Dismantling the Privilege of the Mathematical 1%

[This is an elaboration on a talk I gave at #MAAthfest in Chicago.]

It’s wonderful to be here. I spend most of my days with people who don’t fully get me. Wife, friends, dog – none of them gets me like you get me.

None of them understands the feeling of mathematical epiphany that motivates my professional life, the sudden transition from not knowing to knowing.

One of my earliest mathematical epiphanies was the realization that if you let the number of sides on a regular polygon increase without bound, you get a circle.

And that all the relationships you find in a regular polygon have analogous relationships in a circle. For me, that realization was literally a religious experience. I finished that limit on the back of a church bulletin while a churchlady glared at me.

So on the one hand it’s great to be in this room – I am among my people – but on the other hand it’s really uncomfortable to be here because you all make me really aware of my privilege, and aware of how many people are not in this room.

The economic 1% gets a lot of grief lately and whether we know it or not, whether we like it or not, we are all also in the 1% – the mathematical 1%.

In 2014, 2.8 million degrees were awarded in US universities – bachelors, masters, and doctorates – and 1.1% of them were in mathematics. If you change the denominator to reflect not advanced degree holders but anyone with a high school diploma our elitism becomes even more apparent.

I was on Instagram last night checking out the #MAAthfest hashtag along with The Rich Kids of Instagram. While there are fewer yachts, bottles, and shrink wrapped stacks of bills on the left, and maybe more plaid and elbow patches, there is still the same exuberant sense of having arrived. We have made it.

And just as the economic 1% creates systems that preserve its status – policies like the mortgage interest deduction for homeowners, discriminatory lending policies, and lower taxes on capital gains than income – through our action or inaction we create systems that preserve our status as the knowers and doers of mathematics.

When someone says, “I’m not a math person,” what do you say back? Barring certain disabilities or exceptionalities, everyone starts life a math person. Infants can recognize changing quantity. Brazilian street vendors develop sophisticated arithmetic algorithms before they set foot in school.

It is our action and inaction that teach people they are not mathematical. So please consider taking two actions to extend your privilege to the other 99% of humanity.

First, change the definition of mathematics that people experience.

[Here we explored together Circle-Square, a task that involves questioning, estimation, intentionally declaring wrong answers, recalling what you know about circles and squares, computing an answer, and verifying it. You can watch it.]

Now I don’t want to suggest to you that this is the experience that will change a person’s definition of mathematics and extend our privilege to the 99%. I just want to suggest to you that you just had a very different mathematical experience than the people who encountered that problem in its original form:

Mark an arbitrary point P on a line segment AB. Let AP form the perimeter of a square and BP form the circumference of a circle. Find P such that the area of the square and circle are maximized.

That experience offers people only a certain kind of mathematical work. You recall what you know about perimeter, circumference, and area, compute it, and verify it in the back of the book.

Those verbs are our mortgage interest rate deduction, our discriminatory lending policy, and our tax advantages. Through our action and inaction, society has come to understand that math is a merry-go-round revolving endlessly through those three verbs – remember a procedure, compute it, verify it.

You might think, “Well that’s what math is,” but the definition of math isn’t a physical constant in the universe. It’s defined by people, just as people define the ways that wealth and power accrue in the world. That definition is then underlined, reflected, and enforced in public policy, curriculum, and syllabi.

So, second, let’s change the definition of mathematics in public policy, curriculum, and syllabi.

To begin with, let’s eliminate policies that require intermediate algebra for college study.

The facts as I understand them are that:

  • College completion is increasingly essential to even partial economic participation.
  • College study is generally predicated on a student’s ability to pass a mathematics entrance exam. In the California State University system, that exam is heavily weighted towards intermediate algebra, problems like these, the majority of which depend on the recollection of an obscure and abstract procedure:

  • Students fail these exams in staggering numbers (68% nationally) placing into “developmental math” courses, courses which cost time and money and don’t offer credit towards graduation.
  • Those courses are disproportionally composed of African American and Latinx students.
  • Only 32% of students in developmental math ever take a math course required for graduation.

It’s hard to imagine a machine more perfectly configured for the preservation of mathematical privilege.

