Constructive Elementary Arithmetic

Need More Learning-to-Understand in School Mathematics

For all intents and purposes, the basis for early school arithmetic is memorized small number addition and multiplication. This can be an efficient way to learn an arithmetic for calculation. It also may have a significant downside: (1) it sends a message that it is good enough to know how to do without understanding, and (2) it forgoes the opportunity to start learning how to understand. Sharply focused tools address immediate problems; learning to understand is the base on which to move forward.

To some students mathematics education can be a series of ever higher barriers, and they often appear to come as surprises, as if the student is seeing something for the first time. It should not be this way. Mathematics at any level can reveal modes of thought and understanding that lower barriers to next level.

 

This post intends to show how to understand early arithmetic with a constructive approach. The slogan might be: arithmetic is built, not memorized. It is based on the assertion that speed is not of the essence, understanding is necessary to get you further along. 

Constructive mathematics is close to the programming of arithmetic. It is much like writing in a computer language. I understand this might be a good thing now.

Of course there is nothing new here. It is the result of trying to follow up simple questions about a foundation for school arithmetic. 

Constructing Natural Numbers

A Natural Number is one of the non-negative integers, the numbers with which we all started our mathematics education. Historically they were considered as "natural" and hence without need of further explanation. This view changed in the 19th century when concerns were raised about the foundations for arithmetic and Euclidean geometry. For arithmetic this eventually lead to the understanding of calculation which plays an important foundation role in Computer Science. Foundation work for the natural numbers provided a simple way to construct them, and this leads to simple construction of arithmetic with them. I think it could be a practical way for beginning students to understand their arithmetic; and to begin learning how to understand mathematics.

        (mostly from Halmos, Naive Set Theory)
Define a natural number as a set/collection that contains exactly that number of elements/units. By analogy think of a meter as being defined by a selected object, that on intuitive or practical grounds deserves to be called one meter long. Then "one-meter-ness"  is attributed to another object just in case it has the same length as the selected object.   Our number is such an object for number-ness; e.g. for 2 we take a set/collection with exactly 2 elements, or a set/collection with
2-number-ness. Where do we get an element with number-ness? We need a scheme that creates elements as we proceed in the construction of numbers, a process that starts with a small seed and then proceeds on its own with a simple repetitive process, a bootstrap with a small seed that brings itself into being.

Let's say we already have a number, say 7 defined as a set/collection with 7 elements. Clearly the number 7 itself can not be one of those 7 elements, so we can take it as the eighth element. We can think of the number 7 as the name of the set that defines it.

Now we are in position to specify the step in the repetitive process that constructs all the natural numbers. Before any step the process has at its disposal all the names of all the sets/collections produced up to this step. For any set n the next or successor set [n] is given by

            [n] = n U {n} ,

where U specifies the union of n, the set/collection of the elements of n; and the singleton set {n} whose element is the name of n. That is every natural number is to be the set of (the names of) its predecessors. This is more straight-forward than it might at first appear. It is easiest to see in the construction of natural numbers shown below.

0 is set/collection with no elements, the empty set/collection:

             0 = { } .

For each number to be the set of its predecessors:

            1 = [0] = 0 U {0} = { } U {0} = {0},
            2 = [1] = 1 U {1} = {0} U {1} = {0,1} ,
            3 = [2] = 2 U {2} = {0,1} U {2} = {0,1,2}  ,
            etc.
Here etc. has its usual meaning: you get the drift, and the process can now be left on its own to produce any natural number we should need. "[-]" is the successor of the number between the brackets. With this construction in mind, it is easy to picture the

Axiom of Infinity: There is a set containing 0 and  containing the successor of each of its elements.

Where did those numerals 1, 2, 3, ... come from? We used the familiar ones, but we could have made up our own; all a numeral does is name a set/set/collection that is filled with names we supplied previously. The problem with custom names: this scheme is designed to construct natural numbers that do what the natural numbers did before the scheme was invented. Mathematics did not invent them, but it did make this scheme to build them so they can be trusted because they can be understood.

All this is elegant and efficient; we have a process to construct all of the natural numbers in terms of the natural numbers themselves that is ordered to generate natural numbers before they are needed to generate succeeding natural numbers. On the other hand it seems like a little too much; we never really use all all this structure for much beyond constructing the numbers themselves. It is like scaffolding: it allows us to construct what we need, but then what it allowed us to build can provide what it was designed for on its own. The good news: you can construct the numbers you will use yourself, you understand how they work, and you no longer have to fall back on the authority of "naturality". Just remember what 0 is, and what it means for each natural number to have a successor:

• 0
• 1 = [0]
• 2 = [1] = [[ 0 ]]
• 3 = [2] = [[[ 0 ]]]
• 4 = [3] = [[[[ 0 ]]]] 
•  
• 
  
•  

Defining Constructive Arithmetic on Natural Numbers

We have numbers, how we do arithmetic with them? What is addition, and what is multiplication, with Natural Numbers? Here's the plan:

• Define addition on Natural Numbers. Then show [] and "+1" have the expected relation [n] = n+1 : we can count to construct a natural number.  
• Introduce the "unit" for the element being counted; i.e,  for the unit "ball",                                .
•  Given a unit, we can construct a number of units, or count out the "number of units"; e.g.          4 balls. A "number of units" is constructed in the same way a natural number is constructed.
• We show: a "number of units" can only be added to a "number of the same units" .

The definitions for addition and multiplication found in the Natural Number Wiki appear to be the common ones. Addition must be defined before multiplication; addition is contained in multiplication's definition.  

Definition of addition,  "+" , in  n + m

There are two clauses:

• n = n + 0,
• n + [m] = [n + m] = [n] + m 

The second clause states the equivalence of the successor of different combinations of the operands:

• n + ( the successor of m) is the same as the successor of ( n + m) is the same as (successor of n) + m. 

This will lead to a strong uniformity constraint on, and among, operands and sum; this will come later.

With 1 = [0] : 

• n + 1 = n + [0] = [n + 0]       using the second clause.
• [n + 0] = [n]                          using the first clause, giving
• n + 1 = [n]                             for any n, its successor is equivalent to n + 1.

With addition we can give a second construction for natural numbers:

•  0
•  0 + 1 = [0] = 1,
•  1 + 1 = [1] = 2,
•  2 + 1 = [2] = 3,
• 
  

 Let's look at the pattern of an addition example working itself out. But first we consider just a single number working itself in: 3 = [2] = [[1]] = [[[0]]], showing the expression of a natural number as 0 clothed in the number of its successors that take it back to, in this case, 3. Since 0 is not the successor of any natural number it it not possible to go further. Consider the the addition: 9 + 3:

• 9 + 3 = 
• 9 + [ [ [ 0 ] ] ] =  
• [ 9 + [ [ 0 ] ] ] =                 using the second clause in the definition of addition 
• [ [ 9 + [ 0 ] ] ] =                 and again
• [ [ [ 9 + 0 ] ] ] =                 and yet again, and for the third time.
•    [ [ [ 9 ] ] ]    =                     using the first clause ) 
•     [ [ 10 ] ]      =  9 + 1                     10 is the successor of 9 ,
•       [ 11 ]         = (9 + 1) + 1             11 of 10.
•          12           = ( (9 + 1) + 1) + 1 = (10 + 1) + 1 = 11 + 1 

If we are counting a set/collection of "these" where "these" could be balls, days, elephants, nails, ... then 0 of "these" is an empty set/collection of and for "these", and the successor of 0, [0] is 1 of "these". Let's take ball for 1 of "these". We can now construct the set of ball enumerators:

• 0 ball
• [0 ball] = 0 ball + 1 ball = 1 ball,
• [1 ball] = 1 ball + 1 ball = 2 ball,
• [2 ball] = 2 ball + 1 ball = 3 ball,
• [3 ball] = 3 ball + 1 ball = 4 ball.
•  
•  

4 ball  =  ((((0 ball + 1 ball) + 1 ball ) + 1 ball) + 1 ball) and etc. All those parenthesis are there because we are describing a process, not just describing a relationship, and we only know how to add two numbers. In any case 4 ball can be constructed by 4 consecutive additions of 1 ball to 0 ball.

