Let’s drop down a rung and make abstraction concrete.
You’re walking across a street. This is a photograph of what you see.
This is your context. What is its abstraction? There’s no way to know because you don’t know your purpose here, your question.
You ask yourself, “What colors do I see?” Now you have a question and you’re on the ladder of abstraction.
You start speaking very informally about the context, perhaps comparing one shade of green to another. You ask yourself, “What’s important here?” and decide it doesn’t matter whether the green thing is a car or a tree. All that matters is its greenness. This is abstraction. You’re removing aspects of the context that are inconsequential to your question.
Now you have to decide how to represent the consequential aspects. You could represent them with words:
A lot of grays on the street and sidewalk. Light blue in the sky. Red on the curb. Different shades of green in the trees and on a car.
Different representations are more useful for different purposes. This representation might work if you were writing some prose about the colors. If you wanted a more precise representation, though, you might turn to a histogram of the red, green, and blue values.
Now if the question changes, the entire ladder changes. If your question is, instead, “How do I get home from here?” different predictions are useful, different information becomes consequential, and the representations of that information will look nothing like the histogram we used to examine color.
A useful abstraction of this scene would be an overhead view of the terrain.
Of course, we only care all that much about the roads, not the trees or houses in between them, so we abstract all that away.
If our purpose here is to create some kind of enormous geolocation system, we don’t really care whether or not a road curves. We just care whether or not the road connects one intersection to another, or, abstracting those terms a little, we care whether or not an edge connects one vertex to another in a graph:
An array would be a representation of the graph that’s friendly to manipulation by a computer, though as a human, I miss a lot of the visual information we’ve abstracted away.
Great. But not perfect. This representation will only tell us whether or not it’s possible to get from one point to another –Â whether a route exists. If we want to find the shortest route, we add another useful variable, “abstracting over distance,” at it’s said.
If we want to find the fastest route, we’ll also need to abstract over the speed limits of each of those edges.
That’s a concrete example of the process and ladder of abstraction. The adjectives “concrete” and “abstract” just aren’t all that useful here. Everything is concrete if you think about the rungs above it and everything is abstract when you think about the rungs below it. The photograph that kicks off the post is more concrete than everything that comes after it but it’s also more abstract than the full-bleed, full-audio, moving panorama you experienced as you walked across the street. What matters isn’t the rung itself but how deftly you can move between all the rungs above and below you on the ladder of abstraction.
Checking For Understanding: Give an example of abstraction as it exists in your own life, in the problems you or other people try to solve. Two examples to kick off our list:
- Airplanes landing at night don’t care about the color of the tarmac or the grass on either side. All they care about are the margins of the landing strip, which are therefore lit up by lights.
- Google’s self-driving cars abstract away a metric ton of data that your senses usually take in while driving – the color of the sky, the music in the car, the humidity outside, etc. It also retains a metric ton of data, of course, and the quality of Google’s abstraction of the roadway will determine whether these things will kill us or let us (once again!) text while driving.
Should I call or text? If the message is short and quick, I could just shoot a text, but what is my data plan like, and are there financial considerations? Or I could call, and risk a phone conversation which I perhaps don’t have the time or want for at this point in my life? How important is the message? Or will I possibly see this person soon anyway and it’s all a moot point? Or maybe I could just tweet it? But how sensitive is the message?
One way I can see abstractions is to think of domains where I’m not an expert. Is it an abstraction when two kids who are really into video gaming communicate their solutions to challenges in terms of button pushes rather than the story on the screen? As in, I’m likely to say, “Gee, I wish I could make Mario jump up and do a flip in the air to get that gold coin without being hit by the hammer.” Whereas a Mario expert is likely to tell a fellow expert, “That level’s easy. It’s just right-right-A-left” (or whatever).
When looking at my email inbox, I disregard most of the information presented there. During the busy workday I “see” only the messages that are highlighted as unread and which were sent to me individually rather than schoolwide. This level of abstraction sometimes presents problems, so if I’ve read an important message but haven’t yet resolved it, I have to star it AND mark it as unread, tricking it back into the category that registers.
One thing a great context / question also gives you is the experience of figuring out what information is important and what sort of abstraction is most useful for extracting and using the right information thoughtfully. And that’s a skill a lot more adults will use than factoring.
