My dad figured this out years ago.

The string method work for all matrices, and it is at least ten times quicker to “do” than to “write about”.

My dad figured this out years ago.

The string method work for all matrices, and it is at least ten times quicker to “do” than to “write about”.

Filed under algebra, education, math, teaching, transformations, Uncategorized

I found this on Medium

Gracias to Junaid Mubeen

Oxford Mathematician turned educator. @HGSE ’12. Head of Product @MathsWhizzTutor. Long-distance runner. Anagrams. http://www.fjmubeen.com

18 hrs ago6 min read

Filed under education, math apps, Uncategorized

What exactly are negative numbers?

A reference , from Wikipedia:

In A.D. 1759, Francis Maseres, an English mathematician, wrote that negative numbers “darken the very whole doctrines of the equations and make dark of the things which are in their nature excessively obvious and simple”.

He came to the conclusion that negative numbers were nonsensical.[25]

A minus times a minus is a plus

Two minuses make a plus

Dividing by a negative, especially a negative fraction !!!!

(10 – 2) x (7 – 3) = 10 x 7 – 2 x 7 + 10 x -3 + 2 x 3, really? How do we know?

Or we use “the area model”, or some hand waving with the number line.

**It’s time for some clear thinking about this stuff.**

Mathematically speaking, the only place that requires troublesome calculations with negative numbers is in algebra, either in evaluation or in rearrangement, but what about the real world ?

Where in the real world does one encounter negative x negative ?

I found two situations, in electricity and in mechanics:

1: “volts x amps = watts”, as it it popularly remembered really means “voltage drop x current flowing = power”

It is sensible to choose a measurement system (scale) for each of these so that a current flowing from a higher to a lower potential point is treated as positive, as is the voltage drop.

Part of simple circuit A———–[resistors etc in here]————–B

Choosing point A, at potential a, as the reference, and point B, at potential b, as the “other” point, then the potential drop from A to B is a – b

If b<a then a current flows from A to B, and its value is positive, just as a – b is positive

If b>a then a current flows from B to A, and its value is negative, just as a – b is negative

In each case the formula for power, voltage drop x current flowing = power, must yield an unsigned number, as negative power is a nonsense. Power is an “amount”.

So when dealing with reality minus times minus is plus (in this case nosign at all).

The mechanics example is about the formula “force times distance = work done”

You can fill in the details.

Now let’s do multiplication on the number line, or to be more precise, two number lines:

Draw two number lines, different directions, starting together at the zero. The scales do not have to be the same.

To multiply 2 by three (3 times 2):

1: Draw a line from the 1 on line A to the 2 on line B

2: Draw a line from the 3 on line A parallel to the first line.

3: It meets line B at the point 6

4: Done: 3 times 2 is 6

Number line A holds the multipliers, number line B holds the numbers being multiplied.

To multiply a negative number by a positive number we need a pair of signed number lines, crossing at their zero points.

So to multiply -2 by 3 (3 times -2) we do the same as above, but the number being multiplied is now -2, so 1 on line A is joined to -2 on line B

The diagram below is for -2 times 3. Wow, it ends in the same place.

Finally, and you can see where this is going, we do -2 times -3.

Join the 1 on line A to the -3 on line B, and then the parallel to this line passing through the -2 on line A:

and as hoped for, this line passes through the point 6 on the number line B.

Does this “prove” the general case? Only in the proverbial sense. The reason is that we do not have a proper definition of signed numbers. (There is one).

Incidentally, the numbering on the scales above is very poor. The positive numbers are **NOT NOT NOT** the same things as the unsigned numbers 1, 1.986, 234.5 etc

Each of them should have a + in front, but mathematicians are **Lazy**. More on this another day.

Problem for you: Show that (a-b)(c-d) = ac – bc – ad + bd without using anything to do with “negative numbers”

*******************************************

**References.**

Wikipedia:

Reference direction for current

Since the current in a wire or component can flow in either direction, when a variable I is defined to represent

that current, the direction representing positive current must be specified, usually by an arrow on the circuit

schematic diagram. This is called the reference direction of current I. If the current flows in the opposite

direction, the variable I has a negative value.

