Comments on: The Problem with Multiple Representations

By: Melie

Melie — Thu, 15 Dec 2016 23:02:05 +0000

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?

By: Chen Earon

Chen Earon — Sun, 04 Dec 2016 10:39:45 +0000

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).

By: John Manicke

John Manicke — Wed, 30 Nov 2016 23:20:30 +0000

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.

By: Paul

Paul — Sat, 19 Nov 2016 18:16:27 +0000

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.

By: Dan Meyer

Dan Meyer — Wed, 16 Nov 2016 00:34:08 +0000

In reply to Katie Waddle. 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.

By: Lesley Cowey

Lesley Cowey — Tue, 15 Nov 2016 17:01:07 +0000

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.

By: Katie Waddle

Katie Waddle — Tue, 15 Nov 2016 05:19:35 +0000

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.

By: Dan Meyer

Dan Meyer — Tue, 15 Nov 2016 01:21:15 +0000

In reply to Sue H.. Super helpful analogy, Sue. Added to the post.

By: Sue H.

Sue H. — Tue, 15 Nov 2016 01:06:10 +0000

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.