Those statistics would bother me less if either a) I believed in the value of intermediate algebra, b) better alternatives weren’t available. Neither is true. That intermediate algebra has little value to the majority of college educated professionals hardly requires a defense. As Uri Treisman said, “The most common use of algebra in the adult world is helping their kids with algebra.”

I am sympathetic to the argument, however, that we shouldn’t choose college requirements solely because they’re useful professionally. College should offer students a broad survey of every discipline – a general education, as it’s called. That survey should generate intellectual interest where perhaps there was none; it should awaken students to intellectual possibilities they hadn’t considered; it should increase the likelihood they’ll speak favorably about the discipline after college.

Those goals are served poorly by intermediate algebra. And better alternatives to intermediate algebra exist to serve the CSU’s desire to “assess mathematical skills needed in CSU General Education (GE) programs in quantitative reasoning.”

Specifically, statistics.

When 907 CUNY students were assigned either to remedial algebra, remedial algebra and supplementary workshops, or college-level statistics and workshops, that latter group a) passed their course in greater numbers (earning credit!) and also b) accumulated more credits in later courses.

So we should be excited to see the California State University drop its intermediate algebra requirement for graduation. We should be excited to see a proposal from NCTM that reserves intermediate algebra concepts for elective courses in high school. But we should regard both proposals as tenuous, and understand that as people of privilege, our support should be vocal and persistent.

We can choose action or inaction here. Through your action, the definition of math may change so that it’s accessible to and enjoyed by many more people, so that many more people understand themselves to be “math people.” I want to be clear that our own privilege will diminish as a result, that we will become less special, but that humanity as a whole will flourish. Through your inaction, or through your tentative, private support for initiatives like these, the existing definition will endure, along with the existing distributions of privilege. Choose action.

2017 Nov 14. Please read a follow-up comment from Alexandra W. Logue, one of the authors of the CUNY study:

Three years after the intervention, although 17% of the traditional remedial group had graduated, 25% of the statistics group had done so (almost 50% more students). To graduate, students had to pass, not only their general education quantitative requirement (which could be satisfied by college algebra or statistics), but also their social and natural science course general education requirements. So, for many students, passing remedial algebra was not necessary in order to pass these other courses. Further, there were no differences in our results in accordance with students’ race/ethnicity. Given that Black and Hispanic students are more likely to be assessed as needing remediation, our results mean that our procedure can help close graduation rate gaps between underrepresented and other students.

Challenge Creator & the Desmos Classroom

Briefly

  • At Desmos, we’re now asking ourselves one question about everything we make: “Will this help teachers develop social and creative classrooms?” We’ve chosen those adjectives because they’re simultaneously qualities of effective learning and also interesting technology.
  • We’ve upgraded three activities (and many more to come) with our new Challenge Creator feature: Parabola Slalom, Laser Challenge, and Point Collector: Lines. Previously, students would only complete challenges we created. Now they’ll create challenges for each other.
  • The results from numerous classroom tests have been – I am not kidding you here – breathtaking. Near unanimous engagement. Interactions between students around mathematical ideas we haven’t seen in our activities before.

Read More

One question in edtech bothers us more than nearly any other:

Why are students so engaged by their tablets, phones, and laptops outside of class and so bored by them inside of class?

It’s the same device. But in one context, students are generally enthusiastic and focused. In the other, they’re often apathetic and distracted.

At Desmos, we notice that, outside of class, students use their devices in ways that are social and creative. They create all kinds of media – text messages, videos, photos, etc. – and they share that media with their peers via social networks.

You might think that comparison is unfair – that school could never stack up next to Instagram or Snapchat – but before we write it off, let’s ask ourselves, “How social and creative is math edtech?” What do students create and whom do they share those creations with?

In typical math edtech, students create number responses and multiple choice answers. And they typically share those creations with an algorithm, a few lines of code. In rarer cases, their teacher will see those creations, but more often the teacher will only see the grade the algorithm gave them.

For those reasons, we think that math edtech is generally anti-social and uncreative, which explains some of the apathy and distraction we see when students use technology inside of class.

Rather than write off the comparison to Instagram and Snapchat as unreasonable, it has motivated us to ask two more questions:

  1. How can we help students create mathematically in more diverse ways?

So we invite students to create parking lots, scale giants, mathematical arguments, tilings, sketches of relationships, laser configurations, drawings of polygons, tables, stacks of cards, Marbleslides, informal descriptions of mathematical abstractions, sequences of transformations, graphs of the world around them, and many more.

