CHAPTER ONE INTRODUCTION This chapter encompasses the background of the study, statement of the problem, purpose of the study, research questions, and significance of the study, limitation, delimitation and organization of the study. Background of the study Mathematics is one of the important subjects within the foundation subjects that constitute the core curriculum for basic (i.e. primary and secondary) education in countries throughout the world and Ghana is no exception. Mathematics occupies a privileged position in the school curriculum because the ability to cope with it improves ones social advancement. “It has attained this position since it was made to replace classical languages like Latin and Greek which prior to the early half of this century were used as screening devices for entry into higher education and certain professions” according to (http://wikieducator.org/Review of mathematics in Ghana ) .This emphasize the fact that mathematics is one of the important subjects which is studied in our basic schools. Again, mathematics is of a utilitarian value because it is applied in various aspects such as farming, architecture, music etc (Agyei et al (2013). In view of this developing pupils competency in the subject by teachers is very much applauded. 1 Addition and subtraction are basic operations on which a lot of mathematical concepts are built or formulated. Dexterity in these two operations enhances pupils’ performance in other mathematical concepts. The concept of place value will help pupils to perform addition and subtraction of two digits numbers upward. In an encounter with the primary six pupils of Offinso State ‘A’ on the 6th of November 2015, the researcher observed that the pupils find it difficult to add and subtract six and seven digit numbers. Among factors contributing to addition and subtraction difficulties among pupils are misconceptions on the part of both pupils and teachers about Mathematics, teachers failure to use practical oriented approaches in teaching and lack of interest and motivation. Also, teacher’s failure to use appropriate teaching and learning materials in teaching is a domineering factor. The school of practice of the researcher is located at Offinso Newtown in the Offinso south district. The school is situated near a zongo community called Saaboa. Offinso has farming to be predominant profession in the area. Parents concern on their wards education is quite encouraging with respect to the enrolment of students in Offinso State A primary but there are some hitches on parental concern. The unsupportive nature of some parents in terms of failure to buy Mathematics textbook for their wards is also a contributing factor to pupils’ difficulty in adding and subtracting six and seven digit numbers. 2 It is against this background that the researcher has decided to address pupils’ difficulty in addition and subtraction of six and seven digit numbers using the abacus and place value chart. Statement of the Problem An encounter with the basic six pupils of Offinso State A primary uncovered their difficulty in adding and subtracting six and seven digit numbers. And since addition and subtraction serve as a pre – requisite to other mathematical concepts, pupils may lose interest in the subject as a result of poor performance in the topic and further mathematical concepts. Further inquiry from pupils and teachers by means of interviewing brought to light the fact that teaching and learning materials were not incorporated in Mathematical lessons. Also the researcher found out that pupils’ understanding in the place value concept is very low. A critical scrutiny into the Mathematics exercise books of some of the pupils also brought to light that they encountered the same difficulty in addition and subtraction of five digits numbers. This concretises the alarming nature of pupils’ low understanding in place value concept. Basic six pupils are expected to have an average age of eleven (11) or twelve (12), which means they are at the concrete operational stage according to Paiget’s theory of mental development. Some characteristics of pupils at this stage is that they are tied to concrete materials and learn very well when made 3 to interact with concrete materials because they find it difficult to do abstract reasoning. Information gathered and the psychologists’ views compelled the researcher to assist pupils to overcome their difficulty in addition and subtraction of six and seven digit numbers with the aid of abacus and place value chart. Purpose of the Study The purpose of the study is to identify the causes of the difficulty in adding and subtracting six and seven digit numbers among the basic six pupils of Offinso State A primary and to design appropriate teaching and learning materials (abacus and place value chart) to help the basic six pupils of Offinso State A to overcome their difficulty in addition and subtraction of six and seven digit numbers. Research Questions The following questions were formulated for the study. 1. What are the causes of basic six pupils of Offinso State A primary difficulty in adding and subtracting six and seven digit numbers? 2. How would the understanding of place value concept help pupils of Offinso State A primary six to overcome their difficulty in addition and subtraction of six and seven digit numbers? 4 3. What measures have been put in place to help the basic six pupils of Offinso State A to overcome their difficulty in adding and subtracting six and seven digit numbers? 