Richard Bitgood, via e-mail:
Transformations is one step higher than functions because it is the abstraction of our abstractions.
2012 Aug 26. Of course, money is an abstraction of the value we provide society.
SeanJuly 25, 2012 - 12:48 pm -
My favorite question: “What information is important?”
One frustrating reality for the modern student is how often she’s left out of this conversation. The abstraction must be accepted without any of the abstracting.
That said, they’re not always ready for the leap from, say, image to fastest way home to aerial view to coordinate plane to distance between two points. How do we scaffold this process?
Michael P (@mpershan)July 25, 2012 - 1:20 pm -
Some thoughts provoked by this excellent post:
>You ask yourself, “What colors do I see?” Now you have a question and you’re on the ladder of abstraction.
You’re on the (a?) ladder of abstraction before then. “Colors” is an abstraction. You can’t ask the question without already being able to see the world through the lens of color.
I’ve had kids that really don’t get the concept of rate. When I ask them “which car is faster” they might say things like “that one, because it got there first.” They can’t ask the question “Which car is faster?” yet. They don’t see the world in terms of rates.
You can’t remove details unless you see them there first. Imagine a color-novice who doesn’t see the difference between jade and emerald. The novice would be happy to call both of those shades “green,” but a color-expert might refuse that abstraction when cataloging the colors. True, different purposes call for different abstractions, but the color-novice doesn’t have that option.
I don’t get how an aerial shot is an abstraction of the street scene. If by “abstract” you mean “the removal of information,” then an abstraction should never contain more information. But the aerial shot contains way more information that the on-the-ground scene.
Same issue with the way you present the move graph+distance as a higher abstraction. That’s adding information.
At the same time I agree that the graph+distance shot is a more abstract representation of the problem then the graph. So there might be something to the process of abstracting other than removing information. Maybe the move is that at this stage you’re no longer abstractly representing the streets — instead you’ve moved to abstractly representing the dilemma that you face.
I’m not sure that these points cohere to an alternate view than the one that you’re offering yet.
Bob LochelJuly 25, 2012 - 1:21 pm -
“What matters isn’t the rung itself but how deftly you can move between all the rungs above and below you on the ladder of abstraction.”
Dan, this is my take-away, and one of your best all-time lines.
In algebra 1, I want my students to understand written linear functions, graphed linear functions, and function tables, with applications a thread running throughout. But I find that expert teaching goes beyond that, and challenges students to move “deftly” between the representations and consider the efficiency, elegance, and appropriateness of methods. If we continue to teach Algebra as a strict sequence, we often lose this opportunity.
Here’s my Check For Understanding: should I call of text? When attempting to communicate these days, how is the decision made whether to call or text? If it is short and quick, I could just shoot a text, but what is my data plan like, and are there financial considerations? Or I could call, and risk a phone conversation which I perhaps don’t have the time or want for at this point in my life? How important is the message? Or will I possibly see this person soon anyway and it’s all a moot point? Or maybe I could just tweet it? But how sensitive is the message? Brain freeze….
Steve ThomasJuly 25, 2012 - 6:23 pm -
@Michael P: who wrote:
“I don’t get how an aerial shot is an abstraction of the street scene. If by “abstract” you mean “the removal of information,” then an abstraction should never contain more information. But the aerial shot contains way more information that the on-the-ground scene.”
Well if you take the definition of abstraction as first defined by Dan (via Wikipedia and American Heritage to be fair) as “Abstraction is a process or result of generalization, removal of properties, or distancing of ideas from objects.” or “To take away; remove.” then yes. But I would argue this is the wrong definition or at least it is missing an important piece which is that you Abstract not only to reduces information, but also to facilitate focus (and hopefully understanding) on relevant concepts.
I think part of the goal in teaching the “Powerful Idea” of Abstraction is to make them aware that we use abstractions in various ways to achieve certain goals. Also to teach them that the particular choices of abstractions are important as they affect our “models of the world” and how we view it.
I really like where this post is going (ie: a “Concrete” view of Abstraction) as this is the kind of thing that I believe can help kids get their heads around this to start to “see” it and hopefully to play with the idea.
