Math notation in school

While tutoring pre-algebra today it occurred to me that many school kids may be confused by the notation used in mathematics more than by the underlying mathematical concepts. Correspondingly, I'm wondering whether undue emphasis is being placed on the operational aspects of mathematics (how to manipulate mathematical expressions) and too little on the semantics (what the expressions mean).

As an example, today in math class we were practicing how to manipulate exponents, e.g.:

The rule says, "if you multiply two numbers with the same base, add the exponents."

On the other hand:

because of the rule, "if you raise a number raised to a power to another power, you multiply the two powers."

From my interactions with my students, I had the impression that the students were executing the rules blindly, with no understanding of why they were correct. Didactically I found it helpful (and hopefully they did too) to expand the powers of x:

so that the "addition" rule of exponents emerges naturally.

Room for Improvement

I'm imagining a different way (from what I'm used to) of teaching what exponents are: First off, the reason why you would ever multiply a number by itself should be exemplified clearly (I'll sweep this issue under the rug for now). Second, start multiplying several variables by themselves many times, e.g.,

Then say, "instead of having to write out all these x's, let's create a 'shortcut': when I write x^4, this means xxxx." (The carat ^ symbol was used in place of superposition because the Blogger doesn't allow it; sorry.)

The idea here is to make the students feel the pain of not having the notation - it takes too long to write. Then, show them the shortcut notation and how it saves them time and effort. I think this might help make the computation that is based on the notation (e.g., add the exponents) seem less arbitrary when it's taught.

Comment on Devine's warning of "permanent injury" when youth embrace their homosexuality

Philosophy professor Philip Devine of Providence College wrote a book called The New Fuzziness (pp. 85-86):

Rorty's habit of announcing himself persuaded, without citing an argument, is also a political liability. One issue on which Rorty exhibits this trait is of special interest to educators. Adolescent sexuality is frequently unformed, and many adult heterosexuals have homosexual episodes in their past. Hence some parents are likely to fear that some educators will encourage young people involved in homosexual relationships (or even undergoing homosexual feelings) to label themselves as homosexual, to their permanent injury. Without so much as a hint of an argument, Rorty accepts Eve Klossofsky Sedgwick's inflammatory description of policies designed, well or badly, to prevent such counseling as "child abuse" (PP 2:138). In an area of life where strong feelings abound, the attempt to override some of them requires claims on behalf of reason that he neither makes nor is in any position to make.

The book as a whole actually looked quite interesting, but this particular paragraph I found irritating and somewhat offensive for a couple of reasons:

First, and most technically, I find it hypocritical for Devine to criticize Rorty for making claims "without so much as a hint of an argument," when Devine himself claims that "adolescent sexuality is frequently unformed" without providing any evidence supporting this claim.

Second, and more substantively, Devine implies that a teenager who labels himself as gay is inflicting "permanent injury" on himself. I have several problems with this claim: In my assessment, what people really mean when they say such things is that, if you're gay, then you're going to die of AIDS. Now, there is an undeniably strong statistical correlation between (male) homosexuality and AIDS, but to say that coming out as gay dooms you to a horrible death caused by AIDS is ridiculous given that the disease is highly (nearly completely) preventable. Other legitimately cited problems with modern gayness include drug use and promiscuity-induced sexually transmitted infections. Though there may be a statistical correlation (perhaps even a strong one) of these problems with homosexuality, there is nothing causal in the relationship, and I can only believe that greater societal acceptance of gays -- fostered by encouraging young people to accept their sexuality even when it disagrees with the norm -- will tend to decrease the problems associated with it.

The last possible source of "permanent injury" that I can imagine is the ridicule one risks when coming out. But to attack Rorty for encouraging young people to accept their sexuality for this reason is even more ridiculous because obviously by encouraging young people to accept their own, and others', homosexuality and other forms of non-heterosexuality, this form of "permanent injury" will be reduced and, hopefully, gradually eliminated entirely.

What I find most frustrating about Devine's paragraph, however, is the complete omission from consideration of the negative effects of not encouraging young people to accept their natural sexuality, i.e., telling our youth that they should really try to be straight. From first-hand experience I can say that trying to play the heterosexual role when it is not one's innate predisposition is harmful. In some cases, the injury may even be permanent.

Anti-gay rhetoric in the UCSD locker room

From: Jacob Whitehill, Graduate Student, Computer Science & Engineering, UCSD.

Title: "No, it's not ok to ridicule fags"

Dear Editor [of UCSD Guardian],

I consider it an achievement of America today that it is generally considered unacceptable to maliciously ridicule or intimidate an entire ethnicity in public. Expressing disgust of gays, on the other hand, is merely considered "controversial"; though it may (rarely) raise an eyebrow, it will almost never trigger anyone to object to the speaker's message.

This evening at the RIMAC gym, while changing my clothes in the Men's Locker Room after working out, I overheard a rowdy group of 3-4 guys in the aisle next to me talking. At some point their conversation went as follows:

Person 1: "... is a homo."

Person 2: "Hey, what you got against gay people?" (apparently setting up the joke)

Person 1: "A lot! Hahahah!..." (punchline). Malicious laughter ensues.

To these dear gentlemen:

As perverted, aberrant, and pansy you may think us to be, no, you still can't use homophobic speech in public. Not in general society, and especially not at UCSD. If you replace the word "homo" with any racial epithet, I'm confident that even you would find it unacceptable to make that same joke.

Sincerely,

Jacob Whitehill

Grad student

PS: slamming the lockers unnecessarily is annoying as well.