4. How would the use of abacus and place value chart help Offinso State A primary six to overcome their difficulty in addition and subtraction of six and seven digit numbers? Significance of the Study This study will help the basic six pupils of Offinso State A to overcome to overcome their difficulty in adding and subtracting six (6) and seven (7) digit numbers. Secondly, it will help teachers institute measures to safeguard future occurrence of such a problem by using teaching and learning materials as well as child centred approaches in teaching concepts in Mathematics and other subjects. Furthermore, this study will help create awareness about the need to give pupils difficulty in mathematics a systematic study in order to provide scientific solutions to the problem. Moreover, this study will assist the government and other stakeholders in education to find a solution to the problem at stake in our basic schools. Finally, it will also serve as a reference document to those who in subsequent times will choose to study the same or similar problem. 5 Limitation The study could have been done in all the schools in Offinso south district but due to financial constraints and logistics to carry out extensively, the researcher was compelled to focus the study on only the basic six pupils of Offinso State A primary, Again, the time duration at the researcher’s disposal is very limited, because the researcher has to combine studying of the two distance courses (i.e. Guidance and Counselling and Trends in Education) as well as the teaching practice. Finally, truancy on the part of some students may affect the credibility of the research results and findings that may come out because some students in Offinso State A primary six are habitual truants and researcher seeks to use half the total population of the class. Delimitation Even though the study should have been extended to all the schools in Offinso municipal to help all those at the basic education level, however due to limited time as a teacher trainee with limited resources , the study was confined to only 25 pupils in Offinso State A primary six in order to get ample time for the research. here are many areas of learners’ difficulty in Mathematics but this research work concentrated on pupils’ difficulty in solving problems involving addition and subtraction of six and seven digit numbers. 6 Finally, the researcher confined himself to using abacus and place value chart to help pupils to overcome their difficulty in adding and subtracting six and seven digit numbers though there are a lot of teaching and learning materials which could have been used to help pupils to overcome their difficulty. Organisation of the Study This work is made up of five chapters. The chapter one basically puts the study into perspective and it comprises of the background of the study, statement of the problem, purpose of the study, research questions, and significance of the study, limitation, delimitation and organization of the study. The chapter two deals with the review of related literature on the study, it talks about the views of others which are relevant and related to the study in books, journals and other sources. The methodology aspect of this research is captured in the third chapter. It encompasses the research design, population, sample selection techniques well as the research instruments used. The fourth chapter presents the results of the study as well as discussion of the study and analysis and data collected are expatiated in this chapter. Finally, the fifth chapter gives an overview or summary of the study. Findings and conclusions drawn from the study are also highlighted on and recommendations and suggestions for further studies are presented. 7 CHAPTER TWO LITERATURE REVIEW Introduction This chapter reviews relevant literature on the study. It examines the views of authors that have relevance to the subject of study from both theoretical and empirical perspectives. Relevant literatures were reviewed under the following sub-headings: a. What is Mathematics? b. Importance of Mathematics. c. How children learn Mathematics and their educational implications. d. The use of teaching and learning Materials. e. The place value concept. What is Mathematics? Mathematics is the abstract science of number, quantity and space either as an abstract concept (pure mathematics) or as applied to other disciplines such as physics and engineering (www.google.com/definition_of_mathematics ) Oxford English Dictionary also defines Mathematics as the abstract science which investigates deductively conclusions implicit in the elementary conceptions of spatial and numerical relations and which includes as its main divisions geometry, arithmetic and algebra. 8 Encyclopaedia Britannica also defines Mathematics as the science of structure, order and relation that has evolved from elemental practices of counting, measuring and describing shapes of objects. Mereku (1999) opined in his speech delivered at the 6th biennial delegates conference of the Mathematical Association of Ghana that ‘Mathematics is a language and provides a means of communication because it makes use of symbolic relations which is similar across continents, abstract ideas and concepts.’ Charles Darwin poetically describes a mathematician as a blind man in a darkroom looking for a black cat which (http://en.wikipedia.org/wiki/definitions_of_mathematics isn’t ). there Darwin’s description throws more light on the abstract nature of Mathematics. Eugene Wenger metaphorically defines Mathematics as the skilful science of operations which concepts and rules are invented just for this purpose (skilful operation). From the above definitions it can be deduced that Mathematics is purely a science and it is well structured with sub-branches (i.e. geometry, arithmetic and algebra). Geometry is the visual study of shapes, size, and patterns. Arithmetic is the branch of Mathematics which deals with the properties of the counting numbers and fractions and the basic operations applied to these numbers (http://en.wikipedia.org/branches_of_mathematics ). 