Containerization of Basic Operations: A Way of Constructing Mathematics Knowledge

By Atovigba, Michael VershimaDepartment of Curriculum and Teaching,

Benue State University, Makurdimikeatovigba@gmail.com

AbstractThe work focuses on containerization as a constructiveway of performing basic operations. By containerizationis meant manipulating containers of particles (orobjects) in representation of number and numeration fromwhere patterns are established and abstracted as conceptsand processes of addition, subtraction, multiplicationand division. This prepares pupils for future advancednotions of mathematics, which necessarily have to do withbasic operations and their extensions. The paper arguesthat containerization is a way of introducing basicoperations to beginners who should by this approach areable to construct mathematics knowledge using locallyeasily available manipulative materials.

Key Words Basic Operations, Containerization, KnowledgeConstruction, Manipulative Materials, Number andNumeration

Introduction

The introductory stages of mathematics require pupils to

manipulate concrete materials in active processes that

lead them to constructing mathematics knowledge. The

pupils can be led to do this as a profitable activity in

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line with social constructivism as posited by Vygotsky

(in Nunez, 2009). This activity prepares pupils towards

ultimately doing set theoretic mathematics in the latter

stages of learning. One of the activities suggested in

this paper is manipulating objects and containers in a

process called containerization as a position ( a

proposed theory) taken by Atovigba (2012) which is the

focus of this paper.

Definitions

1. Containerize is to manipulate objects and containers

in depiction of basic binary operations. The basic

operations or processes are (+,-,x,/).

2. Object is a thing. Thus a point, a particle, a set, a

group, a structure, or a galaxy is an object.

3. Container is an object that contains another or other

objects.

4. Contain is to accommodate.

Theory of containerization

The theory of containerization states that: Nature

manifests as ‘object and container’ phenomena and can be

described by containerizing, that is: performing basic

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binary operations with counts of objects and containers

for sums (and differences) and equal distribution or

sharing of objects into containers for multiplication

(and division) from where basic mathematics structures

are abstracted (Atovigba, 2012).

Containerization involves: adequate count of containers,

adequate count of objects, problem solvers performing all

the operations with teachers’ guidance provided towards

realizing set objective (a flow into abstraction or

generalization), and adequate number of exercises and

continuous formative and summative assessments involved

as a principle or strategy aimed at achieving

automaticity.

Basic Binary Operations with Containerization

There are four basic operations, namely: addition (+),

subtraction (-), multiplication (x) and division (/)

from which extensions are made into advanced mathematics.

The basic binary operations are containerized as follows.

Addition

In containerization, addition involving a binary

operation (involving two numbers) would be the gathering

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of objects from two different containers into a third

container to indicate that all the objects are now

grouped into one whole bunch or container. The whole is

now counted and recorded as the answer or solution. An

example suffices.

Problem: To perform the operation 2 + 3.

Solution: In containerization, the pupils would be

presented with two containers, one containing 2 objects,

another containing 3 objects. The contents of both

containers are emptied into a third container (which is

the actual meaning of addition, as pooling together of

existing objects in the operation). The sum is the total

of the number of the objects in the new container which

are counted by the children. The children perform this

operation themselves and record results which turn out to

be generalizations.

Subtraction

Subtraction is performed similarly by having an entire

count of objects in a container, then taking or removing

some requisite count of those objects away from the

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container; what is left is counted and recorded as the

solution or answer. An example will suffice.

Problem: To perform the operation 5 - 2.

Solution: In containerization, a container of 5 objects is

availed the pupils. Their task is to take from the

container 2 objects, and record that 3 objects are left,

which is the required solution.

Multiplication

Multiplication (x.y) in containerization has a requisite

number of containers (x) in each of which an equal number

of objects (y) is shared ( and the roles of x and y could

be shared without having a different result – thus

multiplication connotes sum of the contents of all

containers which were shared equal counts of objects.

This can be exemplified with a problem as follows.

Problem: To perform the operation 2 x 3.

