Category Archives: language in math

Infinity, a place beyond.

That most strange place, infinity,
Is somewhere I don’t want to be.
I’d rather stay with Brouwer
In his ivory tower.

 

and for something lighter try Heavy Man

 

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Lament to the Common Core geometry

Could I move this trapezoid
To that one, in the endless void?
I tried translation and rotation.
Then I had a crazy notion.
I would pass a rigid motion.
Result – a lovely hemorrhoid.

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Fractions, at home and away

In the morning Johnny’s mom
Said “Here’s six candies for your break.
“Give your sister half of them”.
Now Johnny’s brain is on the make.
He gives her one, and then another.
Little sister stamps her feet!
“And the last one!” says his mother.
“Damn” thinks Johnny, “I can’t cheat!”.

Later that day

“Johnny, what is half of six?”.
“I dunno”.
“Well, go get out six lego bricks
“And make a row.
“Now break the row right in the middle.
“That’s half the row.
“Just split the half and count the bricks”.
“I got three”.
“So now you see, three’s half of six”.
But does he know?

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What is an equation? . . . . .What is NOT an equation?

Before starting, a definition: Any combination of numbers and letters and arithmetical operations (including   =   <   >   <=   >=) with more than three symbols is an algebra “thing”.  So, in passing, observe that a “number sentence” is an algebra “thing”.

Equations are neither true nor false: Some examples –

x + 10 = 45
x + 2y = 8
x^2 + y^2 = 4
y = x^2 + 5x + 7
x^2 + 5x + 7 = 0
x = 35
ax + by + c = 0

In each case the equation specifies the value or values of the letter quantities

x + 10 = 45
The value of x is such that if I add 10 to it I get 45

x + 2y = 8
The values of x and y are such that twice the y value added to the x value gives me 8

x^2 + y^2 = 4
The values of x and y are such that the square of the x value added to the square of the y value is equal to 4. This and the one above specify pairs of values.

y = x^2 + 5x + 7      You do these two
x^2 + 5x + 7 = 0

x = 35
The value of x is specified to be 35
and lastly,   ax + by + c = 0
————————————-
Then we have identities, sometimes called equivalence statements.
These are ALWAYS true.
Examples:

3 + 2 = 5
4 = 1 + 3
8 = 11 – 3
(x + 1)^2 = x^2 + 2x + 1
2x(4 + 7) = 2×4 + 2×7 (where x is multiplied by)
6/8 = 3/4
————————————-
Later on, in algebra, we get definitions:

f(x) = 3x + 2
This means “The rule for the function whose name is f and whose input is x is multiplythevalueofhteinputxbythreeandaddtwotoit”
or “The value of the output of the function f for input x is the value of 3x + 2”.

y = f(x)
This means “The value of the output of the function f for input x is to be given the name y”.
These are NOT equations and they are NOT identities.
————————————-
The whole current mess arises from the use of the equals sign for “gives” or “makes”, or “we get”, as in “3 + 5 makes 8”, or  “If we multiply 4 by 6 we get 24”, and we write

3 + 5 = 8, and 4 x 6 = 24
3 + 5 is 8 and 4 x 6 is 24 would be better.

The newfangled term “number sentence” appears to have been invented in order to avoid dealing with the correct mathematical jargon, but I see it as making things EVEN worse,

 

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“Number sentence”, what is this?

Being in complete agreement with  Dan Meyer on the term “Write an expression” I take exception to the vague instruction “Write a number sentence”.

Multiple choice question – Which of the following is a number sentence?

a) 3 + 2 = 5
b) three + two = five
c) three and two makes five
d) 2 + don’t know = 7
e) seven is 5 more than 2
f) they gave him 20 years
g) Mary gave three of her sweets to Jane and was left with 5
h) none of these, although they all have a verb

Answers on a postcard please, addressed to Santa Claus, North Pole

and next time I have much to say about “equations”

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Motherhood now, or should I study math first?

It’s doggerel time again, this time with apologies to Harry Graham, who apparently didn’t write the original “Oh mama dear, what is that mess ……”.  See  allpoetry

         “Oh Mommy dear, what are these sums you can’t do anymore?”
      “Hush,hush my child, just do your best, It’s called the Common Core.”

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Additional Subtraction

Not many do!!!!

Line Of The Week

pi“I don’t math.”

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“Why do Americans stink at math?” Your weekend MUST-READ

I reached this long but well written and quite compelling article by Elizabeth Green via Dan Meyer’s blog. Everyone should read it, especially education administrators and decision makers, all the way to politicians.

http://www.nytimes.com/2014/07/27/magazine/why-do-americans-stink-at-math.html?_r=1

Here are two excerpts:

With the Common Core, teachers are once more being asked to unlearn an old approach and learn an entirely new one, essentially on their own. Training is still weak and infrequent, and principals — who are no more skilled at math than their teachers — remain unprepared to offer support. Textbooks, once again, have received only surface adjustments, despite the shiny Common Core labels that decorate their covers. “To have a vendor say their product is Common Core is close to meaningless,” says Phil Daro, an author of the math standards.

Most policies aimed at improving teaching conceive of the job not as a craft that needs to be taught but as a natural-born talent that teachers either decide to muster or don’t possess. Instead of acknowledging that changes like the new math are something teachers must learn over time, we mandate them as “standards” that teachers are expected to simply “adopt.” We shouldn’t be surprised, then, that their students don’t improve.

Here is Dan’s post

 

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Conjecture……….true?……..false?………undecided?…..

conjecture jpeg1

and now for the dictionary

conjecture jpeg2

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Explain……………………..Prove

Read this, it is an excellent post on the use and meaning of the words in the hexagon (which I borrowed from Michael Pershan’s post), and other relevant words:

http://rationalexpressions.blogspot.com/2014/06/beyond-justify.html

pershan hexagon of proof

 

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