My response one year ago to a commenter who said I was always recommending that math teachers apologize for the abstractness of math:

Abstraction doesn’t make math harder. Abstraction makes math possible.

S.I. Hayakawa, 71 years earlier, in *Language in Thought and Action*:

The invention of a new abstraction is a great step forward, since it

makes discussion possible…

So that’s interesting.

Here is a scan of the eighth chapter of Hayakawa, called “How We Know What We Know.” If this series ever turns into a dissertation proposal, odds are extremely high I’ll pull in one or more of the following excerpts:

Our concern here with the process of abstracting may seem strange, since the study of language is all too often restricted to matters of pronunciation, spelling, vocabulary, grammar and such. The methods by which composition and oratory are taught in many schools seems to be largely responsible for this widespread notion that the way to study words is to concentrate one’s attention exclusively on words.

Is it useful to draw a line from that excerpt to school mathematics instruction? Swap “pronunciation, spelling, vocabulary, grammar, and such” for “calculation and symbolic manipulation.”

We learn the language of baseball by playing or watching the game

and studying what goes on.

I’m drawing a line to this post.

This process of abstracting, of leaving characteristics out, is an indispensable convenience.

As many commenters in the last installment pointed out, “abstract” is both a verb and an adjective, though it’s usually the *adjective* that people complain about. My working theory is that if we help students manage the verb better, the adjective will seem less threatening.

Hayakawa notes that the word *calculate* derives from the Latin word *calculus* which means “pebble.” Sheepherders would put a pebble in a box for each sheep that left the fold.

Each pebble is, in this example, an abstraction representing the “oneness” of each sheep –Â its numerical value.

Hayakawa gets explicit about mathematical abstraction:

Our x’s and y’s and other mathematical symbols are abstractions made from numerical abstractions, and are therefore abstractions of still higher level. And they are useful in predicting occurrences and in getting work done because, since they are abstractions properly and uniformly made from starting points in the extensional world, the relations revealed by the symbols will be, again barring unforeseen circumstances, relations existing in the extensional world.

… and the metaphor of the ladder:

The fundamental purpose of the abstraction ladder, as shown both in this chapter and the next, is to make us aware of the process of abstracting.

So here is another tentative thesis: secondary math instructors are generally less aware of the process of abstracting than their colleagues at younger grades. Students at the secondary level are generally assumed to be comfortable with mathematical abstraction so their teachers spend a great deal of time at higher rungs of the ladder. Secondary math curricula *also* tends to disregard lower rungs on the ladder, instead pointing weakly at concrete representations of other things. (eg. “Here’s a frog. You can use the polynomial function that describes the frog’s motion to predict the time the frog will land. Got that? Okay, now let’s do some work with polynomials.”)

But as the abstraction ladder has shown,

all we know are abstractions. What you know about the chair you are sitting in is an abstraction from the totality of the chair. [..] The test of abstractions then is not whether they are “high-level” or “low-level” abstractions, butwhether they are referable to lower levels.

Everything is abstract. Everything is more abstract than something lower on its ladder and less abstract than something higher on its ladder. The chair you’re sitting in is not concrete.

Hayakawa quotes Wendell Johnson who coined the term “dead-level abstracting,” a term which is so useful it’d get an entry in the official lexicon of this blog if this blog had an official lexicon:

Some people, it appears, remain more or less permanently stuck at certain levels of the abstraction ladder, some on the lower levels, some on the very high levels.

See earlier reference to secondary math instructors.

It is obvious, then, that interesting speech and writing, as well as clear thinking and psychological well-being, require the constant interplay of higher-level and lower-level abstractions, and the constant interplay of the verbal levels with the nonverbal (“object”) levels. [..] The work of good novelists and poets also represents this constant interplay between higher and lower levels of abstraction. [..] The interesting writer, the informative speaker, the accurate thinker, and the sane individual operate on all levels of the abstraction ladder, moving quickly and gracefully and in orderly fashion from higher to lower, from lower to higher, with minds as lithe and deft and beautiful as monkeys in a tree.

He would have helped me out a bunch if he had inserted “math teachers” between “novelists” and “poets,” or at least given us a spot in the tree next to the monkeys.

