Category Archives: abstract

“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.

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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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:

pershan hexagon of proof


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The abstract approach to the abstraction which is “Negative Numbers”

An approach to the formal definition of negative numbers, using the ideas of abstract algebra.

Section 1 – is background. Skip it if you like.

What is a negative number?

1: It’s a number with a “-” in front.

2: It’s the opposite of a positive number.

Well, 1 is very poor, and 2 is no good, as there are no positive numbers until we have  negative numbers, they are just numbers (referred to later as the original numbers).

There is a need to compare numbers, and one way is to ask “What is the difference  between this number and that number. This is easy – the difference between 3 and 7 is  4, and we all learn to write 7 – 3 = 4.

Everything is fine for a while until someone says “But what about 3 – 7 ?”.

“Cannot be calculated. Has no meaning. You cannot take 7 things away from 3 things.  You cannot cut a 7 inch piece of wood off a 3 inch piece.” are the answers.

These original numbers are usable for counting and measurement of quantity, but numbers can also be used to measure position, leading to questions of the form “How far is it  from this number to that number?”. Temperature is the most obvious situation. “How  much warmer is it today, compared to yesterday?”. With numbers we can ask “How far  is it from 3 to 7 ?” and get a response ” 4 “, but we can also ask the question “How far  is it from 7 to 3 ?”. The response is the same, with the extra “but in the opposite  direction”.

Thus there arises a need for numbers capable of dealing fully with this new situation , the  measurement of changes in position. So negative numbers are born (or created), and  we hope they obey the same rules as the original numbers. Playing around seems to  support this position, with a few mysteries, such as (-1) times (-1) equals 1, and two  negatives make a positive.

However, in math we should not be satisfied by “Well, it seems to work OK”.

Section 2

What follows is a formal definition of an extended number system, in which every number has an “opposite”, or an “additive inverse”, and in which every number not equal to zero has a multiplicative inverse, and in which the “properties of operations” are still valid.

The definition only uses brackets (parentheses), commas, and pairs of original numbers. It does NOT use the negative sign, and subtraction  b – a, is only applied where it makes sense, that is when  b>a.

An extended number (or “thing”), written  ( a , b ), is defined as the distance from a to b.

It is immediately obvious that  ( a , b ) = ( a + 1 , b + 1 ) = ( a + 2 , b + 2 ) and so on.

So we can write  ( a , b ) as ( 0 , b – a )  when  b>a, and as  ( b – a , 0 ) )  when  b<a,  using subtraction only  with the original numbers.


Addition needs to match the original number addition, so

(definition)              ( a , b ) +  ( p , q ) =  ( a +p , b + q )                    check it

Zero is the “thing” which when added to anything has no effect, so the zero “thing” is ( p , p ) for any p.

Now we can have the additive inverse of a “thing”, the one which when added to the “thing” gives zero.

(definition)   Additive inverse of   ( a , b )  is   ( b , a )  since   ( a , b )  +   ( b , a )  =  ( a + b , a + b )

Subtraction for “things” can now be defined as addition of the additive inverse.

We can define multiplication of “things” by looking at the product of two differences.

(original number definition)  ( b – a )( d – c ) = bd + ac – ( ad + bc ) , so we have

(definition)              ( a , b ) X ( c , d ) =  ( ad + bc , bd + ac )

For multiplication we need a unit or identity “thing”, and the obvious choice is ( 0 , 1 ), or anything where the second number is one bigger than the first, for example  ( 12 , 13 ).  Using the “multiply” definition we have                                           ( 0 , 1 ) X ( 0 , 1 ) = ( 0 , 1 ),

and                          ( 0 , 1 ) X ( 12 , 13 ) = ( 12 , 13 ) = ( 0 , 1 ),

and                          ( 0 , 1 ) X (3, 7 ) = (3, 7 )

Division is defined as the inverse of multiplication, so to divide a “thing” by another “thing” we multiply the “thing” by the multiplicative inverse of the “other thing”, which we now define.

(original number definition)       multiplicative inverse of  ( b – a ) is 1/(b-a)

(definition)           Multiplicative inverse of  ( a , b ) is ( 0 , 1/(b-a))  if  b>a  and  ( 1/(a-b) , 0 )  if  b<a

If we multiply a “thing” by its inverse we should get the unit or identity ( 0 , 1 ), and so we do:

( a , b ) X ( 0 , 1/(b-a)) = ( a/(b-a) , b/(b-a) ) = (  0 , b/(b-a) – a/(b-a)  ) = ( 0 , 1 )

We have enough here to show that the new operations of + and X have the same properties as add and multiply in the original numbers. Go on, show it !!!!

Now what has this got to do with negative numbers ?  Well, the first thing is that ( 0 , 1 ) has an additive inverse, namely  ( 1 , 0 ), or any of its other representations, say ( 5 , 4 ) for example.

The second thing is that  ( 1 , 0 ) X ( 1 , 0 ) =  ( 0 , 1 ) .

The third, and most important thing is that we have an arithmetic for the “How far is it from A to B” quantities which incorporates direction.  When A<B the direction is one way. When A>B the direction is the other way. These directions are coventionally called “the positive direction” and “the negative direction”.

So, finally, we identify distances in the positive direction with the original numbers, and distances in the negative direction with new numbers, each of which is the “opposite” or “additive inverse” of one of the original mumbers.

Using the minus sign for “the additive inverse of” makes it quicker to write, at a cost of some possible confusion.  We see now that for example  ( 3 , 7) is identified with the original number  4 and  ( 7 , 3 ) is identified with the new number  -4.  Also  ( 0 , 1 ) is identified with the original number  1 and  ( 1 , 0 ) is identified with the new number  -1.

So since we have  ( 1 , 0 ) X ( 1 , 0 ) =  ( 0 , 1 ) it follows that  -1 x -1 = 1, and there is no mystery about it!


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Negative numbers: truth, existence, reality, abstraction.

There are things with names which I can pick up, see, feel, or if too radioactive then at least observe and measure…..this is the reality we have (though some philosophers, and neuroscientists would have us believe otherwise). But what with numbers?

I can see three coconuts, I can count them, and match the count to the quantity, but the number THREE is not real, it is an abstraction from all the occurrences of three object that I have seen or can visualize.

Well, if THREE is not real then MINUS THREE hasn’t got a chance at being real, it is in fact a second stage abstraction, as negative numbers were invented by humans to deal with situations not adequately described with “ordinary” numbers. It gets worse, as complex numbers were invented to get over the difficulties with “real” numbers (the positive and negative numbers). It is a shame about the use of the word “real” in this situation (see above). They should have been called “simple numbers”.

Abstraction is also the basis of geometry. Euclid says “A line is that with extent but no breadth”, which does make it difficult to see! 

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-1 x -1 = 1, but some need convincing

Here is a popular argument:   -1 x -1 has to be 1 or -1

If it was equal to -1 then -1 x -1 = -1 x -1 x 1 = -1 x1 and so dividing both sides by -1 we get -1 = 1, which is not a good idea!, hence -1 x -1 = 1

This argument begs so many questions that it is difficult to know where to start.

Here is a much better one, but it does stretch the idea of area a little :


From the diagram  (a – 1) x (b – 1) = a x b – a – b + 1

Set a = 0 and b = 0 to get  (0 – 1) x ( 0 – 1) = 1, and since 0 – 1 is equal to -1        we get -1 x -1 = 1

This has some connection with evaluating for example  3 x ( 8 – 2)  using the distributive law.

The distributive law is a law for  a(b + c) and says nothing about  a(b – c), but never mind, we go gaily about the common task.

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