Solution: Currently some teachers compel pupils to cram the

multiplication time’s table, which is repugnant to

mathematics education. Some textbooks have published the

approach of continuous addition to multiplication, and

would perform this operation as: 2 x 3 = 2 + 2 + 2 = 6,

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or 2 x 3 = 3 + 3 = 6. This approach is faulty as can be

seen with a few examples. Take for instance, to perform

‘-2x3’ which means adding continuously ‘-2’ up to three

times which rightly gives ‘-6’. However, since addition

of numbers is supposed to be commutative, ‘(-2)(3)’

should be equal to ‘(3)(-2)’ in which case we need to add

‘3’ continuously ‘-2’ times which makes no real life

meaning. This makes the definition of multiplication as

continuous addition, to crash. In containerization,

multiplication is done by first recognizing number as

count of objects at play, thus there are 2 containers in

each of which will be distributed 3 objects, or there are

three containers in each of which will be distributed 2

objects thus ending with a summary pooling of all objects

in all the containers involved and then counting the

total for the solution as 6 objects. Pupils will then be

led to realize that it is in this real life context that

‘(2)(3)=6’ and this means two containers each allotted 3

objects, thus totaling 6 objects, not just a certain

abstract, mysterious number called six. To multiply a

negative number with a positive number is posited in

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containerization as an operation involving a count of a

borrowed object (negative number) distributed into each

of the containers (positive number). The result is a

number of borrowed objects hence their remaining

negative. Thus whether one of either an object or a

container is borrowed will give the result as a negative

quantity although numerically the result remains as if no

quantity was borrowed in the first place. From this

binary operation, algebraic extensions could be made for

cases involving negative numbers in multiplication.

Containerization thus provides basic premises upon which

non-basic premises or initial conclusions are made for

further deductive conclusions in what could be termed

complex arguments.

Division

Division under containerization is done with the

denominator suggesting the number of containers into

which an equal number of objects would be shared, while

numerator suggests correctly the number of objects that

are to be shared. (Another denominator-cum-numerator

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notion under containerization will be discussed when

fraction is inferred.) The share of each container is

then counted as the quotient or solution to the division

problem. The following is an example:

Problem: To perform the operation 6 ÷3.

Solution: Currently some teachers compel pupils to realize

that since 2 x 3 = 6, by reverse thinking, 6 ÷3 = 2. This

is forcing poison down the throats of the students. For,

they could similarly reason that since 3 x 0 = 0, hence 0

÷ 0 = 3, and this has been a source of confusion to

students who believe that everything religiously follows

another in such a reversible fashion. Some textbooks have

done their best to teach the pupils that division is

continuous subtraction of the divisor from the dividend,

such that 6 ÷3 = 6 – 3 – 3, hence the solution as 2

because 3 is taken from 6 two times. This definition also

portends contradictions when such numbers like 0 are

involved as both numerator and denominator; e.g. 3÷0

would then mean continuous subtraction of 0 from 3 i.e.

3-0-0-0…=3 which is not true. Similarly, 0÷3 means

continuous subtraction of 3 from 0, i.e. 0-3-3-3…= -3n if

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n is the number of subtractions of 3 from 0, which is an

incorrect result and a source of confusion.

In containerization, the denominator indicates the number

of containers to be involved in the operation, here 3

containers. Six objects are then shared equally into the

3 containers, and that makes 2 objects for each

container, thus the answer is 2, which the pupils

appreciate as two objects.

Between Multiplication and Division

Containerization identifies the boundaries between

multiplication and division as follows. In both

operations objects are shared equally into containers.

The share of each container is the result of division,

while all the shares pooled (added) is the result of

multiplication.

Other Processes of Arithmetic

Other operations like factors, multiples, and powers

dealt with under the fundamental theory of arithmetic,

can be equally extended from conclusions made with

containerization. Containerization becomes handy as both

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means of solving mathematics problems and as means of

tackling everyday real life problems. Some few examples

of problem solving are mentioned below.

