CHAPTER ONE
Introduction
Background to the Study
Mathematics is indeed very useful in a lot of human activities. One
could not agree more with Kraner, (1993), when he wrote, “Everyone uses the
Mathematics as their language. Others who use Mathematics include
Navigators, Surveyors, Economists, Bankers, to mention but a few. According
to Cockcroft (1982), Mathematics provides a means of communication, which
is powerful, concise and unambiguous. To Nunes (1996), though Mathematics
is seen as a school subject, it also forms an integral part of their everyday
lives. According to him without. Mathematics not only will pupils be ill at
ease at school but in a great many of their everyday activities. For example
when sharing valuables with friends, when planning to spend their pocket
money, when they argue about speed and distance dealing in different
currencies as well as engaging in buying and selling. He explained that though
these acts are not directly seen as Mathematics, but in carrying these activities
out one would inevitably use mathematical principles.
Most students often fear mathematics, the world over and especially in
Ghanaian schools. In their research Cruikshank and Jensen (1988), observed
that children’s feelings affect their ability to learn. They went on to explain
that many children have learned not to enjoy Mathematics. In their view,
children might have experienced so much failure in their bid to learning
Mathematics partly as a result of being asked to learn some mathematical
ideas, that they were not ready to learn or that they might have been pressured
1
to memorize hundreds of unrelated basic mathematical operations. Therefore,
there is the need to use a variety of methods to boost pupil’s interest in all
topics taught in Mathematics. For this reason, the researcher has decided to
use concrete materials to generate pupil’s interest and to boost their
understanding in the addition and subtraction of number bases. It is obvious
that Mathematics deals with numbers. Staszkows and Bradshaw (2004),
defined a number as a quantity that answers the questions “How much?” or
“How many?” They differentiated between numbers and numerals by saying
that numbers are given a name in words and are represented by symbols. The
symbols that are used to represent numbers are referred to as numerals. A
system of numeration was also defined as consisting of a set of symbols and a
method for combining those symbols to represent numbers. Some numeration
system that have ever being used includes the Babylonian (3400BC), Ionic
Greek (450 BC), May an (300 BC) – Roman (200 BC), Chinese (200 BC) and
Hindu – Arabic (825 BC). According to Bassarear (2007), the numeration
system we use today is that of the Hindu – Arabic numeration system. The
number system forms a very important aspect of algebra in Mathematics,
Inarguably, it has even become more essential in this computer age where the
computer, the most efficient tool in our daily life is based on the binary
system.
Furthermore, the number system is the set of symbols used to express
quantities as the basis for counting, determining order, performing
calculations, representing values and so on. A fair knowledge in number bases
therefore will enhance students understanding and ability to master the four
2
basic mathematical operations, which are addition, subtraction, multiplication
and division. According to Smith and Peterson (2008), the binary system is
fundamental to all electronic computers regardless of their size or purpose,
hence learning the binary system will help students to get a clear
understanding of how the computer works. Besides, a good understanding in
base ten and base two will undoubtedly enable students to extend their ideas in
other bases, be it base four, base seven, octal and hexadecimal.
There are several topics being taught in today’s number system. Some of
these include rational and irrational numbers, natural numbers, decimals, and
number bases and so on. Performing the basic operations such as addition,
subtraction, multiplication and division on these number systems has been a
challenge to most students in Ghana. A child cannot boast of having
understood Mathematical concepts without being able to perform simple
operations with these Mathematical concepts. Educationists and psychologists
such as Jean Piaget, Maria Montessori and Zoltan Dienes have done several
researches into Mathematics with the aim of designing suitable methods and
teaching materials to make the teaching and learning of Mathematics easier
and to remove the aura of fear in the subject. Some studies into fractions led to
the discovery of useful teaching and learning materials such as the fraction
board, base ten blocks and the number grid. Further research into addition,
subtraction and place value also gave rise to strategies such as the abacus and
bundles of sticks. Dienes (1960), also emphasized that the use of certain
blocks could also make the teaching and learning of the binary system easier.
These blocks he termed the Multi-base Blocks. According to the World Book,
3
Encyclopedia (2005), the word binary originated from the Latin word meaning
two at a time. In this system only two digits or bits are used, they are 0 and 1.
Statement of the Problem
It is very worrying to see that J.H.S 2 students at Ummul Qura Islamic Junior
High School in Kumasi are unable to do simple addition and subtraction in
base two. As a newly transferred teacher to Ummul Qura Junior High School
in Kumasi, I observed that students in J.H.S 2 performed poorly in their
Mathematics lessons. One of the topics, which seemed easy but students