9 Algebra is the part of mathematics in which letters and other general symbols are used to represent numbers and quantities in formulae and equations. Importance of Mathematics ‘When will I use Mathematics?’ is a question often posed by students wondering how topics like factorization and algebra will play a role in their daily lives. However often without realising it, we use it in our day to day activities like making purchases, tracking cell phone’s minutes and many more. Taylor (2013) in his book ‘How children learn Mathematics’ also asserted that ‘Mathematics equips pupils with tools which are unique and powerful to understand and change the world. These tools include logical reasoning, problem solving skills and abstract thinking. Asafo .A (2002) also stated that mathematics is a service subject applied in physical sciences, social sciences and other related areas and it also serve as the basis for modern technology and scientific development. Again, mathematics is a compulsory subject for pursuing higher education. Failure to attain a credit or higher in this subject creates the problem of advancing into any tertiary institution for students. According to Petty (2001) generic skills are developed through the study of mathematics and these skills can be applied to other areas. Generic skills are 10 developed when young children are introduced to pre-number work. Some of these generic skills are: • Observation • Inferring • Sequencing • Comparing • Sorting etc.(www.geofpetty.com ) Furthermore, physician who has to study biological cells and bacilli need to have knowledge of mathematics if he means to reduce the margin of error which alone can make his diagnosis dependable. To the mechanic and the engineer, it is a constant guide and help and without exact knowledge of mathematics, they cannot proceed one step in coming to grips with any complicated problem. Also, the habit of accuracy and exactitude are developed and it prevents man from being careless and slipshod. Mental alertness is increased as well as sharpening the reasoning powers of man, These outlined importance of mathematics unveil some hidden importance of mathematics and it also emphasise on some practical experiences that need mathematical knowledge before one can go through them. 11 How children learn Mathematics Collins Co Build Advanced Dictionary defines learning as the process of gaining knowledge through studying Learning is also defined as the process of gaining knowledge through reinforced practice. A number of theorists have proposed ideas about how children learn generally and these ideas can also be related to the learning of mathematics. Piaget believes that children construct their own knowledge and understanding through their interactions with their environment. This is called a constructivists theory. Vygotsky (in Atherton 2011) also known as a social constructivist emphasised the need for a child to have guidance from ‘a more knowledgeable other’ and also to be given opportunities to interact socially with peers as a means of learning (Taylor 2013). Mathematical learning is associated with mathematical understanding. Barmby et al (2009) see this as a continuum where children ad to and refine previous understanding. This is built on the work of Bruner (1966) who identified the idea of spiral curriculum where children meet an idea at one level and then later meet the idea again but are able to study it at a deeper level and achieve better understanding of it. Bruner again asserted that children go through three phases when learning. The phases are enactive, iconic and symbolic. The three phases concretise the pedagogic axiom ‘concrete to abstract’. 12 Enactive phase is when a child engages in something concrete in order to explore and manipulate ideas; this could be related to kinaesthetic learning. Therefore at this stage pupils are made to interact with concrete materials when learning. Iconic phase is where the child creates mental images to represent concepts. This can be supported in mathematics by using models and images so that the child eventually visualise some of them internally to assist their thinking The last phase which is known as the symbolic phase is where the child uses abstract ideas as ways of representing mathematics. Liebeck (1981) also in his book ‘How children learn Mathematics’ also postulated that children develop their abstract thought in mathematics by first exploring with physical objects such as toys etc. It is not long before he recognises words to represent them. Later the child will recognise pictures of them and written symbols are then associated with them. Liebeck (1981) again categorise the sequence by which children learn mathematics with an acronym ELPS which expatiated below. • E – Experience with physical objects • L- Spoken language that describes experience • P- Pictures that represent the experience • S- Written symbols that generalise the experience Skemp (1971) described two ways of understanding mathematical ideas that he called relational and instrumental understanding. 