It’s a familiar scene for a math teacher. You’re chatting with a stranger at a party or the guy giving your hair a quick trim or anyone else. Conversation comes around to occupations. You mention you’re a math teacher. No one has a neutral reaction to “math teacher.” You take the tension head-on and ask, “What did you think about math as a kid?” The majority opinion on childhood mathematics is often negative and you notice the same adjective crops up over and over again: “abstract.”

“I liked Geometry. Algebra was too *abstract*.”

“Math was too *abstract*. I liked working with my hands more.”

“I liked Algebra. Geometry was too *abstract*.”

I’m going to try to pound in some fenceposts around the terms “abstract,” “abstraction,” and specifically, “the ladder of abstraction.”

That last term has its deepest roots in the fields of language and rhetoric (Hayakawa, 1940) though Bret Victor recently knocked it out of the park with an interactive essay describing its applications in mathematics and computer science. This fencepost-pounding process may require only a few months and a few blog posts (if you’re lucky) or a few years and a dissertation (if *I’m* lucky). However long it takes, you should help me interrogate the term. Does it mean anything? Does its meaning have any implications for the workings of a math classroom? If we understand it, can we counteract the perception that math is too abstract, or at least understand that perception well enough to manage it?

I’ll finish this brief introduction by describing the personal appeal of the ladder of abstraction:

**Self-study.**In the best classroom experiences I’ve witnessed or orchestrated, I could describe the students as “ascending the entire ladder of abstraction.” I want to know more about that.**It ties a lot of good pure and applied math instruction together.**I’ve done an excellent job pigeonholing myself as some kind of zealot for applied mathematics but some of my favorite experiences in the classroom haven’t involved any applied context at all. Common to all of them (and common to my applied math methods) is their origin at the bottom of the ladder of abstraction. I didn’t hoist students to a higher rung until they’d worked on the rungs below.**There might be a dissertation & career in there somewhere.**Implementing the ladder of abstraction in the classroom requires multiple media. Don’t misunderstand me. I’m not saying, “Good math instruction requires*digital*media – photos and videos, etc.” I’m saying that it’s difficult to fully exploit that ladder if you design a task using only*one*medium. (Paper, in particular, limits your tasks to exactly one rung on the ladder, depending on how strictly we define our terms.) The task I linked above – graph: 3x + 2y = 12 – required two media, none digital: 1) voice, 2) a collaborative writing surface. Tasks that work up and down the entire ladder of abstraction don’t*require*digital media, but holy cow does a digital platform make those tasks easier to implement. As I look ahead to (fingers crossed) finishing this PhD and getting a job doing I have no idea what, I think I could contribute some value to our field by helping people create tasks that ascend the entire ladder. Provided I understand it.

Thanks in advance for your help.

## 33 Comments

## TrevorCashmore

July 16, 2012 - 9:56 am -Sorry in advance if I misinterpret or mess up some ideas. I’m kinda just going off your post, as I’m rushed for time and can’t read the essay right now. (Thanks for linking it!) And I’m not well-versed in math education yet, as I’ve yet to graduate high-school, so I might be messing up some ‘basic terms’.

I’m assuming that we’re going to interpret abstraction as ‘distance from the concrete’, then? A ladder of abstraction would then represent, well, a measuring device for ‘how far away’ you are from ‘the concrete’ (nice metaphor, harhar).

The more immediately applicable it is, the more obviously related to ‘reality’ (as we see it), then the less ‘abstract’ it is.

Of course, then you get into weird -subjective- abstractions. Counting apples is easy, as that’s concrete — so it’s low on the ladder. But should counting non-specified items be slightly higher, as it lacks an immediate application? Or, since it’s so obviously related to “counting apples”, does that mean it’s on the same rung?

I’d argue that could be a method of ‘ascending the ladder of abstraction’ – taking away the guise of the concrete and revealing the workings underneath. First you observe, then you extract — for instance, watching a ball fall and deriving how it’s falling.

## Bruce James

July 16, 2012 - 10:17 am -Thank you Dr., I mean Mr. Meyer! I have been looking forward to these fence posts for some time.