4 ball can also be constructed by 4 successor steps of 0 ball. 

• [[[[ 0 ball ]]]] = 
• [[[ 1 ball ]]]] = 
• [[ 2 ball  ]] = 
• [ 3 ball ] = 
• 4 ball 

And now we have to confront a basic question: what does "4 ball" mean? . We construct "4 ball" by four successor steps from "0 ball" ( we know this is equivalent to four consecutive additions of "1 ball" ). The sense of "4 ball" is it is the thing we constructed in four steps, each step incrementing by "1 ball"the count that started with 0. If we think of "Ball" as the name of a set/collection of balls from which one ball is removed and added to the set of counted balls, just as the above process does, until the ball set/collection is empty, then by doing that process we will have proved, by construction, there were 4 balls in Ball. In this case "4 ball" is a sentence, something like" there are four balls in Ball", and the above process is a proof this sentence is true. And the purpose of mathematics is to write true sentences about significant mathematical objects like numbers, triangles, a space, ... We will return to the notation for a "number with unit" after we have defined natural number multiplication.

An elementary school student knows or learns, in an informal way, this proof and it is the sense of this proof that is the student's earliest number sense. It is their number sense before they learn arithmetic, but when they turn to arithmetic a strange thing happens, or so it seems to me. Instead of building out their intuitive, informal understanding of constructing numbers in order to have the foundation for a constructive arithmetic, school education turns to an arithmetic of natural numbers that is contained in tables that are to be memorized. This is all well, if not good, if the student is just being prepared for speed with simple arithmetic with numbers, but this is not the case; students are expected to apply their arithmetic to problems beyond numbers themselves, to word problems that can ask ask for multiplication of rabbits by cages.

A rabbit and a cage are examples of units, just as ball was.   

Multiplication: n x m

for n and m any Natural mumber. Again there are two clauses in the definition:

• n x 0 = 0,
• n x [m] = n x (m + 1 ) = n x m + n.

Note, the multiplication definition uses addition and the relation between addition of 1 and successor: [n] = n + 1. We also note that n x 1 = n x [0] = n x 0 + n = n. This is expected, but for multiplication of numbers with units will have to be reconsidered.

Here is an example of multiplication of natural numbers:

          9  x 3   =

• 9 x [ [ [ 0 ] ] ] =                          (using clause 2 and [m] = m + 1)           
• 9 x ( [ [ 0 ] ] + 1 )  =  9 x [ [ 0 ] ] + 9   = 
• (9 x [0] + 9) + 9   =  ( (9 x 0 + 9 )  + 9 )  +  9   = 
• 
  
• ( (9 + 9) + 9) =
• ( 18 )  +  9    =           
• 27 .   

This example again shows the expansion of the expression as the multiplication is layed out, followed by contraction as the expression is evaluated by one addition at a time. A multiplication is both expansion and contraction, it is both "a taking apart" and a "putting together". The taking apart uses step-wise dis-assembly of one operand to produce a multiple addition that is evaluated, as it must be, one addition at a time. What else could it involve? To make multiplication we only have what we have already constructed, or at least that is all we have for defining multiplication, and multiplication is defined. 

Unit and Multiplication

We return to the question of notation; what does "1 ball" mean? OK it means "one ball", but then we say "2 ball" is its successor. It feels like we are thinking of it as one word, in fact we are, so from now on we use "1_ball for "1 ball" and "n_ball" for "n ball". Remember numbers can only refer to one "thing"; "one thing" can be "one thing or another thing", but that really counts as "one thing". The two operands and the sum in addition are numbers for the same "thing".

Multiplication is altogether a different matter. The numbers in each operand and the product can be numbers of different "things". For multiplication we must learn how to work with this difference. An addition problem deals with one "thing", and the arithmetic for addition could be considered independently of that "thing". This is the not the case for multiplication.

We need to conform with the common nomenclature: "1_ball" is the counting unit for balls, and etc. To to do useful work with multiplication a student needs unit sense, number sense is insufficient. The secret for solving real problems is to get the units straightened out first, the operations second, and the numbers last.

There are Three Different Multiplications ,

1. Only one operand has a unit. These are multiplications where "x" is often pronounced "times". Examples:                                                                                                                       ** An example is: n x (m_cages) =  m x (n_cages) = (n x m)_cages. These can be derived by arguments you must be tired of.                                                                                                     ** 3 x (6_rabbit) = 3_rabbit x 6 = (3 x 6)_rabbit = 18_rabbit.                                                        
2. Each operand has its own unit, but the two units can be related to produce one: One unit is a for a rate or ratio that acts on the unit of the second to produce a third unit from the rate unit. Examples:                                                                                                                                 ** 3_(rabbits/cage) x 6_cage = 3_(rabbits/cage x cage) x 6 = 3_rabbit x 6 = (3x6)_rabbit = 18_rabbit,                                                                                                                 ** 3_(miles/hour) x 6_hour = 3_(miles/hour x hour) x 6 = 18_mile.                                                 
3. Each operand has its own unit, but now multiplication ties them together. In the previous two cases the product is the number of "thing"s. Now the product is the number of pairings of one operand's unit with the other operand's unit. The meaning of the pairing varies with the problem. Here multiplication acts on the "things" themselves and not just on the number of things; unit sense is central to the problem; e.g. two perpendicular lengths, one dimensional figures, can be multiplied to produce a two dimensional, rectangular-shaped surface. This is not elementary mathematics, but the sense of it is. Now we have a multiplication that goes beyond multiple adding. Examples:                                                                                                          **  6_cage x  3_rabbit = (6 x 3)_(cage,rabbit) = 18_(cage,rabbit) , is a pairing of a cage unit with a rabbit unit. In this case 1 rabbit with 1 cage. Each of the 18 pairings can not exist at the same time, there are not enough rabbits, or there too many cages. There are 18 distinct ways to pair a cage with a rabbit. If we think of a rabbit as well as a cage as points, then a (cage,rabbit) is a pair of points that locates the pair. This is much like a two dimensional Cartesian coordinate system, and indeed the pair is the result of a Cartesian product of sets, in this case one of cages and another of rabbits.                                                                                                                                                                                                                                                           Let me end with an example that shows multiplication producing a measure of  a two-dimensional object, the area of a rectangle, from two one-dimensional objects, the Length side, 6_ftL, and the Width side, 3_ftW. The Length side is perpendicular to the Width side. Each side length has the measure unit, foot(ft). The area is given by                                                        **  6_ftL x 3ftW = (6 x 3) x( 1_ftL x 1_ftW) = 18_square-ft.                                                             

Summary 

1. You can construct the Natural Numbers you need, including a Natural Number count of Things.
2. On the system of these numbers, the Natural Number system, you can define a process for addition and one for multiplication that constructs a Natural Number for the sum or the  product. This also works for Natural Number counts of "Things". In later posts, the Integers, the positive and negative whole numbers, will be defined in terms of the Natural Number system. Integer arithmetic, including subtraction is  also defined in terms of Natural Number arithmetic. Finally the Rational Number system along with division is defined in terms of the Integer Number system; and now we have the complete picture of elementary school arithmetic based on the Natural Number system.
3. Multiplication can be multiple addition for either Natural Numbers of Natural Numbers of Things. It can also be a Cartesian Product that produces pairings. 

