Thank you Joe for deciding to have a bit of fun in my, 'How would YOU simplify arithmetic?' post.
I now think it appropriate to branch out via a new post.
RE: "OK - let's have a bit of fun. Holes and bumps are OK, but '3 holes take away three bumps' still seems a bit recondite."
Please note my arithmetic framework is concrete and is LOLAPANA (LOgic LAnguage PAttern NAture)
My hope is that by 'our' exploring bumps and holes as a model for integers, we can develop a set of self-evident (NA) axioms that are within reach of children via age appropriate activities.
The model of the number line you present is appreciated, yet may add precision, yet may not simplify the 1685 version created by John Wallis.
Joe you (and others in this forum) are instinctively operating at the ABSTRACT and SYMBOLIC end of arithmetic/algebra towards the latter part of elementary education.
Maybe you all come from a degree of expertise that means you cannot possibly comprehend how arithmetic COULD be made simpler! Yet at the moment, 'commutative rings with identity' may as well be Swahili as I have made a conscious effort ONLY to make arithmetic simpler for children and not enjoy the same adventure you have experienced.
Your challenge and fun as a mathematician is more likely to explore more complex forms of mathematics requiring such vivid imagination that five year old children (and me) cannot keep up!
My challenge and fun as a naïve mathematician is to make math (arithmetic) simpler. I love having AHA! moments, whether they be discovering why the diagonal within a unit cube has length root three or new models for multiplication based on geometry. I have read a massive number of books and journals since 1983 yet all along, my single focus has been making simple stuff simpler.
One friend asked if I had aspergers, while moer just know me to be very stubborn... Yet what if all children had aspergers? Would we get away with our abstract models requiring arbitrary beliefs such as right is positive and left is negative?
Your definition of simplicity is abstract and in need of definitions beyond the experience and logic of a child.
I know exactly what you mean by your explanation of a number line. Yet bit by bit it's built upon non-sense from the perspective of a child - the arithmetic customer.
If I was to tell a child about this number line, I would expect it to create more questions than it answers such as:
"Why do we go sideways when we go forward?" "Is it my right or your right?" "When I turn the arrows around do I turn around with the arrow, because then my right swaps with my left and I am not sure which way the numbers go!" "IS you forward the same as my forward or are you looking at me and wanting me to go backwards to my left?"
Taking a child on a row boat is fun. They might be able to copy you and learn to row. Yet I doubt any child would have a clue how to row a boat if it was described à la number line.
The boat goes forwards. You sit backwards. Your right side is on the left of the boat. You pull the oar towards you in the boat to make it go away from you in the water and make the boat go forwards.
I am operating at the CONCRETE end of arithmetic prior to the necessity of involving relative direction. So yes, your model of the number line is fine as is, yet to a child it complicates arithmetic more than simplifies. The amount of dyslexia I have is almost nil, yet other members of my family will be driving a car and say we turn right at the lights and then turn left! Perhaps less emphasis on L-R may be part of the pathway to reducing the incidence of dyscalculia - yet back to the point!
Interestingly John Wallis gave a concrete metaphor to positive and negative that was better than his marching analogy, in which the number line with negative to the left and positive to the right. The metaphor he also gave was taking away the sea created more land. ie -ve x -ve = +ve
Unfortunately, Brahmagupta's logic of arithmetic was ignored and the western world became confused via the illogical translation of Euclid that ignored the subsequent creation of zero as number.
While Brahmagupta gave examples of positive and negative in the form the nouns, fortune and debt. To take away debt is the same as adding to fortune.
Rather than create a balanced integer set with equal and opposite nouns, the - sign took on the role of adjective as well as verb.
Had Wallis placed the object to the left the subject of the march would have resulted in positive being to the left and positive to the right. He may also be switching the 'object' in his marching analogy...
Most line charts in day to day life tend to put time on the x axis and the corresponding quantity on the y axis.