performed poorly in their exercise books was on number bases. I found out
that the teacher only used algorithms in teaching the topics which the pupils
could not grasp. The problem might have also emanated from their disinterest
in the subject which resulted from being inattentive during Mathematics
lessons.
Upon noticing this problem, I decided to use the Dienes multi base blocks to
generate pupil’s interest in the topic and also to help pupils to understand the
topic ‘number bases’.
This means that those students invariably would not be able to convert
between different number bases nor do any other simple calculations involving
number bases.
It is upon finding this problem that the researcher decided to use the multi –
base ten blocks as a strategy to remedy the situation.
4
Purpose of the Study
The purpose of this study is to help J.H.S 2 students at Ummul Qura Islamic
Junior High School in Kumasi in the Ashanti Region to overcome their
inability to do addition and subtraction in number base two.
Research Questions
(i)
How can the use of the Dienes base ten blocks help students convert
base ten numbers to base two?
(ii)
To what extent can the Dienes base ten blocks enable students to add
in base two with ease?
(iii)
How can the use of the Dienes base ten blocks enable students to do
subtraction in base two easily?
Significance of the Study
This study would be beneficial not only to students but teachers, as well as any
other person who is interested in the study of binary numbers.
Firstly, with regards to students, it will help demystify the concept of number
bases, and enable them to master the skill of adding and subtracting in base
two with ease.
The methodology used in this study if adopted by other Mathematics teachers
would help them to teach number bases at their various schools smoothly.
5
Finally, the strategy used in this study could be used by circuit supervisors
during School Based Inserts (SBI’S) and Cluster Based Inserts (CBI’S) to
bring to the knowledge of teachers the importance of using concrete materials
especially the Multi –base Block in helping students to understand the concept
of number bases.
Delimitation
This study is delimited to the use of the Dienes base ten blocks to helping
J.H.S 2 students at Ummul Qura Islamic Junior High School in Kumasi to do
addition and subtraction in base two only.
Limitations
This study is limited to a J.H.S 2 class at Ummul Qura Islamic Junior High
School in Kumasi, mainly due to inadequate resources and time constraints in
conducting the research.
Organization of the Study
The whole project is made up of five (5) chapters. Chapter one looked at the
introduction; this chapter deals with the background of the study which
identified the statement of the problem, research questions, purpose of the
study, delimitations and limitations. The second chapter is the review of the
related literature. This chapter deals with what some scholars, educationists
and other authorities in the field of Mathematics have said or written about the
problem under study and also elaborates methods or suggestions that could be
used to solve the problem. The chapter ends with a summary of the literature
6
review. It dealt with the importance of the use of the binary system in this
technological world especially in computers. Strategies and materials essential
for teaching base two were also discussed.
Chapter three (3) gave a detailed overview of the research methodology. It
discusses the research design. The research design contains the type of
research used, the population and sample selection, method, data collection
procedure as well as data analysis. The fourth chapter discusses the results and
findings which came out during the study.
Chapter five consisted of summary, conclusion and recommendation.
7
CHAPTER TWO
RELATED LITERATURE REVIEW
Research Studies
A lot of research has gone into the various methodologies used in teaching
Mathematics. Some topics in Mathematics, which has caught the attention of
so many researchers are; performing the basic operations in fractions, indices,
surds and number bases. There have been several literature reviews on these
topics, however this chapter reviews literature related to the study.
Teaching Mathematics is a daunting task and every Mathematics teacher must
be very careful and well prepared in order not to kill student’s interest.
In his book, Brumbaugh et al (2006), outlined some meaningful guidelines for
effective teachers.
These guidelines are stated as follows:
1.
Know more than the subject you are teaching.
2.
Motivate your students to want to learn the subject in hand.
3.
Communicate your knowledge to students in words that they can
understand and that are meaningful to their world.
4.
Guide your students to new heights of thinking.
5.
Know what to teach and when.
6.
Perceive where and why students are having difficulties.
7.
Decide when and how to practice skills.
8.
Determine how to make concepts meaningful.
According Suydan, (1984), Children might have already experienced
considerable failure in their attempt to learn concepts and skills. They may
have developed a feeling that success in Mathematics is about knowing a
certain “magical process” that would result in correct answers. He therefore
8
suggested seven measures which the teacher could adopt to free pupils from
such misconceptions about learning Mathematics and also to boost student’s
interest in the subject.
According to Suydan, these measures include:
1. Showing that you like Mathematics.
2. Making Mathematics enjoyable so that children develop positive
