Last year, I decided to take Algebra I in order to get over my math anxiety. I wrote about it a couple of times here and at in the New York Times. In what anyone but myself could have predicted to be a critical error on my part, I began attending class halfway through the year. I missed the first half of the textbook, never learned how to sing the quadratic equation to the tune of some Christmas carol, and I could only attend class three days out of five because of my teaching schedule, so I was doomed to failure. This year, I have no such excuses. I am all in, from day one, four out of five classes a week.
If I was anxious about my performance in math the first time around, imagine the added pressure I am under now. My students are all aware that I am re-taking the class, and while they finished Pre-Algebra last June, I finished it in June of 1982. My teacher, Alison Gorman, calls on me in class, I do the homework every night at the dining room table while my sons do their homework, I hand in review sheets, and I just took my first test. I planned to take the test along with the rest of the class, but an emergency parent meeting took precedence, so I took my test at home. My sons were fighting over some thing or other at the time, so my test-taking environment was not what you might call calm and stress-free.
That said, things are going really well. Alison allows us to hand our review sheets in early for corrections, and I was the first student to earn a perfect score on the first one. Definitely a new Jess Lahey moment, and the review sheet has a place of honor on our refrigerator. I have been doing well on my homework, and was feeling confident going into my first chapter test. I will hand that in tomorrow, with all fingers crossed.
In the meantime, I've been trying to work out precisely what it is about math that freaks me out. Part of it is certainly the habit I've gotten into, the automatic assumption that I can't execute simple math without using my finger to draw the math problem in the air so I can remember to carry the one. Part of it has to do with my frustration that when when my husband or son multiplies 26 * 4, they simplify the whole endeavor to 25 * 4 for an easy 100, then remember to add on those extra four, for 104. I don't naturally think that way.
Then, last week, I isolated the real source of my frustration. I was doing the following problem:
30 - 5 * 3.
Simple, right? Well, here's what irks me. You have to do the multiplication first, because that's the rule according to the order of operations (PEMDAS - parenthesis, exponents, multiplication/division, addition/subtraction). So...do you do 5*3, which equals 15, then remember that the negative sign is there and take 15 away from 30? Or do you multiply a -5 by 3, which automatically gives you a -15, which you then add (or, as it's negative, take away) the 15 from 30?
Turns out, it works either way. Which is my problem. Alison claims that it doesn't matter. Rules are rules.
Rules. Tricky things, those rules. In Latin and English, rules are slippery creatures, best banned from my classroom. As soon as I tell a student they can rely on a grammatical rule, an exception sneaks out of the recesses of the English language to bite me on the butt and prove me wrong. So this whole trusting of rules thing does not lie within my comfort zone. I brought this up with Alison, and she smiled at my discomfort, reassuring me that I can relax. Rules are the rules, and that's the nice thing about math. Certainty.
My certainty lasted until second period Algebra class. Alison was explaining why numbers to the exponent of zero always equal one - it's a rule I did not understand last year, even after a very sweet student explained it to me repeatedly (see her explanation here). This year, I finally got it. In the middle of the explanation, Alison stopped, got a strange look on her face, and looked at me. She sheepishly apologized to me as she began to write the exception to the rule she had articulated not an hour before. Zero to the exponent of zero is not, in fact, one, but...drumroll...undefined. Not definable. Ahem. An exception. See? Bitten right there in the butt.
And then, an angel appeared. My personal math angel, who, as angels are wont to do, managed to make my life simultaneously simpler and much, much more complicated.
My math angel? Steven Strogatz - sorry, Professor Strogatz. He's a kick-ass teacher, bard of all things mathematical, and a lovely person, as I found out last week. He wrote a series of articles a while back for the New York Times that were amazing, and in advance of the release of his new book, The Joy of X (yeah, I know...so great, you know he did a fist-bump with his agent when he came up with that title), he's doing a new series for the Times. The first piece, on singularities, appeared last week, and I wrote to compliment him on it.
I wrote him an email, he wrote me and email, and we got to chatting about math. I explained my whole distrust of rules issue, especially in light of the zero to the zero debacle, and he replied to that email with his thoughts on the subject. My original email is in black, his are in red (I edited out some of his non-math comments):
Sorry to bother you again, but there's a major issue of math you should be made aware of.