Conclusion

There is a straight forward constructive path through school arithmetic based on Natural Numbers.

DRAFT Natural Number Arithmetic

I try to construct a rigorous and relatively simple path from natural numbers to arithmetic for rational numbers. It is pieced together from Wiki articles on the natural, integer, and rational number systems. I would like to think it shows the existence of an accessible, and rigorous basis for all of school arithmetic, accessible for teachers to enrich and inform their instruction.

[I did not intend the font change. In published form it does not follow the word processing instructions.]

Rote learning can be put in its proper place: it can make simple, practical calculation possible. But at the end of the day we can rest assured there is firm support for what we are doing A little exposure to foundation plants the seed to grow mathematical sense; and besides that, seeing how thought leads to understanding is the unique pleasure of mathematics.

I am not qualified to present material in a ready-to-teach-young-students form. I do try to get at the roots of elementary arithmetic. What is presented is certainly sufficiently straight-forward to be inform teachers on the foundation of the material they teach. This material should put teachers in position to evaluate for themselves claims like "multiplication is really addition". It looks to me like arithmetic proceeds in a straight line from a foundation well within the grasp of young students, and certainly well within the grasp of their teachers.

The mathematical approach can only be an abstraction of what? To use the result of that abstraction it is not necessary to know "what" it is, or could be. Teaching mathematics to beginners, on the other hand, requires taking a stand, conscientiously or not. Since I am trying to teach myself, I eventually had to face up to having no plausible story for how the bootstrap could have started for me. This is not cognitive science, it is simply what I see as a possibility when I look at the mathematics: what I could have been doing before I started learning school mathematics?

Here is the story I imagine. I realized one thing came after another thing, it wasn't all one long hum. I heard sound from my most significant other; it came in bursts, one after the other, sometimes little bursts inside big bursts. Eventually I got the hang of this after, after, ... business. At the same time I noticed the leg going out in front of me followed my other leg going out out in front of me, sort of like one sound burst followed another. "after" was like "following". My feet made marks in the sand, one after the other. When I took a step from a rock there was a mark in the sand, when I continued walking more marks appeared. Eventually I came up with a song to hum when I was walking with a word for the first step followed by another word for this next step. I noticed the tree close to the to the rock was ug1-ug2-ug3-ug4-ug5, that is an ug5 walk from the rock.  

Counting

Now I would describe counting as the attribution of "1 more", call it "+1" to the count of the pile before I put a ---- onto it. It is the same pile that was empty, call its count "0"(zero), before I put a --- onto it: Counting produces a sequence of counts:

• starting with:          old-count = 0 
• proceeding with:    new-count = old-count +1
•                                old-count   = new-count
• and continuing       new-count = old-count +1
• 
  
• 
  
•  

Counting evaluates the sequential growth/augmentation of the pile in the process of adding a --- to the pile that was empty to begin with. It is a sequential process, it does carry the idea of progression, of "step after step ..." but it also carries a record of augmentation, the pile is getting bigger. We attend to the progression part, or rather we let Peano and his axioms for the sequence of Natural Numbers take care of it. There is a problem with calling them numbers at this stage because Peano's axioms alone are only requirements that a sequence of words must satisfy in order for

• the sequence to start with a unique word,  
• for each word in the sequence to be different from all other words and for each word to have a unique place in the sequence, and 
• for there to be a simple way to prove something for every word in the sequence.

The set of all the words is written as N = {0,1,2,3, ...}. Each one of the words is called a natural number, but at this stage we need to be careful to remember they are only what we say about them, and we say they are words that Peano calls natural numbers. Peano says:

• 0 is a natural number.
• Every natural number has a successor.
• 0 is not the successor of any natural number.
• if the successor of x equals the successor of y. then x equals y.
• if: [a] something (since we are talking about words that "something is a statement) is true about 0, and [b] if the truth of that statement for a number implies (means) it is true for the successor of that number, then [c] the statement is true for every natural number.                                                                  

Here is a place you have to be careful not to assume "1" or "2" mean what you remember they meant before we decided to get at the truth of things. "2" is not the successor of "1" is not the successor of 0 until we show how Peano's axioms let that be so. We have to prove what we say so we can know, really know the truth of what we are talking about. We are talking about natural number arithmetic. Is it true, and exactly what is it?

First we need to define addition. We do not know what is the successor of what. All we know is 

• that special word 0 is not the successor of any natural number, 0 is a natural number, and therefore 0 is not a successor of itself.
• Every natural number has a successor. 

If we use the usual numerals in the following, we are likely to use the properties we have come to associate with them in the proof that all those properties follow from just the Peano's axioms. For now we take the usual alphabet for the natural numbers. Use "+" for addition, then we define "+" by

• a + the-succ-of b = the-succ-of ( a + b), and
• a + 0 = a.

First notice that we are trying to define + on the left hand side of = by an expression on the rhs that contains + ;  rather like the definition of a word including the word itself. It works in this case because the b is the-succ-of c in a + b , and then  a + b = a + the-succ-of c = the succ-of (a + c), and we continue in this "recursive" manner until we inevitably arrive at a + 0. 0 is not the successor of an natural number. That is why setting a + 0 = a is part of the definition of addition. And finally we have + on the lhs but not on the right hand side, and the recursion stops. This is a general definition, it works for all natural numbers a and b, but recursion is slippery; we will provide some numerical calculation later to show how it all hangs together

We are free to name the-succ-of 0: let's say 1 := the-succ-of 0. 

• b + 1 = b + the-succ-of 0 = the-succ-of ( b + 0 ) = the-succ-of b,

b + 1 is the successor of b, where b can be any natural number. Using only: [a] 1 defined as the succ of 0, and [b] the defined addition for natural numbers, we can name/define them in accord with our expectations for them:

• 0 is not the successor of any natural number
• 1 := the-succ-of 0 ,        definition of 1
• 2 := the-succ-of 1 ,        definition of 2
• 3 := the-succ-of2 ,        definition of 3
• 4 := the-succ-of 3 ,        definition of 4
• 5 := the-succ-of 4 ,        definition of 5
• 6 := the-succ-of 5 ,        definition of 6
• 7 ...
• 8 ...
• 9 ...
• ...

From here on we will denote "the-succ-of" by "succ". The defined addition gives:

• 6 + 3 = 
• 6 + succ (2)  =  succ (6 +2) =  
• succ ( 6 + succ (1)) = succ(succ(6 + 1))  =
• succ(succ(6 + succ(0)) = succ(succ(succ(6 + 0))) = succ(succ(succ(6))) =
• succ(succ(7)) = succ(8) = 
• 9

This is a general pattern for recursion; construct the "program" to produce the result as a sequence of steps, for addition each step is a "succ"; until it comes to a step that can be executed; and then it runs back up the program, executing each step to produce the input for the next until it runs out of steps. Also note the associated "shape" of the program: the width starts out small, reaches a peak, and returns to small corresponding to first building out a counting program and then consuming it as each counting step is executed. 

