Representations are tools for thinking, explaining, justifying. NB: Pape/Tchoshanov, 2001. pic.twitter.com/Nux9jP7Z4b— Matthew Oldridge (@MatthewOldridge) June 7, 2016

The Pape/Tchoshanov paper is instructive for teachers in getting kids to develop their own mental representations, and to get them out into the physical world.

In their words:

"We use the term

*representation(s)*to refer to both the internal and external manifestations of mathematical concepts."This is a huge argument for paying attention to your students' thinking, in order to make it visible. We need to help kids visualize representations of interesting mathematical concepts, in order to imagine them into being.

So we have two complementary aspects of mathematical representations:

-the act of representing (verb)

-the representation itself (noun)

This quite neatly corresponds to both NCTM and Ontario curriculum definitions of representation: as both process and product.

The authors propose a "'zone" of interaction between mental and physical interactions.

The nice example used is of "six"-a kid develops an internal mental image of "'six', then connects it to sets of six objects, the numeral 6, the written word "six", the spoken word "six", and so on.

As kids advance in math, we should encourage representational thinking. That is, building and having a repertoire of ways to externalize math concepts. Tables, graphs, tree diagrams, area models, arrays- these are all examples.

The authors put it nicely:

"the development of students' thinking skills requires a multiple representational approach."

Put simply: representations are tools for reasoning.

Two things to think about.

Kids often mathematize math tasks by drawing pictures with little or no mathematical content. Do these count as representations?

"Students often produce representations that lack meaning."

Manipulatives are generally a good thing, as long as they are accurate mathematical representations of the concept, the mathematics you are working on. Daniel Willingham presents this caveat here.

Manipulatives, if not chosen wisely, can get in the way of mathematical representation. Let the tool fit the task (and the math).

The end goal of representation is good communication. Can the student communicate their thinking, in words, or on paper, using mathematical representations?

Even better, can they start to see how different representations are connected? Do they see a table of values, a pattern made with cubes, a graph, and an algebraic expression as fundamentally the same thing?

How do they show the same math in different ways?

Fluency with multiple representations should be a goal of thinking math classrooms; we can help kids to visualize, conceptualize, and make real their understanding of powerful mathematics.

"The end goal of representation is good communication. Can the student communicate their thinking, in words, or on paper, using mathematical representations?"

ReplyDeleteThis is a piece that speaks to me the most. As I get more experienced in my teaching, especially over the past few years, I've found myself becoming less and less concerned about students getting the perfectly right answer and more concerned about how they show their thinking around their (sometimes) wrong answer.

I've started loading all of the curriculum that applies to graphing functions into the first month of the course I'm teaching and worry about the algebra afterwards. I've found once students have a good picture in their heads of what a particular function looks like or behaves, the tricky math of solving/manipulating/transforming gets easier.

In the end, I want kids to approach problems in a similar way to how mathematicians would approach them...with multiple tools at their disposal and ready to use. When I write solutions to a quiz/test/exam I have paper, calculator, Desmos all ready to use.

To me, a student tries a problem, gets it wrong, and can tell me why their algebra and their graph aren't agreeing...they are on the right track to mathematical fluency.

True fluency is about understand WHY multiple representations work...not just how they can work...I think?