In response to Trevor, I think of abstraction kinda like this:

Pick up an apple. Ask yourself, what am I holding? Say it aloud.

Answer: an apple(a word, a sound, to stand for a thing)

Now write down the letters a,p,p,l,e in that order.

What does it spell? (don’t say it aloud, just let your brain conjure up that thing you are holding in your hand). Another layer. Written symbols to stand for the sound that represents the thing.

Next step, or sideways step or…??? Ask yourself once more what that thing is in your hand, only this time don’t say apple or write apple, instead draw a picture of an apple.

Or something like that.

Eat the apple. Or give it back to Eve.

It gets even fuzzier with numbers which aren’t things at all but represent amounts or relationships between things.

In short, this is sort of how I talk to my 7th graders. Practice problems= eating the apple.

Apologies for not having read the essay. I will chew on that soon.

Again, thinks in advance Dan for pounding in the fence posts. No easy task.

## Jonathan

July 16, 2012 - 10:22 am -I think it goes beyond secondary education as well. Years ago I interned at a small engineering firm that specialized in low-friction gaskets for oil drilling equipment. The engineer I was working under was charged with developing fluid models of their bearings (metal discs that kind of do the same thing). Anyway, we were discussing the project and the best way to come up with some results and I liked the approach of brute testing. Strap the bearings into a machine, apply many many load conditions and generate a pretty chart. He, being more than just an intern, decided that a computational model was best and we set about designing bits of the bearing in a fluid comp program instead. He had also made many remarks that he found differential equation analysis “fascinating” despite the lack of anything tangible.

In there I think our kids have a similar problem. Lots of them want to see real results and build up a library of cases that prove these rules. What too much of math education focuses on is pie-in-the-sky proofs with x’s and y’s that don’t mean a lot to them.

If you’re going to study this, there’s another phenomenon you might consider. To earn spring exemptions, I had my seniors write me an essay about their relationship with math. Not much instruction, just tell me what math was like growing up, good, bad, whatever. In maybe 10 cases they all pointed out that somewhere between elementary school and middle school math went from something they could see and understand to something they no longer got. Every one of them said the same thing, I loved math until middle school. What in the world changes in middle school?

## Michael P (@mpershan)

July 16, 2012 - 10:38 am -I’m really looking forward to reading this series. In fact, I can’t help myself from trying to work out some of what we mean by “abstract” on my own with a series of questions.

Are there any things that are both easy to understand and also abstract? Or is abstract a term whose use implies that something is difficult to understand?

What sorts of things is it appropriate to append the word “abstract” to? Ideas? Sentences? Concepts?

Is “abstract” an objective or is it subjective to learners?

Why isn’t learning an instrument ever described as abstract? Why isn’t the word “abstract” used for the knowledge necessary to write a novel?

What’s the relationship between “abstract” and “very general”? Are there statements that are “abstract” but very specific? Are there statements that are concrete but very general?

What other questions should we ask in thinking about what “abstract” means?

## Bruce James

July 16, 2012 - 11:24 am -What happens in middle school is that we become poor at marketing our services and knowledge. Yes, we are math teachers, but we must market the ideas-make them tangible or sexy, whatever.

I ask my students repeatedly why people work out. Cause they want to get fit and look good are the common answers. Indeed, I tell them. Math is like taking your brain to the gym-the goal being to think clearly and to peel away the fat,i.e. the stuff that does not serve the answer. There is even an app called math workout that the kids love because the marketing of basic operations in this format is more appealing than the same questions on an 8.5 x 11 inch piece of paper.(no, I do not think of this app as great pedagogy, but simply a small marketing device in the great toolbox of math education.)

## Michael P (@mpershan)

July 16, 2012 - 11:45 am -The message that “math is like working out” might not be be such a good marketing move. Americans don’t like exercise either.

## David Patterson

July 16, 2012 - 12:00 pm -Saying math is ‘abstract’ might simply be a way for people to avoid admitting that they didn’t get it…not a statement on whether it was applied math or not. Note that in your examples above, they either ‘liked’ something or it was too ‘abstract’. (I personally don’t see how geometry could be considered more abstract than algebra when it is heavily dependent on visualizing concrete examples.)