perceptions of Mathematics and of themselves in relation to
Mathematics.
3. Showing that Mathematics is useful in both careers and everyday life.
4. Adapting instruction to student’s interests.
5. Establishing short term goals that students have a reasonable chance of
attaining.
6. Providing experiences designed to help children to be successful in
Mathematics.
7. Showing that Mathematics is understandable by using meaningful
methods of teaching.
Cruikshank and Jensen (1988), also agree with Suydan on the basis of
providing experiences to increase children’s success by helping them see
meaning and sense in their Mathematics. They went further to say that
Mathematics is ‘after all the study of relationships or how things are
connected’. The teaching of number is very essential in our schools. We use
number to count things. Life would be a lot harder without numbers as you
wouldn’t know how much something or anything is.
9
Kronecker as cited by Cameron (1994), is often quoted as saying “God made
the integers; the rest is the work of man. By this statement, he was referring to
the natural numbers which are believed to be older than the earliest
archaeological evidence. Numbers are tools that was honed and forged from
interaction of our ancestor’s cultural and social behaviour with our ancestor’s
ability to conceptualize.
Some uses of numbers include measuring distance or length, grouping and
dividing quantities, record keeping, measuring temporal quantities such as
time, relating diverse ideas to properties or spatial relations of numbers.
Various number systems were developed. One of the most essential number
system is the binary system. According to the Collier’s Encyclopedia (2006),
the binary system of numeration was first used by Thomas Harriot. It was later
developed and used by the German mathematician and philosopher Gottfried
Wilhelm Leibniz. Interestingly no practical use was found by mathematicians
for the binary system until the 1940’s when computers were developed. The
word ‘binary’ was coined from the Latin word meaning two at a time. This
system uses only two digits, o and 1. It went on further to state that in the
binary system every positive integer is the sum of distinct powers of 2 in just
one way. For example the numeral 10two is read as one, zero, base two and it
means that we have “1two plus no ones”.
The Binary system is very useful in our life. The McGraw – Hill Concise
Encyclopedia of Science and Technology (1984), lists some uses of the binary
system as:
1. The binary system is useful when representing numbers.
2. It is useful in recording and processing information.
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3. Digital computers invariably use the binary system.
4. Arithmetic in the binary system is remarkably simple. For addition
only 1 + 1 = 10 is needed, while the multiplication table reduces 1 x 1
= 1.
Such simple operations are readily performed electronically with extreme
rapidity and reliability.
Another advantage of using the binary system is that there are only two kinds
of digits, namely 0 and 1 and in this case makes arithmetic very simple as well
as providing a language in which to treat two – valued functions.
Also in this system there is no table of addition to be memorized. There is also
no multiplication table apart from the simple 1 x 1 = 1. Long multiplication is
carried out easily since there is no ‘carrying’ except in the summation. The
importance of teaching number bases especially, base two cannot be over
emphasized. Callahan and Glennon as cited by Cruikshank and Jensen (1988),
compared ‘base ten only and multi-base methods and conclude, “The
hypothesis that the study of other bases systems will enhance understanding of
our own decimal system would seem to be a reasonable justification for its
inclusion as a topic for study in the elementary grades, Evidence is not
conclusive, however, that this is the only or best way of accomplishing this
objective”.
However, one disadvantage in using the binary system is that it requires
almost three times as many digits to represent a given number as does the
decimal system. Another danger to the use of the multi – base blocks is that
the indiscriminate and perfunctory use of Base Ten Blocks can prove harmful
11
to the natural development of constructive mathematical thinking of young
children. Swan and White (2004), indicated that students have little chance to
build their construct of the number system. Again, students may play with the
manipulative (Base Ten Blocks) and not pay attention or even throw the
material around. To curtail this problem Swan and White (2004), suggested
that the teacher should set some simple rules and limit for the use. They
stressed that enforcing these rules early on is very necessary for students to
learn to pay attention and also respect the material.
Methods of Teaching the Base Two System
In teaching number to children, Cruikshank and Sheffield (1988), grouped the
characteristics of children with respect to acquiring the concept of number
under four broad headings as given below.
a.
Children have many number experiences.
b.
Children are active in their world.
c.
Children observe relationships in their world.
d.
Children’s feelings affect their ability to learn.
Piaget (1965), in his book “The child’s concept of Number” agrees with the
assertion that children observe relationships in their world. He explained that
children should be involved in inventing Mathematics and that through
experiences children discover relationships and solve problems.
