## [LOA] London Underground Maps

Here are two maps of the London underground railway, the first from 1928, the second from 1933.

1928

1933

I stipulated earlier that the act of abstraction requires a context (some raw material) and a question (a purpose for that raw material). These are two different abstractions of the same context. So what two different purposes do they serve? Rather, whom does each one serve?

BTW. If you’ll let me troll for a minute: aren’t we doing kids a disservice by emphasizing “multiple representations” rather than the “best representations?” Given that some abstractions are more valuable than others for different purposes, why do we ask for the holy quadrinity of texts, graphs, tables, and symbols on every problem rather than for a defense of the best of those representations for the job given?

BTW. I pulled those maps from Kramer’s 2007 essay, “Is Abstraction the Key to Computing?

2012 Nov 19. Christopher Danielson links up two examples of curricula (CMP) emphasizing “best representations” over “multiple representations.”

Nik:

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.

Sean Wilkinson

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.

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. More here.

1. #### Nik

November 19, 2012 - 1:27 pm -

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!

2. #### Dan Pearcy

November 19, 2012 - 1:35 pm -

In quick response to your first BTW.

It seems fair and appropriate to allow for multiple representations to see how different students construct things from their won reference frame. Then we can go ahead and discuss which representation would be most efficient/beneficial for the given problem.

3. #### eddi vulic

November 19, 2012 - 1:35 pm -

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…

4. #### David Cox

November 19, 2012 - 1:39 pm -

aren’t we doing kids a disservice by emphasizing “multiple representations” rather than the “best representations?”

Can we ask for “best representation” before we have emphasized “multiple representations?” Or do you think asking for best will motivate students to actually learn the different representations?

For some reason, my gut tells me one has to come before the other and that emphasizing multiple representations helps students see there are many ways to tell the same story. However, I don’t think we should ever stop at “show me” when “which one is best?” is an option.

5. #### Timon Piccini

November 19, 2012 - 1:44 pm -

I am curious about your bit of trolling. I’d like to hear it more defined.

I see multiple representations as steps on the rung of abstraction.

Many have seen this video, and I think it shows that yes there are more efficient means, but to throw out the visual representation altogether is forgetting where that particular representation fits on the ladder of abstraction.

I like the physics teachers idea of breaking models. Challenging the efficiency. One thing I hear students say is “There must be an easier way!” And
I think multiple representations open students up for a special motivating factor. How can we do this same process more efficiently?

That all being said I assume that you recognize this, and you mean more particularly that always showing all representations is not worthwhile because students need to learn how to decide which model is most efficient for what particular reasons.

That actually may be an interesting form of assessment. “Discuss why this model is efficent/inefficient for this particular scenario.”

6. #### Dan Meyer

November 19, 2012 - 7:21 pm -

Let’s feature Nik’s comment in the post itself.

David Cox:

Can we ask for “best representation” before we have emphasized “multiple representations?” Or do you think asking for best will motivate students to actually learn the different representations?

For some reason, my gut tells me one has to come before the other and that emphasizing multiple representations helps students see there are many ways to tell the same story. However, I don’t think we should ever stop at “show me” when “which one is best?” is an option.

I’m not going to draw a hard line here. (Like I said, I was mainly trolling.) I think exposing kids to different representations is a good thing. I think asking students questions that require those different representations is a great thing. But both are good. 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.

7. #### Troy McConaghy

November 19, 2012 - 8:26 pm -

If you’re a tourist new to London and you want to get from Bounds Green to Euston, then neither map is the best representation. Both have way more information than you need. What you want is directions like this:

“At Bounds Green Tube Station, take the Picadilly Line south to the King’s Cross St. Pancras Tube Station. Transfer to the Northern Line…”

In other words, you want a smartphone and Google Maps, or similar.

(As it happens, Google Maps says to take the bus!)

8. #### Julia Tsygan

November 19, 2012 - 11:59 pm -

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.

9. #### Tom

November 20, 2012 - 5:30 am -

As someone who was introduced to subway maps in a baptism by fire way (I went to Europe before ever using a subway), I strongly wish I had the first representation in all the cities I visited. As Nik mentioned, it gives you a real scale, real location representation of the city. If I’m deciding between the subway and walking, the first representation helps me more.