13 Instrumental is the shallower form of understanding. For instance we might develop an understanding on how to add, subtract, multiply and divide using set of procedures or algorithm by memorising steps required. However we may not understand how the procedure worked. One major flaw of instrumental understanding is that if our memory of procedure fails, we would be unable to continue. In contrast a relational understanding of these procedures would mean that we understand how and why the procedure worked. The work of Piaget, Bruner and Liebeck emphasise practical activity as a starting point for learning with young pupils. Gifford (2008) also reports neuroscientific support for this approach too. Again, research indicates that desposition is very important in the longterm learning of mathematics (Renga & Dalla 1993). Desposition concerns more than attitudes towards mathematics alone: persistence, risk taking, hypothesis making and self regulation are important to a motivated desposition (Coppley 2010). Gayne R and Skinner B.F propounded the behaviourist theory of learning. They hold that learning takes place through stimulus-response (S-R) mechanism and it is the process through which the child obtains certain desirable behaviours. They also hold the view that the child’s mind is termed to be a tabula rasa (i.e. the child’s mind is thought to be an empty spot) and it is the responsibility of the teacher to fill it with knowledge 14 From the above theoretical framework derived from psychologists with different school of thoughts, the following implications were deduced the classroom teacher in managing mathematical lessons. Firstly, relevant teaching and learning materials should be incorporated in lesson delivery by teachers because of young pupils’ tendency to explore and play with familiar materials. The enactive phase and concrete operational stage of Bruner and Piaget’s theories respectively concretise this fact. Secondly, relational understanding on the part of pupils should be a major concern to teachers. Pupils should be taught to understand how and why procedures worked and do away with drill-oriented teaching. Skemp’s relational and instrumental learning theory buttresses this point. Again, teachers should make the classroom environment simulative enough by giving all pupils equal chance to partake in mathematical lessons. The stimulus-response (S-R) as asserted by the Behaviourists supports this point. Moreover, teachers’ mastery in the subject matter content is very expedient to a successful lesson delivery. Vygostsky assertion that children are to be guided by ‘a more knowledgeable other’ provides the basis for this implication. Opportunities to interact with peers as means of learning should be taken into consideration by grouping pupils in some mathematical lessons. Finally, motivation as means of reinforcement should be used by teachers during mathematics lessons. Not neglecting the principle of individual differences on the part of pupils because it’s on this basis that you can group pupils in case of applying any type of grouping and also assessment. 15 The use of Teaching and Learning Materials Lowe (1993) stated in his book ‘successful instructional diagrams’ that a picture is worth a million words when used in the period of instruction. This shows how effective lessons will be when teaching and learning materials are used. Lowe (1993) further explained that for instructions to be judged successfully, the students must be able to produce the new or improved performance as required. So, if diagrams or teaching and learning materials are used in lessons, they help students to: • Recall the knowledge and skills they have been taught. • Understand the material that is being taught. Handbook on lesson notes preparation and teaching and learning materials in primary schools (2003) by Ghana education service states that teachers who rely on oral presentation of lessons find the pupils frequently unable to understand concepts effectively. There are also visual, auditory and kinaesthetic learners. The visual learner say ‘i see’ to mean ‘i understand’ and respond best to task involving demonstrations or looking at illustrations and diagrams. The auditory leaner also prefers direct instruction hence performs better when they are used frequently The kinaesthetic learner on the other hand prefers direct involvement through games, role playing and interacting with materials in the classroom. Deductions from how children learn mathematics young pupils are mostly 16 kinaesthetic learners, therefore it is very prudent for teachers to use instructional materials in lesson delivery. Young children are tied to concrete materials according to the Piagetian theory and it is also backed by Bruner’s theory on how children learn mathematics. These theorists’ postulations back the idea of young children learning kinaesthetically and there comes in teachers’ incorporation of teaching and learning materials in lessons. Akinyemi also classified teaching and learning materials under six categories. 1. Printed materials- textbooks 2. Graphic materials- posters and diagrams 3. Display materials- overhead projector, video 4. Audio materials- radio, tape recorder 5. Microfilm etc 6. Miscellaneous materials- regalia, exhibit Lowe (1993) again stated that not all instructional diagrams found in teaching materials are equally successful in promoting desired learning outcomes. This is because when a diagram is poorly designed they actual hinder than help learning. Lowe (1993) then made an assertion that even well designed instructional materials may be ineffective if the student is given insufficient support on how to use them. So perhaps the adage ‘a picture is worth a million words when used in the period of instruction’ can be rewritten as ‘a diagram is 17 worth a million words provided it has been well designed and sufficiently supported.’ Pupils’ involvements in the collection of instructional materials are also important because it helps them to be familiar with the environment according to (Methods of teaching science, tutors notes). Some examples base ten materials which are used in the teaching of addition and subtraction of two digit numbers upwards are; I. Abacus II. Dienes material (multibase block) III. Bundles of sticks and loose ones IV. Place value chart etc The place value concept According to Swenson (1973), place value is the value of a digit in a numeral derived from its place or position in the numeral. She illustrated that in our modern notion, each of the (2’s) two in 2222 is a different value depending on its position among the other digits She again explained the place of 2 in the extreme left-hand position as thousand times the 2 in the extreme right-hand position. She again pointed out that the idea of place value was used in written records such as those of ancient Babylonians and Mayons and in mechanical computing like the Abacus. 