Continued from bob.txt
BOB
Regarding your "system", how do you transition to standard notation from it?
JONATHAN
People used both Roman and 'Arabic' until they made their final choice and Roman Numerals
are still used for lists or to hide what year a movie was released!
You are jumping a long way into my ideas when you ask about notation. I teach children
the logic of integers via intuitive 'right-brain' concepts BEFORE they start writing in
symbols and applying algorithms.
I have developed integer axioms that children play with. The model (all mathematics is
just a representation) is based on bumps and holes which children make with a spade and
bucket.
You have ground level zero. Then you dig one hole (eventually the container of a pigit
mone or rotated glyph 1) and you also create one bump (eventually a digit one contained
by its surface).
So while you are jumping ahead, nineteen can also be represented as twoty mone or 2- where
the - in this case is mone or the digit 1 rotated from positive to negative.
All the combinations of digits and pigits in my system can be subitised.
So I start off with smaller bases of bumps and holes that can be subitised. Ultimately kids
know a base to be that number when you stop counting singles and count in bunches or piles.
The choice of a base means that digit disappears from the game and gets replaced by a 1
in the next place.
So converting twoty mone back to nineteen is a visual representation of having two stacks
of ten units with a hole one unit deep to the right.
Dropping one of the stacks of ten into the hole of one creates a stack of ten next to a
stack of nine, or nineteen.
Similary seventeen (or onety seven) is a stack of ten units besides a stack of seven units. To
make this subitisable make it twoty mee or two stacks of ten and a hole three deep. What kid
wouldn't want to dig a hole three deep to 'top up' the seven to a ten?
Once the digits and pigits are viewed as bumps and holes they are easily 'seen' to do the
subtraction before your eyes as kids have already been trained to put blocks in holes pre-kinder.
I gave a student studying media/journalism at university the subtraction...
19
-20
---
She knew that is was -1 yet believe it or not, she had no idea how to work it out with
an algorithm!
Yet 90 seconds later she understood the same expression can be...
2- (where - is mone or the 1 rotated from positive to negative)
-20
---
- or one unit hole, mone, or negative one.
She 'got it' without the visual imagery of bumps and holes.
After children understand integer theory I progress further into the 'physical world'
of for example, engineering (what Aryabhatta and Brahmagupta called logistics) or business,
in which there are no negative numbers. Instead of negative numbers, the unit reverses or
bounces back transformed as an inverse unit upon reaching zero.
You've a job, a home and feel like a hero
Yet if negative numbers are less than zero
Banks shouldn't care if you're negative ten grand
Lose one lose both and you're begging for a plan!
How does mathematics make us feel
False explanations appearing real
Need a number sense to erase our fears
We all need a puppy to lick away our tears
Try some four year old logic for a twist
What if negative numbers did not exist
Wondering what the meaning of this verse is
Numbers just get bigger, it's the unit that reverses
Fourteen trillion is a BIG number! The unit is debt.
For China, the unit for their trillions is called reserves.
The Indians gave the world some secret sauce 1400 years ago. We picked up on the zero,
yet didn't pick up on the dual nature THE UNITS being inherently positive or negative.
I worked out math for myself from scratch and then discovered the Indians had beaten me
to the inutuition - even though to them it was initially implicit (almost hidden) while
I am making it explicit via the set of inverse digits transformed into pigits.
I created stories of bumps (cubes) as units and holes and referred to bases as stacks
(positive) and wells (negative). Then I discovered a Digitised by Google translation of
Brahmagupta's arithmetic and he said (via translation from Sanskrit) the only difference
between the volume of excavations (holes) and stacks is that one is measured by depth and
the other height. ie the numbers used are the same, the unit is what reverses.
Throughout mainstream history, by which I mean Brahmagupta, Wallis, Newton, Euler and others,
the example of debt has been given to explain negative numbers. Some academics then claimed
Brahmagupta SAID debt multiplied by debt equals fortune. He almost certainly didn't as I
can't find that in any early translation from Sanskrit! This is perhaps the first and
most important case of logic being lost in translation. This was classic multiplicand by
multiplicand stupidity that germinated while remaining undetected.
So what is positive and what is negative. Let's define those terms.
Positive means the presence of an attribute being considered.
Negative means the absence of an attribute being considered.
Your bank balance may be negative yet the number gets bigger and the unit is debt. The
absence of money in the container is what is being counted.
Even thousand of years before India, the Chinese had red rods and black rods to represent
positive and negative. They counted two DIFFERENT concepts, not just one.
So in my kiddy logic, dirt is the attribute being considered and that is what is operated on
with unit bucket container and spade.
Initally the holes and bumps appear spread out one unit deep or one unit high. Then the
bumps get stacked and the holes are dug at the bottom of holes!
I admit that some of the benefit my approach creates may be placebo, as the medicine is
given via picture story books featuring puppies. That makes it sweet. Yet the active
ingredients are the integer logic and the creation of embedded attributes of BOTH number
and sign within the glyphs.
Because the dual nature of the 'unit' has been missed John Wallis created a clumsy
directional number line. It would have been better vertical than horizontal so viewpoints
would not flip -ve to +ve or +ve to -ve. That way above and below zero would have been
more intuitive.