So if that's where we end up, let's experiment with the vertical number line going both above and below ground level zero.
The vertical number line operates correctly without any POV bias. It is NAtural as there is no brain disorder in which UP and DOWN are confused. Being left or right handed or being dyslexic is eliminated as a complicating factor in processing number logic.
Your operation of subtraction presumably uses the tools of axioms induction and proof.
My operation of subtraction that children 'just get' use the tools of a square (cubic) bucket and spade.
Give a five year old a bucket and spade, ask for two bumps and regardless of how many holes there are on the flat ground, there will be two more holes after two bumps are taken away.
Similarly, if your ground has neither bumps nor holes, it can be described as ground level zero.
After 15 minutes of fun (you asked for it Joe!) there may be ten holes and ten bumps.
What happens should someone come along and take away* two holes twice? Then there would be ten bumps and six holes.
Simplify the expression (fill in the holes) and you get four bumps remaining. So little children can play - 2 x -2 = 4 with their bucket and spade.
I have never met an elementary school ARITHMETIC teacher that can explain why -ve x -ve = +ve yet I have never played with a child that doesn't enjoy playing with bumps and holes to understand why four bumps remain.
Stacking the unit bumps up one on top of the other and digging unit holes down one under the other becomes a simple way kids can create negatives (holes). If you have a stack of four bumps (+4) and take away six bumps (-6) with your bucket and spade, you will have a hole two deep (-2).
Arithmetic can be made simpler when nouns are used. Count your bumps and holes, play with them and you are experiencing integer logic as play.
So while a model of bumps and holes may appear recondite to some, the model is accessible to children BEFORE they learn how to hold a pencil, let alone determine which hand is easier to write with.
Successor axioms for integers operate on a horizontal mindscape. What I would like to explore, is integer logic that operates on an above and below vertical mindscape.
My experience is that all kids really need is to know when a digit is a noun and when it is an adverb. They understand the verbs of + for add and - for take away or remove.
So -3 x -2 multi-played is linguisticly logical (to me anyway) as:
noun verb adverb
three holes taken away two times
Children know to take away a hole, you fill it in by adding a bump, so six holes taken away means you have six bumps more than previously - even if they are in the ground! If I owed $6 and you helped me get out of this financial hole by paying my debt I will be +$6 better off compared to my previous financial position.
If it was -3 x 2 it would be
three holes added two times = six holes = -6
The confusion comes in when both digits are treated as nouns. This is like multiplying two slices of pizza! The model of plane area created by adjacent factor fences is a different game entirely!
The path to making arithmetic easier comes when negative integers and positive integers are both understood to be nouns and the natural principle of additive inverse applies.
So in John Wallis' example from 1685 + land = - sea and + sea = - land
In my bump and hole model you can play games in which you start with zero and create a lot of equal numbers of bumps and holes so they nett to zero.
Then play away and integer logic is self evident!
So can we explore how an integer axoims might look with a vertical framework and signs as adjectives are no longer required?
Lastly the model of bumps and holes for integers is a sweet spot as there is no L-R presorting required to simplify the expressions.
As any child knows, when you play with your bucket and spade you then want to make a wall! Then you also want to make a channel! You might want to make a pool and you definitely like to build castles!
So the bump and hole integer model starts of as a number surface. (2D)
NAtural exploration of number within the realm of subitising and development of an innate integer sense takes place.
A number line can be constructed by sorting bumps and holes and making them adjacent. A line of square bumps becomes a number line that can be counted. A line of square holes (a trench) can also be created and counted by numbers of adjacent container spaces.
The game of bumps and holes can be played by children and is a simple extension of the innate number sense based on visible surface area we have before the age of 1. (Dehaene et al)
It seems truly bizarre to hide integers from children until they are teenagers when such a simple model exists.
We have plenty of examples where vertical number lines exist such as: thermometers above and below zero, above and below sea level and lifts up from ground level and down to carparks below.
