Check out Dan’s interesting semi-philosophical discussion of the meaning and importance of abstraction:
- [LOA] The Ladder of Abstraction, Part One Of Probably A Lot
- [LOA] The Place Where Language And Math Make Friends
- [LOA] Abstracting Abstraction
- to be continued…
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.
Some of My Favorite Comments
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?
I wonder how much of the discomfort in math, or English for that matter, is due to personal taste, and how much 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.
For me, it begins with pushing students into the verbal – that intersection of math and language. Tom Barrett’s “Fizz and Martina” series pose cartoon-like problems that students enjoy. But I’ve been able to harness the power of the series by stealing one question from them: Explain how you solved the problem. Do not use numbers in your explanation. (This year, I’ll be giving Edmodo badges for this type of explanation).
Once students can verbalize their thinking apart from the numbers, I can ask them to describe patterns they see – and give those names. Suddenly, the “names”, or equations are not abstract. They apply to something concrete that can be applied to the 100th term situation or 1000th term situation.
When my students complain that I’m smarter than them, I counter that I’m just at a higher level of misunderstanding.
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.