Yahoo Answers: Reference direction for potential difference

Best Answer: Potential difference can be negative. It depends on which direction you measure the voltage – e.g.

which way round you connect a voltmeter. (if this is the best answer, I hate to think of what the worst answer is)

********************************************

Filed under algebra, arithmetic, definitions, education, geometrical, math, meaning, negative numbers, Number systems, operations, subtraction, teaching, Uncategorized

Well, actually, she commented on a comment of mine on someone else’s blog, which led me to finding her book, here:

https://ciedieaech.files.wordpress.com/2015/12/why-is-you-always-got-to-be-trippin2.pdf

**Have you seen it yet? Have you read it yet?**

It is a brilliant first-hand account of the school “reform” process from the receiving end, with a logically presented sequence of analyses, intertwined with actual happenings and incidents which make your hair stand on end.

The often believed statement “Corporate school reformers were once open about their belief that public education was hopelessly broken” she argues is simply untrue, but that this was what they wanted others to believe. *They* didn’t have to.

Her story covers the years from 1995 to the present, and shows the full depth of mayhem caused by the “reform” movement.

Her account of the not too imaginary classroom where all the time is taken up following all the edicts and mandates that there is no time to actually do any teaching. It is priceless.

Here is a section on one of the many stupidities encountered:

**Flying Blind**

Frantically written upon demand by an evidently unbounded

wellspring of young hires, a torrent of suddenly created

district exams gushed up in a manner which soon began to feel

truly magical. And, as was becoming rapidly apparent, actually

understanding many of these precipitately manufactured tests?

Called for just a touch of magic as well.

Pushed repeatedly into the role of test graders, it wasn’t long

before a diversely collected school personnel began to comment

upon, and even argue about, not only the point value attached

to student responses but, more and more frequently, to the

tangible intentions behind the intricately worded test questions

themselves.

“Help!” I whispered to a grading partner one afternoon.

“Do you have any idea what this means?”

Sliding a test booklet across the table, I pointed to an essay

prompt so convoluted that I could make little sense of it:

“In what way does this story’s diction create foreshadowing while

working sympathetically inside the author’s choice of syntax?”

My students – well, if we were being very optimistic, at

least a couple of them – possibly knew what diction, foreshadowing,

and syntax meant. But even I didn’t know how to combine

these three uniquely discrete elements in a logical response for

this tortuous prompt. I struggled with my conscience, tempted to

give full credit to the student who had written simply, and I thought most reasonably:

“I don’t know what the fuck this is talking about.”

Another student, less inclined to waste words?

Had printed more succinctly: IDK.

I Don’t Know.

Well damn, kid, me neither.

Holding little patience for those old-school processes so

monotonously tied to a methodically careful (and oh-so-tedious)

analysis, as the years bent to the magic of no-waiting transformations

systematically edged out an educator resistance, it was

rapidly determined that a test question ambiguity (up to and including

plainly misleading typos) did not, actually, invalidate

tests. Nor, subsequently, nullify an endlessly collected testing

data. Specifically hired to address issues of examination, testing

experts were ready to advise; expressly versed in party line, assuredly

and absolutely they always knew the answer. Every single

time.

Oh, it was magical.

They could simply walk over and show you. “See?” Here

they could point with an absolute confidence to the official answer

sheet. “It’s right here,” they could tell you. “The answer is: D.”

Or: Two.

Or: No change.

In years now gloriously imbued with the high brilliance of

an instantaneous reformation, all you ever really had to do? Was

close your eyes. And, then, clicking your heels together: Believe.

Believe, as you took your first frightening step over an unknowable

cliff; believe, as anxiously you began to flap your arms; believe,

as apprehensively you started to fly alongside in a blind

obedience:

Believe, absolutely and without reservation?

In the answer sheet.

