Wednesday, November 10, 2010

Mastery and Memory

Working at the polls last week, I was talking to a fellow old person. Since she was female, it would be impolitic of me to say she's even older than I am, but it would not be false. The topic was young people today, a favorite topic of old people everywhere.

The reason I think my co-worker last week is my elder is because she said when she got vocabulary words wrong, she had to write them 100 times. That sounds more like torture than teaching to me. I think the standard in my day was ten or maybe twenty repeats of each misspelled word.

While I am not against computers and calculators as teaching aids, a lot of kids don't put much effort into committing things to memory. After all, why learn to spell when there is spell check? Why learn basic math when there is a calculator handy?

When I was a lad, I was good at spelling, though you might not believe me when you see the many typos in my blog. My common fault is that I'm a weak typist and lazy editor. On a test this week, my right thumb got ahead of my left hand and I typed "an done" instead of "and one". A spell checker isn't much use when you incorrectly spell the word you wanted but correctly spell something that makes no sense.

This is a significant problem in early education these days. The question "When will I use this?" often expects a specific answer of when the exact skill being demonstrated will be used in a real life situation. Sometimes, the skill the student is actually learning is how to learn.

My strongest memories of grade school are drills learning how to diagram a sentence or knowing the homonyms backwards and forwards. I almost never make a mistake about there, their and they're or yore, your and you're. I am a fully deputized member of the apostrophe police to this day. There was no such thing as spell check in my youth, and it still won't stop someone from typing loose when lose is correct. That is the nightmare of English spelling. If there was a shred of consistency, lose should rhyme with close and loose should rhyme with choose.

Sorry, kid, no consistency here. Learn how to spell or look like an idiot. It's sink or swim in this pool.

A major difference between language and mathematics is how important the foundation is. A good writer doesn't have to be a good speller if that writer can find a good editor. Shakespeare was famously bad at spelling, but brilliant at rhythm, remarkably insightful as a student of human nature and if he isn't the best coiner of new words ever in any language, I have no idea who is. (Examples: Give me precise synonyms for "assassinate" and "apostrophe". I haven't a clue how people expressed those ideas succinctly before Shakespeare made those words up, among dozens and dozens more.) James Thurber, who wrote some great and funny essays about grammar, freely admitted his first drafts were terrifyingly clumsy.

You won't get to be good at math if you can't do arithmetic. I have a lot of students who don't know if 3/4 should be .75 or 1.333... I tell them that if the top number (numerator) is less than the bottom number (denominator), the decimal should be less than 1. It goes in one ear and out the other.

It's pretty well established that children need to learn language early, probably before the age of five, or they will not understand grammar rules, synonyms and context. I have a hypothesis which I haven't seen tested that committing stuff to memory helps you commit more stuff to memory, like exercise makes you stronger in the long run. The other important component of a good memory is the relational end of it, when you can access a memory from multiple directions. I'm not sure exactly why some people are better at that than others. I know I've had times in my life when my relational database failed me spectacularly, but I haven't been able to come up with a testable hypothesis for why it happened and the best way to avoid it in the future.

I'll be blunt here. I sit on top of a mighty mountain of mathematical knowledge and most of my students are barely in the foothills on a cloudy day, completely unaware of the mountains they have yet to climb, or more likely never will climb. How do I teach mastery to them? I can teach them a few cute tricks they may have never seen and maybe, like me, they'll decide they want to see more. I can give them some vocabulary and grammatical rules, but will they have even a vestige of mathematical insight?

When you see it work, when you have a thorny problem and you see the path home, it's like heaven. Those who haven't done it may not believe me, but it's prettier than Indira Varma when the solution to a hard math problem falls into place. It really helps to have a memory that makes getting to the end of a mathematical idea no harder than finding your way to your childhood home from a few miles away. I am convinced you get that memory from exercising it again and again when you are young. I worry that we have a generation where that kind of exercise is getting rarer and rarer.


sfmike said...


student said...

I remember back in statistics a couple of years ago you mentioned finding the square root without a calculator and taught us how to do it. When my mind wanders in class I either resort to drawing or calculating the square root of random numbers with the method you taught us that day. Even last Wednesday in my applied linear algebra class.

Matty Boy said...

Hi, student. Thanks for writing. I have no idea how many of my former students even know about this blog.

I assume I taught you the iterative method to find the square root of k by taking the average of the recent best guess and k divided by the best guess, which is the shorthand form of the Newton/Raphson method.

There's also a way to get the answer digit by digit which was taught in Robinson' New Higher Arithmetic back at the turn of the 20th Century. It's slower and a little less straightforward.

Again, thanks for stopping by. Glad to hear you went forward with more math after your statistics class. For a lot of people, stats is the last time they set foot in a math class.