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?

**BTW**

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**

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.

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.

## 9 Comments

## Sue H.

November 14, 2016 - 5:06 pm -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.

In my experience, traditional teaching puts far too much emphasis on learning the equation with a graph pattern attached to it almost like an afterthought. If the students can memorize these pairs correctly, they move on. If not, they sink. When I say to students that equations are supposed to evoke pictures which in turn represent information or a relationship, they give me the blank look. Whatever level I teach, I now start with the visual — information turned into a picture — and stick with it until the students connect some math language to key points on the graph and then can use that language to build an equation that expresses the same relation or achieves the same set of outputs for given inputs. When they can comfortably predict what changes in the graph will do to the equation and vice versa, then I have some confidence that when they read an equation they can also visualize a particular graph pattern and vice versa.

## Dan Meyer

November 14, 2016 - 5:21 pm -Super helpful analogy, Sue. Added to the post.

## Katie Waddle

November 14, 2016 - 9:19 pm -Just want to point out that CPM (where a lot of the talk of representations originates I think) actually does a lot of work around having students think about which representation is helpful at a given time. Maybe you could look there for more ideas.

In my class we do a lot of work learning how to use color/words/arrows etc to show off the features of a graph/table/equation, since one point of representing something different ways is seeing how the different representations are connected. 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.

## Dan Meyer

November 15, 2016 - 4:34 pm -Love those bridging questions. Not just “How is the table connected to the graph?” but more specifically, “How do you see the y-intercept in both the table and the graph?” Added to the post.

## Lesley Cowey

November 15, 2016 - 9:01 am -I think there is a practical problem here – students can see that the equation is not going to describe the realities of a race, in which people speed up, slow down, fall over, different bits of them are ‘ahead’ at different times … If the maths is tied to a real world context which is not accurate or exact, it’s difficult to justify a more accurate or exact solution.

I have encountered similar problems trying to justify the topic of geometric construction with compass and straight-edge as more exact than drawing because it represents an algebraic process – students can’t imagine a situation in which that degree of accuracy matters, so they may understand but they don’t feel the significance.

## Paul

November 19, 2016 - 10:16 am -As a current college student studying mathematics education, a lot of conversations in my classes have revolved around Universal Design for Learning. These conversations focus around multiple means of representation, so when I am planning lessons and reflecting on lessons I have taught or someone else has taught, I am always thinking about how the information will be or was represented in multiple ways. It wasn’t until reading this blog post that I thought about what students thought of this. Yes, these representations are present to engage students and help them learn, but do they understand that? This post has encouraged me to think through creative ways for students to understand this, and I am interested to read further comments on how current educators have found success in this area.

## John Manicke

November 30, 2016 - 3:20 pm -Hello, I am a student in the mathematics secondary education program at the University of Illinois. I really appreciate the “backwards blue graph” representation idea as a means of pointing out an issue that we have in mathematics. It really opened my eyes to what students may be thinking. However, I think it is a stretch to call it a problem with multiple representations. Instead, I think that it is only a part of a much larger problem in mathematics which is that too often we focus too much on procedures without enough emphasis on conceptual understanding or “the why”. Sure in this case it is different representations of lines, but it could just as easily be the steps of a geometry proof or solving problems with trigonometric proofs. I would be interested to hear what you all think. Do you agree or is there something fundamentally different about multiple representations that I may be overlooking? Thanks.

## Chen Earon

December 4, 2016 - 2:39 am -I also use the analogy of a map in my class. I draw a simple map on the whiteboard with streets names, city sites (Bank, School, etc) and two houses. I then ask my students to show me the way from one house to the other. After sketching a line between the two houses I ask them to describe the way for me in two different ways: using street names (walk along red street and turn right at blue street) and then using sites on the way (you pass the bank and when the school in to your right you turn right and pass the clinic).

We then have three different representation of the same way: one is graphic, another is a collection of points while the third is a description of the line tendency or direction (slope).

## Melie

December 15, 2016 - 3:02 pm -I have a IEP for math and English but this is about math. I’ve needed extra help in math since I was in 1st grade. I’m in Algebra 1 and my teacher pointed out something to me every time I graphed a equation. I always graph the equation backwards. So I have one question. Do I have dyscalculia?