28 Jan 2010 11 Comments
Negative Numbers and Positive Children
My father had a PhD in Math from CalTech. He must have been confused when I grew up unable to deal with numbers. He had to tutor me through High School Algebra, which I barely passed. In my twenties, I decided it was unbefitting an educated person to go glassy-eyed whenever I saw an equation in Scientific American, and started an Algebra class via UC Extension. This lasted until we got to the multiplication of negative numbers. Now I always understood that multiplication was a short cut for addition. 3 + 3 + 3 = 9, or 3 x 3 = 9. And minus 3 is like spending $3, right? But when I add 3 $3 purchases in my checkbook, what I end up with is minus $9.00. So I decided that they had been lying to me all these years about Math (as opposed to English) being Reality and algebra was really magical thinking, and I dropped the course.
So now, all my grandchildren are doing variations of homeschooling. This means that I have much more opportunity to talk to them about what they are learning than families usually get. They are taking algebra, so I asked Evan and Mike to explain to me about negative numbers. My grandchildren are all bright, interesting people, and a lot of fun to talk to, and I was delighted to have a topic of mutual interest.
Various well-meaning people have explained the mystery of negative numbers to me over the years, and I have temporarily understood it. Temporarily. But after a lot of discussion with the kids, it suddenly occurred to me that maybe what we are talking about here is not subtraction, but negation. Negating 3 three dollar checks could leave me with a positive $9. This may be/probably is Really Bad Mathematics, but for the moment, it allows me to get my head around the problem.
On the principle that the best teacher is someone who just learned something, and explaining something is a good way to learn it, I think that maybe it’s time to let the kids teach me. But I’m going to have to have to go back to pre-Algebra to even get up to the point where they are.
Anybody got an old book or know any good on-line courses I could try?
Jan 28, 2010 @ 21:57:32
Actually, your explanation of “Negating 3 three dollar checks..” is from a number theory standpoint and /excellent/ way to look at it.
Jan 28, 2010 @ 22:42:31
Wow! Let’s hear it for intuition. That’s very encouraging.
Jan 29, 2010 @ 00:18:44
Indeed, your intuition captures the essence of subtraction very well. I like to think of negative numbers as “anti-numbers”. If matter and antimatter meet, both will disappear in a flash of light, and it’s the same with 3 and -3. When you spend $3, you add the entry to your checkbook as a “subtract 3″ because spending the $3 makes three of the dollars in your checking account disappear (hopefully producing something more interesting than a brief flash of light in the process). But when you think “subtract three” that’s really the same thing as adding negative three; you didn’t pull out your trusty eraser to erase $3 of your last deposit from your check register, you added a line saying to cancel out $3 of that deposit. Subtraction is just the process of adding a negative, but they don’t explain it that way in first grade, which makes lots of people confused ever after.
That probably doesn’t make much sense, but your intuition is good. Go with it.
Jan 29, 2010 @ 03:12:27
That’s what I was going to add
Specifically, the flip side being that “3 X -$3″ is *positive* 3 times a negative $3, and a positive times a negative is a negative. Double-negative would be *not* spending 3 dollars, three times – saving nine dollars.
Which is exactly where she went, and for exactly the right reason.
–Ember–
Jan 28, 2010 @ 22:41:28
Beginning. Like what they use in grade school to introduce the basic concepts…. I can sort of remember that x and y represent unknown numbers, and that’s about it.
Jan 29, 2010 @ 00:31:57
Oh…I didn’t see this response before I added mine. In that case, I kept my basic math text from last semester…you’re welcome to borrow it if it will help.
Jan 29, 2010 @ 00:30:51
If you don’t find (or someone offers) something better, you’re welcome to borrow my Intro to Algebra text when my class is done.
Jan 29, 2010 @ 02:25:54
Another way to think about negative numbers is using the number line. Positive numbers make your position go to the right, and negative numbers to the left.
Think of addition as a set of commands to your finger (or army man, or ant, or whatever): 3-4 (rewritten +3 + -4) means start at 0, move 3 to the right (+ moves right) then move 4 left. You’ll end up at the -1 spot which is one spot to the left of 0.
Jan 29, 2010 @ 02:29:41
For me I learned/memorized algebra in a language/grammar sort of way. I intuitively gave certain elements in a formula a kind of grammatical or syntactical value…and then after learning the rules I would read a word problem and go “Oh, well such-and-such is this part of that ‘sentence’ and that part is this part, and since the formula goes like this..’that’ is the answer.” Not sure if my right-brained-work-around is helpful…My husband is regularly astounded at how un-mathematically I think.
Feb 12, 2010 @ 19:31:09
Started school last year. I’m taking Finite Math now. I was very dismayed to have to worry about Algebra again. Needless to say, not my best subject.
On the bright side, I’m also taking a class on Black Women Writer and we’re studying Octavia Butler’s Kindred. I LOVE this class!
Feb 12, 2010 @ 19:31:55
BTW, hope to see you at WFC this year!