  1. How can we help teachers and students interact socially around those creations?

So we collect all of those creations on a teacher dashboard and we give teachers a toolkit and strategies to help them create conversations around those creations. It’s easier to ask your students, “How are these two sketches the same? How are they different?” when both sketches are right in front of you and you’re able to pause your class to direct their focus to that conversation.

Today, we’re releasing a new tool to help teachers develop social and creative math classrooms.

Challenge Creator

Previously in our activities, students would only complete challenges we created and answer questions we asked. With Challenge Creator, they create challenges for each other and ask each other questions.

We tried this in one of our first activities, Waterline, where, first, we asked students to create a graph based on three vases we gave them.

And then we asked them to create a vase themselves. If they could successfully graph the vase, it went into a gallery where other students would try to graph it also.

We began to see reports online of students’ impressive creativity and perseverance on that particular challenge. We started to suspect the following: that students care somewhat when they share their creations with an algorithm, and care somewhat more when they share their creations with their teacher.

But they care enormously when they share their creations with each other.

So we’ve added “Challenge Creators” to three more activities, and we now have the ability to add them to any activity in a matter of hours where it first took us a month.

In Parabola Slalom, we ask students to find equations of parabolas that slip in between the gates on a slalom course. And now we invite them to create slalom courses for each other. Those challenges can be as difficult as the authors want, but unless they can solve it, no one else will see it.

In Laser Challenge, we ask students to solve reflection challenges that we created. And now we invite them to create reflection challenges for each other.

In Point Collector, we ask students to use linear inequalities to capture blue points in the middle of a field of points. And now we invite them to create a field of points for each other.

We’ve tested each of these extensively with students. In those tests we saw:

  • Students calling out their successes to each other from across the room. “Javi, I got a perfect score on yours!”
  • Students calling out their frustrations to each other from across the room. “Cassie, how do you even do that?”
  • Students introducing themselves to each other through their challenges. “Who is Oscar?”
  • Students differentiating their work. “Let’s find an easy one. Oo – Jared’s.”
  • Students looking at solutions to challenges they’d already completed, and learning new mathematical techniques. “You can do that?!”
  • Students marveling at each others’ ingenuity. “Damn, Oscar. You hella smart.”
  • Proud creation. One student said, “We’re going to make our challenge as hard as possible,” to which his partner responded, “But we have to be able to solve it!”
  • Screams and high fives so enthusiastic you’d think we were paying them.

At the end of one test of Point Collector, we asked students, “What was your favorite part of the activity?” 25 out of 27 students said some version of “Solving other people’s challenges.”

I’m not saying what we saw was on the same level of enthusiasm and focus as Instagram or Snapchat.

But it wasn’t that far off, either.

Questions We Can Answer

How much does it cost?

As with everything else we make that’s free for you to use now, we will never charge you for it.

Will we be able to create our own Challenge Creators?

Eventually, yes. Currently, the Triple C (Challenge Creator Creator, obv.) has too many rough edges to release widely. Once those edges are sanded down, we’ll release it. We don’t have a timeline for that work, but just as we think student work is at its best when it’s social and creative, we think teacher work is at its best under those exact same conditions. We want to give teachers the best toolkit possible and enable them to share their creations with each other.

Questions We Can’t Answer

What effect does asking a student to create a challenge have on her learning and her interest in learning?

What sorts of challenges are most effective? Is this approach just as effective for arithmetic expressions as laser challenges?

Does posing your own problem help you understand the limits of a concept better than if you only complete someone else’s problems?

Researchers, grad students, or any other parties interested in those same questions: please get in touch.

Putting the “Use” in “Look for and Make Use of Structure.”

This is beautiful, right? Put enough straight lines in the right places and your eyes see a curve.

How many linear equations did the student use to create it? You might start counting lines and assume it required dozens. For some students, you’d be right. They typed 40 linear equations and corrected a handful of typos along the way.

But other students created it using only four linear equations and many fewer errors!

The seventh mathematical practice in the Common Core State Standards asks students to “look for and make use of structure.” The second half of that standard is a heavier lift than the first by several hundred pounds.

Because it’s easy enough for me to ask students, “What structures do you notice?” It’s much more difficult for me to put them in a situation where noticing a mathematical structure is more useful than not noticing that structure.