That said I would prefer a “Concrete” example that students can better relate to and care about. Although I give bonus points to the connection to mathematics, which is the topic of this blog. Perhaps if it was how to get to the ice cream store or a concert (where you could use subway and train maps, for which there are some very well designed abstractions).
Gabe DavisJuly 26, 2012 - 2:48 am -
I think in mathematics we use abstractions in two big ways. The first is to make complex problems easier to solve, which is what you are demonstrating here. But the second way, which unless I’ve missed something you don’t touch on at all, is to make solutions more generalizable. When we use a map to calculate the distance to our house, we’ve solved a particular problem; when we express that map in terms of a graph, we’ve created a tool that can be used to solve a wide variety of problems, some of which are totally unrelated to geography. My question is: how can we bring this second sense of abstraction into the classroom? Is this even a useful thing to be doing?
WilliamJuly 26, 2012 - 4:20 am -
“I think part of the goal in teaching the “Powerful Idea” of Abstraction is to make them aware that we use abstractions in various ways to achieve certain goals. Also to teach them that the particular choices of abstractions are important as they affect our “models of the world” and how we view it.”
There are definitely two levels of mental work in play here. One is being able to construct a new abstraction, the other is being able to choose which abstraction to construct.
Because of that, I’m suspect of the metaphor or a ladder (which implies a well defined ordering of “abstractions”. What if the reality of how we work with math is something closer to an undirected, cyclic graph of consistent representations of a collection of information? We can add nodes to the graph and move between representations:
45 + 32 is another way of representing 87. But I could also represent that information as 40 + 30 + 7 or 0b1010111 or as a picture of forty-five apples stacked next to thirty-two pears, or as a point on the number line, or as a pile of actual skittles.
When Dan says, “Now you have a question and you’re on the ladder of abstraction,” he’s close to something pretty powerful. Michael P is right that you’re already dealing with abstractions before you ask a question. The important thing the question does is it guides you on what nodes on your graph of representations may be useful to finding an answer. I think what Dan’s pointing to with the ladder of abstraction is that we need to teach our students to construct new nodes on the representation graph instead of doing that work for them. That’s assuredly true.
Dan’s suggestion that, “if the question changes, the entire ladder changes,” is insightful. I think it’s even trickier than that. At each step on the “ladder”, there are infinitely many higher and lower (and equally abstract?) rungs to choose from. It’s more like a monkey-bar dome. What’s really important, and really hard to teach is how to pick which abstractions are going to be useful. There’s a lot of research to be done on how grown-ups who use math do this. It’s much messier than we portray, I think. Lots of trial and error, lots of peer review.
James KeyJuly 26, 2012 - 5:30 am -
@Micheal: “I don’t get how an aerial shot is an abstraction of the street scene. If by “abstract” you mean “the removal of information,” then an abstraction should never contain more information. But the aerial shot contains way more information that the on-the-ground scene.”
Here’s how I look at it: the abstraction is a more general setting that *contains* the specific, concrete setting. The concrete universe is a subset of the abstract universe. So the street scene is a subset of the aerial scene.
Math correlation: I give students a linear pair with measures 100 degrees and 80 degrees. That is part of the (more abstract) universe of linear pairs, all of which make 180 degrees.
SantoshJuly 26, 2012 - 5:52 am -
I understand better now what you are getting at with LOA. [placeholder, so I get subsequent comments via e-mail.]
WilliamJuly 26, 2012 - 6:15 am -
James — I too wrestle with an aerial view of a street scene as being a superset. There is definitely information in the street sign missing in the overhead view just as much as the reverse. For example, you can’t see the contents of the bicycle lane sign on the overhead view.
It seems like we *can* say that the two representations are consistent. They don’t contradict each other. They communicate (some) different information. One representation is more useful for mapping a route. The other representation is more useful for reading signs.
Bowen KerinsJuly 26, 2012 - 6:57 am -
This was the opening question / context at PCMI (Park City Math Institute) this year:
When trying to analyze this amazing thing about “perfect shuffles” of cards, it turned out really useful to ditch the card labels and use 1-52 instead … or 0-51 instead … where the decision on what abstraction to use depended on its utility. Others decided to add a variable (the number of cards).