Piaget (1965) also emphasized the use of concrete materials when teaching
mathematical concepts. Students in Basic 8 are generally thirteen years and
over and this means they fall in the formal operational stage according to the
12
Piaget’s stages of development. This means students can think logically about
abstract work, yet the use of concrete materials help pupils in their abstract
thinking.
Dienes (1960) suggested that in order to aid abstraction of the place value
idea, wooden blocks should be made available in a number of different bases.
These wooden blocks he called the Multi-base Arithmetic Blocks (MAB). In
teaching base two, the apparatus include a block, a flat, a long and a unit.
Turnbull (1903),as cited by Cruikshank and Jensen (1988), also recommended
the use of number bases other than ten (10) in order to help children
understand the idea of place value.
Clement, et al (2004), also stresses the use of base ten blocks, as a first and
second grade learning and teaching approach for multi-digit addition and
subtraction and place value. Kamii (1985), advised teachers to use activity
based methods such as the Dienes Base Approach in teaching mathematical
concepts especially those concepts which appear abstract. Cruey (2008),
indicated that Base Ten Blocks can make abstract ideas like place value and
regrouping visible and tangible for primary school pupils. He also observed
that Base Ten Blocks like many other manipulatives help children to see and
touch the ideas they are being asked to cope with in a Mathematics class.
Thompson (1994) also found out that Dienes Block Approach significantly
brought improvement in Eight Grade students’ achievement and interest in
decimal fractions. Aubrey (1994) observed that learning place value and
developing a full working understanding of our number system and notation
used, is one of the basic aim for all primary – aged children, therefore he
suggested that teachers should plan integrated topic work, stressing the
13
importance of play, flexibility and choice with opportunities provided for
practical activities.
Furthermore, Arthur and Dowker (2003) observed that using base ten blocks
to teach Mathematics will enhance pupils understanding. Teaching place value
to students needs careful planning and the use of concrete materials. If
students are able to master the concept of number bases well, definitely
teaching place value will be very easy.
Suggestions for Teaching Addition and Subtraction in Base Two
Cruikshank and Jensen (1988) suggested strongly that since children are active
in their world, there must be opportunities for spontaneous response and
divergent thinking and that students’ should be physically involved in
Mathematics. According to them, materials such as pattern blocks, Cuisenaire
rods, geo-boards etc. should be available in the school.
Aside the uses of concrete materials, some algorithms have been developed for
converting base ten numerals to other bases. Asiedu (1997) and
Gordon et al (2005) all used the repeated division method. For example 19ten
can be converted to base two as follows:
2
19
2
9r1
2
4r1
2
2r0
2
1r0
19ten = 10011two
14
Kraner (1993) also termed the changing of base ten numerals to other bases as
‘regrouping’. It is worth nothing that addition and subtraction in other bases
are done the same way as for decimal numerals, except that the base is number
ten. It is essential to use concrete materials to teach these concepts before
introducing the algorithms. Many a time, teachers do not want to use a variety
of methods or approaches. Aubrey (1994), in his research in British
classrooms found out that the lack of teacher mathematical exposition and
reliance on scheme work signifies lack of subject matter knowledge about how
children learn Mathematics.
Dienes (1960), as cited by Aubrey (1994), strongly believes that children’s
active involvement in the process of learning Mathematics should routinely
involve the use of manipulative materials.
Bassarear (2007), suggested to teachers to observe students as they work with
manipulative materials, such as base ten blocks.
Aubrey (1994) however suggested that to use a manipulative such as the
multi-base blocks, teachers should first teach the students the concept of place
value and how to regroup. In this case, the teachers must ensure that students
have developed a solid understanding of place value before introducing
regrouping (borrowing and carrying). Using the Base Ten Blocks to teach
requires a lot of tact and creativity on the part of the teacher. As Swan and
White (2004), put it
“Creative use of Base Ten Blocks will help form powerful creatively thinking
students”.
15
Summary of the Related Literature
The literature review made it clear that ‘number’ is very essential in every
sphere of human life. The teaching of number bases especially the binary
system also called base two was also seen to be one of the most important
systems which is been widely used in this technological world where the
computer which uses the binary system has become the single most powerful
tool ever to be used by man. Therefore understanding the binary system would
enable one to fully understand how the computer works. As Hallberg (2009),
put it, “The natural numbering for to use would therefore be the base two
numbering system”. Teachers have a lot to do as far as teaching the binary
system is concerned. A lot of strategies were suggested of which the multibase block was found to be very effective in helping students understand the
concept of the binary system.
Also teachers must make sure that they use a lot of concrete materials to help
