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.
7 comments:
You made my head hurt.
Timely! We unschool but lately Young Son has been watching The TEaching Comapny Basic Math videos in prepartion for algebra to dual enroll in college (in a couple of years) and this is what we just saw this afternoon. How algebra is really just solving for unknowns and any regular problem can be expressed that way, division for example. How division is multiplication reversed and subtraction is addition reversed, multiplication is addition repeated, etc.
This is why I love RightStart -- it lays all of this out from Day One.
I was one of those kids who intuitively saw all of these connections; I didn't realize that other people didn't automatically see this stuff until I tried to teach my own child and realized that -- eek! -- the vision in her head was totally different than the vision in mine. No wonder she hated some of the popular math programs that I thought looked fun. RIghtStart gave us the foundation and tools I needed to pass on the vision.
Dawn,
Have you seen Liping Ma's "Knwoing and Teaching Elementary Mathematics"? She has a very clear explanation of several ways of looking at division of fractions, all of which of course lead to the standard algorithm. I found particularly clear her explanation using the “cancellation law”:
Whatever we mean by x divided by y, we want it to be the same as a*x divided by a*y for any non-zero number a.
So, suppose we want to know 3/4 divided by 1/2. That would have to be the same as 2* 3/4 divided by 2 * 1/2.
But, of course, 2 * 1/2 is just 1. So, the answer is 2 * 3/4 divided by 1.
Since dividing by 1 does nothing, the answer is just 2 * 3/4.
Hopefully, that is obviously 6/4.
I won't try to go through the most general case, and Dr. Ma explains this more clearly than I have here. But, with a bit of work, you can convince yourself that the cancellation law means that you invert and multiply.
By the way, if you ever have any doubts about the cancellation law, one way of thinking of it is just as a change of units of measurement: i.e., 2/3 of a foot divided by 1/4 of a foot must be the same as 12 * 2/3 divided by 12 * 1/4.
I.e., you have just changed units from feet to inches.
So, since 12 * 2/3 is 8 and 12 * 1/4 is 3, your answer is 8 divided by 3 or 8/3, the same answer that you get by invert and multiply.
Sometimes, this is actually faster or more intuitive than invert and multiply, though of course it always gives the same answer.
If I can change the subject from math, what is happening with your plans for Greek?
We ourselves are working (slowly) on ancient Greek. We’ve been using “Hey Andrew! Teach Me Some Greek,” but may try out something else soon (“Hey Andrew” gets a bit tough in the later books).
I’ve promised an acquaintance to give her a copy of the simple computer flashcard program I’ve written for ancient Greek vocabulary (you can enter whatever vocabulary you want, although it now has the “Hey Andrew” vocabulary list). I hope to post it on my blog soon for anyone to download, so if you are interested, check my blog now and then.
Anyway, I’d be interested in knowing what you are doing in Greek, what resources you have looked at, etc.
Incidentally, just in case you are wondering, since many homeschoolers studying Greek are fundamentalists, no, we’re not. We know we are descended from fish.
All the best,
Dave
I actually have Liping Ma's book! I read a bit of it, found it wonderful and then neglected to go back to it. Thank you for bringing it to my attention - I'll be grabbing it today to get a firm understanding of what you've explained.
As for Greek, I've stalled a little. I'm using a piece of software, Greek Primer 32 (availible for free here - http://www.triviumpursuit.com/downloads/index.php - under Programs for Windows). It actually sounds like you're ahead of me and Catherine by quite a bit. I'll definitely be checking you're blog!
I've been looking at Hey Andrew! myself but also at some older texts on Google books and Internet Archive. We're in love with a geometry text from 1920 that I picked up at a used bookstore for $5 so I'm wondering if I might have similar luck with an older Greek text as well.
I have to choose something soon though.
Oh! And since with evolutione why not grabbed the revised Darwin fish "Evolved Homeschooler" graphic on my blog and put it on yours too? :)
Gail - I've always thought RightStart looked wonderful but stayed away because anything with too many components and planning tends not to work for me. I even found MUS too much. I've got no doubts about RS's effectiveness though!
Thanks for this great post and link!!! I'm one who failed to make the connection when going through math myself. I'm doing my best to ensure I don't pass that disjointed mentality on to my kids.
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