**2012 Jul 25**: My favorite author came to mind just now. In Shipping Out [pdf] David Foster Wallace abstracts over all the micromanaged comforts of a luxury cruise and finds existential despair at the top of the ladder. He does an excellent job shimmying up the ladder from ground-level to the stratosphere and back down again, sometimes within the same paragraph and all while keeping the reader along for the ride.

**Featured Comments**

When my students complain that I’m smarter than them, I counter that I’m just at a higher level of misunderstanding.

I didn’t really learn to understand abstract-as-a-verb until I got it from the computer programming folks, via the How to Design Programs book (free at http://htdp.org if you’re interested). That process is one of the few times in my adult life when I felt like studying one thing made me significantly smarter.

## 23 Comments

## Joshua Zucker

July 19, 2012 - 6:31 pm -I didn’t really learn to understand abstract-as-a-verb until I got it from the computer programming folks, via the How to Design Programs book (free at http://htdp.org if you’re interested).

That process is one of the few times in my adult life when I felt like studying one thing made me significantly smarter.

I still feel very inarticulate when I’m trying to talk about these issues, though. I tell people things like “It’s important to give names to the broad, underlying themes so that you can use them to unify your work”, but when I read something like what you wrote here, I realize that what I’m trying to say is “It’s important to identify the commonalities in your problem-solving experiences so that you can abstract them”, and giving them a name is one important part of that process.

This is a big part of the power of the mathematical practice standards: we’ve got a few specific abstractions that we can recognize in many different forms throughout our teaching and learning.

## Michael P (@mpershan)

July 19, 2012 - 6:40 pm ->But as the abstraction ladder has shown, all we know are abstractions.

You read this line as something like, “everything that exists is an abstraction.” I haven’t read the book you’re excerpting, but that’s not how I read the quote. I read it as emphasizing that everything that we

knowis abstract.I imagined this working something like this: imagine life without abstractions. For now, I’m going to cache that out as, “Imagine life without concepts.” What would that life be like? It would be a life full of particulars, of sense experiences that are completely disconnected from another. There would be no basis to collect certain sense experiences together under one concept.

That would also mean that we wouldn’t know anything useful. Maybe, depending on what “know” means, we would still know some things — I’m imagining the memories of what a particular sense experience was like at a particular time — but we’d be unable to describe it, and we’d be unable to transfer that knowledge to any other context.

Everything that we know involves abstraction. But I don’t think that everything that exists is an abstraction.

## Michael P (@mpershan)

July 19, 2012 - 6:46 pm -Also, there’s a lot of philosophy that I think you’ll find really interesting as you’re going through these questions. A lot of it is very hard reading, but it’s worth it, I think.

One work that comes to mind is Hartry Field’s book “Science Without Numbers.” Part of what Field is trying to do in that book is explain how it is that math is useful to scientists. His explanation makes important use of a notion he calls the “conservative” nature of math. (He’s using “conservative” in the sense that a field extension can be called conservative.) It’s hard slogging, but he isolates a particular aspect of the abstract nature of math that you might find useful.

## William

July 20, 2012 - 4:12 am -Imagine we have some ladder of abstraction (though, more properly, it’s some sort of lattice–there are multiple rungs above and below every rung, and that’ll be important later); there are three things we can do on our ladder.

We can do work on a rung, we can move up a rung, or we can move down a rung. When we manipulate statements on a particular rung, the work we do obeys the (grammatical) rules of that rung, and it’s concrete. When we do work to move up a run on the ladder, we’re making our statements more general and the work we do is synthesis. When we move down a rung on the ladder, we’re making our statements more particular and the work we do is analysis.

In math classes we spend the enormous bulk of our time teaching concrete operations at a level of abstraction that isn’t motivated to our students. If we spent less time teaching what turn out to be fairly abstruse concrete relations among disconnectedly general ideas, and spent more time teaching our students how to do the tricky and demanding work of analysis and synthesis on a smaller vocabulary of general statements, I suspect we’d serve them well.

## John T.

July 20, 2012 - 4:52 am -This is excellent. I can’t wait to see how you propose to help secondary teachers understand they might be existing on one level and why it even matters to be able to navigate to different levels.

## louise

July 20, 2012 - 7:19 am -Abstraction = generalization?