I liked it = I understood it and did well

It was too abstract = I had no clue and struggled

## Ryan

July 16, 2012 - 12:32 pm -For some background reading, I recommend this paper which compares a grounded representation (word problem) to an abstract representation (symbolic problem): http://pact.cs.cmu.edu/koedinger/pubs/Koedinger%20Alibali%20&%20Nathan%2008.pdf

Spoilers: symbolic form is more useful when dealing with a more complex type of problem. Maybe expected, but this goes into the gory details.

The issue is very deep. It’s not just about how the problem is presented but about how the representation continues to be perceived and interacted with throughout the thinking and problem solving process (a term often associated with this is external representation). I ramble about this and provide some more citations in http://blog.learnstream.org/2012/01/weekly-review-through-january-15/

Of course, symbolic vs grounded is just one dimension of abstraction vs non-abstraction.

## Mathy McMatherson

July 16, 2012 - 12:55 pm -I don’t know if I have much to contribute to your particular interests in abstraction – but I’d like to share my favorite thought relating to this topic and maybe it’ll be food for thought for you. It comes from the book Everything and More: A Compact History of Infinity by David Foster Wallace. The quote appears in his foreword in which he is discussing this very notion of mathematical abstraction (in fact, you can read the whole forward on Amazon as part of their ‘peek inside the book’ preview).

“Here is a quote from Carl B. Boyer: ‘But what, after all, are the integers? Everyone thinks that he or she knows, for example, what the number three is – until he or she tries to define or explain it”. It is instructive to talk to 1st and 2nd grade math teachers and found out how children are actually taught about integers. About what, for example, the number five is. First they are given, say, five oranges. Something they can touch or hold. Are asked to count them. Then they are given a picture of five oranges. Then a picture that combines the five oranges with the numeral ‘5’ so they associate the two. Then a picture of just the numeral ‘5’ with the oranges removed. The children… start talking about the integer 5 per se, as an object in itself, apart from five orages. In other words, they are systematically fooled, or awakened, into treating numbers as things instead of symbols for things.

Sometimes a kid will have trouble, the teachers say. Some children understand that the word ‘five’ stands for 5, but they keep wanting to know 5 what? 5 oranges, 5 pennies, 5 points? These children, who have no problem adding or subtracting oranges or coins, will nevertheless perform poorly on arithmetic tests. They cannot treat 5 as an object per se.”

I think about this quote *a lot* when teaching and especially when remediating. It wasn’t until I reflected on this idea that I realized how often I separate the world of number and mathematics from the physical phenomena it often describes – and, more importantly, that my students need to be ‘awakened’ into thinking this way. That it needs to be taught purposefully and with careful scaffolding. Some students need to start at the bottom rung – the realm of the physical – before they can transition into the world of the abstract and considering numbers as objects of their own with their own rules and properties and beauty. I find myself very conscious of this when introducing new material – always finding a way to ground it into something real before transitioning to the abstract, then making it absolutely clear that the act of abstraction is meaningful and, in most cases, was the entire purpose of the task to begin with.

## mr bombastic

July 16, 2012 - 2:18 pm -Is reading a less abstract process than doing math? Does anyone like abstractions before they become comfortable with the abstraction? I wonder how much of the discomfort in math, or English for that matter, is due to personal taste, and how much is due to being paralyzed with fear as you look down from an upper step on the abstraction ladder and don’t see any steps below.

Also, do people mean “not concrete” when they say abstract. Acclimating to a different environment is like developing intuition around math abtsractions – very similar feeling for me. With time, the street names, parts of the town, etc., become much more concrete to me. I gain a sense of what these things are – they have a certain feel to them.

## Mr. K

July 16, 2012 - 5:36 pm -I’e long suspected that “abstraction” is a key part of the answer to the question: “What is math?”

I don’t think you can get math without getting abstraction. Sure there’s more, but I think it’s just a formalized way of talking about the abstractions, so that we can share them with each other.

The real problem, though, and what kills people from word problems through algebra and all the way on up, is moving from one step of that ladder to another.

Sometimes (like with the problem I’m currently working on) it’s not even clear how far up the ladder you have to go. The fog obscures all but the next rung, and you have to take it on faith that there’s more there behind it.

