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.


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.


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.

NCTM’s Bold and Invigorating Plan for High School Mathematics

NCTM released Catalyzing Change in High School Mathematics last week. I anticipated it would address the gap between the K-8 Common Core State Standards, which feel tightly designed both within and across grades, and the high school standards, which feel loose and shaggy by comparison.

NCTM went about that goal in the second half of Catalyzing Change, enumerating a set of “Essential Concepts” along with two pathways students can take to learn them. I’ll comment on those concepts and pathways in a moment. But it’s worth mentioning first what I didn’t anticipate: a document full of moral ambition, the first half of which is a reimagination of the purpose of a math education along with a high-decibel endorsement of equity in that education.

You should read the latter half of the document if you have any stake in high school math education. But you should read the first half of the document if you have any stake in math education at all, at any level.

While the Obama administration proposed college and career-readiness as the purpose of schooling, NCTM broadens that purpose here to include “Understanding and Critiquing the World,” addressing the question, “When will I ever use this?”, and also “Experiencing Wonder, Joy, and Beauty,” acknowledging the millions and millions of people who love studying math even apart from its immediate application to the world outside the classroom.

NCTM reinvokes its call for equitable math instruction, citing Gutiérrez’s perspective that until it is no longer possible “to predict mathematics achievement and participation based solely on student characteristics such as race, class, ethnicity, sex, beliefs, and proficiency in the dominant language,” we haven’t finished the work. To advance the cause of equity, NCTM pulls precisely zero punches in its condemnation not just of student tracking (which allocates students inequitably to the best classes) but teacher tracking (which allocates teachers inequitably to the most underserved students), also double-year math courses, and other less overt ways in which students are tracked even in elementary school.

This is what I mean by “moral ambition.” NCTM hasn’t merely underlined its existing statements on equity or de-tracking. Rather it lets those statements stand and then opens up several new fronts and runs at them. Catalyzing Change doesn’t arrive pre-compromised.

So again: everyone should read the first half of Catalyzing Change, which addresses much of the “why?” and “who?” of mathematics education. The second half of the document makes several clear and ambitious claims about the “what?”

NCTM proposes that all students take four years of math in high school. 2.5 of those years will comprise “essential concepts,” taken by every student regardless of career or college aspiration. Students may then take one of two paths through their remaining 1.5 years, one towards calculus, the other towards statistics and other electives.

40 essential concepts cluster under five conceptual categories:

  • Algebra
  • Functions
  • Statistics
  • Probability
  • Geometry

If we only examine the number of concepts and not yet their content, this proposal compares very favorably with the Common Core State Standards’ over 100 required standards for high school. Under NCTM’s proposal, students may come to understand a proof of the similarity of circles (Common Core State Standard G-C.1) or a derivation of the equation of a parabola from its directrix and focus (G-GPE.2) but only as an incidental outcome of high school math, not an essential outcome.

Then, as I read the content of the concepts, I asked myself, “Do I really believe every student should spend 2.5 years of their limited childhood learning this?” In nearly every case, I could answer “yes.” In nearly every case, I could see the concept’s applicability to college and career readiness, and even more often, I could see how the concept would help students understand their world and nurture their joy and wonder. (I wouldn’t say that about the derivation of a parabola’s equation, by contrast.)

That’s such an accomplishment. The writing team has created a “Director’s Cut” of high school mathematics – only the most essential parts, arranged with a coherence that comes from experience.

If I’m concerned about any category, it’s “Algebra” and, particularly, essential concepts like this one:

Multi-term or complex expressions can represent a single quantity and can be substituted for that quantity in another expression, equation, or inequality; doing so can be useful when rewriting expressions and solving equations, inequalities, or systems of equations or inequalities. [emphasis mine]

Without any evidence, I’m going to claim that one of the top three reasons students leave high school hating mathematics is because their algebra courses required weeks and weeks of transcribing expressions from one form into another for no greater purpose than passing the class. I’m talking about conjugating denominators, converting quartic equations into quadratic equations through some clever substitution, factoring very special polynomials, completing the square, and all other manner of cryptic symbology, none of which deserves the label “essential.”

NCTM has done much more work here defining what is “essential” than what is “inessential,” which means their definitions need to be air tight. Some of their definitions in “Algebra” and “Functions” leave room for some very inessential mathematics to slip through.

