Month: December 2016

Total 9 Posts

What’s Wrong with This Experiment?

If you’re the sort of person who helps students learn to design controlled experiments, you might offer them W. Stephen Wilson’s experiment in The Atlantic and ask for their critique.

First, Wilson’s hypothesis:

Wilson fears that students who depend on technology [calculators, specifically –dm] will fail to understand the importance of mathematical algorithms.

Next, Wilson’s experiment:

Wilson says he has some evidence for his claims. He gave his Calculus 3 college students a 10-question calculator-free arithmetic test (can you multiply 5.78 by 0.39 without pulling out your smartphone?) and divided the them into two groups: those who scored an eight or above on the test and those who didn’t. By the end of the course, Wilson compared the two groups with their performance on the final exam. Most students who scored in the top 25th percentile on the final also received an eight or above on the arithmetic test. Students at the bottom 25th percentile were twice as likely to score less than eight points on the arithmetic test, demonstrating much weaker computation skills when compared to other quartiles.

I trust my readers will supply the answer key in the comments.

BTW. I’m not saying there isn’t evidence that calculator use will inhibit a student’s understanding of mathematical algorithms, or that no such evidence will ever be found. I’m just saying this study isn’t that evidence.

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Scott Farrand:

The most clarifying thing that I can recall being told about testing in mathematics came from a friend in that business: you’ll find a positive correlation between student performance on almost any two math tests. So don’t get too excited when it happens, and beware of using evidence of correlation on two tests as evidence for much.

[Makeover] Systems of Equations

Here is the oldest kind of math problem that exists:

Some of you knew what kind of problem this was before you had finished the first sentence. You could blur your eyes and without reading the words you saw that there were two unknown quantities and two facts about them and you knew this was a problem about solving a system of equations.

Whoever wrote this problem knows that students struggle to learn how to solve systems and struggle to remain awake while solving systems. I presume that’s why they added a context to the system and it’s why they scaffolded the problem all the way to the finish line.

How could we improve this problem – and other problems like this problem?

I asked that question on Twitter and I received responses from, roughly speaking, two camps.

One group recommended we change the adjectives and nouns. That we make the problem more real or more relevant by changing the objects in the problem. For example, instead of analyzing an animated movie, we could first survey our classes for the movie genres they like most and use those in the problem.

This makeover is common, in my experience. I don’t doubt it’s effective for some students, particularly those students already adept at the formal, operational work of solving a system of equations through elimination. The work is already easy for those students, so they’re happy to see a more familiar context. But I question how much that strategy interests students who aren’t already adept at that work.

Another strategy is to ignore the adjectives and nouns and change the verbs, to change the work students do, to ask students to do informal, relational work first, and use it as a resource for the formal, operational work later.

This makeover is hard, in my experience. It’s especially hard if you long ago became adept at the formal, operational work of solving a system of equations through elimination. This makeover requires asking yourself, “What is the core concept here and what are early ways of understanding it?”

No adjectives or nouns were harmed during this makeover. Only verbs.

The theater you run charges $4 for child tickets and $12 for adult tickets.

  1. What’s a large amount of money you could make?
  2. What’s a small amount of money you could make?
  3. Okay, your no-good kid brother is working the cash register. He told you he made:
    • $2,550 on Friday
    • $2,126 on Saturday
    • $1,968 on Sunday

    He’s lying about at least one of those. Which ones? How do you know?

This makeover claims that the core concept of systems is that they’re about relationships between quantities. Sometimes we know so many relationships between those quantities that they’re only satisfied and solved by one set of those quantities. Other times, lots of sets solve those relationships and other times those relationships are so constrained that they’re never solved.

So we’ve deleted one of the relationships here. Then we’ve ask students to find solutions to the remaining relationship by asking them for a small and large amount of money. There are lots of possible solutions. Then we’ve asked students to encounter the fact that not every amount of money can be a solution to the relationship. (See: Kristin Gray, Kevin Hall, and Julie Reulbach for more on this approach.)

From there, I’m inclined to take Sunday’s sum (one he wasn’t lying about) and ask students how they know it might be legitimate. They’ll offer different pairs of child and adult tickets. “My no-good kid brother says he sold 342 tickets. Can you tell me if that’s possible?”

Slowly they’ll systematize their guessing-and-checking. It might be appropriate here to visualize their guessing-and-checking on a graph, and later to help students understand how they could have used algebraic notation to form that visualization quickly, at which point the relationships start to make even more sense.

If we only understand math as formal, operational work, then our only hope for helping a student learn that work is lots and lots of scaffolding and our only hope for helping her remain awake through that work is a desperate search for a context that will send a strong enough jolt of familiarity through her cerebral cortex.

That path is wide. The narrow path asks us to understand that formal, operational ideas exists first as informal, relational ideas in the mind of the student, that our job is devise experiences that help students access those ideas and build on them.

BTW. Shout out to Marian Small and other elementary educators for helping me see the value in questions that ask about “big” and “small” answers. The question is purposefully imprecise and invites students to start poking at the edges of the relationship.

[Pseudocontext Saturdays] Fish Tank

This Week’s Installment


What mathematical skill is the textbook trying to teach with this image?

Pseudocontext Saturday #9

  • Calculating roots of polynomials (47%, 179 Votes)
  • Calculating mean, median, and mode (37%, 141 Votes)
  • Proving triangles are congruent (16%, 61 Votes)

Total Voters: 381

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(If you’re reading via email or RSS, you’ll need to click through to vote. Also, you’ll need to check that link tomorrow for the answer.)

Current Scoreboard

Team Me: 5
Team Commenters: 3

Pseudocontext Submissions

Kimberly Robertson


Every Saturday, I post an image from a math textbook. It’s an image that implicitly or explicitly claims that “this is how we use math in the world!”

I post the image without its mathematical connection and offer three possibilities for that connection. One of them is the textbook’s. Two of them are decoys. You guess which connection is real.

After 24 hours, I update the post with the answer. If a plurality of the commenters picks the textbook’s connection, one point goes to Team Commenters. If a plurality picks one of my decoys, one point goes to Team Me. If you submit a mathematical question in the comments about the image that isn’t pseudocontext, collect a personal point.

(See the rationale for this exercise.)


This was a nail-biter between Team Commenters and Team Me this week, with Team Commenters narrowly tipping the scales in their favor.

The judges rule that this satisfies the second rule of pseudocontext:

Given a question, the assigned method isn’t a method most human beings would use to find it.

Reasonable people might wonder about the dimensions of a water tank. The judges rule that most human beings would use a tape or a stick or any other kind of measuring device to answer it, not a cubic polynomial.

I can’t think of any way to neutralize this pseudocontext. The number of actual contexts for cubic polynomials with non-zero quadratic and linear terms is vanishingly small.

Here is an activity I would much prefer to use to teach the construction of polynomials. It doesn’t involve the real world but it does ask students to do real work.

Featured Comment

William Carey:

One motif in pseudocontextual questions seems to be treating as a variable things that, you know, don’t vary. I have a funny video playing in my mind of some surprised fish watching the volume of their tank become negative. But happily the volume of that tank is not varying, inasmuch as it’s sides are made of glass.

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.