[personal profile] matthewdaly
I think it all starts with word problems.  This is contrary to the model of American math education, and maybe beyond.  We teach you the skill and then teach you how to apply it.

You know what, though?  I'm not the one who is backwards on this.  Remember Psych 101?  When a person is exposed to a new piece of knowledge, she will attempt to incorporate that fact into her worldview.  If she can't find a schema for it or hasn't developed an advantageous strategy for assimilation, then she's barely better off than if she hadn't been exposed to the fact in the first place.  On the other hand, if she can effectively identify a schema and assimilate the fact into it, then her worldview is expanded and she is embiggened by the process.

Not everyone has had Psych 101, I suppose, so I'll illustrate.  Let's say that a young child sees a German Shepherd, which is the first time that she recalls seeing a dog that is as big as she is.  She doesn't know what to make of it; perhaps she is inclined to think of it as a "big animal" like a pony or a goat, except that she has developed a sense of "dogishness" (that adults would call "canine") that this animal satisfies in many ways except size.  That sense is her schema of "dog", and when an adult tells her that German Shepherd are also dogs, she is able to broaden her schema to understand that dogs can be big and perform tasks beyond companionship. 

However, this model is exactly what we don't do in math education.  The skill comes out of the textbook through the blackboard without giving any sense of how it works into your worldview.  This is tragically short-sighted, since mathematics is actually the formalization of the modelling our ancestors did to solve their real-world problems.  Sometimes we are so busy answering "how" that we forget to ever get around to "why".  Some students can get it anyways, of course, but we're far too comfortable with thinking that when other students can't get it that the problem is with the students and not with the math instruction.  Back when I was in high school, if 80% of students barely understood math and 10% of students mastered the concepts, we'd have enough understanding to sustain society's population of alphas and betas.  We don't live in that world any more; machines are currently doing much of the work that used to be done by Deltas and the Gammas are next in line for reassignment.

Let me give you an example I use at the beginning of one of my pre-algebra lessons.  I am leading a full-day field trip, so I made some packaged lunches for the adults and children.  Each adult lunchbox contains seven crackers and five pieces of sausage, and each child lunchbox contains three crackers and two pieces of sausage.  If there are ten children and four adults on the field trip, what is the total inventory of food in the lunchboxes?  Go ahead and solve that problem: note that it is a moderately difficult arithmetic problem that requires a few steps.  Still, in my experience, it is not beyond the intuition of the sufficiently motivated student to understand that they want to "dump out" all of the lunchboxes and gather all of the similar foodtypes together.  I've got a follow-up question too -- did you come up with fifty-eight crackers and forty pieces of sausage as your solution, or ninety-eight food pieces?  Whichever answer you came up with, why did you settle on that as the better answer?

So, once people have worked out a few problems like this in small groups, I show how you would solve this in "algebra language":

10 ( 3C + 2S ) + 4 ( 7C + 5S )
30C + 20S + 28C + 20S
58C + 40S

See, I don't need to drag students on a trip to appreciate how to simplify algebraic expressions using the Distributive Law and collecting like terms.  They made that trip themselves.  My job as a teacher is to point out some of the highlights they'd have spotted along whatever journey they made, and they're going to be able to retain and apply that knowledge effectively because they had already engaged their schema for solving pre-algebra problems.
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Matthew Daly

December 2012

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