I’d also add “Change the question”, maybe near the top of the ladder; once you’ve worked on a good question it tends to open a lot more, related questions. For the shuffling these questions were pretty wide, including different kinds of shuffle, shuffling odd numbers of cards, shuffling into N piles, etc. Great questions open lots of doors.
One thing a great context / question also gives you is the experience of figuring out what information is important and what sort of abstraction is most useful for extracting and using the right information thoughtfully. And that’s a skill a lot more adults will use than factoring …
BarryJuly 26, 2012 - 7:24 am -
This post strengthens my conviction that the verb “abstract” should either be avoided or should be given a specific, narrow definition and alternative words be found for common usage outside of that focus. Dan, you last used the word “context” back when you displayed the color histograms. I think there are at least two ways of abstracting from a context 1) As you say, the removal of inconsequential information from a context is abstraction 2) The process of transferring from one context to another can be viewed either as abstraction, as “concretization”, or as neither.
I think some of the confusion about moving from street view to aerial map comes from the fact that you transferred to a new context that was neither more abstract nor more concrete. If your entire experience was that street view, you would have no concept of a 3D space in which your house lies, nor of the aerial view you presented. If someone told you that you had a house somewhere but it wasn’t in your singular street-view experience, you would have no launching point for deciding how to get home. You, however, have much broader experience than a single street view. You already have a very abstract mental model of 3D space, where one can place an observer and an orientation and produce a 2D cross-section. Moving from the street view to this (very abstract totally mental) 3D context, you THEN ask the question, “How do I get home”? Removing extraneous information involves simply taking the 2D aerial view snapshot.
Similar, graph + distance is a different context from graph. Viewing both tabularly, the graph table consists of ordered pairs of letters, while the graph+distance table consists of sets of two objects: an ordered pair of letters and a nonnegative real number. The objects themselves have been abstracted, in the sense that you can recover just the ordered pair easily from a set, but going the other way is not so easy.
I would prefer to give the name “generalization”, or something similar, to moving to a new context containing your prior one as a specialization (in this case, you could always return to your street view from full knowledge of the 3D model — moving from the street view to the 3D model is what’s hard). I would definitely distinguish this from the process of abstraction in the context of 1). And I think it is here that you can make the strongest case for being on a “poset of abstraction”, if you like. It should be easier to determine from two contexts if one is a specialization of the other than to determine exactly what information is relevant to a given question about a context.
Here’s a final mathematical example to push this point: take your context to be the positive integers, with everything you know about them already. Ask the question, “in how many ways can a number factor into primes”. Now you can remove facets of the integers you know that are unrelated — you can ignore, addition, and things like the distributive property that depend on addition, you can remove knowledge that positive integers are used to count things, etc. On the other hand, definitions like “multiplication”, “commutative property of multiplication”, “divisor”, “prime”, and even “division algorithm”, “greatest common divisor”, etc. should become more in focus. This seems to be what Dan means by abstraction.
On the other hand, I think everyone would agree that moving from the positive integers to the new contexts of integers, then rational numbers, then real numbers, and then complex numbers (or quaternions!) is a sequence of abstracting of a different sort — moving to a new context after which recovering the old one from the new one is much easier than the initial move. I would distinguish this form of abstraction.
BarryJuly 26, 2012 - 7:47 am -
And BTW, I agree that “What matters isn’t the rung itself but how deftly you can move between all the rungs above and below you on the ladder of abstraction” is spot on, even pertaining to your example. I am pretty good with directions, and when I drive somewhere I’ve never been before, I keep a very fuzzy mental model of an aerial view in my head. I then prefer phrases like “head East” to ones like “turn right”. In personal experience, I’ve found that people who struggle with using compass directions and prefer “right/left” are worse with directions. I’ve asked several right/left people if they keep a mental model of an aerial view in their head — all of them said no. It requires some deftness to focus on all the dangers of driving and your street view while maintaining that mental map. I don’t believe I do both simultaneously — rather, when I need my map, I think I very very very quickly and perhaps deftly transfer back and forth between attending to the road and attending to my mental model. But boy is it easier to use an existing map if I travel to a new street in my hometown than to construct a new one if I travel to a new city for the first time!