How is this useful?

• It shows how mathematics works: it proves things about words that are specified only by simple relations among them. For the set of natural numbers it shows how the definition of an addition of two of these words/numbers leads to a relation between them that can be expressed with the numbers themselves.
• It shows how addition of natural numbers is true: it is the rigorous working out of the implications of a small number of requirements for the set of all the natural numbers.
• It displays the elegance and economy of mathematics. 
• It will turn out that all of school analysis can be related back to natural number arithmetic.  

Multiplication

Given that addition has been defined, multiplication can be defined in an analogous fashion by setting

• a x 0 = 0, and
• a x succ b = (a x b) + a.

For the experience represent the set of natural numbers as N = {0, #, ##, ###,  ...} with # = succ 0, etc. Then

• ###### x ### 
• (###### x succ ##)  = (###### x ##)  + ###### =
• (###### x succ #)  + ######  =  ((###### x #) + ######) + ###### =
• (( ###### + ######) + ######) =
• ( ############ + ###### ) =
•  ##################

We used

• (###### x #) = (###### x 0) + # = 0 +  ###### = ###### .

Sorry for any problems reading this. Actually there is an advantage in being limited to a  small number of symbols that can only be displayed in a straight line, but the advantage is mine; I am allowed to look for a scheme that makes sense without facing diversion by pretty notation. It is not good for you though.

Eventually I noticed word processing a multiplication of natural numbers in the # notation was mostly a matter of copying and pasting. And then it occurred to me that was true because the # notation for natural numbers is a self representation; e.g. pasting a # on to the end of natural number constructs its successor, and that leads to the realization a natural number can be representing by the thing that constructs it: a string of successor "operators" we have denoted by "succ" that extends to the left ( this is  just the way readers of mathematics are used to seeing it) and originates with a 0 on the right. It is best to think of succ as a small machine that pulls in the natural number on its right, makes the successor of that natural number, and puts it directly under itself, and then goes away. Starting from the right it is then easy to see that 6 can be represented by the construction

• 6 := succ succ succ succ succ succ 0

In the # representation of natural numbers: we can have

• for # = succ 0, 
• # + ##...  = ###... for any natural number; i.e,
• succ ##... := prepend # to ##...

We can go back and forth between the mathematical construction of a natural number with a string of succ operators, and a displayed representation of them made by executing a string of prepend instructions. In either case we could write a program to produce

• natural numbers as prepend(prepend( prepend( ... prepend(# to 0) ...))). There are n prepends to create an n length string of #s.
• n + m as prepend(m to n). where the natural numbers n and m are represented as n and m length strings of #s as shown in first bullet. The a correct program that implements the bullet above would produce a string of (n + m) #'s.
• n x m as prepend(prepend(...prepend(n to 0)...))). There are m prepends and again n is a natural number produced as the first bullet indicates. Now a correct program would generate a n x m length string of #s.

This is just an outline. You could work entirely in the # representation by implementing counting as picking a # off a string off the string of #'s that represents a natural number until there were no more. If this doesn't relieve you of some worries, forget it.

I did not start with the intent to show all natural number arithmetic, at bottom, is just fiddling with strings of #'s or succ's, by whatever names these may be known; it just turned out that way. All integer and rational number arithmetic can be expressed as natural number arithmetic, and hence all school arithmetic is, at bottom, just more of this fiddling. At the end of the day that is why computers can do it. What we learn in school is the syntax of working with the names of these strings. For computers this is called syntactic candy. For us, the time we must spend on syntax it is the price we pay for being slow calculators. Still I think we could write computer programs that showed young students how it all really works, we could call it the "The String Theory Game".

Windup 

What have we learned?

• There is a foundation for arithmetic.
• A little of how mathematics works.
• A good basis for going on to develop the foundation for the integer number system which incorporates negative integers, and the rational number system where fractions/ratios live. I hope to tackle this in the next post.

Currently there is a belief, or at least a strong wish, that natural number multiplication is really addition. This is the sort of issue that is difficult to address from the perspective of school arithmetic itself, where apparently a thing can get defined by the practices associated with the signs in it; if an expression only has plus signs it must be addition. The plus signs that appear in an intermediate stage of multiplication do reflect addition in the definition of multiplication; addition must be defined before multiplication can be defined, but the definitions of addition and multiplication are not the same: multiplication is not addition. For one thing thing, with a slight abuse of definition, addition counts by 1 from n, while multiplication counts by n from 0. There should be, at the least, someone available to every teacher who can explain questions at the level of: what is addition? what is multiplication? what is the difference?         

I left high school thinking math was mostly a matter of retrieval of small facts put in memory by rote learning and repetitive manipulation. This may have produced some number sense, but it is not number sense that underlies success in later work in mathematics. It is more a matter of mathematics sense. I came away with very little. Some exposure to the foundation of arithmetic would have been helpful.

Arithmetic Symmetries, Understanding

Arithmetic Symmetries and Dualities

From the way we usually come at arithmetic, subtraction and division can appear to be subordinate to, and even derived from, addition and multiplication. Why is it not the other way around? And why are positive numbers "natural" and negative numbers not? The answer is straightforward: we bootstrap our way to negative and rational numbers from the whole numbers, the non-negative integers, which, strictly speaking, can only be added and multiplied: we start from numbers intended for counting which itself is then elaborated as addition and multiplication, and end up with negative numbers, and division. 

This would present no problem if mathematics instruction looked back from the results of the bootstrap, and cleaned up the mess. With the full set of integers and rational numbers, there is nothing special about "subtraction of (-M) is the same as addition of (+M)", or "division by M is the same multiplication by the M^(-1)".

For each of addition/subtraction and multiplication/division there are symmetries that allow us to pass back and forth from one member of a pair to the other if that is convenient. This can all be derived by "removal" of the term in an expression for N that is responsible for M "being in" N, thereby mathematizing the concept of "removal":

• N remove (+M) == N - (+M)
• is obtained by removing the term + (+M) from
• N = N + 0 = N + (-M) + (+M), to obtain
• N - (+M) = N + (-M).  

This shows "subtracting (+M) is the same as adding (-M)". Since we are discussing symmetries it is convenient to use explicit notation for signed values. Interchanging (+M) and (-M) in the above gives

• N - (-M) = N + (+M),  

the notorious "minus-a-minus is plus-a-plus". Here it is merely one member of a symmetric result: subtracting a positive(negative) is the same as adding a negative(positive), nothing special here. We could have approached this problem as "addition" is the removal of a subtraction operator; there really is nothing special about subtraction (or addition).

We can approach multiplication/division with the same "removal" approach. Now we start with

• N = N x 1 = N x M(^-1) x M(^1),

and remove the operation "x N(^1)" from it to obtain division:

• N remove "x M(^1)" == N / M =  N x M(^-1),

With this we have "N divided by M( the inverse of M) is the same as N multiplied by the inverse of M( M)". And again the same result could have started with multiplication as the removal of a division operation.

