Category Archives: education

Maths(UK) is full of surprises

The radical axis theorem
radical axis theoremradical axis theorem2
How come that after 55 or so years of involment with math this came as a big surprise !!!!!

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Filed under education, geometry

School——>PARCC tweet———>suspension———->and then?????

Thank you Audrey Watters for leading me to this exposure of the behaviour of testing corporations.

These two are MUST READs, and should be passed on to everybody:

http://www.bobbraunsledger.com/breaking-pearson-nj-spying-on-social-media-of-students-taking-parcc-tests/

http://kengreenwood.com/pearson_spying_on_kids/

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Filed under big brother, education, testing

The Future

“How’s your Mary doing?”.

“She’s doing well. She’s 8 now. She’s in Grade 3. She really enjoys the Pre-Algebra and the Pre-Textual Analysis.”.

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Filed under algebra, education, language in math, teaching

Smarter Balanced: Lacking Smarts; Precariously Balanced

It’s a bit long, but it sure takes the lid off the CCSS. Read it now.

deutsch29: Mercedes Schneider's Blog

In this time of  “public-education-targeted boldness,” the Common Core State Standards (CCSS) has made the American public one whopper of a “bold” promise:

The standards were created to ensure that all students graduate from high school with the skills and knowledge necessary to succeed in college, career, and life, regardless of where they live. [Emphasis added.]

There is neither now nor never has been any empirical investigation to substantiate this “bold” claim.

Indeed, CCSS has not been around long enough to have been thoroughly tested. Instead, the above statement–which amounts to little more than oft-repeated advertising– serves as its own evidence.

However, if it’s on the *official* CCSS website, and if CCSS proponents repeat it constantly, that must make it true… right?

Keep clicking your heels, Dorothy.

Now, it is one issue to declare that CCSS works. It is quite another to attempt to anchor CCSS assessments to the above cotton…

View original post 1,321 more words

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Complex Numbers via Rigid Motions

https://howardat58.files.wordpress.com/2015/02/complex-numbers-by-rotations.doc

Complex numbers via rigid motions
Just a bit mathematical !

I wrote this in response to a post by Michael Pershan:
http://rationalexpressions.blogspot.com/2015/02/could-this-introduce-kids-to-complex.html?

The way I have presented it is showing how mathematicians think. Get an idea, try it out, if it appears to work then attempt to produce a logical and mathematically sound derivation.
(This last part I have not included)
The idea is that wherever you have operations on things, and one operation can be followed by another of the same type, then you can consider the combinations of the operations separately from the things being operated on. The result is a new type of algebra, in this case the algebra of rotations.
Read on . . .

Rotations around the origin.
angle 180 deg or pi
Y = -y, and X = -x —> coordinate transformation
so (1,0) goes to (-1,0) and (-1,0) goes to (1,0)
Let us call this transformation H (for a half turn)

angle 90 deg or pi/2
Y = x, and X = -y
so (1,0) goes to (0,1) and (-1,0) goes to (0,-1)
and (0,1) goes to (-1,0) and (0,-1) goes to (1,0)
Let us call this transformation Q (for a quarter turn)

Then H(x,y) = (-x,-y)
and Q(x,y) = (-y,x)

Applying H twice we have H(H(x,y)) = (x,y) and if we are bold we can write HH(x,y) = (x,y)
and then HH = I, where I is the identity or do nothing transformation.
In the same way we find QQ = H

Now I is like multiplying the coodinates by 1
and H is like multiplying the coordinates by -1
This is not too outrageous, as a dilation can be seen as a multiplication of the coordinates by a number <> 1

So, continuing into uncharted territory,
we have H squared = 1 (fits with (-1)*(-1) = 1
and Q squared = -1 (fits with QQ = H, at least)

So what is Q ?
Let us suppose that it is some sort of a number, definitely not a normal one,
and let its value be called k.
What we can be fairly sure of is that k does not multiply each of the coordinates.
This appears to be meaningful only for the normal numbers.

Now the “number” k describes a rotation of 90, so we would expect that the square root of k to describe a rotation of 45

At this point it helps if the reader is familiar with extending the rational numbers by the introduction of the square root of 2 (a surd, although this jargon seems to have disappeared).

Let us assume that sqrt(k) is a simple combination of a normal number and a multiple of k:
sqrt(k) = a + bk
Then k = sqr(a) + sqr(b)*sqr(k) + 2abk, and sqr(k) = -1
which gives k = sqr(a)-sqr(b) + 2abk and then (2ab-1)k = sqr(a) – sqr(b)

From this, since k is not a normal number, 2ab = 1 and sqr(a) = sqr(b)
which gives a = b and then a = b = 1/root(2)

Now we have a “number” representing a 45 degree rotation. namely
(1/root(2)*(1 + k)

If we plot this and the other rotation numbers as points on a coordinate axis grid with ordinary numbers horizontally and k numbers vertically we see that all the points are on the unit circle, at positions corresponding to the rotation angles they describe.

OMG there must be something in this ! ! !

The continuation is left to the reader (as in some Victorian novels)

ps. root() and sqrt() are square root functions, and sqr() is the squaring function .

pps. Diagrams may be drawn at your leisure !

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Filed under abstract, algebra, education, geometry, operations, teaching

Memorization versus Number Sense

This is a link to a post by WatsonMath about Stanford Professor Jo Boaler and her thoughts, opinions and research backed statements of the counterproductive combination of “learn your tables” and “take this (yet another !) timed test on them”.

Definitely worth reading, and worth passing on as well.

http://www.watsonmath.com/2015/02/07/jo-boaler-fluency-without-fear/

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Filed under arithmetic, education, math, teaching

SOLVING PROBLEMS THAT MATTER, NOT GIVING ANSWERS

This is not mine, so read the original at

http://www.base-ten.com/tm/why-i-dont-give-answers/

I used to not give answers to my engineering students. They hated me ! When i pointed out that if they ever got hired it would be because the firm had problems to be solved, and they DIDN’T have the answers.

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A “real life” geometry problem.

While designing a system for connecting “educational” cubes together I figured that the holes in the faces had to be positioned very carefully. To achieve what I wanted the holes had to be positioned with  length a equal to length b, and length c had to be twice the length a.

So what is length a, as a fraction of the side of the cube ?

geometry from real life
There will be eight holes altogether, and all cubes are the same size.

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I didn’t know whether to laugh or to cry,

so I posted it without further comment (too many <expletive deleted>’s required).

metacog say ask check

Click the link for the full thing with theory, explanation and so on. The item is in .doc text format

metacog SayAskCheck complete

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Filed under arithmetic, education, teaching

Testing to destruction.

Testing to destruction.
IIHS_Hyundai_Tucson_crash_test
“IIHS Hyundai Tucson crash test” by Brady Holt – Own work. Licensed under CC BY 3.0

A method used in manufacturing for product testing, where the product is
designed for a single action, and is used in practice as an insurance. The
best example is vehicle air bags. To see if they work a car is driven into
a wall, and the effect on the dummy people in the car is assessed.
Unfortunately some medical procedures can have a similar effect, of course
unintended. The classic case is amniocintesis, a procedure for assessing
the presence of Down’s Syndrome in the fetus. The reality was that the
probability of a fetus having the syndrome was way smaller than the
probability of the procedure itself causing a miscarriage. Initially, and
for quite a long time this was not realised. Eventually the test was only
offered to women who had a higher chance anyway of having a syndrome baby.
Could there be a connection between this stuff and the roll out of high
stakes testing in schools. Think about it.

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