@MatthewMadduxJuly 26, 2012 - 8:08 am -
for Barry (and others):
Mason, J. & Pimm, D. (1984). Generic Examples: Seeing the General in the Particular, Educational Studies in Mathematics, 15(3) p277-290
MaxJuly 26, 2012 - 8:44 am -
Trying to check my understanding:
It’s hard to see some of the abstractions I use to solve problems, since they have become so familiar as to feel like they contain all information in its original form…
One way I can see abstractions is to think of domains where I’m not an expert. Is it an abstraction when two kids who are really into video gaming communicate their solutions to challenges in terms of button pushes rather than the story on the screen? As in, I’m likely to say, “gee, I wish I could make Mario jump up and do a flip in the air to get that gold coin without being hit by the hammer.” Whereas a Mario expert is likely to tell a fellow expert, “That level’s easy. It’s just right-right-A-left.” (or whatever).
Another way for me to see abstractions is to think about when I carry pure information in my head instead of carrying physical objects around with me. So, for example, I needed to buy a curtain rod and material to make curtains. Someone who could abstract nothing at all would either need to bring his house to the store or have the fabric and curtain rod* options delivered so he could hold them up against the window.
But instead, I only cared about the length of the window and the area I wanted to cover with fabric. I could have conveyed that to the store by cutting a piece of newsprint to the exact size of my window, and matching the lengths at the store.
I could use a code to scale down my newsprint cut out — I could make it half the length and half the width. At the store I could scale it up again to make sure I had a long enough curtain rod and enough fabric. Think of it like folding the newsprint in half longways and again shortways, but instead of taking the whole folded piece, just cutting out one quarter and bringing it to the store, then showing the cutters how this was just one of four pieces of my window.
That’s still a bit unwieldy, though, and I don’t really need to represent each square inch of the space I want to cover. So I could also use two pieces of string, one for the length and one for the width. At the store I could lay the strings out perpendicular to one another and re-create the size of my window.
But… I don’t even need to represent lengths and widths with physical objects because of one of the great metaphors of math: the real numbers can be associated with points on a line, and so the length of something can be represented by a single number. I can represent the length of each piece of string I would have brought with a single number.
To make sure they can interpret my measurement at the store, I need to use a common scale. Probably inches. I can measure the length and width of my window in inches, tell the folks at the store, and they’ll know exactly how long of a curtain rod and what dimensions to cut my curtain fabric to.
What’s neat is that when I go to pay for my fabric, they will want me to pay per square foot or square yard, and I won’t even need to use a physical 1 foot by 1 foot square to measure how many square feet I can buy… And studying why the two pieces of string abstraction was a useful one might unlock more abstractions I can use to understand more about measurement, about area, and about how the real numbers can be mapped onto different metaphorical spaces!
*When using this example with middle school students, be sure never to talk about the length of the rod without saying “curtain rod”. #justsayin
SantoshJuly 26, 2012 - 11:30 am -
Dan, and others,
as you mine these ideas further, I wonder if it is time to also start laying out the similarities and differences between a ‘representation’ and an ‘abstraction’ (the product of abstracting).
Dan MeyerJuly 26, 2012 - 1:04 pm -
I think this is correct. There are a lot of reasons why curricula and instruction in secondary levels exclude students from the process of abstraction – many valid, many invalid, most related to the limitations of the print medium. I’m downcast on a lot of implementations of technology in the math classroom but digital technologies are uniquely suited to pull students back into that process. Those are the posts I’m most excited to get to.
You’re only on the ladder of abstraction once you ask the question. As you’ve pointed out, you’re not on the lowest rung of that ladder, though.
This is fair. If the rung beneath it were every possible visual view of the area surrounding you, then the ladder makes more sense. Rather than taking every possible view of that scene, we’re getting rid of all other views but the one immediately above the street.
This is another weak part of the essay. I don’t really know how to conceptualize the process of generalization or the addition of more variables. Intuitively I want to place those tasks at the top of the ladder (the sequence of events makes sense that way, for one reason) but the metaphor gets thinner the higher it goes.
What information is important? How do you represent it? You can’t do the latter without deciding on the former. I can’t think of a counterexample. So the linearity of the ladders seems true to me. Resolving the question, “How do you represent the important information?” requires someone to consider the different possible representations, their advantages and disadvantages. You exemplify that well with 87. I don’t think that example contradicts (or even complicates) the metaphor, though.
