A few weeks back, I was in Washington, D.C., for this year’s National Council of Teachers of Mathematics conference—by far the largest annual math education conference.
Being in D.C., there was one attraction I made a point to see: The National Portrait Gallery. In particular, I wanted to see one work by Kehinde Wiley, an artist mostly known for painting enormous and ultrarealistic portraits of African-Americans surrounded by rich, ornate patterns.
Former President Barack Obama tapped Wiley to paint his official portrait for the National Portrait Gallery, which features every president in history. If you haven’t seen it—and I hope I can say this irrespective of your political leanings—it’s stunning. (View the painting here.)
In one way or another, I have been a math educator my whole life. Looking at this painting, I reflected: “Where is the math?” I first thought of some simple contexts that could make for rich conversations:
About how many leaves are in the painting?
How much paint is needed to complete this painting?
Where it gets more interesting, in my opinion, are in the realistic contexts—that is, where there is no stock textbook lesson to guide you, and someone, somewhere actually has to try and solve the problem:
How can a conservator ensure the colors won’t fade over time?
How much is the painting worth?
In these contexts, mathematics is just a part of the solution, albeit an integral part. (But only Riemann integral.) There is a subtle problem, though. The most realistic problems require a lot of mathematical sophistication to approach and take far longer than a class period to solve, if there even is a solution. My goal for the conference was to get a pulse on today’s teaching, but it seems the real world always has this “simple” vs. “realistic” divide. Teaching addition is drastically different from teaching calculus, and I struggled to find the cluster inside a multidimensional Venn diagram.
When I read over my notes on the train ride home, a spark lit when I realized that many of the conversations I had and sessions I attended boiled down to math problems being a whole lot more subjective than many people think. Mathematics—all of it, no matter your experience—is a bit like a Kehinde Wiley painting: permanent, austere, and forever jazzed up by humanity.
Consider this question, which at first seems more mathematically abstract than human: Why are there 360 degrees in a circle? This question was the focus of a session titled “How Many Degrees Are in a Martian Circle? And Other Human—and Nonhuman—Questions One Should Ask!” The perhaps anticlimactic answer is the Earth revolves around the Sun once every 365.25 days, which humans retooled as degrees in a circle and rounded to 360. We rounded because 365.25 is a hideous number to work with, but 360 is easy. (“OK, it’s not divisible by seven, but nothing’s divisible by seven,” the speaker added, eliciting laughter from an audience of obvious math geeks.)
This tendency for us mortals to scrub the messiness in nature pops up everywhere. A few other times I spotted it over the course of the conference:
- A vinculum is a horizontal bar placed over expressions to section them off, identically to parentheses. There’s a reason we use parentheses over the antiquated—yet arguably easier to parse—vinculum: The human invention of the printing press needed everything to fit on a single line.
- A first-grade teacher presented an activity where she took an abstract equation (e.g. “3 + ___ = 10”) and had students provide the human interpretation (“What story does this equation tell?”). This concept can extend to a high school classroom: “What story does the equation e3x = 1,000 tell?"
- One website appeared in both K–2 and high school sessions: wodb.ca. On the website, the user is presented with four objects and must answer “Which one doesn’t belong?” (hence wodb). The objects are mathematical and include dice, graphs, or numbers. Both speakers praised the site as mathematics in its most realistic form: people debating and supporting their own subjective answers.
- One speaker noted the hypocrisy of using a lecture to explain how bad lecturing is for learning; research supports this claim. However, he also noted why humans will never stop using it: it’s by far the fastest way to share information.
- One speaker noted the importance of distinguishing between larger and greater, which have different connotations (contrast a large painting to a great painting) but are often interchangeable in mathematics, a claim that research partially supports. Encouraging students to compare numbers along a number line using “greater” and not “larger” is a decidedly human endeavor, not a mathematical one.
The most striking example of the humanness of mathematics, I thought, was in a session led by Mike Reiners, who was once a contestant on Who Wants to Be a Millionaire. For those unaware, it’s a game show where contestants answer multiple-choice questions with gradually increasing prize values. At any point, they can pass on a question and take home the money they’ve won so far.
Reiners is a high school teacher who shared different ways that math students can engage with the game show. One example was having students discuss the different prize values in the game.
The left column in the first image below—which displays the prize list from the show's first five seasons—shows how many questions contestants have to answer correctly in sequence, and the right column shows how much money they win for doing so. But why is the 12th rung $125,000 and not $128,000, in effect breaking the doubling pattern? And why do we start at $100 and not $62.50, which would have maintained the doubling pattern for the first few dollar amounts and resulted in an even $1,000? The answer is because we’re human, typically with 10 fingers, and we like numbers with 0s and 5s in them.
During the last session I went to at the conference, I saw a video of children cutting out amoeba-like shapes called Splats, an invention of Oregon math teacher Steve Wyborney. The students placed counters on the floor and covered some of them with a Splat. They told a partner how many counters there were in total, and the partner had to figure out how many were under the Splat.
Two things struck me. First, when I initially saw the image of Splat-covered counters, I thought they were to practice estimation. The fact that they were intended to practice subtraction reveals the versatility of the activity. Second, it warmed my heart to see students making art in math class.
My trip continues next week with “A Learning Architect Goes to Washington, Part 2: Lifelong Kindergarten.”
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As a contributor to HMH’s next generation math program, Into Math, this blogger is creating learning moments for today’s students and tomorrow’s problem solvers.