I'm writing right now about the fact that math "rules" make me nervous. [They should make you nervous. There's a beautiful book called John Stillwell called "Yearning for the Impossible" on this very point. http://www.amazon.com/Yearning-Impossible-Surprising-Truths-Mathematics/dp/156881254X It argues that the great advances in math come when people break the rules. Two simple examples: (1) Rule: you can't divide 5 things evenly among 3 people, because 3 doesn't go into 5. True in the world of whole numbers. But if you "yearn for the impossible" and open your mind, break the rule, and invent fractions, suddenly your system of math becomes more free AND more powerful. (2) Rule: You can't take 6 away from 5. True, in the world of positive numbers. But if you yearn for the impossible and enlarge your concept of number and invent negative numbers, you can now work the math of of debts, which you couldn't even conceive of before. I discuss this sort of thing in the beginning of The Joy of x. It's the great narrative arc that runs through all of early math (and later math too, as Stillwell explains).] I teach grammar and languages and there, "rules" are always in quotations - they are subject to the inevitable exceptions. So, when I come across the homework problem:
I get nervous. Is that -5*3 which is -15 [yes] that I take away [no. You don't take it away. If you're regarding it as -15, you need to add it, not take it away, from 30. You already used the negative sign to make the 15 into negative 15. You don't get to use it a second time to do a subtraction. I'll avoid the temptation to make a bad linguistic joke here about double negatives. Boo!] from 30, or is that 5*3, and the negative is a subtraction symbol rather than a negative symbol - I mean heck, I don't know enough to know the difference, I managed to take linguistics as a substitute for college math…
Anyway, the fact that the answer is the same either way, and the fact that my Algebra I teacher can explain that it will always work out the same way, no matter what, does not ease my worry.
There are always exceptions. My husband explained that this is what is lovely and amazing about math, that there are not exceptions.
Alison Gorman, my lovely Algebra I teacher, explained that this is what is lovely and amazing about math, that there are not exceptions.
I went in to math class the next day and learned why 3^0 is 1. Last year, I wrote about this, about how I had to just accept it as a RULE because I could not SEE it, and math people offered some lovely comments that went over my head.
But Alison explained it so I understood this year, and I was happy.
She then gasped and realized that she was about to make me very unhappy. Just when I had been mollified, when I realized that I could take solace in the rules of math, I could TRUST the rules of math, she offered up the one exception to that rule.
Damn that 0^0 and its exceptive qualities. [yes, 0^0 is worthy of a column. Very tricky.]
Back to square one and my distrust of math rules.
This is what I am writing about this evening. Although, as a side note, I did get an A on my first Algebra I chapter review, and I have decided that my new favorite thing to do is simplify. It's quite satisfying. Although when I was just asking my otherwise brilliant husband what the word I was looking for, "What am I trying to say…I simplify like…terms? variables?" he said, absentmindedly, "You simplify like a 'muthah'" which I assume he meant as some sort of compliment.
Anyway. That's what I'm writing about, and I expect you to fix that whole "exception" issue in math as it appears to be my main stumbling block and source of distrust and unease. [Hope my comments above help. The biggie, which I did not talk about yet, is infinity. The Greeks after Aristotle said the rule is, you can't think about infinity. Not allowed. Too many paradoxes. Math was stuck with that rule for thousands of years, till the 1600s, when the yearning for the impossible got too overwhelming, and the need too great. And the result was calculus, which tamed the infinite and harnessed it to solve all kinds of problems that ultimately made modernity possible.You'll see what I mean when you read the book...]
So you would think that Algebra I would be a breeze, what with Professor Steven Strogatz willing to personally explain the subtleties of my Algebra homework, but I guess not. And now that my teacher, my students, my students' parents, my blog readers and math stud Steven Strogatz are aware of my adventures in Algebra I, the stakes have been raised. Good lord, I hope I did well on my Chapter One test.
P.S. Professor Strogatz's Radiolab segments are some of their best, and they generally produce the best stuff on the radio. His TED lecture on synchronicity in the natural world rocks, too.
P.P.S. After reading this post, he recommended this page, for its discussion of 0^0.
UPDATE: I just got my test back from Alison, and I got a 97/102! Some stupid addition errors and not fully understanding how to simplify some fractions lost me a couple of points, but I'm pretty darn proud of myself today.