We’ve thus far tackled (lightly) the verb “abstract” and the adjective “abstract.” The noun “abstraction” has been out of scope, but that’s an oversight. Totally off the cuff, I’ll say I think the noun “representation” has a static quality that “abstraction” lacks. “Abstraction” calls attention to that which was abstracted in a way that “representation” doesn’t. I probably oughtta check a dictionary, though. Anybody else?
WilliamJuly 26, 2012 - 1:49 pm -
The whole plot of Moneyball is an example of what we’re all talking about. Imagine that you’re running a baseball team. You have a context: the historical data about the performance of all the players in the league. You have a question: Who should my first baseman be?
If you base your answer on mathematical abstractions (as Oakland did), you can wildly outperform people who don’t base their answer on those mathematical abstractions.
Then things get interesting. What set of abstractions you use to answer your question will hugely influence the answer you get. If you think that batting average is the most appropriate abstraction, you’ll hire certain players. If you think that OPS is the most appropriate abstraction, you’ll hire other players. If you’re a VORP fan, that’ll drive decisions too. There’s not broad agreement among teams as to what abstractions generate the best answers to the question.
Steve ThomasJuly 26, 2012 - 7:59 pm -
“What’s really important, and really hard to teach is how to pick which abstractions are going to be useful.”
Yes, and I am thinking that perhaps some formed of worked examples may be useful.
“What set of abstractions you use to answer your question will hugely influence the answer you get. ”
Using some examples do demonstrate your point above (with wildly different answers depending on the abstractions chosen) could be a good case. Now I just need to think of some to use!!!
“45 + 32 is another way of representing 87. But I could also represent that information as 40 + 30 + 7 or 0b1010111 or as a picture of forty-five apples stacked next to thirty-two pears, or as a point on the number line, or as a pile of actual skittles.”
A number is all the ways you can name it, is a good thing to teach. And this could be part of the lessons on abstraction. For instance 3 is fine saying your age or counting how many toy bears you have, 1/2 of 6 is good when you are are sharing 6 cookies between two people or cutting a 6 foot board in half. 10-7 is good when we want to know how much we will have left after spending 7 of our 10 dollars.
“What information is important? How do you represent it? You can’t do the latter without deciding on the former. I can’t think of a counterexample. So the linearity of the ladders seems true to me. Resolving the question, “How do you represent the important information?” requires someone to consider the different possible representations, their advantages and disadvantages.”
Great questions. As for a counter example to the ladder metaphor (which I still like as a starting point for kids, no metaphor is perfect) I am still thinking, but using your questions, depending on what is important,you could get different representations which at the same time hide/reveal (or perhaps allow/disallow control over) different characteristics that matter for your purpose. That said, I’m stumped right now and can’t think of a counter example.
So another way I have been thinking about abstraction, (and how have taught it) is more programming and lego related. Ie: how do you design the proper set of pieces that can be re-used to accomplish multiple tasks. One way I teach this is using parts of Barry Newell’s Turtle Confusion puzzles. Where you have kids program the turtle (or Scratch cat or Etoys object) to draw a square, triangle, and pentagon. You then ask them to look at what they just wrote and ask “Whats the same and whats different?” Every time I have done this the kids get it and see the common patterns and how they can create one set of blocks that can draw any regular polygon. Part of what can help make this work is hiding some of the complexity and providing them a limited set of programming blocks which visible for use. So the question “What’s important” is very useful in deciding what to hide and what to show when designing an activity.
Gary StricklandJuly 26, 2012 - 8:12 pm -
As a high school physics teacher I really enjoyed reading this post. I recently decided to create an introductory unit that essentially discusses this very topic.
I have discovered that in our school we have never laid this fundamental foundation of essentially how math and science are related. I think you have found a very good way of describing how science asks questions, and then we have to find a way of isolating the dimensions that interact that truly define the condition that we are curious about.
It is this process of abstraction where we make the transition from the question that is perhaps qualitative in nature to framing the question in a quantitative manner using mathematics.
