Comments on: [LOA] They Don’t Know Their Own Power

By: dy/dan » Blog Archive » Better Online Math

dy/dan » Blog Archive » Better Online Math — Fri, 07 Dec 2012 11:30:19 +0000

[…] very explicitly why we use abstractions like x-y pairs and a coordinate plane. This satisfies John Mason's recommendation that we become much more explicit about the process of […]

By: cheesemonkeysf

cheesemonkeysf — Thu, 08 Nov 2012 03:18:34 +0000

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)

By: Paul Wolf

Paul Wolf — Wed, 03 Oct 2012 15:45:17 +0000

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.

By: Andrew Morrison

Andrew Morrison — Tue, 02 Oct 2012 16:11:00 +0000

@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.

By: Dan Meyer

Dan Meyer — Tue, 02 Oct 2012 15:37:00 +0000

Gary Strickland:

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.

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?

By: Michael Paul Goldenberg

Michael Paul Goldenberg — Tue, 02 Oct 2012 14:18:48 +0000

John Mason is indeed the bomb. Seen the 2nd edition of THINKING MATHEMATICALLY yet? Great stuff.

By: Gary Strickland

Gary Strickland — Tue, 02 Oct 2012 00:43:15 +0000

“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.

By: Dan Meyer

Dan Meyer — Mon, 01 Oct 2012 23:40:52 +0000

Andrew Morrison:

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.

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:

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.

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:

(I would have been in the camp of people who got your question wrong, I guess.)

There isn't a wrong answer here. It's just interesting, that's all.

By: Andrew Morrison

Andrew Morrison — Mon, 01 Oct 2012 23:13:11 +0000

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.)

By: Dan Meyer

Dan Meyer — Mon, 01 Oct 2012 20:55:57 +0000

@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.