I find myself lying to my students. A lot. I suppose it doesn't much bother me on a moral level. For one thing, my conscience is perhaps less muscular than it ought to be. For another, I'm generally pretty open with my kids. They know, for instance, that I'm divorced. That I'm quitting smoking for my 30th birthday. That for several years I was professionally violent. I go out of my way to let them know that, within reason, I won't shy away from their curiosity. Still, I lie.
"If we have evidence that current teachers are ineffective (and don't international math test scores provide this evidence?), then why not let the non-educators take a shot?"
In reading through the comments, this particular point focused my attention because I suspect its primary sentiment runs deep within the larger discussion about the state of U.S. math education. I.e., our international math ranking is poor; teachers must not be doing such a good job; so why not let somebody else take the wheel?
On the surface, this doesn't seem a particularly unreasonable argument. It does, however, make some tacit assumptions that are questionable, and even downright strange. Namely:
Teacher training is a negative-value-added process.
Any meaningful discourse about education has to address teacher preparation programs. If teachers are truly failing en masse, then clearly something important and fundamental is lacking in their initial training. Now of course these programs aren't perfect. They might not even be world-class. But to suggest that non-educators would be better classroom teachers is to imply that I'm somehow a worse math teacher after year-long stint in grad school and months of practicum work and student teaching than I was the day I left the Marine Corps in search of a new career. Which is ridiculous. How can exposure to current research in pedagogical content knowledge, educational psychology, and legal/policy decisions make me worse? How can active observation time in great, good, bad, and awful classrooms make me worse? Hands-on experience with real students? Regardless of the level of helpfulness of any of those things, I know that it's strictly non-negative. And please let me know if you can prove otherwise, because I'd like to sue for my $25,000 back.
The countries with better outcomes are being taught by non-educators.
If international math test scores suffice as proof of U.S. teachers' ineffectiveness, then those countries with higher scores, one supposes, must provide some sort of evidence about effective practices. Is Finland spanking us because it got its teachers out of the picture somehow? Not quite. Over there "teacher" routinely rates as the most admired profession among high school graduates; after a rigorous screening process, the most qualified teaching candidates are educated at government expense; all teachers are required to hold, at a minimum, a master's degree (source for all the above). Finland isn't great in spite of its teachers; it just does a much better job of screening and training them. Which all sounds easy and practical, but in order to replicate that in the U.S., our society's view of the teaching profession would have to change dramatically. You must believe your tax dollars are well spent paying pre-service teachers' tuition. You must believe teaching is an important and prestigious position. You must believe that teachers shouldn't have to take a vow of poverty to educate our children. In short, the answer seems not to be getting non-educators into the game, but forcing über-educators into the game.
People who devote all their professional time, energy, and resources to teaching can be called "non-educators."
Like any other profession, education is vulnerable to a certain level of entrenchment. Change can be slow and difficult, and new blood can't hurt. But if someone leaves a career in, say, mechanical engineering to spend all her working (and a whole lot of her non-working) life thinking, studying, worrying about, and practicing teaching math to high school students, then she has officially become an educator. Because that's what educators do. But again, if the profession hopes to attract any of the best and brightest from other fields, then there is going to have to be a societal sea change that makes teaching a viable option for people who have gone to a whole lot of intellectual and financial trouble to become the best and brightest in the first place. Those of us who have made those sacrifices in spite of the trouble/prestige ratio welcome you.
If it were physically possible to fold a piece of paper in half 50 times (it's not), how thick would the resulting origami sculpture be? Quick! No fair calculating. What does your gut say?
If you have absolutely no idea, I'll tell you that a standard piece of printer paper, folded six times by high a school student with very little concern for symmetry or crease definition, has an average thickness somewhere between six and eight millimeters. How much will that increase over the next 44 folds? Any ideas?
Let's be clear: American software engineering is in crisis. Thirty years ago our computer programs were the best in the world; now they routinely lag behind those from South Korea, Finland, China, and even...*gulp*...Canada. In fact, a 2009 assessment found that U.S. reading software ranked 17th among the 34 OECD countries, math software a dismal 25th. In the absence of radical reform, our code will cease to be competitive in an increasingly global economy, and we risk losing our preeminent place on the world stage.
In addition to poor test scores, American software has been suffering from increased feelings of alienation and disengagement. In a survey from last year, 61% of programs said that they "strongly disliked" or "hated" compiling, and more than half said they would rather digitize Wuthering Heights than debug. There's no doubt the situation is dire.
But there is a solution.
I recently had a chance to do one of my favorite (and my students' least favorite) things: talk about words in math class. Math words. I also had the opportunity to use one of my favorite math-teacher-type resources: a dictionary. I don't mean the glossary out of a math book, or a page from Wolfram MathWorld, or any one of the approximately 10.5 million web results (as of this writing) that the Google spits out when prompted with "math" + "dictionary." I'm talking about a good, old fashioned English dictionary, one of three left in my room by the previous English-teaching occupant: Webster's Ninth New Collegiate, circa 1989.
Something deeply unsettling is afoot in the land of math education when I'm teaching the same backwards thing in the same backwards way it was presented to me as a high school kid. To wit, combinations.
Here is the current state of the art, according to the big boys of Advanced Algebra publishing:
Fundamental Counting Principle ====> Permutations ====> Combinations ====> Pascal's Triangle ====> Binomial Theorem ====> Celebration
I submit that, when we do it this way, we're double-charging our students for their attention. We bog them down in unnecessary algebraic trifling, and we go out of our way to delay the payoff for just as long as possible. It's bad marketing, and it's bad teaching. And we don't exactly get away scot-free in all this. I know I'm in for a rough couple of weeks any time I have to close my opening lesson with, "Trust me."
So for the past few months I've been telling my kids that, every time they write $latex (x+y)^2 = x^2+y^2$, they kill a puppy. In fact, I will hereafter refer to that equation as the Dead Puppy Theorem, or DPT. Since its discovery in early September, my students' usage of the DPT has accounted for more canine deaths than heartworms.
Thanks to the book Group Theory in the Bedroom, by Brian Hayes, I finally found someone to blame for my geometry students' daily growing hatred of mathematical language. His name is Robert Recorde. For those of you who, like pre-this-week me, have never heard of Robert Recorde, don't worry: you've seen his handiwork. Recorde was a 16th-century Welsh doctor, mathematician, and author of Whetstone of Witte (1557), the book in which the modern equals sign first appears.