience that modeling with math requires.

BTW. I’m very interested in situations like these where the world subverts what seems like a straightforward application of a mathematical model.

One more example is the story of St. Matthew Island, which dumps the expectations of pure mathematics on its head at least twice.

Do you have any to trade?

The Bureau of Non-White Dude Math Education Keynote Speakers

At a workshop in New York City yesterday, I was complimented in the morning for my In-N-Out Burger activity (which was actually created by Robert Kaplinsky) and in the afternoon for my File Cabinet activity (which was actually created by Andrew Stadel). This mix-up will come as no surprise to either Andrew or Robert or anyone who has ever seen the three of us presenting at a conference together. This happens all the time.

Also this week I received an email from May-Li Khoe, a researcher at Khan Academy, reflecting on her experience seeing Fawn Nguyen keynoting CMC-North. Both May-Li and Fawn are Asian-American.

I did not expect to be so affected by having Fawn speak during the keynote. Obviously the content of her presentation made an impression on me, but reflecting back later, I realized that I have never seen anyone remotely resembling myself as a keynote speaker, at any conference, ever.

We want all students to see themselves as people who can do mathematics, regardless of their race, ethnicity, gender, or any other variable. The power of mathematical thinking is good for everybody, and nobody should feel like their identity excludes them from that power.

The project of extending that access will require a diverse corps of teachers, which will require that a diverse corps of teachers sees teaching as a career full of advancement possibilities. Which means, among other efforts, that we need a more diverse corps of teachers speaking in front of large rooms of teachers.

So if you’re organizing a conference, I’m asking you to consider inviting any of the names below to give a talk before you consider inviting another tall, white dude. I’ll personally vouch for all of their abilities to deliver outstanding talks to large rooms of people. I have included Twitter contact information for each of them, along with websites and sample talks. I’m also happy to connect you with any of them personally. Let me know.

  • Maria Anderson. Applying research to instruction. [Twitter, Web, Sample]
  • Harold Asturias. Teaching mathematics & academic language to emerging bilingual students. [Twitter, Sample]
  • Deborah Ball. Teacher development; mathematical knowledge for teaching. [Twitter, Web, Sample]
  • Robert Berry. Formative assessment; equitable experiences for all math students; #blackkidsdomath. [Twitter, Sample]
  • Jo Boaler. Cultivating a growth mindset in mathematics. [Twitter, Web, Sample]
  • Marilyn Burns. Helping students make sense of math. [Twitter, Web, Sample]
  • Ed Campos, Jr. Technology integration. [Twitter, Web]
  • Peg Cagle. Creating engaging mathematical experiences. [Twitter, Sample]
  • Shelley Carranza. Technology integration. [Twitter]
  • Rafranz Davis. Technology integration; creating equitable experiences for all math students. [Twitter, Web, Sample]
  • Juli Dixon. Teaching students with special needs. [Twitter, Web, Sample]
  • Annie Fetter. Mathematical thinking and problem solving. [Twitter, Sample]
  • Kristin Gray. Creating engaging mathematical experiences. [Twitter, Web, Sample]
  • Rochelle Gutierrez. Creating equitable experiences for all math students (and their teachers). [Twitter, Sample]
  • Shira Helft. Instructional routines that promote discourse and sensemaking. [Twitter, Sample]
  • Ilana Horn. Cultivating a student’s mathematical identity. [Twitter, Web, Sample]
  • Elham Kazemi. Understanding a student’s mathematical thinking. [Twitter, Sample]
  • Jennie Magiera. Technology integration. [Twitter, Sample]
  • Danny Martin. Creating equitable experiences for all math students. [Sample]
  • David Masunaga. Mathematical inquiry, particularly in geometry.
  • Fawn Nguyen. Mathematical thinking and problem solving. [Twitter, Web, Sample]
  • Cathy O’Neil. The powerful and sometimes pernicious effect of algebraic models in the world. [Twitter, Web, Sample]
  • Carl Oliver. Integrating social justice and mathematics education. [Twitter, Web]
  • Megan Schmidt. Integrating social justice and mathematics education. [Twitter, Web]
  • Marian Small. Creating engaging and productive mathematical experiences. [Twitter, Web, Sample]
  • Joi Spencer. Integrating social justice and mathematics education. [Twitter, Sample]
  • Lee Stiff. Technology integration; creating equitable experiences for all math students. [Sample]
  • John Staley. Teaching mathematics for social justice. [Twitter, Sample]
  • Greg Tang. Creating engaging and productive mathematical experiences for elementary students. [Twitter, Web, Sample]
  • Megan Taylor. Creating engaging and productive mathematical experiences. [Twitter, Sample]
  • Kaneka Turner. Cultivating a student’s mathematical identity. [Twitter, Sample]
  • Sara Vanderwerf. Creating equitable experiences for all math students. [Twitter, Web]
  • Jose Vilson. Creating equitable experiences for all math students. [Twitter, Web, Sample]
  • Audrey Watters. Analyzing technological trends and their effect on education and society. [Twitter, Web, Sample]
  • Anna Weltman. Integrating creativity, art, and mathematics. [Twitter, Web, Sample]
  • Talithia Williams. Statistics; diversity in higher education. [Twitter, Sample]
  • Jennifer Wilson. Helping students make sense of mathematics; #slowmath. [Twitter, Web, Sample]
  • Cathy Yenca. Technology integration. [Twitter, Web, Sample]
  • Tracy Zager. Literally anything – have her read the tax code. (Also once her book comes out, your probability of getting her for your conference decreases asymptotically to zero. Buy now.) [Twitter, Web, Sample]

Add someone deserving or promising in the comments. Attach the same information you see above.

