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The Problem with Multiple Representations


I propose we add a representation to the holy trinity of graphs, equations, and tables: “backwards blue graphs.” Have a look.


Expert mathematicians and math teachers instantly see the uselessness of the backwards blue graph representation. It offers us no extra insight into or power over the data. But my suspicion is that many students feel that way about all the representations. They’re all the backwards blue graphs.

Students will dutifully and even capably create tables, equations, and graphs but do they understand the advantages that each one affords us? Or do they just understand that their grades depend on capably creating each representation?

At Desmos, we created Playing Catch-Up to put students in a place to experience the power of equations over other representations. Namely, equations offer us precision.

So we show students a scenario in which Julio Jones get a head start over Rich Eisen, but runs at half speed.


We ask students to extend a graph to determine when Rich will catch Julio.


We ask students to extend a table to answer the same question.


Finally we offer them these equations.


Our intent with this three-screen instructional sequence is to put teachers in a place to have a conversation with students about one advantage equations have over the other representations. They offer us more precision and confidence in our answer.

Without that conversation, graphs, tables, and equations all may as well be the backwards blue graphs.

Your Turn

The power of equations is precision. We put students in a place to experience that power by asking them to make predictions using the imprecise representations first.

In what ways are graphs uniquely powerful? Tables? How will you put students in a place to experience those powers?


Principles to Actions is great here:

Students should be able to approach a problem from several points of view and be encouraged to switch among representations until they are able to understand the situation and proceed along a path that will lead them to a solution. This implies that students view representations as tools that they can use to help them solve problems, rather than as an end in themselves.

Featured Comment

Sue Hellman:

I think an analogy here are the 3 ‘representations’ of location/directions provided by Google: a map, written directions, and street view. They all provide similar or at least related information but each offers advantages depending on the purpose and background knowledge of the user.

Katie Waddle:

I’ve noticed that kids who get the click that they all are connected understand stuff down the road a lot better, so I build in explicit teaching around seeing those connections (where is the y-intercept in the table? how can I see the slope in the equation?). It would be neat if kids could color code and write on things in this exercise, but computers are not good at letting you add stuff like that.

What I’m Working on the Day After the Election

Every morning, the four members of the teaching team at Desmos post a note to their Slack channel listing all the tasks they’re working on that day. We use the hashtag #workingon for easy reference. This is what I posted this morning.

I don’t know how any of you voted and I won’t make assumptions. (It’s clear that a lot of people who represented themselves one way to pollsters voted another way, and that likely holds true for our company as well.) But you may have voted like I did yesterday, leaving you bereft today, and struggling to locate some kind of purpose for your work, struggling to participate in the solution to a problem that has many names. If that’s you, then this what I’m telling myself about our work this morning.

If the name of that problem is economic anxiety, if President-elect Trump was propelled to power by people whom globalization, open borders, and free trade have left behind, I encourage us to locate political and social solutions to their problems, definitely, but also to help those people (and their children, particularly) learn better math better. Capitalists continue to automate routine manual jobs, leaving behind more and more non-routine cognitive jobs. Non-routine math tasks are difficult to design, difficult to teach, difficult to learn, and increasingly essential to full economic participation. We can help design them and we can give teachers tools to make them easier to teach.


If the name of that problem is bigotry, then we should help teachers facilitate constructive arguments, cultivate empathy, and emphasize patience. One dimension of bigotry is impatience, a sense that “I know everything there is to know about a person based on his or her most easily observed characteristics.” The traditions of many math classes – completing short problems resulting in simple answers that are easily verified in the back of the textbook – only exacerbate this problem. Christopher Danielson’s “Which One Doesn’t Belong,” by contrast, invites students to realize that all of those objects don’t belong for one reason or another, that we can negotiate those reasons productively, and that we can understand the world through the eyes of another.


Obviously we have lots of work to do in our neighborhoods, our churches, our social networks, our local and state governments, and in ourselves, work that is probably larger than anything we’ll do at Desmos today. But if yesterday’s election left you wondering what work you can do at Desmos to help solve a problem with many names, this is what I’m #workingon.

BTW. I’m watching Twitter for examples of math teachers helping their students understand where they live today. I’ll continue to update this post throughout the day.

Matt Enlow:

When you learn mathematics, you also learn a lot of other things. Here are three of those things.

John Golden:

We did Elizabeth Statmore’s talking points for Math Mindsets Chapter 7 (tracking), then for the election, then we looked at Megan Schmidt’s Social Justice Math slides.

wwntd offers her classes some words of consolation.

