Category: mathematical philosophy

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Hey, you droogs,

There was an interesting post on the Whallah! blog about an article in the Associated Press,
http://news.yahoo.com/s/ap/20080626/ap_on_go_ca_st_pe/teaching_math;_ylt=Apnza3sjYQ1Rx08Q5.nf0IOs0NUE
concerning the education of math teachers in public schools.

Apparently the National Council on Teacher Quality has done a comprehensive study to come to the conclusion that everyone who is not an “expert” has known for years: Teachers are not being taught math adequately, and generally fail to teach it well to their students. (Do tell…)

Isn’t it funny that the “establishment” will never admit that? It takes an expensive academic “study” to show what is already known, yet Universities (in general) will not do anything about the way they teach teacher how to teach math. They will try some new, expensive methods that some textbook company has lobbied for, of course. But they won’t try anything that might actually work.

That’s why homeschooling and afterschooling are becoming more and more important. Taking an interest in your own child’s education is more important than ever, as public schools tank in their ability to actually teach, thanks to the natural entropy of society, and the idiotically simple-minded ways some people like to deal with it, as with the subtly(?) sardonically named “No Child Left Behind” act.

According to the AP article:
Author Julie Greenberg said education students should be taking courses that give them a deeper understanding of arithmetic and multiplication. She said the courses should explain how math concepts build upon each other and why certain ideas need to be emphasized in the classroom.
Teacher candidates know their multiplication tables, but “they don’t come to us knowing why multiplication works the way it does,” said Denise Mewborn, who heads the University of Georgia department of math and science education.
This is the key to most of what every student needs to know - how multiplication works. Addition is almost intuitive. It is an extension of counting. Once you extend addition to multiplication, though, you need a good understanding of how the base ten system works, and the commutative, associative, and distributive laws. You don’t need to know the names of those laws, of course, but you need to understand how to use them in order to understand multiplication.

That’s the big issue. Just being able to recite multiplication tables is not actually being able to understand multiplication. And just going through the motions and repeating math steps that a teacher has “taught” you by show-and-tell methods, so you can prove that you can jump through the hoops for the big test at the end of the year usually does more damage to your understanding that anything.

So what is there to do about it? First, as a truly concerned parent or teacher, make sure you, yourself understand some of the nuances of multiplication. Like why when you multiply by a fraction, the product is smaller than the multiplicand. (Did I get you with that one? Leave a comment below requesting the Math Mojo take on that one, and I’ll cover it in a new post).

Second, make sure you have at least two ways of explaining to your students how multiplication works. Not just how to do it, but how it actually works. I’m working on a video series about this now. Send me a nudge (again, in a comment below) to make it a higher priority to get it done and available to you faster.

Third, make sure you have a way to assess if your child or students understand what you taught them. The assessment doesn’t have to be a test. Tests are more about beating kids over the head. Asking questions and asking to demonstrate, in a non-threatening way would be my first strategy. If you must beat someone over the head, start with someone in an administrative position.

Here’s one of the reasons why:
According to the AP article:
Since states oversee the preparation of the nation’s school teachers, the report recommends they set tougher coursework and testing standards.
Why is does the solution always involve browbeating the learners? Why are the words “tough” and “testing” so often involved? How on earth does that teach or inspire? The problem isn’t that, “those who can’t do, teach.” The people who run those studies and teach university level education courses usually can do the math they are supposed to teach quite well.
The problem is that “those that can’t teach, teach.” Then they “train” teachers, instead of teaching them. No wonder those teachers have problems teaching. And no wonder the
As I always say, look up when you look for where the problem lies. You can’t blame a third grader for not learning (unless there is neurological damage, of course). If it’s behavior problems, there might be an issue beyond the teacher’s scope, but most behavior problems are dealt with by good teachers.
But beyond those things, start looking up the chain for someone who needs the butt-kicking. If the teacher can’t teach, were they taught well? (Are they even allowed to teach well in that school?) If the teacher’s teacher can’t teach, were they taught well? Is their administrator constantly putting monkey-wrenches in their teaching techniques? Is something going on at the School Board mucking up the school? Is the State requiring more tests, but providing less resources for teachers and students?
Keep looking up. Here’s a hint: Besides the handicapped, who’s got the parking spot closest to the school entrance? Start with him/her.
Remember, when things are looking bad, begin to look up.
I hope to hear from some of you soon,

Brian (a.k.a. Professor Homunculus)

Someone wrote in to ask:

4⁰ * 5³ is 125. Why isn’t it 0?

On the Math.Com website, problems such as 4 to the zero power times 5 to the third power have an answer of 125 as correct. Shouldn’t the answer be zero. If not, why? Thank you!

Professor Homunculus’ response:

The answer actually should not be zero, and here’s why:

Because 4 to the 0 power is 1, not 0.

So 4⁰ * 5³ would be 1 x 5³ which is 125.

Any integer raised to the zero power equals 1.

That is hard for most people to believe, so I wrote a little piece to explain why it makes sense. Here it is:

(more…)

Well, believe it or not, using the “math” they taught you in school, you can “prove” that is true.

Part of the math curriculum of schools is estimating, or rounding up. This is a legitimate and important concept, when it is taught by competent and interested teachers. Man, is that a big “when.”

This brings us to what is one of my main peeves about traditional math-ed. They never mention the consequences and “stuff” concerning what they are teaching you. They teach you how to estimate, and even sometimes what estimation is good for, but they never tell you the interesting stuff about it. In this case, it is the why not.

(more…)

“Common sense is the collection of prejudices acquired by age eighteen.” - Albert Einstein

Towards the end of making math more meaningful, I’d like to discuss something in recent news that resonates with that theme.

While listening to NPR, I heard an interesting story about how political candidates affect each other. You can hear a podcast of the same story here. You can also read the Washington Post’s story (which broke first).

The story concerns what is called “The Decoy Effect” or “Asymmetrical Dominance Effect” in psychology.

In simple terms, the Decoy Effect suggests that if you are faced with two popular choices, the outcome of your choice can be subtly affected by the introduction of a third, less popular choice (the decoy). But the outcome may not be affected in the way you might expect. The introduction of the third choice would have you lean towards choosing the popular choice that is most like the decoy.

The above-mentioned article concerns itself with front-running candidates for the 2008 presidential race. (more…)

“Considering the postmodernist argument that mathematics is nothing more than a game invented by mathematicians, Ben-Ari compares math with chess in the following thought experiment. Supposing we came into contact with an advanced extraterrestrial civilization — would you expect these extraterrestrials to know the Pythagorean Theorem? Of course we would, even though it would have a different name and notation. On the other hand, would we expect any of these extraterrestrials to know how to play chess? No. “

- From eSkeptic, Friday, March 24, 2006 in a review of Moti Ben-Ari’s book Just a Theory: Exploring the Nature of Science. (Prometheus Books, 2005, ISBN 1591022851). The review was written by David Ludden.

Of course, the ETs would probably not have the same name for the Pythagorean Theorem as we do. Neither do people who speak different languages. But they would probably understand the concept.