Average: A look at one of the textbooks predominantly used

in Nigerian primary schools is Understanding Mathematics by

David-Osiagwu (2009). It defines average in the formula:

√9=3.This definition is misleading, ambiguous and a source of

confusion to pupils, as it is rather abstract and compels

pupils to learn by rote. For instance, it raises the

questions: what is ‘sum of objects’ and what is ‘number

of items’? Thus Understanding Mathematics (David-Osuagwu,

2009) presents an example of finding the average of

6,7,8,9,10 as:

This is a source of confusion to pupils who are at an

elementary stage of mental development. Such unclear

statements make pupils to conclude that mathematics is

hard to understand. To clarify this operation,

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5109876

Average

containerization takes the issue of averages as an

operation of division as follows, using the above

example. The pupils have 5 containers initially

differently having 6,7,8,9,10 objects respectively which

are pooled together into one container as 40 objects.

These 40 objects are then shared equally into the 5

containers, which makes 8 objects for each of the five

containers. That is, 8 is the answer (8 taken in the

context of 8 objects in each container).

The Concept of 0

One of the problems that confront the student of

mathematics is trying to understand the concept of zero

and the operations that surround it. One of the questions

is: why is it that if zero multiplies any number, the

answer is zero? Containerization answers this question

with so many containers and zero object to distribute in

each of them. The sum of all the distributions is

practically seen to be zero. This explains why, if zero

multiplies any number, the result is zero. This is

demonstrated as follows:

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To perform ‘3 x 0’: Three indicates the existence of 3

containers in each of which is to be distributed 0 count

of the object in view. Each basket goes with 0 count of

the object. When the contents of all the baskets are

pooled together and counted, behold! the result is 0

count of the object. The reverse of ‘0 x 3’ is similarly

handled with 0 count of containers (say, baskets) and

three counts of the object in view to be distributed into

each basket: Behold! There is 0 basket, hence nothing

distributed and the total distribution is 0 count of the

object. Thus 3 x 0 = 0 and 0 x 3 = 0, and this is taken

in context as 0 number object summed from all the

containers in the operation. More numbers are used in the

experiment and each time the result is 0, and the

required generalization is reached, that the zero

multiplier of any number gives zero as product.

Division by Zero

Another question is: what results if a number is divided

by zero? Traditionally, pupils are simply told never to

divide by zero without explanation. Containerization

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provides the explanation behind this notion, in the

following process. Pupils are availed a number of

objects, say 5 stones, to share into zero containers,

i.e. no container is provided, but the sharing of 5

stones is to be done. Pupils find out it is impossible to

share since there is no container to put any stone in.

More such operations are permitted, and pupils always

find out there is no container to share the objects into.

Conclusion: it is impossible to divide by zero, hence

division by zero is not done.

This exercise with zero using containerization is a

milestone, because division by 0 constitutes an

astounding turning point in advanced mathematics. In

analysis it is rightly called a point of discontinuity,

and as can be seen in the foregoing experiment, the

division operation becomes discontinuous with the zero

divisor. Also a set endowed with properties qualifying it

to be a commutative ring with unity and for which every

non-zero element is invertible is called a field.

Invertibility is reference to attribute of a number ‘a’

having a multiplicative inverse a-1. If a = 0 then we have

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a zero-divisor on ‘a’ being inverted, which will be an

indeterminate quantity and a = 0 is hence not included

among the properties of a field. Thus this zero divisor

operation under containerization helps to lay the

foundation for pupils to later flow without hindrance

into advanced mathematics thought especially in real and

abstract analyses.

Discussion

Containerization thus questions claims that the zero

divisor leads to infinity as quotient. This is because,

by containerization (and naturally) it is impossible to

divide by zero. Therefore there is no known result of

such a division which has not taken place in the first

place. Hence the usual mathematical (Cauchy) result of

infinity if some number were divided by zero, is false.

The fact that the denominator is dynamically tending

toward zero and making the quotient to tend to an

uncontrollably large number called infinity does not

really make the zero divisor to take place; hence it is

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not admissible that the zero divisor ever took place as

‘nearly does not kill a bird’.