I also think that I’ve flailed quite a bit in trying to address this in my teaching. It’s something that looms as a large dark hurdle, both for my kids, and by proxy me.

I’m looking forward to this series, if for no other reason than that it might provide me a framework to address that mush.

## Ben Blum-Smith

July 16, 2012 - 9:59 pm -Just an initial contribution to the conversation, something I’ve been thinking about for a long time:

Part of the idea of abstractness is experiential: something

feelsabstract. Insubstantial, airy, not completely tangible, maybe not entirely there. Contrasted to anexperienceof concreteness: something that feels solid and real to think about. For some people, God is an abstract idea and for others God is entirely concrete.What I believe strongly is that the experiential part of abstractness is 100% relative to what you understand and can do. To a third grader (and many high schoolers I’ve taught), “1/5” is very abstract and “x” is hopelessly abstract. To probably everybody reading this, both these concepts are real and work-with-able as the pen and paper you write them with. To me 15 years ago, groups were abstract but to me now they’re extremely concrete. As of this fall, differential forms were abstract but now they’re pretty concrete, though not as much so as groups. Natural transformations are still very abstract to me. Schemes hopelessly so, at least that’s how it feels. But they won’t be forever. The point is this: from an experiential point of view, nothing is intrinsically abstract or concrete. Concreteness is a matter of what you already understand and can do, and abstractness is a matter of what you can’t.

Relatedly, the presence of the experiential form of concreteness is a necessary condition for a problem to seem exciting and compelling. (Perplexing ;) I think that if a third grader looks at a problem, and a professional mathematician looks at a different problem, and both of them want to solve the problems equally much, then from an experiential point of view the problems are roughly equally concrete. It’s just that different things are concrete for the two of them.

## Andrew Stadel

July 16, 2012 - 10:36 pm -Abstraction v. extraction?

That’s my question.

When you reference your ladder of abstraction, many times I feel like you are requesting teachers to encourage or assist their students to remove information, derive solutions, or obtain reasoning in order to complete a task. I think this is wonderful. I agree, working on those low rungs is vital.

Most likely, it’s my lack of understanding, but abstraction and extraction can be two very similar and different components of learning.

I think your examples of meeting common place people and their view of math is spot on. Could they have a disdain for math because they couldn’t extract info from a problem (or situation) as a kid? Is it because they couldn’t think abstractly, or couldn’t follow the ladder of abstraction (either because their brain wasn’t fully developed or because they were introduced to in their math class(es). Does extracting information precede abstract thought? I don’t know. I’m simply throwing it out there.

I look forward to more of these posts.

## Bowen Kerins

July 17, 2012 - 5:47 am -We have an activity at the beginning of our Algebra 2 book asking students to find functions that fit tables of data. There’s no context, just inputs and outputs.

We interviewed students about this activity during our field test, and several said they thought the math was “more realistic” than what they had done in previous books. This was confusing until we got some clarification: it was the experience, the thinking, the style of work, that made it realistic. Going through the experience made that concrete for those kids, and their teachers could keep referring back to that experience as a starting point for generalization and abstraction.

In my opinion, without those experiences a lot of students don’t see the big picture of mathematics or see any purposes to abstraction.

## mr bombastic

July 17, 2012 - 9:03 am -@Blum-Smith, you express more eloquently a lot of what I was trying to say. Also, your example had me thinking about which abstractions cause problems. Many kids feel comfortable with the abstraction, “a fifth”, but do not feel comfortable with the abstraction “1/5”.

I am wondering if many of the issues with bi-lingual education (student is taught content in a language they have not mastered) also apply to math education.

Language is highly abstract, but math language is not the same as math. Are the issues surrounding abstraction about math or about the language of math? To me, the example you provide is really more about language acquisition — seeing that equation and thinking, among other things, that represents a graph. As you have pointed out, many geometry questions are easily understood if you strip away the notation (math language) and pose them in everyday language. Often, the language is the barrier to entry, not the math. Using math language requires less writing, but it does not necessarily make the problem easier.

There are plenty of interesting math problems and topics that require very little formal math language. But, they interrupt the flow that seems primarily designed to teach the language of math, not actual math.

