[Pseudocontext Saturdays] Fish Tank

This Week’s Installment

Poll

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

Rules

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.)

Answer

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.

[Pseudocontext Saturdays] Spaghetti Bridge

This Week’s Installment

Poll

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

Pseudocontext Saturday #8

  • Identifying supplementary angles (42%, 186 Votes)
  • Calculating angle measures in a regular polygon (36%, 158 Votes)
  • Proving triangles are congruent (22%, 99 Votes)

Total Voters: 443

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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

I’m kicking the number of options back up to three. Two options simply doesn’t give y’all the challenge I know you need.

Team Me: 4
Team Commenters: 3

Pseudocontext Submissions

John Gibson

I don’t know if this is pseudocontext, but I for sure don’t know under what circumstances anyone would wonder about resultant momentum. In my head right now it’s like wondering about the middle names of the people who manufactured that car. It feels like trivia! I’m not saying it is trivia, but I am wondering if someone can put me in a position where knowing how to calculate resultant momentum would feel like power rather than punishment.

Rules

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.)

Answer

The commenters took this one right on the nose. The pseudocontext was in the last place they looked.

The judges rule that this violates the first rule of pseudocontext:

Given a context, the assigned question isn’t a question most human beings would ask about it.

Moreover, I just don’t see any congruent triangles in the picture. None. I know I’ll see some if you widen the camera’s angle, but there aren’t any in the frame right now, which makes this a uniquely poor context.

The only way I can think to neutralize this pseudocontext:

Show students four spaghetti bridges. They have to decide which ones are fragile and which ones are strong. Understanding congruency somehow (waves hands) makes them more accurate in their decision-making.

Featured Comment

Dick Fuller:

I like physics. And math. One without the other is school.

Great Classroom Action

Tracy Zager illustrates a key feature of some of my favorite math tasks: their constraints are simple, but they create paths for complex thinking and ever more interesting questions:

I think my name is worth $239. Beat me? Haven’t figured out my $100 strategy yet.

Lisa Bejarano is a recipient of our nation’s highest honor for math teachers, so when she admits “I have no idea what I am doing” and starts sketching out a blueprint for great classrooms, I tune in:

Now, beginning with the first day of school, I intentionally work at building a unique relationship with each student. I make sure to find reasons to genuinely value each of them. This starts with weekly “How is it going?” type questions on their warm up sheets and continues by using their mistakes on “Find the flub Friday” and through feedback quizzes. I also share a lot of myself with them. When we understand each other, my classes are more productive. I still make plans, but I allow flexibility to meet my students where they are.

David Cox describes “a difficult thing for students to believe”:

Once students begin to believe that the way they see something is the currency, then our job is to simply help them refine their communication so their audience can understand them. Only then does the syntax of mathematics matter.

“Help me understand you.”

“Help me see what you see.”

Kevin Hall thoughtfully deconstructs his attempts to teach linear function for meaning, and includes this gem:

Once you introduce the slope formula, slope becomes that formula. It barely even matters if today’s lesson created a nice footpath in students’ brains between “slope” and the change in one quantity per unit of change in another. Once that formula comes out, your measly footpath is no competition for the 8-lane highway that’s opened up between “slope” and (y2–y1)/(x2­-x1).

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