Frank Noschese recommended Podolefsky.

Christopher Danielson recommended Sfard.

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

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

Michael Pershan posts several questions about abstraction that may light a few useful paths through the issue for us. This one, in particular, matters to me:

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

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:

For some people, God is an abstract idea and for others God is entirely concrete.

Again, I wonder if this describes someone’s familiarity with an abstraction of a thing and not the abstractness of the thing itself.

David Patterson notes 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 to the picture of the five oranges which goes up to the representation of the five oranges as a 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:

I wonder how much of the discomfort in math … 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.

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.

Barry urges 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.