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    

 

Semantics of Elementary Arithmetic Expressions

Musing on the Semantics of Arithmetic Expressions

What is meant by the arithmetic expressions that we read and write? I certainly do not intend to present anything new, just report some observations made while working out what arithmetic can mean, with an eye toward the possibilities when numbers denote more than just themselves. I want to look at elementary mathematics as an active endeavor where you start by writing an expression for something you want in terms of what you know, and whose evaluation plays as a movie in your mind. 

For addition and multiplication what actions flow out from the form of their expression? In the infix notation used at this level, expressions are of the form: "(operand operator operand)"; the parentheses are just my notation to improve blog readability. It has long been noted that the semantics are ambiguous: do we read (2 + 3) as instruction to calculate, or is it just another way to write (5).  If the meaning is calculation, we use a process that takes two numbers and produces a third. If the expression is to just to be rewritten, then 
we look in a big table we carry in our heads. This is what most of us do for small numbers

These are real questions when you are trying to instruct a computer; they are also real questions whenever their answer is not clear from the context. It was clear in elementary education before we tested students on material long after it was taught.  For small numbers elementary math instruction first teaches a calculation process and later assumes a student has the big look-up table.

For our purposes, the arithmetic expression shall mean the operator denotes calculation with operands: the expression (2 + 3) means "add 3 to 2 " and produce the sum. How do we know it does not mean "add 2 to 3"? In a sense we do not care, the sum produced by any process that we know about produces the same result, and in fact we could have chosen the reverse order, but the two processes are different: they treat their two operands differently, and we must choose one. This is reflected in our usual interpretation of the expression for addition as "add N2(1) to N1(2)" and not "produce the sum of the two operands"; or for multiplication: "multiply N1 by N2" and not "produce the product of the two operands".

The classical words for the two operands in both addition and multiplication are different from each other and are suggestive of the calculation process:

math.stackexchange.com/.../what-are-the-formal-names...

addition: Augend + Addend = Sum

subtraction: Minuend - Subtrahend = Difference

multiplication: Multiplicand x Multiplier = Product

Division: Dividend / Divisor = Quotient

...contain a wealth of Latin. If you happen to know Latin, you will understand these meanings more deeply. For example "minuend" comes from a form meaning "about to be lessoned" and "subtrahend" comes from a form meaning "about to be taken away". In general, "-nd" will carry the meaning "about to be --ed".

Interesting! What appear to be exotic labels suggests a practical way to look at the calculation process. I choose to read "Augend" as the operand about to be augmented, with "Addend" about to be the one doing the augmenting. The "Multiplicand" is the operand about to be multiplied, and "Multiplier" is the one that will do the multiplying. 

This is consistent with the operational meaning: choose one operand to be acted on and the other to act on it. For addition one operand plays the passive role, waiting to be augmented by the second operand. In multiplication, again one operand plays the passive role, waiting to be multiplied ( stretched either lengthwise or crosswise in an amount determined) by the second operand. The classical nomenclature suggests actions and actors.

Any suggestions for different words for the operands? Surely you can improve these: ("abt" is "about to be", "at" is "about to")

1:

Addition:             (atbAdded-to + atAdd) 

Multiplication:    (atbMult-ed + atMultiply)

or

2:
Addition:            (atbAdd-edTo + atbAdd-ingTo)
Multiplication:   (atbMultipli-ed     x  atbMultipli-er) .

These position-specific "words"  for "operand" are all adjectives, each one now modifies an unvoiced "operand"; e.g.       "[the] atbAdd-edTo [operand]" in
"(the atbAdd-edTo 2 + the atbAdd-ingTo 3)" for the expression    (2 + 3).

A more practical notation might forgo the "about to be":

3:
Addition:            (To 2 + Add 3)
Multiplication:   (Mult 2 x By 3)

and this suggests the more elegant

4:
Addition:            (To 2 , Add 3) means "add 3 to 2", or               "with 3 augment 2".
Multiplication:    (Mult 2 , By 3) means "multiply 2 by 3",              or "with 3 multiply 2".

None of these feel quite right; but the real "problem" is now apparent. "add" and "multiply" are not symmetrical. "Add" refers to what is is going to be added, not to what is about to be added to; "multiply" refers to what is about to be multiplied, not to what it is going to be multiplied by:

• (2 + 3) means add 3 to 2, 
• (2 x 3) means multiply 2  by 3.

In short, one adds "to", while one multiplies "by".

Things might be less confusing if we could use symmetric terminology for (operand , operator, operand). For addition: 

• the name of the operator "+" could be "augmentation":
• the left operand is the "augment"-this-operand,
• the right operator is the do-it-"with"-this operand; ie,

Addition expression: 

(augment-this-operand  +  by-this-operand)

Multiplication expression: 

(multiply-this-operand  x  by-this-operand).

