Dan,

We are currently studying graphs and networks in one of my classes and we started out by looking at the London tube network, which we discovered was actually a graph not a network, but never mind, (some of them were familiar with it as I teach about an hour from London).

We have since looked at turning other transport infrastructure such as rail and motorways into networks so we can start to understand how computers work out shortest routes and similar.

I will ask them what they thing the different versions are useful for later this week to see what they think, but my intuition is the first (‘real’ scale, ‘real’ layout) is more useful to anyone who cares about how far it is between locations that are not connected, or how they relate to things not shown on the graph, while the second is for those who only care about connections.

Or alternatively; the second is a better abstraction that the first is we only care about connections because it may show the information we need more clearly; but the first is a necessary step on the ladder to get there?

Question: Why does the first not include the river, but all later ones do? I’m not sure!

Dan, I always emphasize the “which representation works in this situation?” question. I imagine the emphasis on the quadrinity came from trying to break students and teachers from automatically going to the equation or graph to get an answer. Now we’ve swung too far in the other direction. Here, let me draw you a symbol…

Another viewpoint when looking for the best, is that there are no best practices: http://www.satisfice.com/blog/archives/27

Multiple representations means linking ideas (graph, table, equation, sentences) to teach other and to a central concept. Cognitively, that is what creating meaning is all about: concepts are only meaningful when related to each other. Asking for the best representation is a step further, but you cannot have a “best” without having other representations to compare with. Whether your “give me four representations” handouts are efficient for your students must depend on whether your students need help understanding multiple representations, or if they already have a variety of them in their mental toolbox, in which case asking for the best, or worst, or yet another representation might be a better way to go.

While the first representation tries to remain true to distance, it is too complex in the middle of the city. It’s difficult to figure out which label goes with each stop because they are so close together.

The new representation makes the density of stops more even allowing more space for labels. This clarifies the center of the map which is probably theist used portion of the map. In addition the label colors match the color of the lines which helps with clarity as well.

That seems to be the biggest benefit of the new representation.

I wouldn’t have thought twice about these maps. Thanks Dan for helping us pause and reflect on things we often take for granted.

Choosing the “best” representation strikes me as a great way to enhance math learning with critical thinking. To make this productive, I’ve found that it’s really important to define criteria explicitly with students – ideally with their input. As Tom points out “the whole ‘what is better’ debate’… is always enlightening.”

Rather than using “efficient” or “best” for general qualifiers, I have worked with “which representation is most effective?” Criteria then include such things as efficient, accurate, builds my understanding, fits the situation. Working these out takes time but the set of criteria can be used over and over and becomes a personal “tool” for students who would otherwise not know where to focus their thinking.

At the Critical Thinking Consortium (TC2), we have been working on finding connections between critical thinking and math. I would love to hear from anyone who is interested in this aspect of math learning and teaching. Please email me – cynthia.nicolson@tc2.ca

As an aside, I lived in London for two years, and my internal map of London has been greatly skewed by the second subway map. It was very useful for understanding how to travel on the tubes, but not so useful for actually understanding where things were.

Really, what would be most useful is if London were willing to take advantage of the digital medium that they can share this map in (when it is shared online) and provide both options, and an overlay of the street-map, etc…

In print form, you could just then choose which model you preferred, and in digital form, you could switch between them seamlessly.

Great post and discussion!

I want to push a bit on one of Dan’s original questions:
> Aren’t we doing kids a disservice by emphasizing “multiple representations” rather than the “best representations?”

One way to resolve the tension here is to unpack the learning objectives we have for kids at any given time. The knowledge we are trying to instill is complex and layered. If we just want them to be able to solve a well-defined type of problem where the optimal representation is known, we can teach them to recognize those situations and to use the optimal representation in those cases. That is a perfectly legitimate approach.

If we want the students to be able to *define* a problem faced with an ambiguous or novel situation, however, then we probably need to teach them the skill of generating representations and evaluating which representation(s) is best given the problem parameters. Also a perfectly legitimate approach.

The key is to be clear about what we are trying to accomplish. It sounds to me like a lot of teaching is designed to do the former (teach specific approaches to specific problem types where they are known) while trying to simultaneously hedge against the latter (by giving kids practice with all four types of representations Dan mentions by default, for example).

People tend to think of education as having only benefits, but there are costs as well. Trying to accomplish too many objectives at once (as Dan hinted at) can create confusion, inefficiency, and can undermine student motivation. So I think a key question to consider in this dialogue is “what are the tradeoffs from the standpoint of learner experience and learning outcomes of pursuing one or the other or both of the learning objectives described above?”

@Chris Shore: Scaffolds that support learning to ride a bike are fine. But I read Dan’s original point about “texts, graphs, tables, and symbols on every problem” as being more akin to a situation where every time a kid wants to ride their bike to the end of the block we make them also practice riding their scooter, roller skates, and Big Wheel during the same session.

Thanks. And agreed (as long as the teacher is making the call (“multiple” or “best”) intentionally – with some specific end in mind – and not just driven by a vague uneasiness that “it can’t hurt and it might help” to expose the kids to a bunch of extraneous material along the way…)

@Nik: Returning to Dan’s original insight (as I understand it), it’s not that teachers shouldn’t ever use more than one representation in a problem. The point is that when multiple representations are used, they should be introduced strategically instead of “by default” for every problem. In the latter case, kids might learn to create the four different types of representations but have no clue how to choose among them when solving a novel problem. That’s the trap of inert knowledge.

Another way of saying this is that in addition to the “what” that is being taught – different representations, in this case – we need to make the strategic knowledge of “how” to choose intelligently between them an explicit part of the curriculum and not just teach the “what” and hope they all figure out the “how” on their own with enough repetition.

@Michael @Cynthia: There is a difference between teaching how and when to use tools (hammer vs screwdriver) and using tools to teach (training wheels on a bike). I used Christopher Danielson’s banner to demonstrate this principle http://christopherdanielson.wordpress.com/ in a blog that was inspired by this discussion. http://mathprojects.com/2012/11/22/multiple-or-best-reps/

@Chris: “There is a difference between teaching how and when to use tools (hammer vs screwdriver) and using tools to teach (training wheels on a bike).”

Good point.

I’m not sure that I agree that both maps are same-level abstractions of the real-world subway system. I would argue instead that the second map is an abstraction of the first.
In order to abstract away the lengths and shapes of the curves that connect the nodes, we need to have already interpreted the subway system as a network of curves and nodes – as the first map does – rather than as a three-dimensional physical structure.

Similarly, I would argue that graphs and tables-o’-values do not occupy the same rung; rather, a graph is an abstraction (and infinite extension) of a table-o’-values.

What I found myself doing for awhile there was passing out worksheets that had a section for all four representations. By default. I’m not sure that’s very good.

[emphasis mine]

I’m sitting here right now staring at CCSS Ratios and Proportions for grade 7. I see how this domain progresses from determining if quantities are proportional –> calculating constant of proportianality (using tables, graphs, equations, verbal, etc) –> representing relationships as equations to graphing –> identifying unit rates as per the relationship between the points (0,0) and (1,r).

Seems to me that 7th grade kids may be well served by a default worksheet that gives one representation and asks for the rest of them. Too often, kids are taught the representations in isolation and don’t see the connection which may be necessary before they can make a judgement as to which is best.