Trying out the Two-Number-Line

It is always a good idea to confront any process/procedure/scheme

you are trying to understand with the simplest problems you can find. I would like to see how the Two-Number-Line provides the values of the addition expressions:

• (N + 0) : The relative coordinate system for 0 is the same as the fixed coordinate system for N except its origin is at the far-end of the N-number-segment. The sum can be read off as the coordinate of this origin point.
• ((-N) + 0) : The same as above except the sum is on the other side of the origin of the fixed coordinate system.
• (N + (-N)) : In the relative coordinate system for the first problem, the far end of (-N)-number-segment is at the origin of the fixed coordinate system. The sum is the coordinate of this point in the fixed coordinate system, 0. 

and the multiplication problems:

• (N x 1) : In multiplication the relative and fixed coordinate systems have the same origin point. The far-end of the N-number-segment is the far-end of the unit-segment in the relative coordinate system, ie the point at the far-end of the 1-number-segment in the relative coordinate system has the coordinate N in the fixed coordinate system. This is the value of the product 
• ((-1) x N) : The point at far end of the (-1)-number-segment in the fixed coordinate system is the end of unit-segment of the relative coordinate system. Now the relative coordinate system is oriented oppositely to the fixed one, and the far-end of the N-number-segment is on the opposite side of the origin and has the coordinate (-N) in the fixed coordinate system. 

It is good to know the value of each of these expressions, because we can then concentrate on how that value arises from the appropriate process on the Two-Number-Line( may be called the TNL in what follows). See the immediately prior post for a detailed discussion of the TNL.

The Difference between Knowing the Answer and Understanding the Problem.

We can see a process on the Two-Number-Line(TNL) that produces a representation of the answer. In playing that back in our minds, we are visualizing a process that evaluates arithmetic expressions; we have something to hang an understanding on, a way to see how the mathematics works. For those of us who use mathematics, this is the sort of understanding we need; we have discussed this before. Arithmetic on a TNL is just one way to see arithmetic. Expect to find others or maybe just pieces of others, and to eventually discover that you are working in a web of understanding. 

There is another reason to think about understanding as a necessary part of the foundation that supports learning. From what I can tell it is generally accepted, unquestioned as far as I can tell, that memorization of the facts of small number arithmetic is an absolute requirement for future success. It may be necessary for executing the algorithms taught to do arithmetic, but that is an academic issue. When and if you come to do calculation, my experience is that you will put it all together for your own purposes with no great anxiety about what you have forgotten from elementary school. You may regret not coming to understand the significance of understanding at an earlier stage, I certainly did.

Let me state this baldly, and wait for enlightenment: it is fraud to state that there is only one route to any of the mathematics any of us is likely to use, and pernicious fraud at that. It somehow suggests mathematics is this dainty, fragile flower that drops its pedals if you approach it with anything less than immaculate preparation.  Relax. Mathematics is robust. break it into pieces, put pieces together, understand what happens. 

Any structure presented to have single points of failure is scary; it is rational to be anxious if you think success requires you retrieve the one fact of arithmetic or the one piece of nomenclature that can solve that problem. Math anxiety is not pathological.

Let me back up a bit on one-digit whole-number arithmetic. I do not think you should rely on a calculator for small jobs. That is silly. I do think small number arithmetic should be thought of as part of our everyday language, not something esoteric that exists only for mathematics. A smart phone has a good dictionary, but you do not look up every word.  

Geometrical Arithmetic on a Two-Number Line.

A Geometry of Arithmetic

We'll work toward a geometrical description of arithmetic. In the last Post a number-segment on a Straight Line Segment(SLS) represented a number. A coordinate system is located, oriented, and scaled the segment. Representation of arithmetic on a SLS requires the representation of the second number in the arithmetic expression.  How does the number segment for the first segment locate, orient, and scale the number segment for the second number? That requires a second coordinate system, and it is built with the first number's number segment.

This may get a bit tedious, but I have to lay the basis out once and for all to make sure it works. At the end we will have a foundation to build subtraction on addition and division on multiplication. We also will have a strong footing for the physical model we introduced in a previous Post; we want a way to "see" arithmetic as an active process. 

Here is the outline of the geometrical process:

• Represent one number using a coordinate system picked for its convenience.
• For addition, take the outer end point of this number segment for the origin of a second coordinate system. The scale and orientation of the two coordinate systems are the same. 
• For multiplication, take the outer end point of this number segment for the outer end point of a second coordinate system's unit number-segment . Here the two coordinate systems have the same origin. 
• Represent the second number as a number-segment with the second coordinate system.
• Read out the coordinate of the outer end of this number-segment. It is the value of the sum or product.

What is traditionally executed as a numerical calculation is now a geometrical process in a one-dimensional space. It produces a numerical result from a geometrical process. Although somewhat disguised, this approach takes the number line apart to get at the tool used to construct it: a coordinate system. The geometric process then attaches a second coordinate system to the SLS. The result is then retrieved from this Two-Number-Line. 

For addition our approach superficially looks like the usual one on a number line, but ours is a unified treatment of addition and multiplication that follows from the two possible ways to represent the combination of two numbers in a superposition of two number systems on one line. I would argue it is the inevitable extension of the ideas underlying the traditional number line to incorporate both addition and multiplication. It is the representation of 

A more intuitive interpretation sees arithmetic as taking the representation of one number to make another representation of the other number. It is arithmetic on a Two-Number-Line.

There are many representations of addition and multiplication on the Number-Line. In all of them that I know about, multiplication relies on a representation of addition. This suggests addition  is somehow intrinsic to multiplication, confusing what multiplication does how computational algorithms work. It is a confusion of mechanism with the meaning and purpose.

On the other hand, on a Two-Number-Line, addition and multiplication are independent variants of one process; and together they exhaust the possibilities of that process; they neither overlap each other, nor do they leave room for any other independent variant in the process: it is not an accident that we learn two arithmetic operations, and that those operations are addition and multiplication. Arguably, once we want the one number that represents the possible results of the commutative interaction of two, it is inevitable. And finally it is easy to draw a picture of the process, and even to make a physical model of it. 

Notation

From the Post Semantics of Elementary Arithmetic Expressions

Addition: 

(to-this-Operand + add-this-Operand = Sum =

(to-this-N1 + add-this-N1)  x common-unit =
N3 x common-unit.

Multiplication: 

(by-this-Operand) x (multiply-this-Operand) = Product =  

(by-this-N1 x multiply-this-N2) x product-unit  = N3 x product-unit.

N1 and N2 are number operands.

• For addition N3 is the number result of evaluating the addition expression (to-this-N1 + add-this-N2).
• For multiplication N3 is the result of evaluating the number expressions:  (by-this-N1 x multiply-this-N2) .

In short we are now addressing the number side of the problem.
We are going to represent a number by a segment. For this we have to set a relative scale, the ratio of the length imposed on the Straight-Line-Space SLS) by coordinate system to the common-unit for addition or the product scale for multiplication. We shall assume it is 1; e.g. 1 inch on the SLS corresponds to 1 square foot, and etc.

Cartesian Coordinate System

For Co a Cartesian Coordinate System(CCS), with N the coordinate of P on a Straight Line Space Sp: 

• Co(P) = N, 
• invCo(N) = P .

That's it for what a coordinate system does.

For a CCS the points on Sp whose coordinates differ by a fixed amount are evenly spaced. 

Co is fixed on Sp by two points: P0 and P1 with coordinates 0 and 1 respectively.

• P0 is C's origin,
• One of: (a) P1 is to the right of P0, the points to the right of the origin have positive coordinates, those to left, negative.  (b) P1 is to the left of P0, it is oppositely oriented.

The distance from P0 to P1 is the scale of the coordinate system: e.g if P1 is one centimeter from the origin then any two points separated by one centimeter have coordinates that differ by 1.