[Photos by Cathy Yenca and Kristin Hartloff.]

2016 Dec 14. The commenters have already caught a bunch of my really embarrassing omissions. Thanks for picking up my slack, everybody.

2016 Dec 16. In response to this critique from TODOS, I’d like to clarify that, yes, this list is incomplete, and my hope was that it would be made more complete in the comments. Additionally, my process in constructing the list is inherently biased towards a) speakers who have already given addresses to large rooms, which likely reflects the institutional biases of organizations who rent large rooms, b) speakers I have already seen, many of whom probably don’t challenge my privilege in ways I’d find uncomfortable, c) speakers who address secondary educators on themes of technology and curriculum design, themes reflective of my own disciplinary interests, d) speakers whom I could remember, which reflects my own lousy memory.

In spite of all those biases, I decided it was better for this list to exist than to not exist. I’m interested in hearing from TODOS (or anybody else) how this project could have done a better job advancing the interests of students and teachers of color.

Featured Comment

Elham Kazemi:

I was in graduate school before I had my first Persian teacher (if you exclude my education in Iran). It was an amazing experience, and I did every ounce of work possible in that class.

Shock and Disbelief in Math Class

Reader William Carey via email:

Last year I realized that Pre-Calculus is really a class about moving from the particular to the general. We take particular skills and ideas students are comfortable with — like solving a quadratic equation — and generalize them to as many mathematical objects as we can — solving all polynomial equations. As we worked our way through polynomials, we wanted to move from reasoning about particular quadratic equations like y = x2 + 2x + 1 to reasoning about all quadratic equations: y = ax2 + bx + c. For homework, the students had to graph about twenty quadratics with varying a, b, and c.

Then we got together to discuss the results in class. They remembered that a controls the “fatness” or “narrowness” of the parabola and sometimes flips it upside down. They remembered that c moves the parabola up and down. They weren’t totally sure what b did. A few students adamantly maintained that it moved the parabola left and right (with supporting examples). After about fifteen minutes of back and forth, we decided to go to Desmos and just animate b.

Shock and disbelief: the vertex traces out what looks like a parabola as b changes. Furious math and argument ensue. Ten minutes later, a student has what seems to be the parabola the vertex traces graphed in Desmos. Is it the right parabola? Why? Can we prove that? (We could and did!)

Previously: WTF Math Problems.

“All the time.”

David Cox:

Yesterday, a student gave me step-by-step directions to solve a Rubik’s Cube. I finished it, but had no idea what I was doing. At times, I just watched what he did and copied his moves without even looking at the cube in my hands.

When we were finished, I exclaimed, “I did it!”, received a high-five from the student and some even applauded. For a moment, I felt like I had accomplished something. That feeling didn’t last long. I asked the class how often they experience what I just did.

They said, “All the time.”

Featured Comment

Lauren Beitel:

Is there an argument to be made that sometimes the conceptual understanding comes from repeating a procedure, then reflecting on it? Discovering/noticing patterns through repetition?

Great question. I wrote a comment in response.

Engagement in Math: Three Places to Start

Mark Chubb, today on Twitter:

If a teacher sees students as disengaged and not liking math, what would be one good thing to watch, one good thing to read, one good thing to try?

Watch: Beyond Relevance & Real World.
Read: Why Don’t Students Like School?
Try: Estimation180.

Andrea Davis, later today in the comments:

Will you please give me the top three pieces of advice you have for the teachers of our youngest learners? We are K-6 and want to start now.

One, ask informal, relational questions (questioning, estimating, arguing, defining, etc.) as often as formal, operational questions (solving, calculating, simplifying).

Two, pose problems that have gaps in them – look up headless problems, tailless problems, and numberless problems, for three examples – and ask students to help you fill in those gaps. The most interesting problems are co-developed by teachers and students, not merely assigned in completed form by the teacher.

Three, before any explanation, create conditions that prepare students to learn from that explanation. These for example.


What are your suggestions for Andrea and Mark?

Featured Comments

Tim Teaches Math:


Let’s try to describe a big number using a small amount of syllables (Berry’s Paradox). For example, 777777 takes 20 syllables, but saying “777 times 1001” takes 15. For a number like “741” which is seven syllables, “Nine cubed plus twelve” is much better. More complicated expressions test our perception of order of operations. Have students come up with a scoring system to rank abbreviations.

Sarah Giek:

Read: Mathematical Mindsets
Watch: Five Principles of Extraordinary Math Teaching.
Try: Number Talk Images