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.One Way to See What Addition and Multiplication Do, and Why that is Important
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:
This is all coordinate systems do.]
- 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:
A drawing to illustrate the relation of a Cartesian coordinate system:
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.
- 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 |
Construction of a Number Segment in an SLS
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. |
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.
(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.
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.
No comments:
Post a Comment