Mathematicians are often sloppy when it comes to logic and don't care because arithmetic is SO simple. We assume people know that there is a 1 in front of x and act surprised when people don't get it.
Arithmetic has been OVERSIMPLIFIED and left incomplete for centuries.
Bumps and holes are consistent with the laws of gravity which means a bump of 4 put in a hole of 6 leaves a holes of 2. Switching to a bump/hole number surface model and a vertical number line with zero at the surface in which L-R is irrelevant is part of the way to make arithmetic simpler.
So Joe what I hope you and others can do is explore how this might be formalised in math speak. Can this be converted into a minimal set of axioms that would satisfy an expert audience?
Maybe this surface based integer algebra could be hieroglyphic?
TOWARDS A MODEL OF 'NATURAL' INTEGER LOGIC
STRAW MAN DEFINITIONS = means same idea + means add - - means take away ? = bump (unit cube) ? = hole (eg. a unit container or box in which a bump can reside) zero = a nett absense of variance from a surface
STRAW MAN 'AXIOMS' All variance as defined by bump and hole is from a surface with zero initial variance.
? + ? = zero ? + ? = zero ? - ? = zero ? - ? = zero zero + ? = ? zero + ? = ? ? - ? = ? ? ? - ? = ? ? ? + ? = ? ? ? + ? = ? ? (? + ?) + ? = ? + (? + ?) ie associativity of addition ? + ? = ? + ? ie commutativity of addition
The bump/hole integer model becomes commutative over subtraction in the sense that 2 + (-3) = (-3) + 2
This means children have no problem with 4 - 7 as they can imagine this as a stack of 4 bumps dropped into a hole 7 deep to leave a hole 3 deep, or -3. In the concrete model, it's a case of WYSIWYG! Of course 4 - 7 cannot be the same as 7 - 4 as you ONLY have one stack of bumps 4 high and one hole 7 deep to play with!
Combining bumps and holes requires no sorting or awareness of number and kids have been putting blocks in holes pre-kinder, so why not extend the model?
As for '3 holes take away three bumps' seeming recondite, the solution is simple. Just make more bump/hole pairs with your imaginary bucket and spade!
Whenever you cannot take away a bump or take away a hole because none are visible, add an equal number of bumps and holes until such time as sufficient bumps or holes exist to be taken away. Then simplify the expression by combining bump/hole pairs to minimise the variance from surface level.
When no further bumps and holes can be balanced the expressions have been completed and equation made ASAPANS. (As simple as possible and no simpler)
Joe I truly appreciate you (and others) reading these ideas that need refinement. I hope we can do something original to help make people realise how simple the bump hole elementary school model of integers (into goes) can be.
This approach may help even more kids understand line measurement (1D) above and below zero as well as variances from bases, area (2D) as measured by square units and volume or capacity measured in cubes.
>From bumps and holes the natural integer game/model can played with gaps being a noun for negative distance and standard wooden block manipulatives. How do you take away a gap? By adding a length of wood! A gap of -1 taken away twice is the same as 2 units added! ie -1 x -2 = 2
(Metric countries also have the -ve x -ve = +ve example of adding heat = subtracting cold on either side of zero degrees centigrade.)
So are you curious? Will you tyhe reader help by contributing 'positive' suggestions? If you feel a need to fault my ideas, then that is the price of admission in this forum and I will cop all that is thrown my way and hope to improve. Constructive criticism that I have a chance of understanding would be great :)
Thank you all,
Best wishes Jonathan Crabtree P.S. Would 'you' be open to using a Google Doc to collectively draft something offline before it might end up in this forum? From my pespective, it would be awesome to end up co-authoring/creating a paper worth peer review and possible publishing. If it doesn't get that far then at least I will have tried. On my own I won't be able to achieve this goal, yet collectively we might do something really cool!
* They could put rocks in the holes, sand bags in the holes to stop beach umbrellas flying away or just import some sand from somewhere else and fill two holes in two times.