Enter Match My Picture, my favorite activity for illustrating my favorite feature in the entire Desmos Graphing Calculator and for helping students see the use in mathematical structures.

First, we ask students to write the linear equations for a couple of parallel lines.

Then four lines. Then nine lines.

It’s getting boring, but also easy, which are perfect conditions for this particular work. A boring, easy task gives students lots of mental room to notice structure.

Next we ask students, “If you could write them all at once – as one equation, in a form you made up – what would that look like?” Check out their mathematical invention!

Next we show students how Desmos uses lists to write those equations all at once, and then students put those lists to work, creating patterns much faster and with many fewer errors than they did before. With lists, you can create nine lines just as fast as ninety lines.

What are the four equations that created this graph? Personally, I find it almost impossible to discern by just looking at the graph. I have to write the equation of one of the lines. Then another. Then another. Then another, until that task becomes boring and easy. Only then am I able to notice and make use of the structure.

Free Online Courses Start Today: Mathematical Play & Mathematical Anthropology

The final two ShadowCon courses start today. They both feature awesome presenters offering important ideas, and they’re both free!

Mathematical Play
Kassia Omohundro Wedekind

Kassia’s ShadowCon talk was such a blast, integrating several different bodies of scholarship all arguing for the mathematical and social value of play. Her course has insightful readings, illustrative classroom video, and Zak Champagne and Mike Flynn as teaching assistants.

Enroll!

The Art of Mathematical Anthropology
Geoff Krall

Geoff has loads of experience with innovative assessments as a coach in the New Tech network of schools. In his course, he’ll help you understand what portfolio assessments offer students and how to develop them. You’ll find me in Geoff’s course as a teaching assistant.

Enroll!

The Bet I Made with Teachers All Around the United States Last Year

Last year, I made the same bet all around the United States with every crowd of math teachers I met:

I’ll pick a number between 1 and 100. I’ll give you ten guesses to figure out my number. And every time you guess, I’ll tell you if my number is higher or lower.

I always wagered whatever cash I had in my pocket – generally between $2 to $20. The math teachers, meanwhile, owed me nothing if they lost. I had no trouble finding people to take the other side of that wager.

Watch one of the wagers below.

I pick my number.

She first guesses 61. I’m higher.

Then 71. I’m higher.

Then 81. I’m higher.

Then 91. I’m lower. She’s got me trapped. Six guesses left.

Then 86. I’m lower. Five guesses left. I’m an injured gazelle.

Then 83. I’m lower. Between 81 and 83. Four guesses left, but she only needs one. The crowd smells blood.

Then, with a trace of sympathy in her voice, 82. The crowd thinks it’s over.

But I’m higher.

Aaaand the chase is back on, y’all!.

Tentatively now: 82.5. I’m still higher. One by one, members of the crowd are wise to my scam.

Then 82.75. I’m lower. She has one guess left.

Then 82.7. I’m higher, at 82.72.

I asked her what I’d ask any crowd of sixth graders at this point:

If I offered you the same wager again, what follow-up questions would you have for me?

“What kind of number are you picking?” she said.

My point in all of this is that math teachers have names for their numbers, much in the same way that ornithologists have names for their birds. And much in the same way that ornithologists haven’t given me a reason to care about the difference between a Woodlark and a Skylark, math teachers often fail to motivate the difference between rational numbers and integers and whole numbers and imaginary numbers and supernatural numbers.

The difference is that ornithology isn’t a course that’s required for high school graduation and university enrollment and labor market participation. Kids aren’t forced to study ornithology for twelve years of their childhood.

So I’m inviting us to ask ourselves: “Why did we invent these categories of numbers?” And if we agree that it was to more effectively communicate about numbers, we need to put students in a place where their communication suffers without those categories. If we can’t, then we should confess those categories are vanity.

Before we give students the graphic organizers and Venn diagrams and foldables designed to help them learn those categories, let’s help them understand that they were invented for a reason. Not because we have to.

There are always ways to make kids memorize disconnected, purposeless stuff.

But because we should.

Featured Comment

Via email:

Did you ever lose?

I never once lost. I was never once asked to specify the kind of number I was picking.

Me, holding up the number I wrote down in nine different cities.