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. 10 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 REFERENCE Asiedu .P (1997) Core Mathematics for S.S.S in West Africa-Aki-Ola Series. Baroody, J. A., Dowker, A. (2003). The Development of Arithmetic Concepts and Skills: Constructing Adaptive Expertise. Lawrence Erlbaum Associates, Inc., New Jersey, London. Bassarear,T. (2007). 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Building Up Mathematics Hutchinson Educational 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