Thanks for a great post. I’m already thinking about asking my physics students to bring a photo of an event that interests them and then asking them through a series of abstractions to reduce their point of interest to a relationship between two defining dimensions (one of the seven fundamental SI base units).
WilliamJuly 27, 2012 - 3:57 am -
I like the idea of using programming to teach students how to abstract. I think I can offer a couple examples of places where the choice of abstraction influences the answer to a question.
The first example (which I like more and more) is baseball statistics. The question is which baseball players are worth paying to play for your team. The “concrete” context for the question is the historical performance of all the players.  On top of that concrete data you can construct abstract representations. One batter might have a batting average of 0.325, another 0.221. One pitcher might have an ERA of 2.2 and another 4.5. We compare baseball players at that level of abstraction. If we construct other abstractions (say, OPS or VORP, or WHIP, or whatever new contraption we use to talk about fielding skill), we’ll make different decisions about which baseball players to hire.
@Gary, your talk about physics reminded me of another important example: mathematical astronomy. The question is “how do the things in the sky move around?” The context is observations of the things in the sky. The ancienct mathematical astronomers collected surprisingly good concrete data about the positions of the various celestial objects. They then constructed an abstract representation of the world to explain the movements of those things. The abstraction they chose was one with the Earth at the center of the universe and a stack of celestial spheres rotating around it. It turns out that that’s a surprisingly good abstraction to pick. You can do fruitful mathematical work in it and it allows you to predict where the celestial objects will be really well. You pay a price in scary geometrical theorems that are very hard to read today, but the work stands. The Copernican revolution was, in large part, a change in abstracted representation. The Copernican astronomers were trying to answer the same question, and used much of the same data as the Ptolemaic astronomers, but they constructed a different set of abstractions on top of the concrete data. Tycho Brahe proposed an abstraction where Mercury and Venus orbited the Sun while the sun itself orbited the Earth. Copernicus proposed a set of heliocentric abstractions. 
To take a Nolanesque turn to the meta, I’m suggesting that we have a concrete context of students in math classes and a question, “how do we effectively teach them to enjoy mathematics?” One abstracted representation of how we do mathematics is traversing a ladder of abstraction. I think that produces a particular pedagogy that’s way, way better than most. So, as far as it goes, I think the ladder of abstraction is a good systemic metaphor. I’d suggest, though, that if you abstract the process of mathematics differently, say as constructing and traversing a cyclic, undirected graph of representations or as traversing a lattice of abstractions, you’ll get a subtly (!) different pedagogy that will better serve students.  You’ll get a pedagogy that not only teaches students to move up and down the ladder (hugely important and generally missing from our curricula), but also teaches them to deftly choose which abstractions will get them from where they are to where they want to go.
It turns out that this too is an abstracted representation. When you keep score for a baseball game, you record particular, concrete events using a symbolic language that regularizes them. You convert the batter ripping a hard ground ball down the left field line past the diving third baseman to “2B”. You convert the pitcher buckling the knees of the batter with a 103 mile per hour fastball to “K”. That translation itself requires significant synthetic reasoning.
 The same thing happened with Newton and Einstein. Relativity is a new abstracted representation of the concrete context of mechanics and the Michelson Moreley experiment. Atomic theory, too, has had its fair share of different abstractions.
 You definitely don’t want to tell students that you’re teaching them to traverse an undirected graph in so many words, not least because mathematicians brutally equivocate “graph”.
Chris SJuly 30, 2012 - 5:22 am -
I’ve been working on a STEM initiative in higher ed–one in which math is identified as a gatekeeper for STEM careers. This concept is very helpful…and a great complement to this Op Ed in the NYtimes, Is Algebra Necessary (Hacker)?
I see some points in that argument that your post answers… unfortunately, many of his critiques stand because ‘in practice’ the teaching of algebra doesn’t connect the representational and abstract.
lesannoAugust 1, 2012 - 8:13 am -
An abstraction in my life: When looking at my email inbox, I disregard most of the information presented there. During the busy workday I “see” only the messages that are highlighted as unread and which were sent to me individually rather than schoolwide. This level of abstraction sometimes presents problems, so if I’ve read an important message but haven’t yet resolved it, I have to star it AND mark it as unread, tricking it back into the category that registers.