I am wondering the extent to which we use an immersion approach for teaching the language of math as opposed to a translation approach, and which is better. I am also wondering how much of this movement on the abstraction ladder is really just translating from math language to everyday language.

Interesting topic — hard to know what to think.

## Barry

July 17, 2012 - 10:09 am -It is clear above that people are using many definitions of abstraction. “Abstract” does have meaning, but unfortunately many and I would choose more precisely defined terms in favor of it. I think it is well worth a look through a dictionary before deciding on your own usage:

http://www.wordnik.com/words/abstract

In chatting with people as you describe, I usually assume they mean one or both of “abstruse” or “impersonal”. A student who views abstract mathematics as “abstruse” has experienced only the pain, not the power, provided by abstraction. Dan is a master at making mathematics personal.

Neither of these connects to the “ladder”, because to define this one must change from consideration of the absolute term “abstract” to the relative term “abstraction”.

One of the many different ways students experience extraction is through quantification. Difficulty with letting `x’ stand for some unspecified number (perhaps in a non-explicitly specified set) stems (at a fair level of abstraction) from not internalizing “universal generalization” as a valid rule of inference. How many students really understand that when you solve Sqrt[x-3]-Sqrt[x]=3 and arrive at x=4, what you’ve really done is prove the statement “for all real numbers x, if x is a solution of Sqrt[x-3]-Sqrt[x]=3, then x=4” and that you must check the solution x=4 because the converse might be false? This is the top of a particular ladder of abstraction — the abstract ladder of abstraction might refer to some systematic process of building up an intuitive understanding of universal generalization. Most of our students need not completely ascend one of these ladders, but the routine sweeping of quantification under the rug hides a big source of abstraction.

Ladders also appear in chains of mathematical definitions. Sometimes ascending means going toward more generality, but would not most people consider a nowhere differentiable continuous function a more abstract notion than just a “function” (in the “abstruse” sense)? I would also refrain from nailing down “ladder of abstraction”, which probably also means different things to different people, and instead use specific terminology.

## blaw0013

July 17, 2012 - 12:01 pm -[this reply (er, essay?) follows well on Barry’s comments, I believe]

[Here’s the link. –dm]

Lets make this–what is abstract and what is concrete–a problem, trouble it as a friend would say. To do so, rather than define those terms through the eyeballs of an omniscient observer, or any observer, I suggest the terms may be most useful for a (math) teacher if considered through the eyes of an other, for the teacher–a child.

A child may prefer not see the abstract as superior to the concrete. And that may very well be true for any one, not just the child. I believe Seymour Papert and his partner Sherry Turkle make an excellent case for this in their seminal paper “Epistemological Pluralism and the Revaluation of the Concrete” or . Here, in case the title wasn’t clear enough ;^) they argue to re-value the concrete.

Concrete is often understood as tangible, real. Or, instead of focusing on the object, but the manner of thinking about the object, the more we are able to visualize (or sensorize) an object from a description, the more concrete it is. For example, describing my pen as a “Papermate ballpoint pen” is more concrete than just the description “pen”. In this sense, the more general is equivalent to the more abstract. “Writing utensil” and “communications tool” ascend in levels of abstraction. Summarized quite simple than, this standard view of the concrete follows the logic, “the fewer the number of objects in the world that fall into the description, the more concrete.”

From this perspective, it makes sense to want learners to move away from the confining world of the concrete to a level where what they learn can be applied more widely and generally. However, we see that attempts to teach abstractly leave students bored and with brittle, non-useful knowledge.

Furthermore, digging in further to this idea about what may be concrete reveals serious flaws in the reasoning. For the Southerner, snow may be considered a rather concrete notion — it is the cold, slippery stuff that provides an excellent excuse to stay home from work for 2-3 days. However, for an Alaskan, the word “snow” may connote a large category of many types of snow, each with particular sensory categories. Snow is a vast generalization, an abstract concept for some people. The faulty assumption? That each person’s ontology is identical.