pupils understand the process of converting between number bases, adding
and subtracting in base two before introducing the algorithms. If teachers fail
to generate understanding of concepts by using appropriate methods, pupils
will totally lose interest in all topics in Mathematics and thus perform poorly
in their exams
16
CHAPTER THREE
METHODOLOGY
This chapter deals with various methods employed to improve or solve pupils’
inability to add and subtract in base two. The chapter also discusses the
research design, research instrument (s), pre – test, intervention and the post –
test design.
Research design
To collect data, for the research a total of six weeks was used. The first week
was used for the pre test. After that the intervention was carried out for four
weeks after which a post test was conducted in the last week.
Population and Sample Selection
Ummul Qura JHS 2 class was purposively sampled for the study because the
problem that the class exhibited met the criteria for the study. The JHS 2 class
is made up of a total of sixteen pupils. Out of this, eleven are boys with
percentage of 68. 75% and only five are girls making up the remaining
31.25% of the class population. However, only fifteen (15) pupils actually
took part in the study. This is because one boy who is a truant was absent from
class throughout the pre-test, intervention and post-test period. The average
age of the class is fourteen years.
17
Research Instrument
A teacher made test was used as an instrument to collect data for the study.
The ten questions posed comprised converting from base ten to base two and
addition and subtraction in base two numerals.
Data Collection Procedure
In all six weeks was used for the study. Week one was used for the pre test.
The number of weeks selected for the study was short due to the short deadline
given for submission of the project.
Pretest
During the first week a pre – test was conducted to know pupils entry
behaviour on which the intervention would be based. In all, pupils were given
ten questions as test items. The questions centered on the following;
converting from base ten to base two , addition and subtraction in base two .
The pupil’s scores from the pre test were used as the base line data.
Intervention
A period of four weeks was used for the intervention. It covered the concept of
the multi base blocks and how they are used, how to convert from a base ten
numeral to base two numerals and how to add and subtract base two numerals
using the multi base blocks.
18
Intervention Activities Week One
Lesson One
Topic: Using the multi – base blocks.
Objective:
The objective of the lesson was to help pupils to learn how to use the multi
base blocks.
Teaching learning materials: The multi – base blocks.
Procedure
The multi – base blocks were displayed on a table for pupils to observe. Pupils
were allowed to group the various blocks according to their sizes. They were
then asked to determine values of the various blocks. The single square blocks
represent units or cubes, the double squared blocks represent longs, four units
represent a flat and eight units also represent a long flat. This is represented
diagrammatically as:
Unit
Long
Flat
Long -flat
Pupils were then called to pick at random any of the blocks mentioned. For
example, pupils were called to pick a long, a flat, two units, two flats,
a long -flat and so on. They were also made to exchange say two units for a
long, two longs for a flat and two flats for a long-flat. The activities were
repeated until all pupils had taken their turn and have become conversant with
the various blocks and how to exchange smaller units for larger ones.
19
Week Two
Lesson (1)
Topic: Converting from base ten numerals to base two numerals.
The Objective of the lesson was to help pupils to be able to convert from base
ten numerals to base two numerals.
Teaching learning materials: The multi – base block
Activities:
By way of revision, pupils were called to identify the various blocks. The
teacher demonstrated how to change a number in base ten to base two as
follows;
First of all to change for instance the number 3 to base two, first let pupils find
out how they can regroup 3 using the highest power of two, which is, let
pupils find out which two different blocks will combine to give 3, starting
from the biggest block to the smallest respectively.
Pupils found out that the possible combination using the biggest block was 2
(a long) and 1 (a unit). Therefore 3 in base ten is equal to a long and a unit or
11 (two). Similarly, for 5 in base ten, pupils will pick 1 flat, no (zero) long and
1 unit which is represented as 101 (two). For the numeral 7, the biggest block
pupils can pick is 4 which is the same as 1 flat, then 2 also 1 long and 1 unit.
Therefore, 7 becomes 1 flat, 1 long and 1 unit. This is written as 111 (two).
Pupils were then given other numbers such as 8,9,10,13 and so on to change
from base ten to base two as teacher went round to supervise pupils work.
20
3=
+
1 Long
5=
= 11 (two)
1 Unit
+0+
Flat
0 Long
= 101 (two)
1 Unit
Week 3
Lesson 1
Topic: Addition in base two
The objective of the lesson was to assist pupils to do addition in base two
correctly.
Teaching learning materials: The multi – base blocks.
Activities:
Pupils were asked to change units for longs, longs for flats and flats for long –
flats and vice versa. Pupils were guided on how to perform addition in base
two numerals as follows; for an addition question like 101 (two) + 11 (two),
first of all pupils were guided to arrange them vertically as
101two
+11
Pupils were also shown the place value of both numerals. For example, in 101
