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+Adam Liss Here's one for your "why do I need math?" files.

Here's the NEA's report for "estimated average teacher salaries". Note the word "average", which of course can mean "mean" or "median".

The artifacts that mostly interest us today are tables C-5 and C-6. The careful eye will notice that the numbers are the same on these two tables for ten states and Washington DC (by my count).

We are also interested in tables C-1 and C-2, where we note that the numbers are NOT the same, as well as the glossary, which says that "average" means "mean", that "teacher" means, well, teacher, and that "instructional staff" means teachers as well as consultants, supervisors, principals, librarians, social workers, and others.

What is the statistical likelihood that means will be

http://www.nea.org/assets/docs/2017_Rankings_and_Estimates_Report-FINAL-SECURED.pdf

Here's the NEA's report for "estimated average teacher salaries". Note the word "average", which of course can mean "mean" or "median".

The artifacts that mostly interest us today are tables C-5 and C-6. The careful eye will notice that the numbers are the same on these two tables for ten states and Washington DC (by my count).

We are also interested in tables C-1 and C-2, where we note that the numbers are NOT the same, as well as the glossary, which says that "average" means "mean", that "teacher" means, well, teacher, and that "instructional staff" means teachers as well as consultants, supervisors, principals, librarians, social workers, and others.

What is the statistical likelihood that means will be

**identical**in ten cases like these? ;)http://www.nea.org/assets/docs/2017_Rankings_and_Estimates_Report-FINAL-SECURED.pdf

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A couple of observations:

1. No Republicans other than Bob Corker, who is retiring, voted against this tax bill. Not John McCain, not Susan Collins, not Jeff Flake. The principled centrists, the champions of normal order, the conservatives-with-conscience who can't stand Trump--except for one, who will be gone soon and has nothing to lose, they all got bought off with some targeted concessions in the end. This isn't some aberration of Trumpism; this is the whole mainstream of the Republican Party doing what they do.

2. No Democrats or Democratic-caucusing independents voted for it. Not Joe Manchin, not Heidi Heitkamp. Clare McCaskill and former senator Ben Nelson (a famous centrist) were furious about it. The fabled squishes in our own camp who liberals love to castigate--they all stood firm. There's only one reason they couldn't stop this bill, and it's that there are only 48 of them in the Senate.

There isn't a clearer indication of the distinction between the parties. All the Republicans have to go, and that means everyone votes in every election. Kick them out from City Council and the State House of Representatives up to the Presidency. No "vote the person, not the party". No excuses for the nice, sane ones; they jumped on this train and they're riding it out of town.

1. No Republicans other than Bob Corker, who is retiring, voted against this tax bill. Not John McCain, not Susan Collins, not Jeff Flake. The principled centrists, the champions of normal order, the conservatives-with-conscience who can't stand Trump--except for one, who will be gone soon and has nothing to lose, they all got bought off with some targeted concessions in the end. This isn't some aberration of Trumpism; this is the whole mainstream of the Republican Party doing what they do.

2. No Democrats or Democratic-caucusing independents voted for it. Not Joe Manchin, not Heidi Heitkamp. Clare McCaskill and former senator Ben Nelson (a famous centrist) were furious about it. The fabled squishes in our own camp who liberals love to castigate--they all stood firm. There's only one reason they couldn't stop this bill, and it's that there are only 48 of them in the Senate.

There isn't a clearer indication of the distinction between the parties. All the Republicans have to go, and that means everyone votes in every election. Kick them out from City Council and the State House of Representatives up to the Presidency. No "vote the person, not the party". No excuses for the nice, sane ones; they jumped on this train and they're riding it out of town.

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This is awesomely useful. It's for typing in the Microsoft Word/Powerpoint equation editor. Here are two new things I learned from it just this evening:

1. To do a cube root, type \cbrt

2. To write a fraction as 2/3 instead of stacked, type "2\/3" (both slashes, side by side). This is particularly useful for rational exponents, where the stacked fraction looks confusing.

http://www.iun.edu/~mathiho/useful/Equation%20Editor%20Shortcut%20Commands.pdf

1. To do a cube root, type \cbrt

2. To write a fraction as 2/3 instead of stacked, type "2\/3" (both slashes, side by side). This is particularly useful for rational exponents, where the stacked fraction looks confusing.

http://www.iun.edu/~mathiho/useful/Equation%20Editor%20Shortcut%20Commands.pdf

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A bit of complex number fun, based on my Algebra II class today.

http://teaching.paulhartzer.com/hamtramck/multiplying-complex-numbers/

http://teaching.paulhartzer.com/hamtramck/multiplying-complex-numbers/

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This has relevance to math education. Consider these two problems.