Rather than look for concreteness in the object, look at the person’s construction of the object. Concreteness is not a property of snow in and of itself, but of the quality of relationship the knower has with snow. In this way, objects that are not mediated by the senses — such as mathematical objects — can be concrete provided the learner has multiple modes of rich engagement with them.

As a consequence of this new view of the concrete, it is evident that rich, interconnected ways of knowing this new concrete implies is to be valued. In fact, most learning experiences begin with an introduction of (or “bumping into”) an abstract, formal object. A learner’s relationship to this object develops through interaction, becoming more intimate and concrete. It can be seen that knowledge actually moves from the abstract to the concrete. Since more advanced objects of knowledge remain abstract for so many of us, we are fooled into believing the process of knowledge development moves from the concrete to the abstract.

“Relational thinking puts you at an advantage: You don’t suffer disaster if the rule isn’t exactly right” (Turkle & Papert).

## Mr Macx

July 17, 2012 - 2:46 pm -I think that “abstract” as vague is really the opposite direction of where we are trying to go. Math is a language with which we can talk about how the universe works. Being precise can cause abstraction. Like Bertrand Russell said, “Everything is vague to a degree we don’t realize until we try to make it precise.” (maybe a bit of a paraphrase.

For example, if I asked a student which would hurt more, getting hit by tackle or getting hit by a bus, they would say bus. This is getting at the concept that bigger things have more force. But how much more?

If we want to really get precise, we need to introduce variables to get the exact relationship F=ma.

Then we could use calculus to get more precise, thinking of velocity and acceleration as derivatives of position. But that is even more abstract.

I think this guides my thinking about the ladder of abstraction, because we shouldn’t get more abstract until we have motivated our students to get more precise.

## Mrs Mckenzie

July 17, 2012 - 4:23 pm -Sometimes we forget that the number 3 is abstract not just the quadratic formula. I think that US schools do not pay attention to helping students bridge the gap between concrete and abstract. Singapore and Japan are famous for multiple representations. In Singapore, the transition from counting things to numbers to drawing pictures to rectangles to algebraic to graphic is clear and delineated. Many times I see teachers who try numbers to algebra with the promise of one day it will all make sense just memorize it now. I recently saw an idea under development by Rob MacDuff, CIMM, of using dots to bridge between concrete, numbers and symbols. It seems to be highly effective for fractions and relational reasoning. I teach both 9th grade Precalc and 12th grade algebra 1. Both of these groups have students who lack true number sense because they have tried to memorize their way through.

## Santosh

July 18, 2012 - 11:09 am -1. Is there a difference between “layers/levels of abstraction” and “the ladder of abstraction”?

2. Dan, is there a way to sign up for email notification of comments on a blog post? All blog posts?

I suspect that what school students mean when they say math is “too abstract,” is that the level at which it is talked about by their teachers (& textbooks) does not match their own familiarity (visual mapping) of the terrain.

Imagine giving directions to get from one part of town to another.

Of course it depends on the modality – foot, bike, bus, car, etc. Then it depends on the level of familiarity of the user with the neighborhoods and the connecting routes. (Navigation user-interfaces are catching on that people dislike 14 turn-by-turn instructions on how to get out of their neighborhood.)

“Take 5 north, get off at U.District, head east, and catch 15th north.” might be perfectly adequate instructions for some people.

Turn-by-turn instructions works for everybody, but they don’t help build mental models in any way.

## Tyson

July 18, 2012 - 11:53 am -Not sure if it’s exactly what you are trying to get at, but I’ll just throw it out there. Jerome Bruner had the idea of learning moving through stages of representation: enactive, iconic, symbolic. It might be worthwhile to explore more…

http://mennta.hi.is/starfsfolk/solrunb/jbruner.htm_3.htm

## John

July 18, 2012 - 11:27 pm -Can you say a little more about why you include the Victor interactive essay? What is that telling you about LOA?

All I get from Victor is that Proportional controllers are notoriously dependent on the gain, and can’t anticipate control you need or correct longer term deviation from a setpoint.

That’s why PID (proportional, integral, derivative) controllers were invented and it is a lesson in control theory for designing those with appropriate gains that can respond correctly to different inputs, i.e. the “initial bend in the road” from Victor.