However, then we can ask the next question (I think students would also agree that the first is better): If we think the first representation is more useful, why have all the major cities turned to the second representation. What features does it have that make is “better”?

Then, of course, we get into the whole “what is better” debate, which is always enlightening.

10. #### Raj Shah

November 20, 2012 - 8:05 am -

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.

11. #### James Key

November 20, 2012 - 8:29 am -

Dan wrote: “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.”

Dan, you actually passed out *worksheets?* You mean like using *real paper?* And here I thought that was beneath you…

Weighing in on multiple representations: I think we can all agree that powerful users of math have some facility with words, graphs, equations, and numbers. So we should emphasize these as often as possible. But do we need to use all of them every time? Nahhh.

12. #### Cynthia Nicolson

November 20, 2012 - 8:42 am -

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

13. #### Breedeen

November 20, 2012 - 9:16 am -

Full disclaimer: I am a subway map junkie. I love subway maps. I can still remember the sweatshirt one of my students had in my first year of teaching that had a subway map as its design. How I coveted that garment… I spent many moments trying to determine which city it was from.

At least for me, if I know where I am and I know where I want to go in terms of subway stops, the second map is better. It is graphically simpler and clearer and therefore easier to read. Now, if I don’t know where I am in terms of subway stops, I’m kind of at a loss, but that is true for both maps.

I suppose if I wanted to estimate how long it would take me to go from stop HERE to stop THERE, knowing the relative distances would be an asset. Although, knowing the difference between the local and express, and whether the train is above or below ground and all sorts of other variables come into play for that question.

It seems to me that the problem that many are pointing out with both maps, is that there is no real overlay of the subway system with the city streets. The first map is a slightly better model with which to compare your street map with the subway map. However, there is still a translation that has to be made to connect my subway stop to the fancy restaurant I’m trying to find, or the tourist trap, or my mother in law’s new apartment building.

As Julia Tsygan mentioned, using multiple representations is about making connections. Subway maps are a good (dare I say “real world”) example of having to connect different pieces of information in order to figure out how to get to the place I want to go. Of course, as Troy McConaghy points out, I could just use my smartphone.

14. #### David Wees

November 20, 2012 - 12:36 pm -

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.

15. #### Lucy M

November 20, 2012 - 4:31 pm -

I hope I’m not being redundant here, but I’d like to comment on “multiple representations” versus “the best representation”… your notion of the “best representation” is intriguing but I also beg the question “Best for whom?” My school prescribes to and uses Gardner’s theory of Multiple Intelligences in our approaches to teaching, and while there’s no arguing to those of us reading this blog (most of which are likely strong mathematical logical thinkers) that there is a BEST representation of the four multiple representations we are all so used to. But which one is best for our students, I believe, will vary from student to student. I’ve always believed the “best” approach is the one the student understands with the most clarity and can perform with minimal risk of error.

16. #### Michael Connell

November 21, 2012 - 5:53 am -

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?”

17. #### Chris Shore

November 21, 2012 - 6:15 pm -

If the goal is to have the students apply their skills to solve a problem, then yes, they don’t need both a hammer and a rock to embed a nail. If the goal is offer students one representation as an access point that will eventually lead to another representation higher up the ladder of abstraction, then I pose this question, “Are we doing the kids a diservice by offering training wheels when learning to ride a bike?”

18. #### Michael Connell

November 21, 2012 - 6:36 pm -

@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.

19. #### Chris Shore

November 21, 2012 - 6:54 pm -

Nice visual. Agreed. We don’t want them starting at the bottom of the ladder everytime we want them to reach for the next rung. My point is that it is up to the teacher to determine when “multiple” or “best” is the better call.

20. #### Michael Connell

November 21, 2012 - 7:14 pm -

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…)

21. #### Nik

November 22, 2012 - 3:38 am -

@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’?

22. #### Michael Connell

November 22, 2012 - 5:13 am -

@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.

23. #### Cynthia Nicolson

November 22, 2012 - 8:50 am -

@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.)

24. #### Nik

November 23, 2012 - 4:19 am -

“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.

25. #### Michael Connell

November 23, 2012 - 4:35 am -

@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.

26. #### Amanda

November 25, 2012 - 1:12 am -

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?

27. #### Sean Wilkinson

November 27, 2012 - 10:24 am -

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.

28. #### David Cox

November 28, 2012 - 10:55 am -

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.