Negative numbers, the minus sign, abstract algebra.

I was pondering the reality of negative numbers and after figuring out that a sequence of dots on a line can be extended in each of the two directions, and then arbitrarily selecting one dot as “the zero”. The line can be further labelled as 1, 2, 3, … to one side and -1, -2, -3, … on the other side.
(better to label the 1, 2, 3, … as +1, +2, +3, … and consider the lot as “signed numbers”)

Soon proceeding towards arithmetic I concluded that 7-3 is 4, and also 8-4 is 4, and therefore 13-9 is 4, and then 3-7 is -4, and -2-2 is -4. It was then observed that if a-b=c then a-y-(b-y) is also equal to c, regardless of the signs of the specific numbers involved.
This of course is stunningly obvious when looking at the signed difference of the first and the second number as an extended number line diagram.

The outcome of all this was an arithmetic for 0, 1, 2 modulo 3, and  the signed difference x-y is a binary operation diff(x,y) with table:

…x  … 0     1     2
y
0         0     1     2
1         2      0    1
2         1      2    0

Example: 1-2 is -1, which is 2 modulo 3

So a non abelian, non associative algebra with a not quite identity satisfies the conditions, where A=1, B=2 and C=0
—————————————————–
There are three objects and an operation called “doesn’t have a name”.
Two are similar, and the third is a bit different
They are paired to yield a single object as follows:

AA = BB = CC = C
AB = BC = CA = B
AC = CB = BA = A

Notice that BC and CB are different, so non-abelian.
Worse is that (AC)A = C and A(CA) = B are different, so non-associative.
And consequently A and B and C are different.
—————————————————-

Interestingly, and maybe separately, the minus sign behaves very differently from the plus sign:

a-(-b) is a+b, but there is no way of writing a-b using only addition.

This means that all expressions can be written with “minus” alone.

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Minimal Abstract Algebra

This is a challenge !!!!!

There are three objects, A, B, and C, and an operation called “doesn’t have a name”.
Two are similar, and the third is a bit different
They are paired to yield a single object as follows:

AA = BB = CC = C
AB = BC = CA = B
AC = CB = BA = A

Notice that BC and CB are different, so non-abelian.
Worse is that (AC)A = C and A(CA) = B are different, so non-associative.
And consequently A and B and C are different.

Your job is to identify (model, in current jargon) the objects and the operation.”

The next post will be “the solution”.

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Ed Reform 2.0: If You Can Train a Pigeon to Detonate Bombs…

The video is scary:

Save Maine Schools

Back in 1977, Superintendent James Guines of Washington D.C. explained his district’s competency-based education pilot program like this:

“The basic idea is to break down complicated learning into a sequence of clear simple skills that virtually everyone can master, although at different rates of speed.If you can train a pigeon to fly up there and press a button and set off a bomb, why can’t you teach human beings to behave in an effective and rational way?”

“We know we can modify human behavior. We’re not scared of that,” he added. “This is the biggest thing that’s happening in education today.”

laughing-gif1.gif

HA! Those crazy 70’s! Boy did they have some crazy ideas back then.

But wait…

Now check out this recent video from Angela Duckworth and Katherine Milkman:

Here’s what Milkman has to say:

“If you repeatedly reward good behavior, and pair it with memorable cues, positive routines become…

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Scary, and not just mathematically speaking …

Try this for size:

https://mystudentvoices.com/how-old-is-the-shepherd-the-problem-that-shook-school-mathematics-ad89b565fff#.a7llwy3mv

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The greatest myth about math education ?

I have to go along with this.

You can get the conclusion, but the rest is quite compelling.

“In conclusion, it would be wonderful to be able to get all students competent in Excel and arithmetic, and a little bit of algebra, statistics and programming. Higher mathematics should be offered and taken by those who need it, or want it; but never required of all students.”

https://fee.org/articles/the-greatest-myth-about-math-education/?utm_medium=popular_widget

The article is from the “Foundation for Economic Education”

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The “One of Many” Fallacy

Nice one!

mathbabe

I’ve been on book tour for nearly a month now, and I’ve come across a bunch of arguments pushing against my book’s theses. I welcome them, because I want to be informed. So far, though, I haven’t been convinced I made any egregious errors.