A Number-Segment

A representation of a number, N1, on Sp, a SLS, is a geometrical figure that relates properties of the Sp to the number. With Co a CCS on Sp, the number can be represented as the geometrical figure: the line segment on Sp with its inner end at the origin P0 of Co, and its outer end at the point PN1 on Sp whose coordinate in Co is N1. Refer to it as a "number-segment" N1:Co : the representation of the N1 on Sp built with Co. It relates a number to the oriented distance of a point from the origin of a coordinate system.

The Geometrical Process

Three steps are needed in the process to evaluate (N1 op N2) for op one of  "+" an "x". The first and last steps are common, the second is the isolated step particular to op.

  

1) For one of the two numerical operands, say N1, with a preexisting coordinate system C0 and the coordinate N1 = N1, construct the number-segment N1:Co on a SLS, Sp. Its outer end is the point PN1 in Sp.

2) One variant for this step is for addition; the other, for multiplication. In both cases it produces a point PN2 at the outer end of a number-segment  N2:COPo, This number-segment is constructed with a coordinate system COPo particular to the operator op.

3) The coordinate of N2 of the point PN2 in Co is the Sum or Product N3 = N2.

Step 2 for addition.

• construct the coordinate system COPo for N2 = N2 as the translation of Co; ie, the origin 0:COPo = PN1 with the scale and orientation unchanged.
• PN2 is the outer end of the number-segment N2:COPo.

Step 2 for Multiplication.

• the origin of COPo and Co are the same point P0.
• the scale and orientation are fixed by N1 as 1:COPo = N1:Co; ie, the scale for COPo is N1 multiplying the scale that N1 was represented by in step 1. The relative orientation is determined by the sgn of N2.
• PN2 is the outer end of the number-segment N2:COPo.

Wrap Up 

It all worked out pretty much as advertised. Now that we have the notation, we can show another way addition and multiplication diametrically differ from one another.

For addition, the relation of the coordinate systems is a difference:

• Co(PN1) - COPo(PN1) = N1 = N1 , PN1 is the origin of COPo
• Co(P) - COPo(P) =  N1 , for all P  , <= Cartesian CS  
• Co(PN2) - COPo(PN2) =  Co(PN2) - N2 = N1 ,
• Co(PN2) = N1 + N2 .

For multiplication, the relation is as a ratio:

• Co(PN1) / COPo(PN1) = N1 = N1 , COPo(PN1) = 1 ,
• Co(P) / COPo(P) =  N1 , for all P  , <= Cartesian CS ,
• Co(PN2) / COPo(PN2) =  Co(PN2) / N2 = N1 ,
• Co(PN2)  = N1 x N2 .

Difference and ratio are the two ways numbers are compared in elementary education. One provides the additive comparison and the other is the multiplicative comparison. These are the two ways coordinates are compared in the Two-Number Line. Further,they reflect the two ways two Cartesian Coordinate Systems relate the points that define them.

For addition:

• PN1, the outer end of the number segment for N1 in the preexisting coordinate system Co, is the origin of coordinate system COPo used to construct N2's number-segment.

For Multiplication:

• The origin points of Co and COPo coincide.
• Now PN1, the outer end of the number segment for N1 in the preexisting coordinate system Co, is the outer end of the unit number-segment 1 in COPo. This fixes both the scale and the orientation of COPo. 

A coordinate system is determined by its origin, its orientation, and its scale. These three possibilities are exhausted by addition and multiplication. 

We conclude:

• two-operand operations can be represented on the Two-Number Line
• the two two-operand, operations, addition and multiplication, exhaust the possibilities;
• addition and multiplication are independent of one another; ie. either one can be defined independently of the other.
• subtraction and division can respectively be defined in terms of addition and multiplication. 

It is not an accident that addition and multiplication, along with their variants subtraction and division, command so much attention. Understand them. 

What Does Arithmetic Do?

One Way to See What Addition and Multiplication Do, and Why that is Important 

Early education teaches you to do arithmetic. One way or another, traditional rote or Common Core understanding, the emphasis is calculation by you. In real (or word) problem solving, before you can calculate, you must first construct/design a mathematical expression (call it a formula ) whose value, obtained by calculation, is what you want. This construction/design is the hard part; you must understand what the arithmetic does before you can use it in a formula. That understanding is the part of problem solving left for us; calculation is more a craft, and like many traditional crafts is now largely mechanized; there really has been a sea change.

This Post develops a representation of arithmetic on a Straight Line Space(SLS). It presents the mathematical model; an earlier Post initiated a discussion of a concrete model that uses Tape Measures to insert and read number segments off a "slotted-slat arithmetic platform" that models the SLS. [ Refer to the Post Number Lines, Straight Line Spaces, and Coordinate Systems for a discussion of the SLS and a CCS. ] We will return to the concrete model in a later Post.
Our motto might be: See arithmetic happen - Understand what arithmetic does

Of potentially greater interest is an attempt to separate numerical and symbolic components in the expression of a problem. It is just possible that the inability to separate them is the major barrier to problem solving. For a real problem the symbolic issues come before the calculational, in early education it is the other way around; looks to me like a recipe for immense frustration. It is not a question of checking or labeling the results of a numerical calculation; it points the way to expression of the problem, to the formula that expresses the problem. We will be brought back to all this, but first lets try to make a geometric approach to calculation show us how numerical addition and multiplication work.

How to Make a Number You  Can See - Make It Big.

At bottom, a number is a (measure of a) deviation from zero. In a SLS the deviation from zero can be represented as the directed segment from the origin to the point whose coordinate is the number. 

Construction of a Coordinate System on the SLS

A number is represented by a special segment in an SLS. For the number N, the number-segment is denoted by N. Construction of that representation requires a Cartesian Coordinate System(CCS), call it C, on the SLS;

• C locates any point on the SLS with a coordinate-number address, and 
• for any point in the SLS, C provides a coordinate. 

One end of a number-segment is at the point whose coordinate is zero: the origin. Zero is a number, it is represented by a zero-length number-segment, C:0, both of whose end points have the coordinate zero: C:0 is both a point and a number-segment.  

For C a Cartesian coordinate system, equally spaced points have equally spaced coordinates, and it is only necessary to specify one segment, say the unit number-segment C:1 with near end at the origin and far end at the point whose coordinate is 1. For C a CCS 

C:0 = {C:0 , C:0} ,

C:1 = {C:0 , invC(1)} , 

[ notation for C a particular CCS:

•  invC(N) takes the coordinate N of a point P and points at P.
• C(P) works the other way around, shown the point P in the SLS, it provides its coordinate. 

This is all coordinate systems do.]

For C a particular CCS:

• positive coordinates locate points on the C:1 side of the
• origin at C:0. In addition to providing the orientation of C
• C:1 is the scale of a number-segment because its length is the "unit" in which the length of any segment is expressed; ie, it relates a difference in coordinates to a length in the SLS.

A drawing to illustrate the relation of a Cartesian coordinate system:

Black line is the SLS. Red line is the coordinate syustem

This is a good time for me to vent on a problem I have with the name commonly given to a plane with a coordinate system on it. A "coordinate plane" seems to imply a plane with a coordinate system defined on it is a plane with numbers in it. It is not. The numbers in a picture of a plane are, for a particular coordinate system, coordinates of the points that compose the plane, they are not part of the plane. A coordinate system on a plane; Cartesian, polar, or whatever, is a construct built on the plane, not a property of it;  a plane, or line, exists independently any coordinate system(s) placed on it, and more than one coordinate system can be placed on a plane, line, or etc.  