I am blessed with a daughter who was “bad at math”, who observed that those “good at math” generalize in a specific way that she did not “get”. So, using triangles to calculate the height of a flagpole – yes, seen that, can do all flagpoles; the height of a building -yes, seen that, all buildings no problem; the height of a tree -? Because she did not group all of them into a “tall thing” class. Once the “tall thing” connection was explicitly made, she could do it, also street lights, statues, (some difficulty with flying airplanes, modification of tall things to include things where the top is high off the ground) but she was unable to make that leap herself, because a tree does not look like either a building or a flagpole. The specificity got in the way. Without that generalization, math becomes ridiculous.

In math we talk about looking for patterns, are we really talking about looking for “abstractions”? And how do we teach looking for generalities? Do we really do that at all, or do we somehow expect kids to pick this up along the way? I suspect the latter.

So then, off on my own wild tangent of thoughts, is it much more difficult for kids with disrupted lives to predict with patterns? If your life is unpredictable, you stop looking for a pattern to predict what will happen next. Life is in fact a series of disconnected events playing at random.

I also wonder whether this is tied to reading difficulties. The word “table” looks nothing like a table, but it has to generalize to all tables – 4 legs, 3 legs, one center post, high, low, cupboard underneath, a cardboard box… and then, for crying out loud, we use the exact same word to mean something totally different in math!

Someone else’s area of expertise…

## Jason Dyer

July 20, 2012 - 9:20 am -My issue with the “ladder” metaphor is the feeling (to me, at least) that the lower levels are “lost” in the ascent.

Quite often my abstraction is done simultaneously with a concrete level. It is, for me, more of a superimposition: I can simultaneously think of the side of a triangle as a tree, a building, or a flagpole.

This is true even between abstract levels: the proper way to think of the unknown

xis as the potential for all numbers simultaneously. One of the big abstraction issues for algebra students is that in an equationy= 2x+ 1 you could makeyany number and that would fixx, or you could makexany number and that would fixy. There is not one set “answer” to either variable.## Timon Piccini

July 20, 2012 - 10:15 am -Any philosopher’s out there? I think Aristotle talked about this same process with knowing in general. Rather than Platonic ideals that were held in our head, Aristotle said we observed, and abstracted generalizations.

Interesting that humans have witnessed this for thousands of years, and we are still trying to figure it out.

## Janet Abercrombie

July 20, 2012 - 1:45 pm -I’m at a wonderful grade where students move from, what Piaget would describe as, concrete operational thought to formal operational thought. These are the grades where students can begin to construct the need for “x”s and “y”s.

For me, it begins with pushing students into the verbal – that intersection of math and language. Tom Barrett’s “Fizz and Martina” series pose cartoon-like problems that students enjoy. But I’ve been able to harness the power of the series by stealing one question from them: _Explain how you solved the problem. Do not use numbers in your explanation_. (This year, I’ll be giving Edmodo badges for this type of explanation).

Once students can verbalize their thinking apart from the numbers, I can ask them to describe patterns they see – and give those names. Suddenly, the “names”, or equations are not abstract. They apply to something concrete that can be applied to the 100th term situation or 1000th term situation.

My school math was “abstract” because I first learned to manipulate equations and then learned to dread the application of equations to “story problems.” Start with stories. Find patterns. Give the patterns names (equations).

## Dave

July 20, 2012 - 5:18 pm -Dan: My favorite location is the intersection of language and mathematics. There’s a nice donut shop right on the corner that I like!

Similar to the use of the ladder of abstraction, I use the lattice of specificity (my phrase) with students, which is akin to the power set, P(S), of set S, which I do not share with my students.

When discussing mathematics with my students, and they name something using a less specific descriptor, like ‘expression,’ instead of ‘quadratic binomial equation,’ or ‘number,’ instead of ‘leading coefficient,’ I frequently explain the need for more specific language to disambiguate their contribution. I illustrate this frequently using a taxonomicesque representation like “mammal – human – male – student – name.”

While this may not exactly be what you envision when you discuss abstraction, the parallel practice of specificity may help students better understand the abstraction rungs on any specific ladder.

## Steve Thomas

July 20, 2012 - 8:47 pm -So I have been struggling with how to teach kids about abstraction (the verb) and how to walk up and down the ladder (as in http://worrydream.com/LadderOfAbstraction/).