Here’s an example of an argument I’ve seen consistently when it comes to the defense of the teacher value-added model (VAM) scores, and sometimes the recidivism risk scores as well. Namely, that the teacher’s VAM scores were “one of many considerations” taken to establish an overall teacher’s score. The use of something that is unfair is less unfair, in other words, if you also use other things which balance it out and are fair.

If you don’t know what a VAM is, or what my critique about it is, take a look at this post, or read my book. The very short version is that it’s…

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Denis Ian: Competency Based Education and the End of Education

Kids need hammers.

Diane Ravitch's blog

Denis Ian warns that “competency based education,” online teaching and assessment, spells the end of education and of childhood. It is not just a threat to public education. It is a mortal threat to education of any kind.

He posted this comment:

Competency based education isn’t a mirage anymore. It’s here.

Beyond the view of skirmishes now underway across an array of states, is an emerging reality that … in a very short while … this destroying reform will have razed an American institution to a mound of rubble.

And in its place … for as far as the eye can see … will stand drive-thru learning centers offering kiosk-educations from a B. F. Skinner touch-screen that will supply the finger-pointer with all they need to succeed in a life of rich monotony.

That’s what your now titling schools are going to look like. And that’s your child’s purgatory. Dante…

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Split a length into 5 equal pieces – fractions

The parallel equally spaced lines

and the  desired  length HI, of ribbon, wood, anything non-elastic.

parallel-lines-guide-2

DONE !

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1/5 is one fifth of the length of a line segment of one unit – but how?

This comes from the Common Core

Develop understanding of fractions as numbers.
1. ……
2. Understand a fraction as a number on the number line; represent
fractions on a number line diagram.
a. Represent a fraction 1/b on a number line diagram by defining the
interval from 0 to 1 as the whole and partitioning it into b equal
parts. Recognize that each part has size 1/b and that the endpoint
of the part based at 0 locates the number 1/b on the number line.
b. Represent a fraction a/b on a number line diagram by marking off
a lengths 1/b from 0. …….

….but how do you do it ?????

The mystery is solved…….

Here is the line, with 0 and 1 marked. /You chose it already !

download20

Here is a numbered line, any size, equally spaced, at intervals of one unit.

It only has to start from zero.

download22

Now construct the line from point 5 to the “fraction” line at point 1, and a parallel line from point 1 on the numbered line.

download23

The point of intersection of the parallel line and the “fraction” line is then 1/5 of the distance from 0 to 1 on the “fraction” line.

download24

1/5, 2/5, 3/5, 4/5 and 1 are equally spaced on the fraction line.

L cannot be moved in the static picture.

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Adding fractions – phew!

Who needs LCM ?

First, three views of LCM with no comments :

1: Change them to equivalent fractions that will have equal
denominators. As the common denominator, choose the LCM of
the original denominators. Then the larger the numerator, the
larger the fraction.

2: Jun 26, 2011 – If b and d were same it was easy to find LCM
since if denominators are same, we just need to find LCM of
numerators, hence LCM of (a/b) and (c/b) would be LCM(a,c)/b.
So we have to first make denominators of both the fractions same.
Multiply numerator and denominator of first fraction by LCM
(b,d)/b.

3: The GCF and LCM are the underlying concepts for finding
equivalent fractions and adding and subtracting fractions, which
students will do later.

 

Now we can do fraction addition without LCM. It just needs the use of the distributive law, and the result shows the way in which the divisors combine.

fraction-addition-png-1

And now using 3/4

fraction-addition-png-2

But the best one is via multiplication ……

fraction-addition-png-3-easy

Now for multiplication and division.

fraction-multiplication-png

fraction-division-png

 

 

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