Where L represents Last Week's Hours
Chloe is considering putting in some time at the high school next year. She's curious about what it's like. So, Friday night, she said she wants to learn some more paper math in preparation for this. She mentioned reading one of the "Princess Diaries" books in which Mia is struggling with an algebra problem. Chloe said she found the problem as confusing as Mia did.
It turned out to be about order of operations, and the phrase Chloe stumbled over was "solve exponentially." So, we talked about exponents, and a little lightbulb went off because of the similarity between 2x2x2x2 and the 2+2+2+2 that Chloe does in lieu of memorizing the multiplication tables.
And then I explained about the order of operations. I forgot (as I am wont to do) how
physical Chloe's understanding of math is, so I scribbled out these equations:
3+5*7=38
(3+5)*7=56
and started explaining about parentheses first, then multiplication and division, then addition and subtraction. She was frowning at me and clearly scornful that those parentheses should make all that difference. And then I said, "Just imagine that last week you worked 5 hours a day for seven days, and then on the eighth day, you worked 3 hours. And for the second one, last week you worked 3 hours on Tuesday and 5 hours on Wednesday, and you made $7 per hour."
Oh!
Click.
It's something that has come up again and again in our unschooling, and I believe it's the root of most kids' struggles with math as it's presented in school. Most kids are physical, hands-on people. Working with numbers in the real, physical, hands-on world is natural and intuitive. But throw those numbers on paper, make them abstract, and it just doesn't make sense.
Also, Chloe will never take mathematics on faith; she won't memorize the rules and "just do the steps." She always wants to know
why. Since I was a champion at memorizing the rules and just doing the steps and it never occurred to me through all those years of math classes to ask why, we are often at cross purposes when discussing math.
Thank goodness for Frank. We call him Mr. Encyclopedia Brain around here. He almost always knows why.
Anyway, our little algebra conversation continued with my showing her a variety of elementary algebra problems. She started reminiscing fondly about a little alegbra conversation we had a couple years ago, where I'd used little drawings of Christmas trees and zombies instead of numbers. I explained that I'd
had to do that, because when the numbers were in there, the girls kept just giving me the answer. I'm sure you can imagine the conversation.
"Okay, so if you have x+2=7..."
"It's five."
"I know, but with algebra, you're supposed to show your steps and..."
"But it's five."
"Right, but right now we're less concerned with the answer than the process, so..."
"IT'S FIVE!"
(pause)
"Okay, so if you have x plus 2 equals Christmas tree..."
Chloe laughed over this little story and listened very attentively as I explained about the equal sign being the fulcrum of the teeter-totter and you must keep the teeter-totter level, and here are the steps for solving for x.
And then it was her turn to work one.
x-4=8
"It's 12."
I think maybe she's an accountant instead of an algebraist.
But it was a fun conversation.