Do you think the order of the operands should be reversed? In the order shown above the operands on the right are those most likely considered prior to those on the right.

For both addition and multiplication there is generally the connotation that after the calculation the left operand has been changed, and the right one did the changing; however, neither the sum nor the product depend on the operand order: addition and multiplication commute. To reduce confusion in subsequent work, however, I will try to keep the changed operand on the left.

There is a major difference between addition and multiplication: In multiplication we must first consider the changing operand because Multiplication distributes the change it specifies over the entire changed operand while, for addition, the augmentation can be effected by one "attachment".  In a sense multiplication is melding; and addition, an add-on. At this juncture all this is premature; perhaps you can see it. It will come up in later posts, and I may go back and change the first post where I think some of it appears more naturally. 

Semantics in Elementary Math

Where I'm Coming From

1. Have not taught mathematics.
2. Did theoretical physics, and taught some.
3. Did engineering at Bell Labs. Retired from there.
4. Did research in the Dept of Civil and Env Eng, Princeton.
5. Started to wonder why most of us eventually have trouble applying the mathematics we learned in elementary and even secondary school; the initial exam results in my first undergraduate physics course indicated I did not not have a clue.
6. Read, talked, thought, and wrote over the past year.

Where I Am Now

I assert: 

1. The mathematics taught in elementary school is both too abstract and too concrete. It is largely based on teaching an arithmetic of abstract numbers which is then used in concrete algorithms whose utility is largely superseded by electronic computation.  
2. Common Core is more a matter of rationalizing the traditional core, less a matter of getting at a core of problems in it.
3. Within the framework of current instruction questions arise that can not be addressed because there is no explicit consideration of the semantics in the mathematics taught; what is the meaning of the numbers and expressions that appear? In the recent discussion of Multiplication is Addition, it was necessary to go way beyond elementary mathematics to argue that multiplication is not addition. Consideration of the semantics of addition and multiplication is a path all of us can follow to understand why it is not a good idea to teach that it is.
4. Semantics is the bridge between numerical calculation and word problem solving.
5. Those who teach the use of mathematics in subjects like engineering, physics, etc., should be in the discussion. Mathematics is too useful, too meaningful, to leave its explanation entirely in the hands of mathematicians. 
6. Manipulables in support of an active approach to simple arithmetic can be used to make a concrete semantics. I will discuss this in follow-on blogs.

My experience suggests it is more effective, and usually more efficient, to assert and apologize than to hedge and explain. Those for whom I am writing know the subject far better that I do; I can only hope you work the tension between cutting me some slack and giving your best shot. I also know I have a responsibility to argue for what I assert. That will follow in this or subsequent blogs.

Semantics of Addition and Multiplication

Outside the classroom and aside from pure mathematics the semantics of operands and expressions is the primary point of mathematics, or at the least understanding the semantics is the primary value added by us. Computers can take care of both numerical and algebraic calculation; i.e. reducing to a useful value or form the numerical and algebraic expressions that encode our understanding of the semantics of the problem.

Of course we also must be able to do simple calculation problems.  We must learn how to do simple and easy syntax because it can be easier to do it than it is to set it up for a computer to do; computers do the complex or onerous syntax.
In any case bringing consideration of meaning into the elementary learning of calculation by addition and multiplication shows how mathematics deals with real problems.

Addition

N apples + M oranges = (?). As apples and oranges, a member of the sum collection N+M is not defined; what is an (apple AND orange). If instead of apples and oranges we consider a member of each of the collections N, M, and (N+M) to be an (apple OR orange ), or perhaps just a fruit, then the addition is well defined: (?) is (N+M) fruits.The point is simple: only a common, additive, identifying characteristic that adheres to each element of the operands is summed by addition. Since it is common to all of the terms in the addition expression, it may be factored out and then the numerical expression alone, N+M= (N+M), describes the addition of the N individuals and M individuals. For this reason ignoring semantics in elementary addition education is not greatly problematical. After all, any rewriting of a numerical addition expression as another numerical addition expression does not change the meaning of the terms in the resulting expression. At some level the semantics of any addition problem is the same as that of any other: you might as well leave the semantics implicit. And finally, from what we have said, it is not possible to chose to draw an apple(orange) from the sum; the element drawn is either an apple or an orange. I think this is the case for a sum in Type Theory, but not for Category Semantics or Set Theory. Obviously this is not elementary issue, but I would appreciate some help here.      

