Algebra Redux

Basic Algebra Review (by StarbuckGuy)

When I entered the 9th grade, age 14, I left behind the boring world of arithmetic and entered the new world of algebra. I was transfixed. It switched on something in my brain that validated all my thoughts about studying math and science. I felt I had arrived.

The experience was heightened by lucking into a great teacher at just the right time. My algebra teacher, Roger Buikema, was fresh out of university on his first teaching assignment. He was late-50’s, early-60’s cool, sporting a crew cut, black-rimmed glasses, and a fondness for the Kingston Trio. His enthusiasm for math and science was infectious.

Algebra was the most elegant thing I had yet encountered in school, and even now, fifty years later, the study of algebra, then advanced algebra, remains as one of the highlights of my education. I was never a math wizard or math precocious, but I liked math and did moderately well at it.

I later studied plane and solid geometry, and trig, which I also loved. I dipped my toe into calculus, but by the time I reached that point in my study of math, my primary interests had shifted to folk music, girls, and literature. I always regretted I hadn’t taken math a little further.

A few weeks ago it occurred to me (I’m a slow thinker) that there was nothing preventing me from restudying math. I was already doing puzzles — crosswords and sudokus — and math exercises are the same: puzzles to be solved. I thought I’d start with trig, but when I looked at the subject anew, I realized my math infrastructure had rusted out with age and disuse. I needed to rebuild the scaffolding in my brain by backing up and starting with algebra.

In my brief one-year stint as an engineering student, I had discovered the Schaum’s Outlines series of publications on math and engineering topics. They were excellent as supplementary material to the main textbooks, and served as inexpensive tutors. I liked them because they were clear, but terse. They didn’t muck about with long explanations of things. They gave the essential information then presented a bunch of exercises and review exercises. They also provided the answers so you could check your work as you went.

Just the thing, I thought. I checked Amazon.ca and the Schaum’s Outlines are still going strong, so I sent for Elementary Algebra. The covers are much slicker than they were in the mid-60’s, but the content is much the same.

As I started my review, I’ll admit outright that my brain hurt from working on the exercises. I could no longer “think math” the way I once did. I kept mucking up signed number operations and even simple arithmetic. But I’ve kept at it, relying on brain plasticity to work up some new neurons or perhaps reassign some old ones to get me back into math think. (Note to brain: I really don’t need the complete lyrics of all those 50’s rock songs in my head. Perhaps you could reassign them?)

Slowly, it’s beginning to work. Each lesson becomes a little less strained. I picked up a nice Texas Instruments scientific calculator for $20 to use to check my work, as well as to learn its higher functions. Back in the day I used a Post Versalog slide rule. Calculators hadn’t yet been invented.

As of today I’m working on the review exercises for chapter five, “First Degree Equations.” Chapter six, “Formulas,” appears to be a review of plane and perhaps solid geometry.

I won’t say it’s been easy, but it’s been worthwhile. Some of the excitement of a long-ago fourteen year old is returning. Each exercise is another puzzle to solve. They feel good when I get them right with no errors. They make me think hard when I get them wrong and have to backtrack through my work to see where I went off the rails. That’s good exercise too.

My brain’s capacity for math is improving daily. Not that it’ll help me tie my shoelaces in the morning, but I’ve recovered a lost love. I have no idea where this might lead, if anywhere, but the journey is stimulating.

Basic Algebra Review (by StarbuckGuy)

Sudoku & Mental Exercise

Tools of the Sudoku Trade

When my family doctor asked me what I did to keep my mind active, I mentioned reading (which didn’t seem to impress him very much) and crossword puzzles (which he liked). Then he said, in his blunt, but friendly, way: ‘Take up sudoku. It’s very good for the brain.’

I was skeptical because, like others I’ve talked to, I thought sudoku puzzles involved math. Curiously in my younger years I studied a lot of math and used it often in my science and engineering classes. I was a B or B+ type math student — competent, but not a math wizard. I liked math okay, but not as much as science and English.

But somewhere over the years I lost touch with math and I was never fond of arithmetic, the daily math most of us use. I’ve never been able to do arithmetic in my head. On paper I’m okay and I can wield a calculator with the best of them, but increasingly my mind doesn’t process or conceptualize math very well. Perhaps it’s an age-related thing. So I avoided the sudoku craze, thinking it was okay for the mathematically inclined, but that I was not among them.

Marion, who had done a few sudoku puzzles herself, and didn’t like them, told me differently: ‘It’s not about math at all. It’s pattern matching and solving.’ With that encouragement I bought a Dell book of Easy sudoku puzzles that included some 6×6 grids at the start for raw beginners. Marion was right — there wasn’t any math involved. The numbers could as easily be icons of chess pieces or Babylonian goddesses — they’re nothing but convenient place markers.

And so I began. I didn’t get it right away; easy 9×9 puzzles seemed to me impossible to solve. I tried the 6×6 puzzles and eventually got the hang of it, after working at them for a couple of days (I’m not a quick learner). After being able to solve 6×6 puzzles routinely, I promoted myself to the big leagues, tackling the easy 9×9 ones. A couple of times I almost succeeded, but discovered I have a brain condition I call ‘spatial dyslexia’. Even when I’m being careful and checking my work, I can unknowingly transpose rows and columns, both in my logic when I’m solving a space, and physically when I’m putting in my answers. To be nearly finished with a puzzle only to discover two 4’s or 8’s in the same row or column is disheartening.

I had to agree with my doc though — this was exercising my brain in a way that was new and rigorous. I stuck with it. I remember very clearly the evening I solved my first 9×9 puzzle: it took me 60 minutes. It was the motivator I needed, and within days I was getting down to 30 minutes, then 20, and often 15. I was (and still am) subject to my peculiar form of dyslexia, but I’ve become a little better at checking my work. Any time I try to solve a sudoku puzzle quickly, I make errors. If I take my sweet time and double check all my logic and then my answers when I enter them, I can usually solve a puzzle without errors. Easy ones, of course.

I got hooked. And in helping me learn, Marion got hooked too, though she still prefers crosswords. I now work on anywhere from two to five puzzles a day. After a couple of months of easy ones, I moved up to medium puzzles. I’ve tried some of the hard ones but I’m rarely able to solve them. I get to a point where, with the logic techniques I know, I run out of numbers I can puzzle out, and I refuse to guess.

I thought I’d be clever and look up some tutorials on the Internet. They’re there all right, but they make my head hurt. All that talk about hidden triplets and quads and x-wing solutions is about as clear to me as particle physics. So I decided ‘the heck with hard’. I do the puzzles for fun after all. Medium, with occasional forays into hard, seem just right to me, at my skill level. They’re challenging, but solvable.

I don’t make notations in my puzzles. All the tutorials recommend it, but the clutter bothers me and makes it more difficult, rather than easier, to see the patterns. Notation also takes a long time to do. I can solve medium level puzzles without notations so I’ve opted to keep it simple.

I must admit though, that it discourages me when I see my son’s girlfriend whiz through the hardest sudoku puzzles in ten minutes or less, completing them perfectly nearly every time. She uses no notation either. Some people are gifted. The rest of us plod.

Sudoku

From my journal, 29 Nov 2007:

I solved a sudoku this morning. I’m definitely getting the hang of it for easy ones. I’ll continue along this path until I feel more confident of trying the medium and harder ones. Doc Khanna thinks this is good for my brain. We’ll see.