Construction of a Number Segment in an SLS

The representation of a number N on the SLS is a number-segment located, oriented, and scaled by a coordinate system C:

C:N  defn {0 , C:invCCS(N)} , [ "defn" says that what follows is                                                        the definition of what preceded ] 

is a segment in the SLS extending from the origin to the point whose coordinate is N. The number makes a representation of itself in the SLS using a coordinate system on the SLS.

The point whose coordinate is N in the coordinate system C is defined by 

C:N  defn  C:invCCS(N) .

"You" made the coordinate system C, that is what you do. The number made the segment that represents itself using your coordinate system (or, as it will turn, out one made made for it within some process), that is what a number does on a SLS. The figure below shows number segments for positive and negative numbers.

Positive Number and Negative Number Segments.
The coordinate system is oriented with positive coordinates
to the right.

We now have constructed a representation of a number as a realized segment on a SLS. First "you" built a coordinate system using your instructions that takes coordinates provided to it to locate points in the SLS. A number, in arithmetic an operand, you provide then represents itself on the SLS as a segment anchored at the origin, with a length equal to its magnitude using the scale the coordinate system provides, along with an orientation that reflects the sign of the number.


An active view could picture the number as an entity carrying its identity as a signed number, emerging into the SLS from a hole at the origin, looking in each direction along the SLS for the direction a number with the sign of its ID number should go; then, pulling its tail along behind it, proceeding in that direction until the coordinate of the point on the SLS at the tip of its nose is its ID number. This picture will carry over to the two active numerical operands in arithmetic; the action of arithmetic is their action on one SLS, guided by the different views of the world in their own coordinate system. 

Construction of Arithmetic in an SLS

[ from the Post Semantics of Elementary Arithmetic Expressions ]

Our format for arithmetic expressions:

addition
(to-this Operand + add-this Operand) = Sum
(OP1 + OP2) = (N + M) where OP1 is one of N or M, OP2 the other .
multiplication
(by-this Operand x multiply-this Operand) = Product
(OP1 x OP2) = (N x M) where OP1 is one of N or M , OP2 the other.

In the  discussion that follows we use the above convention and nomenclature for an arithmetic expression, an Operand, and a Sum or Product.

• "to-this", "add-this", "by-this", and "multiply-this" are adjectives modifying operands.
• Any calculation starts by picking OP1.
• (N + M) is the Sum, (N x M) is the Product.

The Unit in Evaluation of Addition and Multiplication Expressions

In addition each of the operands and the sum have the same unit: each's unit is one of the same thing, or at least one of the thing being added is the same:

(OP1 + OP2) = Sum = 
(N1 + N2) x common-unit = N3 x common-unit
The unit common-unit is common to each term and hence
The addition, (N1 + N2),  to produce N3 is a numerical calculation, but the meaning of (OP1 + OP2) includes the way the "unit" works it way through its evaluation; in short, the meaning of the evaluation is more than just the result of a numerical calculation. Following the units through an adition evaluation provides the semantics, the meaning, of the evaluation. Consider an addition problem:  

If there is 1fruit-per-apple and 1fruit-per-orange, how many fruits are there in 6apples and 3oranges? 
Since 
1fruit = (1fruit/1apple) x 1apple = (1fruit/1orange) x 1orange = 1fruit,
and
6apples = 6 x 1apple, and 3ornages = 3 x 1orange :
(1fruit/1apple) x 1apple x 6 + (1fruit/1orange) x 1orange x 3 =
6 x 1fruit + 3 x 1fruit =  (6 + 3) x 1fruit = 9fruit

Of course this is overly elaborate, but it is just a careful working out of the implications of the simple formule
1fruit = (1fruit/1apple) x 1apple, etc. 
for the relation between an 1apple and 1fruit. The lesson: to express the meaning of a problem is to understand the meanings of the problem's components and the relations among the meanings. Numerical calculation, itself, is a separate issue and generally a simpler one. 

In multiplication each operand and the product are generally different from one another: each's unit differs from that of the others; e.g. (speed x time = distance), or (L x W) = A where each of L and W may have a length unit as foot, but for a rectangle they denote lengths along different perpendicular lines, and A has has its own unit "1foot x foot" pronounced as "foot squared" or "square foot". In either case; however, the unit of (Operand1 x Operand2) is the same as that for Product:

(OP1 x OP2) = Product =
(N1 x unit-O1) x (N2 x unit-O2) =  N3 x unit-Product =
(N1 x N2) x (unit-O1 x unit-O2)  = N3 x unit-Product where:
(unit-O1 x unit-O2) = unit-Product; ie.
the unit of a Product is the product of the units of its Operands. We now have
(N1 X N2) x unit-Product = N3 x unit-product, or finally

(N1 x N2)  = N3 and 
(unit-OP1 x unit-OP2) = unit-Product
for
OP1 = N1 x unit-OP1 and etc.

We have separated a single equation with numbers and units entangled into two equations that isolate numerical and symbolic issues. The symbolic equation contains the meaning of the problem; it leads to a meaningful mathematical expression of it. I think this separation is the key to a principled approach to problem solving. At the level of mathematics in elementary education it looks easier to me than calculation.

For the distance evaluation we could encounter 1mile-per-hour for the unit of speed which multiplies 1hour for the unit of time to make the product 1mile for a unit of distance. The symbolic equation is
(1mile/hour) x 1hour = 1mile.  

Wrapup

This is getting too long. Representation of a number as a number-segment is the foundation of a geometrical calculation, and I set out to develop that, but I let myself get diverted by the separation of a problem into numerical and symbolic parts. Then and now that seemed more interesting. Geometric calculation will be in the next Post.

  

Why Make Arithmetic Real

Why Look for Active Arithmetic?

Mathematics is powerful. It gets at the core of the matter by removing as many of the particulars as possible in order to get at bare essentials. Even at the elementary level, the resulting abstraction is embraced; those whole numbers stand alone awaiting their interaction to produce an inevitable result that has long ago been abstracted from what it described. Alone, without further meaning, an arithmetic result holds for everything it has been abstracted from, and that is its power; we can reliably use the facts of arithmetic because they abstract facts of the world. 

We all stand on the shoulders of giants who themselves stood on the shoulders of giants. We are a long way from the ground, and that is where the problems that confront us are, even those pesky word problems we remember from elementary school. How do we help students climb down from mathematics to to level of real problems?

A young student's goal is not the ability to do mathematics, it is rather to have the ability to use it; not to create abstractions for the advance of mathematics, but to relate an understanding of mathematics to problems in their world.

In first becoming competent in arithmetic, students learn how to evaluate mathematical expressions before they know where they could have come from, more significantly, before they know how they they relate to the real world?  They can do arithmetic problem that stands in for a large class of real world problems before they can solve any one those. Eventually they must "de-abstract" the mathematics, get at particulars, and solve the one problem sitting before them

Make it Real. Real is Active

At least that is how I remember reality in my youth, the fun and useful parts of it anyway. This is not the way I remember my early formal education. I do remember beginning to see reality in undergraduate mathematics and science. I also remember the stark realization in my freshman year that I just did not get the relation of mathematics to real problems. Up to that point I could more or less fake it since the problems appeared next to the presentation of the mathematics used to solve them: extract the numbers, plug them in, and turn the crank.  Although I worked on and around lots of farm machinery, I never made the connection between it and problem solving. I did not see the corruption inherent in my faked learning; I pretended to learn and school pretended to teach something useful. 