Multiplication

Unlike addition, semantics is an active participant in multiplication. Generally the meaning of one operand differs from that of the other, and the meaning of the product is the result of the interaction of the meanings of the operands; in a real sense the meaning of the product is the product of the meanings of the operands Ignoring the semantics of multiplication and just relying on syntax is dangerous: for the whole number M = 3 consider

3 x N = (1+1+1) x N = (1 x N) + (1 x N) + (1 x N).

If one ignores any meaning of 1 (or 3)  beyond its property as a whole number and takes (1 x N) = N, then 

3 x N = N + N + N,

and one is led to infer multiplication by a whole number is equivalent to multiple addition. This can be true for an N that only carries the meaning of a whole number, but most of us have little truck with such problems, and it misleading to suggest equivalence in our problems where multiplication is not addition because what is being added is not the same as what is being multiplied. We illustrate the issue with exemplars of two broad classes of multiplication problems: rate/ratio and area calculations.

Rate as Speed

The distance D traveled in a Time T at a constant speed R for T = N hours and R = M miles/hour is given by the evaluation of the expression M miles/hour x N hours. For M =3 and N = 2, D follows from

3 miles/hour x 2 hours = 

(1 mile/hour x 2 hours) x (1 mile/hour x 2 hours) x (1 mile/hour x 2 hours) =

2 miles + 2 miles + 2 miles.

The semantics is apparent in 1 mile/hour x 2 hours = 2 miles; it is not the case that three Ms =  M plus M plus M  because 1 x M does not equal M. In the real problem "1" is a speed that acts over time that results in a distance. ( I would go further and say it is not a good idea to decompose 3 miles/hour because it suggests there are 3 parallel trips at speed 1 mile/hour while the original problem concerned one trip at 3 miles/hour. If a problem in elementary education is the application of mathematics to word problems, then the mathematics should be taught as following from words not numbers. Your thoughts?)

Area of a Rectangle

For length L feet and width W feet, say L = 3 feet and W = 2 feet, the area A is

3 feet x 2 feet = 6 square feet.

Here each operand is a length measured in a unit of 1 lineal foot, think of a foot-long ruler; while the area is measured in a unit of 1 square foot, think of a square tile one foot on each of its sides. This is the common refrain in multiplication problems: the meaning of a product is the product the meanings of the operands.

Wrapup

Multiplication is not addition. Addition is not equivalent to multiplication: it has different semantics, and the addition process is not equivalent to the multiplication process. The product resulting from multiplication can always be rewritten as a multiple addition, but each of its terms the product in an already performed multiplication: the multiple addition is simply a rewrite of the product, a different way to express it, not an equivalent way to obtain it.

It is the case that virtually all algorithms for multiplication start by decomposing the operands into parts that can be easily multiplied with each other to produce numbers whose sum is the desired result. Now that computers can do the actual arithmetic, teaching the multiplication algorithm can be taken as an opportunity to show/see how big problems are broken up into smaller ones, and the results of the small ones are then swept up into the solution of the big one. Eureka! they have the keys to the kingdom.

Looking Ahead

Here I've written mostly about the semantics of problems using arithmetic. I want to turn to some ideas on making school mathematics a subject with a more active approach.  If semantics/meaning is to go anyplace, ways have to be found to make mathematics an active endeavor involving things you can touch, feel, and name while at the same time raising good mathematics questions. The useful heart of mathematics holds meaning not manipulation. I believe two strands of science, physics and computer science, have developed useful methods and concepts that can be suggest ways to go forward in the effort. Your nominations? Of course my dream is to entice you to join.

I want to discuss ideas for a hands-on coordinate system to make calculation on a number line self-contained, calculation by measurement if you will. It should be in a follow-on blog.

Addendum

I want to revisit the semantics of addition and multiplication to look for better, or at least other, ways to encapsulate their meanings. This discussion is intertwined with the one in the next post. It looks for visceral understanding of the difference between addition and multiplication.

Multiplication spreads the effect of one of its operands over the other to produce a third that generally differs qualitatively from each. In a process of multiplication, and here I picture the physical process: speed acting over time (or time over speed ) to produce distance: 1 mile/hour x 2 hour = 2 miles, extending along a line produced by velocity converting time into distance.
Multiplication of length by width produces area, it distributes width(length) along length(width) to produce area by extending the entire the one dimensional length (width) by the one dimensional width(length) to produce a two dimensional area. 1 foot-length-rectangle x 2 foot-width-rectangle = 2-square-foot-rectangle. Here the extension by one operand is transverse or crosswise to the other operand, sweeping out an area, not "extension along" as it was in the previous conversion calculation.

Addition  produces a sum with the same qualitative features as those of the operands whose addition produced it. There is no connotation of extension, rather addition augments or appends one operand to the other with no conversion of units or change in dimension.

Elementary Thoughts