One of the things I love most about learning at home with my eight-year-old, Eliza, is exploring math in ways that -- for me -- are new and different. I have about as much natural mathematical ability as a groundhog -- or maybe it's just that I was mistaught. Or both.
I can't think of any subject that's taught more badly than math. Think about it -- how much time did you spend, during your formative years, being drilled in addition, subtraction, multiplication and division? This form of teaching probably made sense in an era when children scribbled their lessons on slates in a one-room schoolhouse. But today, when you can buy a pocket calculator for the price of a cup of coffee? And when we're told daily that our children will be facing ever-evolving, complex problems that require flexibility and creativity?
What's with all the computation drill, people? Mind you, I don't claim to know how math
should be taught, only how it
shouldn't.
I do "get" the fact that "real" math is about
patterns. But in terms of my grasp of what math is really about, I'm like a non-reader with vague sense that there's something to writing beyond constructing a grammatical sentence and knowing how to use a semi-colon. Meanwhile, people are out there reading
War and Peace. ;-)
My first formal experience with math occurred when I was five years old
(about to turn six) and had just started first grade. 1972. Students
taking to the streets protesting the Vietnam War. Nixon about to be
re-elected. Yes, I'm getting old.
I was presented with a colorful workbook and told to complete the first few pages. It looked easy enough. I was shown a set of 4 shapes in one circle and a set of 2 shapes in another circle along with the equation 4+2=___ and asked to fill in the blank. I answered all the questions, unaware that I'd made all my numbers backwards. (This is developmentally normal at that age, by the way.) When I cheerfully turned in my assignment to the student teacher, she told me they were
all wrong. "Go back to your seat and see if you can figure out what's wrong with these."
I dutifully went back to my seat, and I tried again and again to work out what I'd done wrong. But I could plainly see that my numbers added up. Either the laws of the universe had been suspended, and 2 and 4 no longer made 6, or my teacher was clinically insane.
No matter. She eventually clued me in to my mistake. And my first grade teacher clued my parents into
their mistake in encouraging me to learn at home during my preschool years. I'd learned to read and write, and was having a marvelous time creating my own stories, but was making all my letters backwards and using capital and lowercase letters indiscriminately. Mothers and fathers: that's why you shouldn't try this at home. Best to leave the teaching to the professionals who
know what's really important.
I quickly decided that math wasn't worthy of my time. I'd complete my work at the other learning centers, conveniently skipping the math center. My teacher readily solved that problem by threatening to paddle me with a ruler if I didn't do all my assignments. So I did. Thus began a 12-year relationship with schooled math that ended with my flunking algebra in my senior year of high school. I drove by the school, a few days after graduation, to pick up my report card. Each subject had its own slip for recording grades, and on the way home I ... umm ... accidentally rolled down the window and let my algebra grade fly out the window. My young, impressionable brother was riding shotgun, and his moral development was irreparably warped. He's never forgotten it. Ask him. He'll tell you.
I could tell you a lot about how badly I mistaught math to my two younger children, who are now teenagers, probably killing their interest in the subject beyond any hope of resurrection. And I am happy to get on my soap box, at any time, about my oldest child's math experiences in her four years in public school. But I won't go there right now. You're welcome.
However, I am now having a
great time exploring math with my youngest. I've completely abandoned all learning standards and expectations and just introduce a topic here and there, following her lead. I've finally figured out that formal teaching of math, at least during the early years*, isn't really necessary anyway. It's in everything: board and card games, video games, cooking, constructive play, art, music,
and the natural world around us, like God's fingerprints on our planet. I'm not a fast learner, but I am learning.
As I posted earlier, I recently "formally" introduced Eliza to multiplication through the "Real Estate Game." Yesterday, I decided to give
multiplication circles a shot.
I'd done this earlier with the older kids.
You start with a circle like this. ( Image lifted from
this post on Mathrecreation)
Then you use multiplication facts, or skip-counting if you prefer, to connect the dots, as shown in
this video and
this blog post. When you get into 2-digit numbers, connect to the
second digit. For example:
the 1s:
1,2,3,4,5,6,7,8,9,1
0 ... (connect the dots between 1,2,3,4,5,6,7,8,9, and 0)
the 2s:
2,4,6,8.1
0,1
2,1
4,1
6, (connect the dots between 2,4,6,8,0 ... then 2,4,6, ... the pattern repeats)
and so forth. If you do this for numbers 1-9, you'll easily see that there is a symmetrical pattern in the shapes created in the circles. (5 makes its own shape, 4 and 6 are the same, 3 and 7 are the same, 2 and 8 are the same, and 1 and 9 are the same.
I love, LOVE this activity, because it doesn't just show how multiplication and skip-counting work -- it illustrates that math has an intrinsic beauty, logic and symmetry. O.K., to a "real math person," this might be amateur hour, but I think it's COOL. ;-) Maybe if I'd been taught this way, my first-grade teacher wouldn't have needed to threaten to hit me with a ruler.
The thing that thrilled me to pieces was the way Eliza took to this activity, stuck with it, and insisted in solving all the math problems herself (through she's never been "taught" how to multiply or learned math facts). Most of the way through, she was devising her own techniques to solve problems, including expanding on what she already knows (e.g. 2 x 7 O.K. ... 2 sixes is 12 ... so (add 2) ... 14!) and using manipulatives:
As I've written before, it's her problem-solving process that fascinates me --
and her perseverance and joy in solving difficult problems! -- not the answers or the specific skills mastered.
As we went along, I encouraged her to notice the shapes we were making and predict what might come next. By the end, she'd worked out the pattern, and was using the shape and pattern to figure out the math facts, not the other way around. (*Happy Dance*) She quickly figured out the "Nines Tables" by looking at the patterns in the circles.
I think this is a great way to introduce math facts -- I don't know whether it is used in schools, but it was new to me when I ran across it on the internet.
What are some of your favorite ways of learning and teaching basic math concepts through exploring patterns?
*I do make my teens do some "book math" -- they'll tell you I subject them to an onerous amount of grueling work, but I call bullshit on that. :-P