(two), 1 on the extreme right represents units,0 represents longs and the1on the
extreme left represents flats. For 11(two) 1 on the extreme right represents
21
units and the 1 on the extreme left also represents longs. Now pupils were
made to represent 101(two) and 11(two) as 1 flat, no long and 1 unit and 11
(two) as 1 unit and 1 long respectively. Pupils then added units to units, longs
to longs and flats to flats. After adding, they got 2 units, 1 long and 1 flat.
Using the idea of exchange, pupils were made to exchange 2 units for 1 long
and after that added it to the longs column. They again got 2 longs and also
exchanged it for 1 flat. Adding the new flat to the old flat, they got 2 flats and
also exchanged it for 1 long-flat. Pupils realized that they have now got 1
long-flat, no (zero) flats, no (zero) longs and no (zero) units. This is
represented as 1000 (two).
Pupils were guided how to add in base two using different questions until
pupils could add numerals in base two by themselves. The above example is
represented by diagrammatically as:
22
101 two + 11 two =
+0
no long
+{
1 Unit
}
+
1 Long
1 Unit
1 Flat
+
1 unit
Exchange units for longs
no Unit
1 Flat
1 long
1 unit
(2 units)
Exchange longs for a flat
+ 0
+0
Unit
1flat
1 long
1 long
Exchange flats for long- flat
+ 0
2 flats
No long
+0
No unit
1 long-flat
+0
1 long – flats
no flat
Week Four
23
+0
no long
+0
no unit
Week Three
Lesson 1
Topic: Subtraction in base two numerals:
Objectives:
The objective of the lesson was to help pupils to subtract numerals in base
two.
Teaching learning materials: Multi-base blocks
Activities
Pupils were guided to perform subtraction in base two using the multi-base
blocks. For example, to subtract 11 (two) from 101 (two), pupils were made to
arrange them vertically with the first addend on top and the second addend
arranged from right to left under the dividend. They were then asked to use the
multi-base blocks to represent 101(two) and 11(two) respectfully. Pupils got
for 101(two), 1 flat, no (zero) long and 1 unit and then 1 long and 1 unit for
11(two). The pupils were then guided to subtract the second addend from the
first addend from right to left. They took away 1 unit from 1 unit and got no
(zero) unit. They realized that they could not subtract 1 long from no longs,
therefore they were made to exchange 1 flat from 101(two) for 2 longs. Pupils
then took away 1 long from the 2 longs to get 1 long. Finally they would get
no (zero) and 1 long, which is represented in base two as 10(two). This is
represented diagrammatically as:
24
101two =
+0
1 flat
11two =
+
no long
1 unit
+
1 long
1 unit
101two – 11two =
+0
+0
-
++
+
Subtract 1 unit from 1 unit
1 unit
= 0 unit
1 unit
Exchange 1 flat for 2 longs and subtract 1 long from it
O
+
No flat
2 longs
=
1 long
1 long
Therefore 101(two) – 11(two) = one (1) long and one (0) unit.
The algorithm is written as;
101(two)
-11(two)
10two
Pupils were guided to solve more questions on subtraction until they mastered
the concept.
Intervention
Post-Test
After the intervention stage, a post- test was conducted to ascertain whether
the use of the multi- base Blocks has improved pupils understanding about
25
addition and subtraction in base two. This test was the same test given to
pupils at the pre-intervention stage.
26
CHAPTER FOUR
RESULTS FINDINGS AND DISCUSSIONS
This chapter deals with the data collection by the researcher from the pretest
and post-test scores.
These scores are presented in tables and bar charts as follows:
Table 1
Pre-test score
Marks
No of pupils
90-100
0
80-89
0
70-79
0
60-69
0
50-59
0
40-49
0
30-39
0
20-29
7
10-19
6
0-9
2
TOTAL
% score
0
0
0
0
0
0
0
46.7
40.0
13.3
15
100
Fig 1: A bar chart showing pre-test scores
100
90
80
Marks
70
60
50
40
30
20
10
0
1
2
3
4
5
6
Number of Pupils
27
7
8
9
10
From the given table and the bar Chart one could see that only two pupils
representing 13.3% scored in the range 0-9
A total of six (6) pupils also representing 40% scored in the range, 10-19
whilst, seven (7) pupils making up the remaining 46.7% scored in the range
20-29. A total of 40% scored within the range, 10-19 whilst seven (7) pupils
making up the remaining 46.7% scored in the range, 20-29. This indicates
clearly that pupils could not grasp the concepts of converting from base ten to
base two and also doing addition and subtraction in base two. It is also clear
that no pupils scored in the range, 30-39 and above.
The post-test results
Table 2
Post–test score
Marks
90-100
80-89
70-79
60-69
50-59
40-49
30-39
20-29
10-19
0-9
TOTAL
No of pupils
10
0
3
0
2
0
0
0
0
0
15
% score
66.7
0
20.0
0
13.3
0
0
0
0
0
100
4. Fig 2: A bar chart showing post –test scores
28
100
90
80
Marks
70
60
50
40
30
20
10
0
1
2
3
4
5
6
7
8
9
Number of Pupils
The table and bar chart presented shows that ten (pupils) in the range, 90-100.
These ten pupils scored all ten given test items correctly, indicating that they
have now mastered the concepts of converting form base ten to base two and
doing addition and subtraction also in base two. However only three (3) pupils
scored in the range, 70-79
and two pupils scored in the range, 50-59 representing 20% and 13.3% of the
student’s population respectively. One could also see clearly that no pupils
scored in the range, 40-49 and below. This indicates that there has been a
drastic improvement in pupil’s performance and that the use of the multi-base
blocks has enabled pupils to understand the concepts of converting from base
ten to base two and also performing addition and subtraction in base two. This
