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As I mentioned previously, I find the verb “abstract” way more interesting than the adjective “abstract.” The adjective is often used critically and defensively (“Ugh. Algebra. Too abstract.”) whereas the verb represents a milestone in human cognitive development and a skill that grows more and more precious in the modern workforce. How many skills have a shelf life of thousands of years? (A: Not typesetting.)

So let’s define the verb:


Abstraction is a process or result of generalization, removal of properties, or distancing of ideas from objects.

American-Heritage Dictionary:

To take away; remove.

Let’s stipulate, then, that abstraction requires a context and a question.

If you’re going to remove stuff, there has to be stuff to remove. (A context.) If you’re going to remove stuff, you have to have some purpose that tells you to remove this stuff but not that stuff. (A question.)

Then you’re on the ladder. As you go up the ladder you turn the context into something that excludes the noisy richness of the context but which is much more useful for answering your question. As I looked at all the times my students have abstracted in math class, I saw that the tasks and questions we confront and their order look a lot like this:

We debate the context on the level of experience and intuition. We make predictions. We compare different examples of the context until we understand which of its aspects are common and consequential to our question and which aren’t. We give those aspects names. We decide how to represent them. We decide what to do with those representations. And then we abstract other things in the same context.

If my mind were “light and deft and beautiful” as a monkey in a tree, I’d stop abstracting abstraction here, step down a few rungs on the ladder, and concretize this abstraction of the process of abstraction with an example. That’s next.

2012 Sep 26. William Carey passes along this succinct definition from Barry Mazur (2007):

This issue has been with us, of course, forever: the general question of ab-straction, as separating what we want from what we are presented with. It is neatly packaged in the Greek verb aphairein, as interpreted by Aristotle in the later books of the Metaphysics to mean simply separation: if it is whiteness we want to think about, we must somehow separate it from white horse, white house, white hose, and all the other white things that it invariably must come along with, in order for us to experience it at all.

7 Responses to “[LOA] Abstracting Abstraction”

  1. on 25 Jul 2012 at 4:24 amMARY KIM SCHRECK

    Your mind is indeed “light, and deft and beautiful”…this was a delight to read and consider and apply to my field of English literacy as well as to math. I am currently an educational consultant and author. I give workshops all over the country on engagement, creativity, and literacy. I always refer the math teachers I meet to your blog as well as your TED presentation. As someone who has only agonizing memories of Algebra II in high school, I find your work an oasis.

  2. on 25 Jul 2012 at 5:56 amJames Key

    Two contrasting attitudes:

    non-math person: “Math is so abstract.” i.e. “hard to understand”

    math person: We abstract *in order to understand.*

    Part of our job is to teach people this latter mentality.

  3. on 25 Jul 2012 at 3:09 pmSteve Thomas

    An Alternative definition: “Abstraction reduces information and detail to facilitate focus on relevant concepts.”

    In there they also list learning objectives (See some below):
    Learning Objective 5: The student can describe the combination of abstractions used to represent data.

    Learning Objective 7: The student can develop an abstraction.

    Learning Objective 8: The student can use multiple levels of abstraction in

    Learning Objective 9: The student can use models and simulations to raise and answer questions. [P3]

    Evidence for Learning Objective 9: Student work is characterized by:
    9a. Use of models and simulations to generate new understanding and knowledge.
    9b. Use of different levels of abstraction to represent phenomena.
    9c. Use of models and simulations to formulate, refine, and test hypotheses.
    9d. Use of simulations to facilitate testing of models.

    In thinking about this I am thinking of using some “worked examples” with the kids where I walk them up and down the ladder. And then perhaps ask them to explain (or facilitate their explanations) of the abstractions used in video games.

    This is from Learning Objectives and Evidence Statements (found here:

  4. on 27 Aug 2012 at 5:33 pmDetective Work « make math

    […] mostly triangles and quadrilaterals – in a way that is harder. Harder in a good way.  Whole ladder of abstraction […]

  5. […] stipulated earlier that the act of abstraction requires a context (some raw material) and a question (a purpose for […]

  6. […] [LOA] Abstracting Abstraction […]

  7. […] My students have recently discovered the convention of describing silicon diodes as having a forward voltage of 0.7 V.  They know that this is not always true — or even usually true, in their experience.  The way they reconciled the difference made for an interesting conversation about abstraction — the verb, not the noun. […]