I was at South Dakota State University last week and I asked some future math teachers to define the word “abstract” in a sentence. All of them defined it as an adjective, not a verb. They were more aware of “abstract” as something you are, not something you do.
- A thought or idea that cannot be made tangible or concrete.
- Abstract is something that is different, non mainstream, and requires higher level thinking.
- Anything that is out of the ordinary or requires creative thought.
- A concept or idea that is not easily or not able to be put into concrete or physical terms.
- Beyond the logical ways of thinking about problems and ideas.
- Not concrete. Imaginary. Out of the box thinking.
John Mason, in a great piece called “Mathematical Abstraction as the Result of a Delicate Shift of Attention“:
When the shift occurs, it is hardly noticeable and, to a mathematician, it seems the most natural and obvious movement imaginable. Consequently it fails to attract the expert’s attention. When the shift does not occur, it blocks progress and makes the student feel out of touch and excluded, a mere observer in a peculiar ritual.
If they don’t understand their own power, how will their students?
BTW: Also great. Frorer, et al:
… we rarely find [abstraction] explicitly discussed let alone defined. You can pick up a book entitled Abstract Algebra and not find a real discussion of abstraction as a process, or of abstractions as objects …
14 Comments
Austin Mohr
October 1, 2012 - 11:30 am -Did you ask them to define “AB-stract” or “ab-STRACT”? For me, at least, the former is an adjective and the latter a verb.
Dan Meyer
October 1, 2012 - 11:35 am -Fair question. I wrote it on the board and said, “Would you do me a favor and define this word in a sentence.”
Christopher Danielson
October 1, 2012 - 11:48 am -You will, of course, repeat this experiment with practicing teachers (who do not read your blog, and therefore have never heard of the Internet-an impeccable sample to be sure). And I doubt you’ll find much different results.
Also, John Mason is the bomb diggity.
Piers Young
October 1, 2012 - 11:58 am -Agree one has to spell it out for students.
Beginning to look at abstraction with 9/10+ year old students through programming and computational thinking.
http://games.thinkingmyself.com/ turns the long, tricky, frightening word into something obvious for a lot of them. They tend to have an “is that all there is?” moment. (And then of course they forget what abstraction is).
What’s nice, though, is that they are beginning to see abstraction in IT, maths, English and more areas. They can’t remember what it’s called but they recognise it with a “hey, it’s that thingy, you know, where you pull things back and swap other things in”.
Dan Meyer
October 1, 2012 - 12:55 pm -@Christopher, yeah, that’d be a fun experiment. I’m not convinced at all that vet teachers are all that aware of this practice that comes so easily to them.
@Piers, thanks for the link. The abstraction tutorial is pretty great.
Andrew Morrison
October 1, 2012 - 3:13 pm -Is the point then that teachers should be aware of how to build the abstraction? I mean, that’s what I’ve been taking away from your LOA posts.
Posing a question about what “abstract” means (even when it is spelled out on a board) doesn’t necessarily clue in a teacher to what you’re actually trying to get at.
I think you want teachers to be cognizant of what it takes to be able to climb the ladder of abstraction, and recall that learners are not all ready to reach the next rung at exactly the same point. Why couldn’t the question you pose to teachers be “What does it mean to abstract?”. Then you’ve cued them that you’re looking for a verb. The discussion could lead naturally into what it means to be able to assist learners in their climb up the ladder.
Also, the examples you gave above refer to “abstraction” (the noun) and not “to abstract” (the verb). (I would have been in the camp of people who got your question wrong, I guess.)
Dan Meyer
October 1, 2012 - 3:40 pm -Andrew Morrison:
One point of this series has been, yes, there’s this process called abstraction that many math teachers do quite effortlessly –Â unconsciously, even – and we need to be explicit about that process with our students.
Andrew Morrison:
I wasn’t curious how the students would define the process of abstraction. I was curious how many of them would cite the process at all, rather than the adjective.
Andrew Morrison:
There isn’t a wrong answer here. It’s just interesting, that’s all.
Gary Strickland
October 1, 2012 - 4:43 pm -“If they don’t understand their own power, how will their students?”
I have been following the LOA discussion for a while and I have to admit I had never clearly defined the definition of abstraction and what it meant to me as a teacher.
I teach physics and moving along the ladder of abstraction makes sense to my students – now. I took a couple of days at the beginning of school to discuss the process. We take concrete events and then dissect them into abstractions of language, diagrams, graphs, and algebraic expressions. The students now begin looking for ways to complete that process.
It is also pretty interesting now that if we begin somewhere in the middle – with an equation perhaps – they understand that they must look for the other components as well as a concrete application.
I’m not sure this is addressing the point of hour post, but I would say that it takes some metacognition and reflection to understand what abstraction means as a teacher. I know I am a better instructor and my students are better learners after spending some time defining what abstraction means.
Michael Paul Goldenberg
October 2, 2012 - 6:18 am -John Mason is indeed the bomb. Seen the 2nd edition of THINKING MATHEMATICALLY yet? Great stuff.
Dan Meyer
October 2, 2012 - 7:37 am -Gary Strickland:
No joke. I haven’t mentioned it enough (if at all) but I’m approaching this #loa feature less as a way of telling you what I already knew about math teaching than as a report of what I’m learning. The more I dig, the more I’m surprised there wasn’t more explicit talk about abstraction when I was a math student or a preservice math teacher. I mean WTF, everybody?
Andrew Morrison
October 2, 2012 - 8:11 am -@Dan, thanks for the followup to both my comment and Gary’s. I had originally read your anecdote of asking the teachers the question and assumed you were attempting to elicit a particular (and maybe less useful?) concept of abstract in order to jumpstart the conversation. I think I was having a bad day when I read that, which clouded my view of what you were trying to say.
This whole issue of metacognition has been interesting to follow in the physics education research community. I see a lot of parallels with your posts about abstraction. Thanks again.
Paul Wolf
October 3, 2012 - 7:45 am -I love this LOA thing.
People who want to Meyerize (you’re welcome) their classrooms but are too afraid or constrained to do it can look at this idea and make little changes to what they already do. Even a person hell-bent on lecturing for a whole class period can take the idea of a ladder of abstraction and improve their teaching by a mile.
cheesemonkeysf
November 7, 2012 - 7:18 pm -I have been nuts about John Mason’s synthesis of thinking about semiotics and mathematical abstraction for a long time. Most competent mathematical thinkers are asleep to the fact of that tiny shift of unconscious attention. Learning how to notice it does seem to be a key to helping students who find it opaque and difficult.
I’m interested to see where all of this takes you!
– Elizabeth (aka @cheesemonkeysf on Twitter)