I may use the “atoms Cow wealth” image in this post and I like Brian Harvey’s explanation in his CS10 class “Beauty and Joy of Computing” http://youtu.be/ok_KcxqVrOk

But that talks about cars and the kids I teach are 10 to 16 and may not get all the references.

Do you have any suggestions on how to teach this?

## Steve Thomas

July 20, 2012 - 8:52 pm -FYI, if you look at the Brian Harvey video he starts talking about abstraction at ~8:45.

I really like his comment “an abstraction is about controlling complexity”

## K morrowleong

July 21, 2012 - 8:37 am -There is a great deal of personal pride tied up in this ladder of abstraction of mathematics. In my work I ask people (and by this I mean secondary teachers, engineers, mathematicians, and others who appear able to function confidently at the higher abstraction rungs of mathematics) to connect this knowledge to models at lower rungs of abstraction. The request is often met with derision, almost as a violation of the sanctity of abstraction: it’s a “dumbing down.”

This post gives words to something I have been contemplating. I have been wondering if the derision comes from genuine pride and confidence. Or is it designed to hide the individual’s inability to move among different rungs of abstraction? If this is the case, as I suspect it is, the goal of encouraging “lithe and deft” movement among the rungs of abstraction is just as important as the goal of teaching others to abstract, the verb. I’m a little out of my comfort zone, but It seems to me that derision never comes from a sense of self-confidence!

## Iain Mackenzie

July 22, 2012 - 4:44 am -Have a look the work done by David Tall On cognitive development (http://homepages.warwick.ac.uk/staff/David.Tall/themes/cognitive-development.html)

His body of work is vast in trying to understand how we acquire mathematical thinking. I find it speaks a lot to the process of moving up and down the ladder and acquiring the ability to connect abstraction at different levels. His whole work is inspiring and helps me to understand why some students progress and others get stuck on a specific rung.

## Bruce James

July 22, 2012 - 12:02 pm -When my students complain that I’m smarter than them, I counter that I’m just at a higher level of misunderstanding.

## Steve

July 23, 2012 - 9:55 pm -Since you mention the abstraction of a chair, you might enjoy the musings of an interesting philosopher named J. Krishnamurti, who used to ask his students to point out a chair in the room, and then having done so, would complain that they didn’t point to a chair they pointed to some strangely constructed wooden object. A chair is an abstraction. Then he would ask again, and they would refrain from pointing, and he would say that any child could see that was a chair over there, and any fool could sit on it.. I think he was trying to get at a similar point that you are — that abstractions sit at different levels in some ladder of consciousness.

You can read some of his conversations here: http://www.jkrishnamurti.org/krishnamurti-teachings/view-text.php?tid=41&chid=1

## blaw0013

July 24, 2012 - 7:05 am -@Bruce Brilliant!

## Joe

July 24, 2012 - 1:58 pm -I can’t help but be reminded of the “Van Hiele” levels of abstraction which deal specifically with geometry. I was exposed to this interesting research because it is cited as a basis of Michael Serra’s Discovering Geometry curriculum. In that context, he tried to address the assumption made by the standard high school geometry curriculum that all students were ready for the abstraction known as proofs. I’m not aware of a direct analogue to algebra, but I’d be interested to hear about it.

## Aaron C.

July 25, 2012 - 2:03 pm -so mathematics is a language and abstraction is a key in any language. In math, quantity -> number -> numeral … in a spoken language like English, object -> sound -> letters/words … same basic process (concrete -> concept used for reference -> symbol ???) right? you move up the ladder of abstraction even further by noticing these human activities seem equivalent …

not that you all aren’t busy or anything, but here are some books that I feel like are related and interesting (edutainment purposes not scholarly journals)

The Math Gene: How Mathematical Thinking Evolved And Why Numbers Are Like Gossip by Keith Devlin {changed the way I thought about math}

Stuff of Thought: Language As A Window Into Human Nature by Steven Pinker {had some applicable bits if I recall correctly}

Where Mathematics Comes From: How The Embodied Mind Brings Mathematics Into Being {currently reading}

It looks like significant previews (if not the entire book) of each of these is on Google Books. I would have loved an undergraduate course where something like this was the reading list; and yes, I like book subtitles. I’d be interested in other people’s thoughts on these works or other suggestions … not that I’m trying to perform a hostile takeover of the comments here or anything.