Binary Operation with Negative Numbers

In containerization, the international convention

regarding basic arithmetical binary operations with

negative numbers is done as follows:

Addition of Two Negative Numbers (-x, -y):

-x + -y = -(x + y)

Here, two baskets are involved containing respectively x

borrowed number of objects and y borrowed number of

objects. Thus x and y numbers of the objects are normally

added but tagged with a negative sign as signifying

borrowed counts of objects. Borrowed could mean here a

shortfall, or a debt, or negative account or what is

expected but yet to be realized. For example, -2 + -3

has one container containing 2 borrowed objects and the

second having 3 borrowed objects. The objects are

normally pooled together and added and the result is 5

borrowed objects, hence -2 + -3 = -5.

Multiplying One Negative with One Positive Number: -x(y) = -(xy)

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In containerization, -x(y) implies some borrowed x number

of objects in each of y containers which is just as

sharing x number of objects in each of y number of

containers and the counted total is xy; this can be

demonstrated by letting k = -x taking –x to some k

object. In that case each of y containers has k object

and the result is a total count of k’s in all the y

containers: that is ky = -xy as the result.

Hence

(-x)(y) = -xy (1).

This result supports the claim that

(-1)(1) = -1 (2)

as we let k = -1 as an object in one container with the

obvious result of k as total count of the object in view.

That is: k(1) = k = -1.

With the forgoing, pupils can confidently return

(-2)(3) = -6, for example.

Binary Multiplication Operation involving Negative Numbers:-x(-y) = xy

In containerization, we take the product (-x)(-x) by

regarding (-x) as an object say k (i.e. k = -x ). Then,

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(k)(K) is having k counts of objects distributed equally

into k counts of containers and the result is 1. Thus

(-x)(-x) = 1 (3).

This result prepares the ground for the proof, that

(-1)(-1) = 1 (4).

This is done by replicating the foregoing steps, letting

k = -1, and the result follows.

Consequently,

(-x)(-y) = (-1)(x)(-1)(y) = (-1)(-1)(x)(y) (5)

owing to the associative law, which renders

(-x)(-y) = xy (6)

with (-1)(-1) = 1 as already demonstrated in (4).

Remark

The fact that objects or containers are in negative

quantities does not change their material status as

objects or containers of uniquely defined attributes;

hence they would be counted as objects and containers

just as if they were not borrowed. That is why the

results are the way they are. Thus the operation respects

the natural law that matter cannot be destroyed or

created as the objects and containers remain objects and

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containers except that they were borrowed and borrowing

does not destroy their character as objects and

containers.

Remark

Containerization takes after real life resource-based

distributing or sharing process or state which exhibits

set and subset relationship. It appears that nature

presents itself in form of either an object or a

container of objects. This truth has shown itself in two

or three quarks glued with glue-balls (as objects) to

form a neutron or proton (container). Furthermore,

neutrons with protons and electrons (as objects) form an

atom (container). Many atoms (as objects) bond to form

molecule (container). Molecules (as objects) group to

form plasma (container). The object-container phenomenon

goes on till more visible objects or containers like

planets, stars, galaxies and universes are formed. To

express this truth is containerizing objects. As language

of scientific expression, mathematics must therefore be

done by containerization. Furthermore, containerization

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is justified with the following arguments which seem to

support the theory.

The container-object mathematical theory can be hinged on

the philosophy by Bertrand Russell as it presents

mathematics in contextual phenomena as recommended by

Bertrand Russell (in Nolt, Rohatyn, & Varzi, 2004) such

that, number is not just number but a quantizing symbol

for the quantum of objects in view. In this connection,

the number ‘6’ means six objects that are similar (or not

similar but resolved to some common domain) and this

context has to be made clear to the audience: hence 6 is

not just six but six counts of an object which could be

stones, pebbles, eggs, balls, sticks, or biros, and

could be extended to any material of any magnitude like

points, particles, bags, rooms, houses, families, wards,

nations, planets, stars, or galaxies.

