Tom Loveless: Why the Common Core Won’t Make a Difference


This much is so obvious, except to those who have no idea about teaching and learning. Pity they are the ones making the decisions.

Originally posted on Diane Ravitch's blog:

The odd theory of the Common Core standards is that if everyone has exactly the same curriculum and the same standards, everyone will learn the same “stuff” and progress at the same rate; and as a result, everyone will have the same results, and the achievement gap will close. If this were true, every child who had the same teachers and the same classes in the same school would have identical outcomes, but they don’t.

In 2012, Tom Loveless of the Brookings Institution wrote an analysis of the Common Core standards and concluded that they would have little effect on achievement. Not because the standards are good or bad, but because standards alone don’t raise achievement, nor, I might add, do tests, which measure achievement, as thermometers measure body temperature without changing it.

Loveless summarizes his 2012 findings here.

He writes:

“The 2012 Brown Center Report on American Education…

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What is Algebra really for ?

An example tells a good tale.

Translation of a line in an x-y coordinate system:

Take a line  y = 2x -3, and translate it by 4 up and 5 to the right.

Simple approach : The point P = (2, 1) is on the line (so are some others!). Let us translate the point to get Q = (2+5, 1+4), which is Q = (7, 5), and find the line through Q parallel to the original line.  The only thing that changes is the c value, so the new equation is  y = 2x + c, and it must pass through Q.  So we require  5 = 14 + c, giving the value of c as -9.

Not much algebra there, but a horrible question remains – “What happened to all the other points on the line ?”

We try a more algebraic approach – with any old line  ax + by + c = 0, and any old translation, q up and p to the right.

First thing is to find a point on the line – “What ? We don’t know ANYTHING about the line.”

This is where algebra comes to the rescue. Let us suppose (state) that a point P = (d, e) IS on the line.

Then ad + be + c = 0

Now we can move the point P to Q = (d + p, e + q)  (as with the numbers earlier), and make the new line pass through this point:  This requires a new constant c (call it newc) and we then have  a(d + p) + b(e + q) + ‘newc’ = 0

Expand the parentheses (UK brackets, and it’s shorter) to get  ad + ap + be + bq + ‘newc’ = 0

Some inspired rearranging gives  ‘newc’ = -ap – bq – ad – be, which is equal to -(ap + bq) – (ad + be)

“Why did you do that last step ?” – “Because I looked back a few lines and figured that  (ad + be) = -c, which not only simplifies the expression, it also disposes of the unspecified point  P.

End result is:  Translated line equation is  ax + by + ‘newc’ =0,  that is,  ax + by + c – (ap + bq) = 0

and the job is done for ALL lines, even the vertical ones, and ALL translations. Also we can be sure that we know what has happened to ALL the points on the line.

I am not going to check this with the numerical example, you are !

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Needs, and “Buying In” versus “Selling Out”


Ouch !

Originally posted on Throw Out The Maybe:

(We havethe-street-sweeper-asan artist & a composer & a poet and, with a touch of transitivity, we may now wish to return and consider the-student-asa street sweeper!)
Needs, informally
Our use of needs does not belong in a nuclear family of self-descriptions, though perhaps it can be thought of as a distant cousin. We use the word precisely with respect to that which needs to be done in the world. The relation to the self, if we wish to articulate as much, is charged via the individualized nature of each person’s worldview.
Do streets need to be swept?
Let us suppose our answer is Yes, and the student’s answer is No.
And let us impose a further, concrete constraint: Teaching the student to factor quadratic expressions.
“Buying In” versus “Selling Out”

We use the term student buy-in as relates to the construction of an…

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Mercedes Schneider Transcribes Yong Zhao’s NPE Speech


This speech is a stunning analysis of the wrong path the US is following, educationally.’Just one quote:
” I’ve always said American schools do not teach creativity better than any Asian systems; we kill it less successfully. “

Originally posted on Diane Ravitch's blog:

Mercedes Schneider has transcribed Yong Zhao’s wonderful speech to the second annual conference of the Network for Public Education. This is the last of five posts; it includes links to all the previous transcritions.

If you enjoy the speech, be sure to watch the video (link included), so you can see Yong’s ingenious use of visuals.

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Angle between two lines in the plane……Vector product in 3D…….connections???????

So I was in the middle of converting my geometry application Geostruct (we used to call them programs) into javascript

 get it here with the introduction .doc file here

when I decided that the “angle between two lines” routine needed a rewrite. Some surprises ensued !

angle between two lines pic1

Two lines,  ax + by + c  = 0  and  px + qy + r = 0

Their slopes (gradients) are  -a/b = tan(θ)  and  -p/q = tan(φ)

The angle between the lines is  φ – θ,

so it would be nice to know something about  tan(φ – θ)

Back to basics, where  tan(φ – θ) = sin(φ– θ)/cos(φ– θ),

and we have the two expansions

sin(φ– θ) = sin(φ)cos(θ) – cos(φ)sin(θ)   and

cos(φ– θ) = cos(φ)cos(θ) + sin(φ)sin(θ)

So we have  tan(φ – θ) = (sin(φ)cos(θ) – cos(φ)sin(θ))/( cos(φ)cos(θ) + sin(φ)sin(θ))

Dividing top and bottom by  cos(φ)sin(θ)  and skipping some tedious algebra we get

tan(φ – θ)   =  (tan(φ) – tan(θ))/(1 +  tan(φ)tan(θ))

This is where the books stop, which turns out to be a real shame !

Going back to the two lines and their equations, the two lines

ax + by = 0  and  px + qy = 0

have the same angle between them (some things are toooo obvious)

Things are simpler if we look at these two lines through the origin when they both have positive slope.

