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.
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.
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.