Most of my early formal education was self referential: it taught what you needed at its next stage; and you could never see outside of it. Problem solving was in the service of learning academic mathematics. Real problem solving, even for "real" problems in an academic setting, was never addressed. 

Have things improved, or has the current testing regime made them worse? I do not see how it could make early education less inward looking. My fear is that school education has chosen rigor to aid in the prediction of results it itself determines. How much time does it spend confronting problems as problems where you have to figure out how to understand because you can not fall back on what you have done before? 

Whatever it is, if it is does not work, little room has been left for students to learn and create on their own. Creativity has historically been our strength; I do not think we ever knew if it resulted from the time schools took from children, or from the time it left to them.

Lest I give the wrong impression: I think I learned a lot in my childhood. My early formal education was not pretentious; it did not act like it was the only way to learn and therefore should occupy most of my time. I had time to work, play, and read on my own, and my teachers gave me most of what I ultimately needed from them. 

Who knows, perhaps it was just that there was no TV in my home town until I was in the sixth grade. Or maybe it was because my mother had been a teacher, and my father read in much of his free time. I do know my parents thought learning was important. They did not think sports were, or at least they did not do much to encourage my participation. They did let me solve many of my own problems.

I think I could ultimately relate mathematics to real problems because I had the opportunity to live where problems were active, where imagination could play on an active stage based on the real world. I am not sure that world is still commonly available. I want to present some results of thinking about making arithmetic active, to render active something all students now experience. Failure is an option.  

   

Number Lines, Straight Line Spaces, and Coordinate Systems

A Number Line is widely used to represent a dense number system as an image of uniformly spaced numbers at tick marks on a straight line, think of a long ruler. Students are invited to think of the numbers as operands participating in arithmetic operations related to the geometry of the line, an analytic geometry of the straight line figure.

I want to do arithmetic, both addition-subtraction and multiplication-division, by arrangement and measurement of two line segments. The location, orientation, and scale of the second operand's segment can differ from those for a (it's our choice) first operand. The difference is determined by the segment for the first operand whose location, orientation, and scale are ours to fix. For addition-subtraction the two segments are constructed on the same Straight Line Space (SLS). For multiplication-division they can be in different SLSs.

In what follows it is necessary to distinguish numbers-as-coordinates from numbers-as-operands, from numbers as results.    ( see page 92 in Adding it Up, National Academy of Sciences, 2001 for a different approach to addition on the Number Line where this distinction is made. I got much of my understanding of arithmetic on the number line from Understanding Numbers in Elementary School Mathematics by Hung-Hsi Wu, 2010). In what follows I will try to to refer to numbers as one of: "operands", "coordinates", or "results" ( or sums or products, ideas for something better that "results"?).

I like to think of a SLS as (an idealization as of) a structure in the world, waiting there for our investigation of it with tools like coordinate systems. Of course the power of our tools increases together with our understanding of the SLS. Children can see the mathematics they develop helping them understand their world; and, conversely, come to a deeper understanding of mathematics by seeing it act in the world. 

Straight Line Space:

a collection of points densely arranged in an otherwise structureless

straight line that extends forever in both directions. No point can come to know where it is, and there is no way to distinguish one point from another.

Cartesian Coordinate Systems

A coordinate system names and locates any point by naming it with its address expressed as a coordinate (one number for a one dimensional line).

A Cartesian Coordinate System(CCS) for the SLS is the entire collection of ordered, uniformly spaced pairs of coordinate, tick-marks. It performs two functions:

• You put the tick-mark at the point that you chose on the SLS, the CCS provides its associated coordinate.
• You provide the coordinate, the CCS provides the tick-mark pointing at the SLS point whose coordinate it is.

Our calculation scheme requires both these CCS capabilities, so does traditional Number Line use.

Three choices are made to construct a coordinate system for the SLS. You or the calculation (but of course ultimately you) choose:

• the point on the SLS whose coordinate will be "0", it locates the origin of the CCS at that point. This sets the CCS location.
• the  direction away from the origin where points will have "positive" numbers for their coordinates. This sets the CCS orientation.
• the length of a segment on the SLS, the coordinates of whose end points differ by "1".  It sets the "unit" of the coordinate system. Since the coordinates are uniformly spaced, it is sufficient to specify the unit as the distance from the origin to a point whose coordinate is "1". The unit sets the CCS scale.

Having specified a Cartesian coordinate system, any point on the SLS has a unique coordinate; inversely any coordinate designates a unique point on the SLS. The same can be said for any other coordinate system, and there is no restriction on allowed coordinate systems; since the SLS is an infinitely extended space the location of an origin is arbitrary, so is the direction from it of the half-line of points with positive coordinates, and so is the unit of length from this origin. 

To reiterate, a different one of the three coordinate system parameters

• locates the origin of a CCS in front of us, another
• sets a scale to accommodate the range of operand values, that interest us, and the third
• sets a convenient orientation.   

Make it Real

We can model a CCS with a tape measure and lay out a SLS along a long, narrow, and flat slat. 

My slat SLS: 8 feet long, slat top is a 2.5 inches wide surface 26.5 inches off the floor.

The provides a section of an SLS where a tape measure can serve as a CCS. Here addition(multiplication?) will be performed by location and measurement of segments by tape measures on the slat.  

Tape measure CCS

Uniformly spaced numbers and their associated tick marks are imprinted on the tape, which is single ended and has a zero tab at the end withdrawn from the case. A Tape Measure(TM) is "handed", the commonly available ones are "right-handed":

• intended to be worn on the user's right hip, 
• retrieved from there with the right hand holding the case, 
• left hand pulls the zero tab out to clip it over an edge of the object being measured, 
• tape extended by pulling the case with the right hand , and 
• opps! you now have your left hand to mark the measured object at the intended location on the tape. 

At least you would if your intent was to locate a point on the SLS from its coordinate. Given the limitations of a right handed TM, you might guess someone is making a "left handed" model , and you would be right. Search for one on the internet, and look at the photo:

Lower TM is right handed: its coordinates are right-side-up when the tape is extended to the right. Upper TM is left handed: its tape is the mirror image of the lower's with right-side-up coordinates when extended to the left.

• The TM on the right(left) provides the coordinate system for points in the SLS to the right(left) of the origin.
• A transverse slit in the slat SLS receives the zero tabs of the left and right located the CCS.
• The numbers are the coordinates for the points their associated tick marks point to.

Numbers, tick marks, and zero tab on a left-handed tape measure. 

For the TMs shown the scale unit is one inch.

Discussion  

Looking down on the middle of the slat SLS. Postive coordinates to the right, negative to the left. The zero tabs of both TMs mark the origin.

The figure shows positive coordinates in their traditional position on the right. The 0 coordinate is neither positive nor negative. At this stage "positive" and "negative" are just words we use to distinguish coordinates on different sides of the origin. 
At this stage they have no other meaning.

The figure can be seen as a number line; or perhaps better as a number line under construction with the numbers and their tick marks not yet imprinted on the line

The numbers (or is it "numerals" here?) are coordinates. Our discussion of the semantics of addition says their sum should also be a coordinate; but there is no simple sense to be made of that. And what could a product be? It will turn out that sense can be made of arithmetic with segments in different coordinate systems; this will be the topic of another post