Comments on: [LOA] London Underground Maps

By: David Cox

David Cox — Wed, 28 Nov 2012 18:55:02 +0000

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.

By: Dan Meyer

Dan Meyer — Tue, 27 Nov 2012 23:08:46 +0000

Useful comment, Sean. I've highlighted it above.

By: Sean Wilkinson

Sean Wilkinson — Tue, 27 Nov 2012 18:24:46 +0000

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.

By: Amanda

Amanda — Sun, 25 Nov 2012 09:12:22 +0000

The same applies to many areas of teaching any concept in maths, the common question/answer being ‘why do I need to learn another way to do something when it already works my way.’ With regards to the tube map…..could you imagine tourists in London trying to figure the first one out? I live here and see them puzzled looking all the time. Maybe the current representation is the not the best anymore with all the new additions?

By: Michael Connell

Michael Connell — Fri, 23 Nov 2012 12:35:08 +0000

@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.

By: Nik

Nik — Fri, 23 Nov 2012 12:19:32 +0000

“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.”

Agreed.

By: Chris Shore

Chris Shore — Thu, 22 Nov 2012 17:36:48 +0000

@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/

By: Cynthia Nicolson

Cynthia Nicolson — Thu, 22 Nov 2012 16:50:07 +0000

@Michael Connell: 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.

Yes! – and I think this ability to “choose intelligently” is all about critical thinking, i.e. making reasoned judgments. We can support this “strategic knowledge” by helping students develop criteria for the choices they are making. For instance, if they are using different representations (diagram, list, table, graph, etc.) to help grasp and solve a problem, they could be considering whether a particular representation helps build their own understanding and fits the situation. Accuracy and efficiency might also come into play.

(Check out http://www.tc2.ca/ for more about teaching critical thinking.)

By: Michael Connell

Michael Connell — Thu, 22 Nov 2012 13:13:16 +0000

@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.

By: Nik

Nik — Thu, 22 Nov 2012 11:38:04 +0000

@Michael Connell: Presumably though, you eventually want the students to choose which way is ‘best’ for them to get to the end of the road, but without trying all the ways, and seeing them at the same time, they cannot make an informed decision.

It is for sure on the teacher to make certain they are showing multiple representations for a reason; but without using more than one in a problem how can you see which is ‘best’?