But maybe that is your point, that the LOA would make a student rip for the introduction to PID since we see the limitations of just Proportional controllers in the car driving problem.

## Kevin Martz

July 19, 2012 - 1:56 pm -This happens with physics teachers as well. People either loved it or hated it. I have never met someone who said that they were “ok” or average at physics.

Both math and physics require you to think and use ideas to answer questions.

I would also say that most of physics is abstract and that is just one part that makes it hard.

## Dan Meyer

July 19, 2012 - 2:18 pm -I’m grateful for the syllabus you folks have developed for me here. A colleague at Stanford recommended via email a piece on Early Algebraic Thinking by Radford, which I’ll also add to this pile. I’ll be digging through these links as I try to nail down the concept.

A few other useful remarks I’d like to highlight:

Michael Pershanposts several questions about abstraction that may light a few useful paths through the issue for us. This one, in particular, matters to me:Here’s an assertion I may regret later: the adjective “abstract” is an objective term as it’s perceived by people. A road map isn’t more or less abstract depending on whom you ask, though one’s familiarity with and comfort around that particular abstraction will vary from person to person. I’ll try to back that up in another post.

Here’s

Ben Blum-Smith, disagreeing with me:Again, I wonder if this describes someone’s

familiaritywith an abstraction of a thing and not the abstractness of the thing itself.David Pattersonnotes that the term “abstract,” as it’s used in conversations with people who didn’t enjoy math as a kid, is often just a defensive description, with no inherent meaning.Daniel Schneider(Mathy McMatherson) offers us several different rungs on the ladder in his quotation of Boyer. The physical five oranges goes up the ladder tothe pictureof the five oranges which goes up to the representation of the five oranges asa numeral.This points in the direction of a definition of abstraction: when we abstract we voluntarily ignore details of a context, so that we can accomplish a goal.

mr bombastic:Bowen Kerins‘ report of student interviews ought to give all curriculum designers pause before they attempt to make math concrete by throwing in wacky “real-world” experiences. Notice also that Bowen refers to abstraction as a verb, not an adjective. I’m coming around to the idea that abstraction is a difficult target to hit as an adjective and much more interesting as a verb.Barryurges clearer definitions and brings in several dictionary entries. Here, again, I’ll say I find the verbs much more interesting and useful pedagogically than the adjectives.@

santosh, you can subscribe to all posts by email at the top of this page (or here) and you can subscribe to comments by clicking the subscribe button at the bottom of the comments page when you comment.@

John, your last paragraph, I think, summarizes a motivation for moving up the ladder – it’s more interesting up there, and more useful for a given purpose.## Kevin Hall

July 19, 2012 - 2:23 pm -You could try studying the Van Hiele levels as a taxonomy of abstraction in mathematics.

## Barry

July 19, 2012 - 2:30 pm -At first, I also thought that the verb “abstract” was more useful pedagogically than the adjective. After further thought, I decided that the verb form is more accessible to reasoning about with regard to pedagogy, but I think we have little chance to teach a student anything if they think math is abstract. The adjective speaks to a type (several types?) of psychologically block, and removing such blocks is of paramount importance. But this is obvious even to novice teachers, and semantics won’t do anything to help.

## blaw0013

July 19, 2012 - 2:34 pm -Briefly, for Radford, what is knowing? learning? understanding? if thinking is focused on as an activity, occurring in inter-relations?

My sense is this metaphor for thinking serves a sociologist well. Probably is not useful to a psychologist. An educator?

## Barry

July 19, 2012 - 2:52 pm -Did I use whiny language? I am sorry if I did. If the discussion seemed pointless, I wouldn’t have contributed again. To the contrary, I thought my post made clear that I agree that it is more worthwhile to study the verb form of “abstract”, but that my own thoughts had gradually diverged from yours over why the adjective form was less useful.

## Dan Meyer

July 24, 2012 - 4:22 am -Note to self:

Christopher Danielson recommended Sfard.

## @MatthewMaddux

July 26, 2012 - 12:05 am -http://www.math.kent.edu/~edd/ICMIPaper.pdf

## Dan Meyer

July 29, 2012 - 8:22 am -Note to self:

Frank Noschese recommended Podolefsky.