| Things I Wish I Had Been Taught in Math Class |
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| Monday, 09 February 2009 05:36 | ||||
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…but I had to figure out on my own: Some of the following missing facts are just particular quirks of my own education, and some are common across all mathematics education. Why subtracting a negative number is adding. There are numerous explanations in the literature, but here’s the simplest: Is subtracting a debt a good thing or a bad thing? Why division by zero is bad. Here’s a proof that 1=2:
Division by zero happens between steps 4 and 5, since a-b = 0 and you’re dividing both sides by a-b. Why dividing by a fraction is the same as multiplying by the reciprocal. 3 divided by 1/2 means “how many times can we fit 1/2 into 3?”
So for 1/x, each whole part gets divided into x smaller pieces. Then we just multiply the number of smaller pieces by the number of whole parts; hence we’re really multiplying our original number by x. For dividing something like 3/4, it can be considered two separate problems: first divide by 1/4 (that is, multiply by 4) and then divide by 3 (that is, multiply by 1/3). Division is distributive just like multiplication. All examples given in textbooks of the distributive law look like 2(4+5) = 2 * 4 + 2 * 5. But the same thing applies to division! That is (for example) if you insist on using the multiplication form division can still be written like this:
The “canceling curse” that so bedevils high school teachers. would be much alleviated by explaining
but no teacher ever explained it to me; it was just marked wrong. And concurrent to that… What canceling really means. It’s just use of the identity property. The Greek alphabet. The version I use with my students is here. Teachers started using “phi” and “beta” as if I knew what they were talking about; equations that used them looked like blurry nonsense.
Where logarithms are used. No teacher explicitly stated to me that solving an unknown in an exponent Your browser may not support display of this image. needed a logarithm. I’m still not sure why. How you treat an expression and an equation differently. Now, my teachers did mention the difference between an expression and an equation but only in a purely identificational sense. None ever stepped forward said, if you have an equation you can do anything you want to one side as long as you do it to the other side
But expressions require much more delicacy, so for example if you want to mess with a fraction
you can use the identity operation of multiplying by 1.
Any obvious ones I missed? What gaps were in your mathematical education?
POSTED ON HOTCHALK.COM Comments (3)
PI
written by Charles, April 01, 2009
I wished that I learned that pi is "approximately equal" to 3.14 and not "equal to" 3.14. Any comments?
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... written by Megan Montgomery, March 12, 2009
How about the borrowing rule in subtraction!
91 -89 As a child I didn't conceptually understand why I crossed out the number added ten and crossed out the other number and subtracted one. This was all procedure to me! Not until I was teaching the NEW ways to subtract did I gain a DEEP understanding of subtraction. I don't know how I ever made it through calculus! report abuse
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Jason Dyer holds degrees in Fine Arts Studies and Math and teaches at Pueblo High School in Arizona. His school mascot is the Warriors and his other blog of residence is 
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As a constructivist, I believe that students should make sense of math in addition to learning algorithms. In a stations/centers based classroom, students can use manipulatives.
How many algebra teachers have found success with the algebra blocks?