I read a post at Kitchen Table Math the other day which linked to a presentation (in pdf) by H. Wu called From Arithmetic to Algebra.
It's an excellent read and led to the revelation (for me anyway) that Algebra isn't about letters stuck in math equations or just about solving for an unknown or variables but, "...what is true in general for all numbers, all
whole numbers, all integers, etc." (from linked pdf). You can use x and y in place of numbers in algebra precisely because the operations you perform can be perform with all numbers. Huge light bulb moment and it fed my growing belief that algebra is totally cool and enormously important.
Anyway, that had been simmering on the back burner of my brain for almost a week when Catherine started geometry today. While I was in Ontario we visited a used bookstore and there I found the neatest little geometry textbook from 1920. It's straightforward, clear and compact and I thought I'd give it a try. I've been up in the air about whether to continue with Singapore 6A this year or take a break and do some geometry (Catherine's preference) so I took finding this little text as a sign.
Chapter one involves drawing lines, marking points and measuring distances. One exercise required that, given how many inches are in a metre, how many centimetres would be in 2 inches. Catherine and I got to work. As we did, she struggled. We had to work with decimals and she couldn't remember how multiplying decimals, in terms of how to move the decimal point, worked.
It was after we'd plowed through this that the Wu pdf came back to me. Algebra being about what's generally true about numbers. I realized that, of course, x and y in an equation could stand for a decimal or fraction as well and then that, of course, that meant there was really no special way of multiplying decimals. There is only multiplying numbers and how we multiply whole numbers is absolutely the same way we multiply decimals. The only difference is that we hide the decimal point with whole numbers and then need to introduce tricks and rules about multiplying decimals later on.
This is the same thing with fractions. Take diving fractions for instance. You don't divide, you multiply the reciprocal right? That's what I taught my daughter and it turned the operation into some kind of arcane formula that had to be remembered with tricks and was thought of as something special you do with fractions. But it's not! You can do the same thing with whole numbers and it's just as true and valid. 5 divided by 3 can also be 5 times one third and the answer will be the same. x divided by y = c can also be x times one over y = c, right?
So I did it wrong in a way. I fell into a trap of teaching fractions and decimals as if different rules applied when really I should have spent a little more time pointing out that the decimal point in whole numbers behaves in exactly the same way as the decimal point in decimal numbers when being put through an operation or that the tricks we do with fractions are in fact, the tricks we do with whole numbers. It's simply that we like to hide those things for simplicity's sake when we work with whole numbers.