The real solution is what I call the integer lines.
Dig a line of bumps and holes. These integer lines when combined sum to zero.
Zero is a container too! it can be seperated into an infinite number of matching bumps and
holes.
So for the expression 7 - 4 you just have seven bumps in a row and below them (left aligned)
you have four holes in a row.
Let gravity do the work and imagine the bumps fall into the holes and disappear! What
remains from this 'andition' are 3 bumps.
OK let's do 4 - 7 (as andition 4 + -7) which is never taught at elementary school.
With the integer lines model you simply have 4 bumps in a row and 7 holes in a row below
the bumps.
Let gravity do the work and four bump fall into four holes and disappear leaving 3 holes.
At this stage the numbers are verbal (auditory) and not symbols.
The pedagogical point here is to have the upper integer line like stepping stones and the
lower integer line like stepping holes. The intuition works because kids can enjoy dropping
bumps into holes before they imagine dropping bumps into holes.
OK let's do -7 - -3 read by adults as negative seven minus negative three. (How bad is it when
math professors say i is the square root of take away* one instead of negative one. * ie minus)
-7 - -3 in this model means you have 7 holes and 3 holes are removed. How many holes remain?
Four holes remain! Soon the digits can be rotated to pigits to embed the adjectival negative
signs and it can also be read meven minus mee equals mour.
These ideas are simple to children yet many adults struggle. They are just too set in their ways.
Teachers will struggle too, both with the simplicity of these concepts and the fact it isn't on
the curriculum. For this reason, my initial market will be adults who are bad at math and have
math anxiety and developing nations.
I believe a lot of people diagnosed with dyscalculia are made to feel their brain is at fault
when it was actually bad instruction and faulty logic. Did I fail math because I was too
clever? I don't have aspergers, yet I am stubborn! When other kids just accepted what the teacher
said, I challenged it. When the teachers gave me what I now know to be BS responses I guess I
failed to see the point...
So when I point out that Euclid's definition of multiplication used for 2300 years is wrong and
always was wrong, professors say yeah, but we know what he meant and teachers would be confused
if a correct definition was established. OK, how about arithmetic be precise and not sloppy? Is
the child the customer? Does he or she deserve truthful explanations of numbers while being told
about the Easter Bunny and Santa? We carry our number confidence with us for life long after we
learn the truth about rabbits, tooth fairies sleighs and chimneys. Yes I know The Definition in
Euclid's Book VII #15 is pythagorean. And yes I know that just as the definition of multiplication
in the Elements has been wrong for 2300 years, the geometry of universe we live in is also
non-Euclidean! And yes, I own a real copy of The Elements as well as many digital editions. OK
enough of the rant and back to numeracy.
The evolution of numeracy relating to multitude has entailed the embedding or more and more
logic into the system with fewer symbols being required.
The dual nature of the - sign as BOTH negative and subtract and the dual nature of the + sign
as BOTH positive and add are confusing. Much logic is lost when the same symbol describes two
distinct and context dependent ideas.
So as the glyph came to embed the count into a single symbol and place embedded the magnitude
of that symbol, I have simply embedded the sign attribute into the symbol and created
pedagogies/methods that supports the reduced mental ram requirements of arithmetic.
Mathematics exists to simplify our life and help us optimise happiness and wellbeing.
There may be an additional mental step of thinking with this model. Yet it's a faster way to
operate as the old dual nature of + and - was a clumsy workaround to the issue of the dual
nature of units that is at the heart of why kid hate and fail math and governments rack up such
tiny numbers less than zero that austerity measures are called for.
With the old faulty number line the further left you go from zero the smaller the number. Nup!
The further left you go from zero the bigger the number of the opposite unit to the right of zero.
a) With $3 - $7 = $-4 the smallest number is -4. Or is it?
b) From -7 take away a bigger number in -4 and you end up with a bigger number again in -3!!!
What if 'debt' carries the sign for us instead of a pigit? We get
c) From a debt of 7 trillion we take away a debt of 4 trillion and a debt of 3 trillion remains.
Digits/numbers describe counts of units and those units carry the sign!
Above below, surplus deficit, more than enough, less than enough, hot cold. OK you don't think hot
cold fits? On a hot day you turn UP the air conditioner and add positive mental units of cold. On
a cold day you say turn up the heater to add positive mental units of heat.
The brain can visualise and/or experience numbers of units and flip the unit automatically.
we have a hard time seeing what isn't there - negative numbers.
The model above is concrete or 3D. After a stage of kinaesthetic discovery you can migrate to 2D
pen and paper and incorporate intuitive self evident geometry as the model.
At this stage I refer to holes as gaps and introduce a new algorithm for the times tables. I wonder
if your son might be able to decode...
Gap times gap the unit goes,
Add your digits, take away toes.
...without the geometry.
You can also read about my axioms of natural integer logic at
http://www.jonathancrabtree.com/about/?page_id=166
If you have more questions you can email me or post them in the forum. I will answer them here
so I retain copyright to my 'essays' rather than hand it over to Drexel. The message is of course
intended eventually for the public domain.