demonstration of understanding which was manifested in their outstanding
performance attest to the assertion made by Arthur and Dowker (2003), that
using base ten blocks to teach Mathematics will enhance pupils understanding
29
10
comparison of the pre-test and post-test score
100
90
80
Marks
70
60
50
40
30
20
10
0
12
1
2
3
4
5
6
7
8
9
Number of Pupils
Key: pre-test = blue
Post-test: Red
Comparing the pre-test results to the post-test, one could easily see that pupils
performance has improved significantly. This is because whilst majority of the
pupils scored in the range, 20-29 and below in the pre-test. No pupil scored
even in the range, 40-49 and below. This indicates that many pupils have
understood the concepts now and therefore performed above average. This
also suggests that pupils were able to use the concrete materials given to them
and that the multi-base blocks helped them to understand the concepts better.
One cannot doubt this very important observation made by Cruey (2008), that
base ten blocks can make abstract ideas like place value and regrouping visible
and tangible for primary school pupils. In this case, since children manipulated
30
10
the multi-base blocks, concepts taught became tangible and this aided their
understanding and thus their high performance in the post test.
A cursory look at the graph also reveal a total decline in the pre-test results
from the range 20-29 to Zero whilst the post-test results rose sharply from the
range(50-59) to (90-100)
Discussion of Findings
(Pretest)
The pretest results shows that with respect to converting from base ten to base
two, pupils performed averagely, though pupils could not grasp the concept
well, they managed to cope with the algorithm used by the teacher, however
pupils performed abysmally with regards to addition and subtraction in base
two. In the case of subtraction, only one pupil scored one out of four questions
posed whilst only two pupils scored one out of the four
given addition
questions, this clearly indicates that the pupils could not understand these two
concepts at all. This poor performance by pupils could have emanated from a
lot of factors.
In the first place it was found out that the Mathematics teacher did not use any
concrete material when teaching the pupils .He rather resorted to using only
the algorithm which in practice should have been introduced after pupils have
grasped the concept, since they are abstract. That is why Kamii (1985),
suggested that in teaching concepts that appear to be abstract, teachers should
use activity based methods such as the Diene’s Base Approach.
It could also be seen that most pupils computed addition and subtraction in
base ten rather than base two, for example, pupils added I and 1 and wrote 2
31
instead of 10(one, Zero). This is a clear indication that pupils could not grasp
the concept of addition and subtraction in base two.
The problem might be that the concept of base two appeared abstract and
therefore pupils were confused and were struggling to understand it. In order
not to encounter such problems, Cruey (2008), advised that base ten blocks
like many other manipulative help children to see and touch the ideas they are
being asked to cope with in Mathematics. Again, it was observed that pupils
could not arrange the given questions which were written horizontally in a
vertical form. For example, instead of arranging 1010two+110two as:
1010two
+110two
1010two
A majority of the pupils arranged it wrongly as
1010two
+110two
+ 110two
This indicates that pupils also lacked the concept of place value. In view of
this, Aubrey (1994) suggested strongly that teachers should first teach the
students the concept of place value and how to regroup. With these problems
that pupils have there is no wonder they performed poorly in a test involving
addition and subtraction in base two.
Discursion of Findings
Post –Test
The post –test scores shows that ten pupils forming the majority scored in the
range, 90-100. This is an indication of a very high performance on the part of
pupils. Even for the average students, three (3) scored in the range, 70-79 and
the remaining two scored in the range, 50-59. This high achievement can be
attributed to the teaching material used by the researcher.
32
Using the base ten blocks demystified the whole concept and pupils no more
found the concept abstract, in this case pupils learnt the concepts as they
manipulated the materials and with continual practice the concepts became
easy for them. Therefore one would definitely agree with Aubrey (1994) when
he stressed the importance of play, flexibility and choice, with opportunities
provided for practical activities
This reiterates the fact that in teaching such abstract topics, concrete materials
should be used to make the activities practical, so that pupils could understand
it easily. This is also embodied in the statement given by Dienes (1960),
“That children’s active involvement in the process of Learning Mathematics,
should routinely involve the use of manipulative materials”
33
CHAPTER FIVE
SUMMARY
The main focus of this study was to find out how the multi-base blocks could
be used to remedy the inability of pupils to add and subtract in base two in a
J.H.S. two class at Ummul Qura Islamic Junior High School.
This action research comprised a population sample of 16 pupils who were
purposively sampled for the study.
A pre-test was conducted, followed by an intervention period and then a
post-test which were used to obtain data.
Findings
Pre-test
Pupils performed poorly in the pre-test and this could be attributed to a
number of factors such as:

