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I believe that millions of children each year, as well as almost that many adults, are not clear about what the whole numbers are. It’s not just the whole numbers, either. Fractions, decimals, algebra, even math itself. What are they? Do you have a good working definition? Or do you maybe think there is only one “right” definition for any of them? [...]
 Can I choke him now?
Photo by foxphotograpy (Edited by Brian)
I have a book sitting in front of me called, Introduction to Mathematical Thinking by Friedrich Waismann; the foreword is by Karl Menger, both of whom I admire greatly.
This book has opened my eyes to something very important about math education. And it’s not because the book is so good (which it is). It’s because after about my fifth attempt at getting through the book I finally realize what it has been that’s impeding my progress.
The impediment is the same impediment that has kept me from learning math and many other things that I have considered beautiful and important, but difficult in my life.
While reading the first chapter of the book again, it finally hit me. There is some really sloppy explaining at a very basic level that, if you are the kind of person that takes things seriously and wants to really understand the deeper meaning, throws a tremendous roadblock into one’s understanding.
This problem is so pervasive in the way so many things are explained in school, at work, or in the real world, that I’m sure you’ve come up against it time and time again. But nobody really calls anyone on it. Or at least not often enough. We let this problem slide by again and again, that we hardly notice it, yet it had a detrimental effect on society, probably since the first caveman tried to explain to his neighbor how to hunt the mastodon (if that’s what they hunted).
The problem is this:
Continue reading Explaining Math Terminology →
There was a wonderful article in the Wall Street Journal today about mathematics in the former Soviet Union. It is worth reading for anyone interested in finding out a little about the inner beauty of math.
Here’s a short except:
what mathematics really is: “It was a wonderful education… Gelfand amazed me by talking of mathematics as though [...]
Hey, you droogs,
There was an interesting post on the Whallah! blog about an article in the Associated Press, 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 [...]
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:
Continue reading Exponents of the Zero Power →
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.
Continue reading 2+2=5? →
“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. Continue reading The Decoy Effect →
“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 [...]
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