Take b and q as positive and write the equations as   ax – by = 0  and  px – qy = 0

Then the point whose coordinates are (b,a) lies on the first and (q,p) lies on the second.

angle between two lines pic2

Also, the slopes of the two lines are now  a/b , tan(θ)   and  p/q , tan(φ)

Let us put these into the  tan(φ – θ)   equation above, and once more after tedious algebra

tan(φ – θ)  = (bp – aq)/(ap + bq)

which is a very nice formula for the tan of the angle between two lines.

This is ok if we are interested just in “the angle between the lines”,  but if we are considering rotations, and one of the lines is the “first” one, then the tangent is inadequate. We need both the sine and the cosine of the angle to establish size AND direction (clockwise or anticlockwise).

The formula above can be seen as showing  cos(φ– θ)  as  (ap + bq) divided by something

and  sin(φ– θ)  as  (bp – aq) divided by the same something.

Calling the something  M  it is fairly clear that    (ap + bq)2 + (bp – aq) 2 = M2

and more tedious algebra and some “observation and making use of structure” gives

M= (a2 + b2)(p2 + q2)

and we now have

sin(φ– θ)  = (bp – aq)/M  and  cos(φ– θ) = (bq + ap)/M

and M is the product of the lengths of the two line segments, from the origin to (b,a) and from the origin to (q,p)

It was at this point that I saw M times the sine of the “angle between” as twice the well known formula for the area of a triangle. “half a b sin(C)”, or, if you prefer, the area of the parallelogram defined by the two line segments.

Suddenly I saw all this in 2D vector terms, with bq + ap being the dot product of (b,a) and (q,p) , and bp – aq as being part of the definition of the 3D vector or cross product, in fact the only non zero component (and in the z direction), since in 3D terms our two vectors lie in the xy plane.

Why is the “vector product” not considered in the 2D case ??? It is simpler, and looking at the formula for sine , above, we have a 2D interpretation of the “vector”or cross product as twice the area of the triangle formed by (b,a) and (q,p). (just as in the standard 3D definition, but treated as a scalar).

So “bang goes” the common terms, scalar product for c . d  and vector product for  c X d

Dot product and cross product are much better anyway, and a bit of ingenuity will lead you to the reason for the word “cross”.

This is one of the things implemented using this approach:

gif rolling circle

Anyway, the end result of all this, for rotating points on a circle, was a calculation process which did not require the actual calculation of any angle. No arctan( ) !

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John Oliver Demolishes Standardized Testing Industry


Just watch this.

Originally posted on J. Giambrone:

My Posts  |  Reblogs  |  Films

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Reformers Save Schools from Parents


I love satire.

Originally posted on lacetothetop:

Federal agents raided the homes and classrooms of hundreds of parents, teachers, and educational “advocates” today because of their involvement with the “opt-out movement.” This illegal act of defiance has cost millions in tax payer’s money and now the feds as well as state education officials are cracking down.

Among the charges these people face are endangering the welfare of minors, insubordination, and churlishness.

Mr. Glynn of Brookhaven Elementary School was removed earlier this morning from his school’s playground in which he was wasting valuable time for rigorous learning to play soccer with his class. It took officials a while to find him as they assumed he would be in the teachers’ lounge during his lunch break.

Arne Duncan, Secretary of Education was outraged over the number of parents who refused the tests in New York. “We made it clear that schools are failing,” he said. “We had scores, scatter plots…

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Rigid motions are actually useful !

I am currently reconstructing my geometrical construction application, Geostruct, to run in a web page using javascript.

One of the actions is to find the points of intersection of a straight line with a circle. Here is a gif showing the result:

gif line and circle

The algebra needed to solve the two simultaneous equations is straightforward, but a pain in the butt to get right and code up, so I thought “Why not solve the equations for the very simple case of the circle centered at the origin and the line vertical, at the same distance (a) from the centre of the circle

line vertical circle at origin

Then it is a simple matter of  rotating the two points (a,b) and (a,-b) about the origin, through the angle made by the original line to the vertical, and then translating the circle back to its original position, the translated points are then the desired points of intersection.

The same routine can be used for the intersection of two circles, with a little bit of prior calculation.

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Words And Mathematics


Instructions: Read at least twice !

Originally posted on Mathemagical Site:


How do you measure a line?
In foot or meter, yard or rhyme?
Do rhythms or do logarithm
better keep your thoughts with them?

Could you draw figures of speech
if you had a compass and ruler in reach?
If armed with compass, quotes, and quips,
would you make an ellipsis or an ellipse?

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Subtraction and the “standard” algorithm

CCSSM talks about “the standard algorithm” but doesn’t define it – Oh, how naughty, done on purpose I suspect, since there are varieties even of the  “American Standard Algorithm”. Besides, if it is not defined it cannot be tested (one hopes!). I checked some internet teaching stuff on it, and as presented it won’t work on for example 403 – 227 without modification.

Anyway, I was thinking about subtraction the other day (really, have you nothing better to think about?), and concluded that subtraction is easiest if the first number ends in all 9’s or the second number ends in all 0’s. So, fix it then, I thought, change the problem, and here are the results

.Two simple algorithms for subtraction

I am quite sure that some of you can extract the general rule in each case, and see that it works the same in all positions.

While I am going on about this I would like an answer to the following-

“If I understand subtraction, and can explain the ideas to another, and I learn the standard algorithm and how to apply it, and I have faith in it based on experience, WHY THE HELL DO I HAVE TO BE ABLE TO EXPLAIN IT?”

I guess this post counts as a rant!

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