Pupil’s inability to arrange the given addends in a vertical form.

Pupils adding in base ten instead of base two.

Previous teacher’s failure to introduce the concept of place value
before introducing the concept of number bases.

Non usage of concrete materials when teaching topics which appeared
abstract by teacher.
Post-test
After the intervention, there was a remarkable improvement in pupil’s
performance. The outcome confirms that research was very successful in
34
achieving the project goals and objectives. Pupil’s motivation to work and the
high performance demonstrated could also be attributed to the following:

The use of concrete material, such as the multi-base blocks which
demystified the concept of base two and enabled pupils to work with
ease.

Teaching the pupils the concept of place value before introducing the
concept of number bases. This enabled pupils to arrange the given
questions correctly in the vertical order bearing in mind the place value
of each digit of each numeral.

Pupil’s ability to recognize that in base two only two digits are
essential that is 0 and 1. Pupils then realized that they would obtain 10
(one, zero) for 1+1 and not 2

Pupils’ motivation to learn since learning was like a play activity.
Conclusion
From the discussions it can be concluded that the use of the multi-base blocks
was very effective in addressing pupil’s challenges in addition and subtraction
in base two.
Recommendations
Base on the outcome of the study, the following recommendations were made:

Basic school teachers should use a lot concrete material in teaching
mathematics, since this will foster understanding and make the subject
real.

The multi-base block can be used effectively in the teaching of number
bases such as base two, five and so on.
35

The Ghana education service must encourage headteachers of basic
schools to organize in- service training regularly in their schools so
that new methods of teaching as well as preparation and use of
teaching learning materials are introduced and taught by experts in the
various fields of education especially in mathematics.

The findings of this research must be made available to teachers and
other stakeholders in Education so that they could assist to improve the
study of mathematics in basic schools.

Teachers who have gone through the college of Education and are well
experienced should be made to handle mathematics both at the primary
and junior high school. This is based on the pretest that if pupils
experience understanding continuously, they would love mathematics
but if they experience constant failure they would hate doing
mathematics forever.

The Ministry of Education should make effective supervision of
schools contingent or their line of action in order to ensure that the
right methods of teaching and appropriate teaching learning materials
are used in the schools.

The ministry of Education and Ghana Education service should liaise
with Government, Non- Governmental Organizations and all other
stakeholders of education to help to retrain teachers through
programmes such as the distant learning so that all teachers are
upgraded to a higher level where they can exhibit a high standard of
professionalism and expertise in the field of Education.
36
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39
Appendix A
Convert the following numerals from base ten to base two
1. 25
2. 39
Add these numerals in base two
3. 1011 two and 111two
4.11two and 101 two
4. 1011two and 1001two
6. 101 and 11two
Perform the following in base two numerals
7. 1010two and 110two
8. 1101two and 110two
9. 1001two and 101two
10. 110 and 11two
40