The Uses of Abstraction: Andrew Hacker's Insight

My readers by now will know one of my most annoying characteristics is beating dead horses into the ground. Howls of despair arise whenever I mention "The Wire" as if it was the only piece of literature ever written and now I'm back on to Andrew Hacker's "Math Myth." 

So, dear reader, you have been warned. This is another posting on thoughts provoked by my nth re reading of Professor Hacker, and some stories herein are not new to long time readers of the blog. 
(I know of only three, actually, but one thing about a blog is it gives you information about your audience, from which I conclude there may be as many as 200, unless the three re read the posts multiple times--I can see the number of "hits" and the countries from which they come. I'm still big in Ukraine, inexplicably.)

One of the many excellent points Professor Hacker makes is that algebra is unnecessary to solve many problems for which it can be used to solve problems. 
He also makes the point that some problems which can be solved by algebra do not even have to be addressed because you can use the information without knowing how it's derived.
He cites as an example the formula for calculating the consumer price index:
PC=[(IX{t+M}/IX{t}) {{12m}}-1]*100

(I have no way of doing super or subscripts so I substitute {} and {{}}.)

As Hacker points out, you don't need to be able to parse this equation to understand and use the CPI--the formula shows how it's calculated and kept up dated--any more than you need to know how chips and circuits work to use a computer.

When I was in medical training, the man who ran the cardiac catheterization lab insisted his fellowship trainees derive all the equations which applied to fluid flow dynamics before he allowed them to slip catheters into the coronary arteries of the patients. Of course, he was not very good at the actual catheterizations, or using that data to make clinical decisions, but it was a way he could intimidate people, and it staved off his dismissal for a few years, until people realized he wasn't very good at catheterizations. 

And you don't need to teach students algebra to get them to use abstraction to solve problems. Algebra is just one way to get kids using abstraction. Hacker isn't against teaching kids abstraction with algebra; he just says there are other ways to teach abstraction and you don't need to throw kids out of school because they can't use that particular  tool.

All this stirred two memories: 

#1  When I was 12 years old I got put into algebra. This was one of the early forms of "tracking" in our school system. Only 1/3 of the kids in 8th grade got put into algebra.  My parents arranged for me to be placed with a very charismatic teacher and I struggled and limped along, getting B's and C's but I did well enough to take the second year of algebra in 9th grade.  In one of the few acts of defiance of my early life, I refused. I insisted on repeating algebra 1, this time with the old, withered lady in whose class  I had originally been placed.  That second go round was amazing.  Algebra seemed so simple and obvious. I never got anything less than an "A" on any test in any grading period. I always assumed I simply had a better teacher. Until....
#2 Preparing one of my sons for his algebra test, age 12, I could see he was struggling with "word problems" and we went over one specific word problem multiple times, until he got it. 
It was one of those time/rate/distance problems: If two trains are traveling at different speeds away from each other and one is going at 40 mph and the other is going at 60 mph, how far away from each other are they after 30 minutes?  After much effort, he got it. 
(I should mention this was a kid who was discovered to be unable to read as a six year old--he had memorized the Dick and Jane books listening in group,  and he simply repeated the stories in reading group until his teacher caught him. He had and still has a prodigious memory.)

Anyway, the next day he returns having failed his algebra test and I look at it and has two trucks moving away from each other, the identical question, and another where the two trains are mentioned, but moving in the same direction but different speeds. 
"We went over this last night," I expostulated. "How could you miss these questions?You know this stuff!"
"But it wasn't the same question!" he objected, face reddening. "It was trucks, not trains."
I told this story to a patient of mine, a middle school math teacher, who until that day I had not much liked, because he reminded me of my own math teachers, with his pocket protector stuffed with pencils and protractors and his shirt buttoned up to the top button and his checkered socks and pants with legs several inches too short.  He was there about his diabetes, but he patiently listened to my story--he should have charged me for that office visit, rather than the other way round--and as he listened, a faint smile grew into a broad grin. When I finished, he laughed.
"How old is your son?"
"Twelve."
"He cannot abstract," the teacher said. "He memorized the solution to the problem without understanding the underlying principle. Give him a year. Let those brain cells mature a little more. He'll get it."

And that is likely what happened to me as well. Of course, the teacher was correct--a year later the kid did fine in algebra.

I now use simple algebra frequently, simply because it provides a faster way to get solutions to certain problems, but I use perhaps 5% of the "algebra" I was taught. What is taught under the rubric of "algebra" is a very wide net.

The other fascinating thing about the capacity for "abstraction" is that some people can do it in one realm, but not another. 

I know a man, one of the brightest human beings I know. He learned calculus with ease--where I could not get past page one.  He passed every part of his CPA exam the first time he took it, a feat, I'm told is quite rare.  His capacity for abstract thought with numbers and in certain types of problem solving is not just competent but superior.  His career depends on it, and he is soaring on the wings of that capacity.

On the other hand, we were talking about the issue of the Born Again Christian baker who refused to bake a wedding cake for a gay wedding and he refused saying he thought homosexuality was an anathema in the eyes of God and he would have no part in a gay marriage.

"I don't see why he couldn't have just said he was too busy," my friend suggested, looking for a "cheat route" around the problem.

"He could have," I said, "But he did not take the cheat route. He was making a point that he should not be required by law to do something he found morally objectionable. He could have said he was too busy with other work, but he didn't."

"Well, but it's his business. He can take what business he wants."
"Not if you advertise your business as serving the public. If you then refuse to serve some customers because those customers have certain characteristics, characteristics which they cannot change, like the color of their skin or their own sexual preferences, then that's illegal discrimination, by law."
"Well, but he can only make so many cakes."
"What's the difference between the baker and the guy who owns the lunch counter but refuses to serve Blacks at the lunch counter? Or the guy who owns a motel but refuses rooms to Blacks? 
The principle is that you cannot tell one customer you won't serve him because of his race or because of his sexual preference. 
You can say you won't serve him at your lunch counter because he is not wearing a shirt or shoes or socks--but that's something he can change. 
The principle is non discrimination on the basis of something you cannot change, and by the law, specifically race or homosexuality."

"Well, how can you even know why he refused to bake the cake?  He could give all sorts of excuses. I don't see why he couldn't do that."

This man could abstract in the world of numbers but he was having trouble wrapping his mind around the abstract principle of non discrimination on the basis of certain characteristics which society has voted to disallow.

 What I was waiting for him to say was:  The baker  could  post a sign in his store saying, "Cakes made for godly, heterosexual weddings only." Then he's not saying who he won't make the cake for only who he is willing to make the cake for

That would have been a tougher argument to crack. Then the refusal is not overtly against homosexuals; the hostility was only implied, linking the opposite of heterosexuality with ungodliness. 

It's not that he could not understand the principle, not that he could not go from the specific to the abstract, it's simply he had not been trained to apply a principle in that particular way. That would have been another course, not algebra.