My other concern with Catalyzing Change is the bet NCTM makes on technology, modeling, and proof, weaving that medium and those habits of mind through every category, and claiming that they have the greatest potential to enable equitable instruction.

I don’t disagree with that selection or NCTM’s rationale. But add up the bill with me here. NCTM proposes a high school course of study premised on:

  • modeling, which students most often experience as pseudocontextual word problems,
  • proof, which students most often experience by filling in blanks in a two-column template,
  • technology, which students most often experience as a medium for mealy, auto-graded exercises,
  • to say nothing of joy and wonder, which most students typically experience as boredom and dread.

This is a multi-decade project! One that will require the best of teachers, teacher educators, coaches, administrators, edtech companies, assessment consortia, policymakers, publishers, and parents. It will require new models of curriculum, assessment, and professional development, all supporting modeling and proof and eliciting joy and wonder from students. It will require a constant articulation and re-articulation of values to people who aren’t NCTM members. That is, changes to the K-8 curriculum required articulation to high school teachers. Changes to the high school curriculum will require articulation to college and university educators! Does anybody even know any college or university educators?

I’m not finding fault. I’m identifying challenges, and I find them all energizing. Catalyzing Change is an invigorating document that makes a clear case for NCTM’s existence at a time when NCTM has struggled to articulate its value to members and non-members.

If you haven’t heard that case, let me try to write it out:

Hi. We’re NCTM. We want to restore purpose, joy, and wonder to your high school math classrooms. We know that goal sounds ambitious, and maybe even impossible, but we have a lot of experience, a lot of ideas, a lot of resources, and a lot of ways to help you grow into it. We’re here for you, and we also can’t do any of this without you. Let’s do this!

Free New Desmos Activity: Transformation Golf

[cross-posted to the Desmos blog]

We’re excited to release our latest activity into the world: Transformation Golf.

Transformation Golf is the result of a year’s worth of a) interviews with teachers and mathematicians, b) research into existing transformation work, c) ongoing collaboration between Desmos’s teaching, product, and engineering teams, d) classroom demos with students.

It’s pretty simple. There is a purple golf ball (a/k/a the pre-image) and the gray golf hole (a/k/a the image). Use transformations to get the golf ball in the hole. Avoid the obstacles.

Here’s why we’re excited to offer it to you and your students.

Teachers told us they need it. We interviewed a group of eighth grade teachers last year about their biggest challenges with their curriculum. Every single teacher mentioned independently the difficulty of teaching transformations – what they are, how some of them are equivalent, how they relate to congruency. Lots of digital transformation tools exist. None of them quite worked for this group.

It builds from informal language to formal transformation notation. As often as we ask students to define translation vectors and lines of reflection, we ask them just to describe those transformations using informal, personal language. For example, before we ask students to complete this challenge using our transformation tools, we ask them to describe how they’d complete the challenge using words and sketches.

The entire plane moves. When students reach high school, they learn that transformations don’t just act on a single object in the plane, they act on the entire plane. We set students up for later success by demonstrating, for example, that a translation vector can be anywhere in a plane and it transforms the entire plane.

Students receive delayed feedback on their transformations. Lots of applets exist that allow students to see immediately the effect of a transformation as they modify it. But that kind of immediate feedback often overwhelms a student and inhibits her ability to create a mental concept of the transformation. Here students create a transformation, conjecture about its effect, and then press a button to verify those conjectures. Elsewhere in the activity we remove the play button entirely so students are only able to verify their conjectures through argument and consensus.

Students manipulate the transformations directly. Even in some very strong transformation applets, we noticed that students had to program their transformations using notation that wasn’t particularly intuitive or transparent. In this activity, students directly manipulate the transformation, setting translation vectors, reflection lines, and rotation angles using intuitive control points.

It’s an incredibly effective conversation starter. We have used this activity internally with a bunch of very experienced university math graduates as well as externally with a bunch of very inexperienced eighth grade math students. In both groups, we observed an unusual amount of conversation and participation. On every screen, we could point to our dashboard and ask questions like, “Do you think this is possible in fewer transformations? With just rotations? If not, why not?”

Those questions and conversations fell naturally out of the activity for us. Now we’re excited to offer the same opportunity to you and your students. Try it out!