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 ,
- 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.
- 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.
- 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
- You can construct the Natural Numbers you need, including a Natural Number count of Things.
- 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.
- 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.