Dianna Hazelton asks her class:

What does the word empathy mean? How do you show empathy?

How I’m Learning to Step into Math Problems

The biggest inhibitor to my development as a math teacher is that I don’t teach or do math enough. That should make plenty of sense.

I’ve ramped up my teaching since fall with regular (okay, monthly) sessions at a local San Francisco high school. Opportunities to do math are bit easier to find and a bit easier to wedge into empty corners of my day than classroom teaching.

I was grateful, for example, that Jennifer Wilson built plenty of time for doing math into her workshop today at North Carolina’s state math education conference. She posed this problem (source unknown) and I experienced two insights into how I experience mathematical insights.


First, I approximate an answer. I recognize that the diameter of the circle will be larger than the side of the square. That’s because I can draw the diameter in my imagination and compare the lengths, and also because I know that chords in a circle are never longer than the diameter. I’m guessing the diameter is around 25 units, not more than 30, and not less than 21.

Second, I try to figure out what makes the thing this thing rather than some other thing. I don’t have any details about how the square was constructed. The circle could be any circle, but what makes that square that square? I need to construct it myself. I start changing the square’s location and scale in my head, asking myself, “Is this square legal? What about this square?”

Here is what I see in my head:


When the square becomes legal at the end, I hear an actual “ding” inside my brain. That’s when all the constraints make sense to me and I can start writing down variables and relationships.


That last “20 – r” was only possible because of the exercise of mentally making different illegal and legal squares.

From there, I trotted over to Desmos with a Pythagorean relationship in my hand.


Because I had approximated right and wrong answers earlier, I knew that 12.5 was too low. I realized that was the radius so I doubled it for the diameter.

I think these techniques are what Piggott and Woodham call “stepping into the problem.”

Here visualisations are used to help with understanding what the problem is about. The visualisation gives pupils the space to go deep into the situation to clarify and support their understanding before any generalisation can happen.

At least that’s the best term I can coax from the Internet. I don’t know if Polya’s work on problem solving speaks to that practice directly.

  • If you have another name for that process, let us know it.
  • If you’ve made mathematical problem solving a part of your development as a teacher, let us know how.
  • And if you have an interesting problem to share, let us know about that too.

I’ll leave you with this awesome little number from Brilliant. I promise you can solve it.


Featured Comments

William Carey:

If you ask a literature teacher what book they read most recently for pleasure and they don’t have an answer that’d be really worrying. But I bet it’s pretty rare. If you ask a math teacher what math problem they most recently worked through for pleasure, I bet the results are much scarier.

Lori M:

For decades, there has been a focus in ELA classes around a push that teachers who read, know how to teach (and reach) readers. Let’s start a similar movement among math educators.

Dick Fuller:

With some over simplification, real problems are not about mathematics, certainly not about arithmetic. The problem is the formulation of the problem. To suggest a problem is particular to values of parameters points toward evaluation as the critical component of its solution. It is not. Evaluation is particularization of a general formulation. A bald assertion: this is at the root of the difficulty students have with “real” problems.

Simon Gregg offers his own solution and then links to a fascinating question about perimeter.

[Pseudocontext Saturday] Mazes

This Week’s Installment



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

Pseudocontext Saturday #2

  • Using properties of symmetry (33%, 134 Votes)
  • Calculating the area of quadrilaterals (28%, 114 Votes)
  • Completing the square (20%, 80 Votes)
  • Identifying rational and irrational numbers (19%, 76 Votes)

Total Voters: 404

Loading ... Loading ...

(If you’re reading via email or RSS, you’ll need to click through to vote.)


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 four possibilities for that connection. One of them is the textbook’s. Three 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.)

Current Scoreboard

Team Me: 1
Team Commenters: 0



The judges rule this pseudocontext because, given that awesome square maze, it’s very unlikely that anyone would wonder about the side length of the maze and unlikelier still that anyone would wonder if the side length was rational or irrational. An exhaustive search for a 1,225 ft2 square maze in Dallas, TX, produced no results, exacerbating the judges’ sense that the textbook is exploiting the world for the sake of math. That’s pseudocontext.

How I’m Voting in NCTM’s Upcoming Election


Here is how I’m voting in the upcoming NCTM board election. Ballots close 10/31. You should vote too.