1. You have a stack of cards. Each card has a letter on one side and a number on the other. One rule is that, if a card has a "D" on one side, it has a "3" on the other. You have deal out four cards and see: D F 3 7. Which cards do you need to turn over to check the rule?

2. You are a bouncer in a bar where the drinking age is 21. You see four patrons: One is drinking beer, the second is drinking Coke, you know the third is over 25, and you know the fourth is 18. Which patrons do you need to check on, to make sure they're not doing something illegal?

Most people get the first question wrong and the second question right even though they're logically equivalent questions. This is because context allows us to interpret situations, while abstract symbols get in the way.

https://www.youtube.com/watch?v=1PXy3vWZiJo

1. You have a stack of cards. Each card has a letter on one side and a number on the other. One rule is that, if a card has a "D" on one side, it has a "3" on the other. You have deal out four cards and see: D F 3 7. Which cards do you need to turn over to check the rule?

2. You are a bouncer in a bar where the drinking age is 21. You see four patrons: One is drinking beer, the second is drinking Coke, you know the third is over 25, and you know the fourth is 18. Which patrons do you need to check on, to make sure they're not doing something illegal?

Most people get the first question wrong and the second question right even though they're logically equivalent questions. This is because context allows us to interpret situations, while abstract symbols get in the way.

https://www.youtube.com/watch?v=1PXy3vWZiJo

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Today we covered matrix multiplication in Algebra II. I wrote this up for my students who wanted a review guide, and figured I'd share it with this community as well.

http://teaching.paulhartzer.com/hamtramck/multiplying-matrices/

http://teaching.paulhartzer.com/hamtramck/multiplying-matrices/

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Paul Hartzer commented on a post on Blogger.

"but that only works with whole numbers"

This is why I hate the terms "whole numbers" and "natural numbers":

-- Some sources define "natural numbers" as being the positive integers, others define it as being the non-negative integers. Set theory sources often suggest using two symbols for N, such as N^0 and N^+, to differentiate these.

-- Some sources define "whole numbers" as being the non-negative integers (i.e., the natural numbers plus 0), while others define it as being the integers.

At the high school level, we tend to tell students they have to learn specialized mathematics terminology because it's universal and precise, and then we literally start Algebra II classes with "natural numbers" and "whole numbers".

</vent>

This is why I hate the terms "whole numbers" and "natural numbers":

-- Some sources define "natural numbers" as being the positive integers, others define it as being the non-negative integers. Set theory sources often suggest using two symbols for N, such as N^0 and N^+, to differentiate these.

-- Some sources define "whole numbers" as being the non-negative integers (i.e., the natural numbers plus 0), while others define it as being the integers.

At the high school level, we tend to tell students they have to learn specialized mathematics terminology because it's universal and precise, and then we literally start Algebra II classes with "natural numbers" and "whole numbers".

</vent>

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Watching this reminds me of the 0.999... = 1 discussion. It's not exactly the same, but it's a similar mathematical principle. On an analog clock that runs infinitely smoothly (which analog clocks don't), the hands overlap at 0 + 0/11, 1 + 1/11, 2 + 2/11, 3 + 3/11, and so on up to 11 + 11/11. On a clock, though, 0 + 0/11 and 11 + 11/11 is expressed by the same time, i.e., 12:00.

https://www.youtube.com/watch?v=HcqdqsQq-6M

https://www.youtube.com/watch?v=HcqdqsQq-6M

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Paul Hartzer commented on a post on Blogger.

I've seen math teachers insist (implicitly or explicitly) that functions are necessarily numeric, and that it's a distraction to discuss functions in other terms. I've long held the opinion that a function is something which takes any sort of input and returns any sort of output, that the key detail is that the output has to be predictable from the input. So I really like the chart you linked to, which reinforces that. :)

One of my arguments against the sheer numeric nature of functions is that there are common mathematical functions that take objects with one sort of measurement and return objects with a different sort. The most obvious example is the trigonometric functions, which take an angle measurement and return a (unitless) ratio. I had a long wavering a while ago about whether sin(sin x)) makes any sense, and at the time I allowed myself to be convinced that it does, but I eventually decided that it doesn't really. :)

One of my arguments against the sheer numeric nature of functions is that there are common mathematical functions that take objects with one sort of measurement and return objects with a different sort. The most obvious example is the trigonometric functions, which take an angle measurement and return a (unitless) ratio. I had a long wavering a while ago about whether sin(sin x)) makes any sense, and at the time I allowed myself to be convinced that it does, but I eventually decided that it doesn't really. :)

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Mathematical fact: 0.999... = 1

Do you find this counterintuitive? Does this bother you? If so, why?

(I'm exploring a thought process here, and I'm curious what people say.)

Do you find this counterintuitive? Does this bother you? If so, why?

(I'm exploring a thought process here, and I'm curious what people say.)

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