Filed under big brother, education, horrors, school, teaching, testing, tests, Uncategorized

Forecasting the future of “education”, by Eli Horowitz

Prescient ??????

Filed under education, future, horrors, humor, satire, school, transformations, Uncategorized

“Mathematics, take a break, the doggerel has come back !”

Today I start to fill my pail.

Through the next test I will sail.

Common sense must then prevail:

I’ll clear my brain of what’s now stale

To make some space inside the pail.

For competence, the holy grail,

Ensures that I will never fail,

Though moving forward like a snail.

* * * * * * * * * * * * * * * * * * * * *

The whole damn thing’s to no avail.

Filed under education, testing, tests, Uncategorized, verse

.The future: I\(I love this cartoon)

.Here are two posts worth a read, showing the rheeformed way forward, and consequences

reading-between-the-lines:Obamas-testing-action-plan/

sciencedaily.com/computers_math

Filed under competency based, education, future

Thanks to intense use of the internet I finally found a simple, understandable way of implementing Save and Fetch operations, enabling the keeping and reusing of any construction.

Here is a reminder of the application (app, program, software, whatever), with the file handling operations:

The user panel and a simple example of three points on a circle, with the bisectors of the pairs of points.

The history panel showing the actions that have been carried out

The Save popup,and, below, the resulting text file.

There is now a not quite finished Spanish option – just click “ESPANOL”

Also a modified “move object” procedure for use with a tablet,or even a smartphone.

The whole application is constructed as a web page, and to run it just click this link: geostruct

The full url is http://www.mathcomesalive.com/mathsite/geostruct/geostructforbrowser1.html

I was on the virtually powerless governing body of the local primary school in the UK when the first National Curriculum came out, some time in the early 80’s. Very “New Math”y. Reworked a few years later. Here is some stuff from the UK Dept for Education about the latest rewrite. The old “Back to Basics” brigade are in the ascendant, but at least the UK is not drowning under High Stakes Testing. Have a look:

Key stage 1 and 2 (ages 5 to 10)

https://www.gov.uk/government/uploads/system/uploads/attachment_data/file/335158/PRIMARY_national_curriculum_-_Mathematics_220714.pdf

Key stage 3 (11 to 13)

https://www.gov.uk/government/uploads/system/uploads/attachment_data/file/239058/SECONDARY_national_curriculum_-_Mathematics.pdf

key stage 4 (14,15)

https://www.gov.uk/government/uploads/system/uploads/attachment_data/file/331882/KS4_maths_PoS_FINAL_170714.pdf

ans about assessment

https://www.gov.uk/government/uploads/system/uploads/attachment_data/file/304602/Assessment_Principles.pdf

Only the dedicated study math in the last 2 years.

You might find this interesting as well, just look at how little time is spent taking tests, and then only in three of the years.

http://www.theschoolrun.com/ks2-sats-in-2015-what-parents-need-to-know

*The main aim is to raise standards, particularly as the UK is slipping down international student assessment league tables. Inspired by what is taught in the world’s most successful school systems, including Hong Kong, Singapore and Finland, as well as in the best UK schools, it’s designed to produce productive, creative and well educated students. *

*Although the new curriculum is intended to be more challenging, the content is actually slimmer than the current curriculum, focusing on essential core subject knowledge and skills such as essay writing and computer programming.*

Filed under education, INTERNATIONAL, math

I found this today. It’s worth a read.

Author: Gary S. Stager

Link: reinventingmath.com/?p=87

Here’s an extract:

*Conrad Wolfram estimates that 20,000 student lifetimes are wasted each year by school children engaged in mechanical (pencil and worksheet) calculations. *

*Expressed another way, we are spending twelve years educating kids to be a poor facsimile of a $2 calculator. *

*Forty years after the advent of cheap portable calculators, we are still debating whether children should be allowed to use one.*

*We are allowing education policy and curriculum to be shaped by the mathematical superstitions of Trump voters. *

*Educators need to take mathematics back and let Pearson keep “math.”*