There are few issues in mathematics education that both matter a lot and that NCTM can directly affect. One issue in that subset matters most to me:

I care how well NCTM accesses the capacity of its members to help each other develop continuously as educators.

NCTM has the largest store of teaching knowledge of any math education organization in the world. Its 70,000 members comprise hundreds of thousands of years of math education experience. But NCTM accesses that capacity only sporadically. Fewer than ten times yearly at face-to-face conferences. Twelve times per year in its five journals. Occasionally in books and blog posts.

The only medium that will allow an NCTM member in Scranton, PA, to help another member develop continuously in San Diego, CA, is the internet. My tweeting and blogging colleagues know exactly what I’m talking about. They know the exhilaration of asking a question from a veteran and getting an answer in minutes. They know what it’s like to read someone’s interesting idea one day, try it out the next, and then offer the originator some useful feedback.

They’re developing, and developing each other, continuously. They don’t want to wait for conferences, journals, books, or blog posts.

So how am I voting? A few years ago, I’d vote for any candidate who even mentioned the internet in her candidacy statement. Now I’m looking for people who have a plan for helping NCTM’s members develop each other continuously. I’m looking for people who seem receptive to the experiments in online professional development Zak Champagne, Mike Flynn, and I put together annually under the name “ShadowCon.” I’m looking for people who understand that NCTM’s membership is underutilized for most of the year.

Here are promising excerpts from the candidates’ statements.

Robert Q. Berry III (President-Elect) [Twitter]:

Membership is a major challenge facing the Council. NCTM must rethink its membership model, working to ensure that longtime members continue to value NCTM while showing potential members the value of associating themselves with NCTM. This can done by tapping into their interests in social media and other digital technologies to promote interactive communities of professionals. Such efforts broaden the Council’s space for professional learning while maintaining meaningful engagement with the membership.

Nora Ramirez (President-Elect):

NCTM has the knowledge, experience, and skills to support both national and state affiliates in developing the abilities to advocate effectively for issues that are critical to them. Affiliates interested in this initiative would meet both face-to-face and online to learn, plan, and collaboratively develop or identify resources.

David Ebert (Director, High School):

NCTM needs to consider all forms of professional learning, including electronic learning opportunities, sustained yearlong professional learning, and joint professional learning opportunities personalized for the needs of the teachers within an affiliate.

Jason Slowbe (Director, High School) [Twitter, Web]:

NCTM should develop an online platform offering members a living portfolio for their professional development. NCTM already attracts top-notch speakers; now it should empower speakers with tools for building a following and facilitating year-round development. Attending sessions should be the beginning, not the end, of the conference experience. NCTM should enable attendees to pin, share, and discuss resources from within and beyond NCTM, including conference handouts, blog posts, articles, and student work. Integration with affiliate conferences and other stakeholders would connect teachers and grow membership organically. NCTM should leverage both the power of collaboration and its unique position as the world’s largest math education organization to influence more teachers and students.

Rick A. Hudson (Director, At Large):

Teachers today communicate in very different ways from the past, and NCTM must make use of the new media while building on its current strengths to reach a wider audience. For example, the quality of NCTM’s conferences is one of the Council’s greatest strengths, and we must think proactively about ways to share content from conference sessions virtually to reach a larger group of the membership and to extend the conference experience for those in attendance.

DeAnn Huinker (Director, At Large) [Twitter]:

A task force on building the next generation of teachers can consider resources, tools, and innovative ways to reach out to prospective teachers, such as providing access to blogs and online mentorships.

Daniel J. Teague (Director, At Large):

NCTM should take the lead in creating online and downloadable video courses (see Jo Boaler’s How to Learn Mathematics and Scott Page’s Model Thinking) to be used by individual teachers and departments for extensive work in these areas.

Desha L. Williams (Director, At Large):

Maintaining and expanding membership is a challenge for NCTM. The age of technology has created avenues for teachers to access information that was once available only within NCTM resources.

Vanessa Cleaver (Director, At Large):

Although I am a huge fan of Facebook, Twitter, LinkedIn, and other social media, I believe that these sources are to some extent now meeting the needs of educators for interaction with one another and exchange of information in non–face-to-face settings.

That’s what matters to me and how I’m voting. What about you?

Featured Comments

  • Steve Weimar outlines NCTM’s current efforts towards helping teacher develop continuously online.
  • Cal Armstrong wants to see current or recent teachers in leadership positions
  • Brandon Dorman would like to see NCTM accredit its members using technology like Mozilla’s Open Badges.