To Immanuel Kant (in McCormick, 2005), knowledge is found

within the confines of mathematics and the natural,

empirical world. Both mathematics and science take

bearing from number, numeration and statistics. Kantian

philosophy postulates also that the sole feature of an

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action’s moral worth is located in the motive behind the

action, the motive having moral value if it arises from

universal principles discovered by reason. Similarly,

containerization is a function of reason arising from the

principles of orderly packaging of objects into

containers, a feature drawn from the universal nature of

functions, number bases, and set theory which is glaring

in the constituents of matter, beginning from elementary

microscopic particles of quarks wrapped up in the

container called neutron or proton, then neutrons and

protons and electrons wrapped into the container called

atom, atoms wrapped into cells or molecules until stars

and galaxies and then the entire universe are arrived at.

Indeed, mathematics, science and technology, and the real

world, is captured within containerizing or assembling or

sharing of objects according to their kinds or mixtures

there-of. For instance, a function is simply an

assemblage of terms. Terms are an assemblage of factors.

Factors are an assemblage of numbers of objects and

containers which could be particles or molecules which

are in turn an assemblage of atoms. Atoms are an

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assemblage of subatomic particles like neutrinos, quarks,

electrons, and gluons (Friedberg, 2007). Any such

assemblage is as gathering of objects into a container

and numeration takes after this universal order.

Containerization has potential to achieving the broad

objectives of mathematics instruction which, as Sidhu

(2006) observes, are classified into knowledge and

understanding, skill, application, positive attitude,

appreciation and interest. The objective of knowledge and

understanding leads pupils into acquiring a good mastery

of mathematics language, ideas and nature.

Containerization tries to give meaning to number and

numeration since these processes are done with reference

to objects rather than sheer abstractions, thus leading

to learning of the very underpinnings of mathematics. It

thus enhances knowledge building with understanding and

exposes the reality of basic operations and their

application in solving everyday problems.

Containerization can be localized to any part of the

globe, as it uses simplistic locally fetched resources

like, in a traditional African setting, calabashes or

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gourds containing stones or pebbles or nuts, bags

containing fruits, barns containing sacks of yields.

These resources are cheap to fetch and easily manipulated

by both teacher and pupil.

The theory of containerization can be shown to have

satisfied Gottlob Frege’s three properties of a

mathematical theory. The three properties of a

mathematical theory are: consistency, completeness and

decidability, and can be verified as follows. Consistency

is the condition that a theory does not contain

contradictory statements. The theory of containerization

is consistent as it does not contain contradictory

statements, since nature shows itself from the smallest

known basic building blocks of quarks and electrons to

large things like galaxies as objects and containers, and

there exists no known phenomena that is neither a

container nor an object.

Completeness on the other hand means, that a statement is

either provable or refutable. The theory of

containerization is provable rather than refutable. This

is because all of nature known so far shows as objects

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and containers, which is manifest from the basic building

blocks, like quarks and electrons through the galaxies to

the entire universe. The basic binary operations done by

containerization too have each been shown to be provable

rather than refutable. Finally, decidability is existence

of a decision procedure for verifying any statement in a

given theory. The theory of containerization is

decidable in form of expression of any form of existence

or phenomenon as object or container. Every existence or

phenomenon can be expressed only as an object or a

container. Similarly, every basic binary operation can be

shown to be a containerized process. Advanced

mathematical operations are only extensions of the basic

operations.

References

Atovigba, M. V. (2012). Selected Topics in Mathematics, Logic and Research Methods. Makurdi: Benue State University

David-Osuagwu, M. N. (2005). Understanding mathematics for Nigeria 4. Onitsha: Africana First.

Freidberg, J. (2007). Plasma physics and fusion energy. Cambridge, United Kingdom: Cambridge University.

McCormick, M (2005). Immanuel Kant: metaphysics. Retrieved from: www.iep.utm.edu/kantmeta. (Accessed on: January 22, 2012.)

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Nolt, J., Rohatyn, D. and Varzi, A. (2004). Logic (2nd ed.). New Delhi: Tata McGraw-Hill.

Nunez, I. (2009). Activity theory in mathematics education. Joubert, M. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 29(2) 53-57.

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