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.

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?

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.)

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

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.

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.

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.

@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.

[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).

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.

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

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.

You could try studying the Van Hiele levels as a taxonomy of abstraction in mathematics.

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?