But *Arthur* nails the nuance in “Sue Ellen Adds It Up,” and reports three important truths about math in ten minutes.

**We are all math people. (And art people!)**

Sue Ellen is convinced she isn’t a math person while her friend Prunella is convinced there’s no such thing as “math people.” You may have this poster on your wall already, but it’s nice to see it on children’s television. Meanwhile, Prunella is convinced that, while she and her friend are both “math people,” only Sue Ellen is an “art person.” Kudos to the show for challenging that idea also.

**Informal mathematical skills complement and support formal mathematical skills.**

Sue Ellen says that she and her family get along fine without math everywhere “except in math class.” They rely on estimating, eyeballing, and guessing-and-checking when they’re cooking, driving, shopping, and hanging pictures. Prunella tells Sue Ellen, accurately, that when Sue Ellen estimates, eyeballs, and guess-and-checks, she *is* doing math. Sue Ellen is unconvinced, possibly because the only math we see her do in math class involves formal calculation. (Math teachers: emphasize informal mathematical thinking!)

**We need to create a need for formal mathematical skills.**

Sue Ellen resents her math class. She has to learn formal mathematics (like calculation) while she and her family get along great with informal mathematics (like estimation). Then she encounters a scenario that reveals the limits of her informal skills and creates the need for the formal ones.

She’s made a painting for *one* area of a wall and then she’s assigned a *smaller* area than she anticipated. She encounters the need for computation, measurement, and calculation, as she attempts to crop her painting for the given area while preserving its most important elements.

Nice! Our work as teachers and curriculum designers is to bottle those scenarios and offer them to students in ways that support their development of formal mathematical ideas and skills.

[h/t Jacob Mehr]

]]>This is my best attempt to tie together and illustrate terms like “intellectual need” and expressions like “if math is aspirin, how do we create the headache.” If you’re looking for an elaboration on those ideas, or for illustrations you haven’t seen on this blog, check out the video.

**The Directory of Mathematical Headaches**

This approach to instruction seriously taxes me. That’s because answering the question, “Why did mathematicians invent this skill or idea?” requires a depth of content knowledge that, on my best days, I only have in algebra and geometry. So I’ve been *very* grateful these last few years to work with so many groups of teachers whose content knowledge supplements and exceeds my own, particularly at primary and tertiary levels. Together we created the Directory of Mathematical Headaches, a collaborative document that adapts the ideas in this talk from primary grades up through calculus.

It isn’t close to complete, so feel free to add your own contributions in the comments here, by email, or in the contact form.

]]>**Knowledge About Teaching**

Lisa has read widely from sources online and offline and has a great memory. So when she asks herself, “How am I going to teach [x]?” she can quickly summon up all kinds of helpful posts, essays, books – even the mental recording of previous classes she’s taught on [x].

**Knowledge About Students**

I stopped to think about how this would work with my class.

Lisa has taught long enough and knows her students well enough that she can test each of those resources out in her head, *all during the lunch break before class*. You can see her swiping right and left on each of them – “Yeah, maybe this idea. Definitely not that one.” – as she sees her students in her imagination. I’m sure Lisa is open to the possibility that her flesh-and-blood students will differ in surprising and awesome ways from her *mental model* of those students. I wouldn’t bet against her intuition, though.

**Knowledge About Math**

She ultimates decides to start her precalculus students with the elementary school analog of their lesson, turning an abstract fraction division problem into a more concrete one.

Then, as her students acquaint themselves again (or in some cases for the first time) with helpful models for that division, she builds back up to the abstract version of her task.

Lisa is only able to move up and down the ladder of abstraction like this because she *knows* a lot of math – specifically where it builds from and towards. If she *doesn’t* know that math, her options for helping her students basically shrink down to “let’s solve a few together.”

**Finally**

I don’t know if it’s possible to *practice* what Lisa is doing here. It’s *knowledge*, the tightly connected kind you get when you spend thousands of hours in math classes, reflect on those observations, write about them, talk with other people about them, and then use them to inform what you do in *another* math class.

It’s possible, even easy, to spend the same number of hours *without* acquiring that tightly connected knowledge.

It’s something special to see it all put to use.

**BTW**. My guess is a lot of those knowledge connections were tightened because Lisa is a dynamite blogger. On that theme, let me recommend The Positive Effects of Blogging on Teachers, an article which does a great job describing ten reasons why teachers should think about blogging.

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.

- 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

outsideof class and so bored by theminsideof 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:

- 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.

- 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.

]]>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.

**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.

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.

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 expressionsand 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!

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

**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!

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