18 Crowder and Wheeler (1968) also described the concept of place value as the value of each symbol which is determined by its position in the numeral. They continued to suggest that the concept of place value and positional notation means that the symbols can be repeated within a number and the value of each number is determined by its position its position in the symbolised number. Crowder and Wheeler (1968) further stated that because of place value concept, the Hindu and Arabic decimal system needed on only ten different symbols to represent any number. They illustrated that basic number has 2 values by saying that the numeral ‘333’ means 3 ones 3 tens 3 hundreds or in expanded form which is (300) + (30) + (3). According to Kramer (1971) children should be taught place value with the use of play activities for better understanding. He again emphasised that base ten materials like Abacus, place value chart, Dienes materials etc should be used to help pupils to understand this concepts From the preceding, most writers agree on a sequential approach to the teaching of place value concept while many activities are involved. Regrouping, grouping and conversion should be inclusive to facilitate the skill of carrying over one value to the next step as in doing addition and subtraction. 19 Summary From the above reviewed literature, it is obvious that Mathematics is very essential considering some of the importance captured. The theoretical framework made with postulations and assertions by some psychologists encourages the use of relevant concrete materials when teaching young people mathematics. Finally, pupils’ mastery in place value concept is very expedient in their comprehension of addition and subtraction of two digit numbers upwards. 20 CHAPTER THREE METHODOLOGY Introduction This chapter discusses the methods used to gather data on the topic under study. It talks about the research design, population and sample selection, research instruments, pre-intervention data collection/analysis, intervention, post-intervention data collection and data analysis plan. Research Design The design used in the study was action research. Action research is the study carried out in course of an activity or occupation, typically in the field of education to improve the methods and the approaches of those involved. Wikipedia encyclopaedia also defines action research as the research initiated to solve an immediate (en.wikipedia.org/wiki/Action Research). Action research has some strength and weaknesses. Kerry Dyke in the chapter three of his manual “Action Research” wrote that action research helps educators to use data rather than hitches to guide the improvement of efforts. This makes the process of action research more scientific in nature proposing ideas and theories that can be backup by data. Action research gives teachers something more concrete to work with instead of just relying on the principles the teachers have used in the past. It 21 also helps to address the quality of student’s education and the progressive growth of teachers. In spite of the numerous advantages, action research is having several lapses. One major disadvantage of action research is that, it is much harder to write up because you probably can’t use a standard format to report your findings effectively. Moreover the cyclic nature of action research to achieve its twin outcomes of action and research is time consuming and complex to conduct. Population Population is defined as a group of people that conforms to a specific criterion and to which a researcher intends to generalise the results of the study. A population which constitutes the target population or the group under study is the basic six pupils of Offinso State A primary and the teachers in the school. There are 850 students in the school consisting of 420 boys and 430 girls. There are also 22 permanent teachers in the school of which some were randomly selected and interviewed on these bases: • The poor performance of pupils in mathematics. • Method suitable for teaching mathematics. • The failure of teachers to use TLM’s. 22 A sample size of 25 pupils was selected from 50 students in Offinso State A primary six. Among the 25 students were 13 girls and 12 boys and 10 teachers were also randomly selected from the school to be interviewed. Sampling Procedure In as much as the researcher wishes to work with the entire target population and teachers, he could not do so because it could have been extremely difficult to handle all pupils and teachers. Simple random sampling was selected and used in the selection of pupils from the target population i.e. basic six pupils of Offinso State A primary. The respondents were selected to ensure fair results. Pieces of papers with numbers written on them from 1-50 were put in a container. All the 50 pupils who form the target population allowed to pick one slip of paper at a time. All the pupils who picked slips of papers with even numbers were selected, which constituted a total of 13 females and 12 males. Research Instruments Research instruments are testing devices for measuring a given phenomenon, such as paper and pencil test, a questionnaire, an interview or set of guidelines for observation. After the researcher had made a consideration of factors that determine the appropriateness of the instruments for research, observation, interview and tests were found appropriate to be used to gather relevant data for the study. Observation 23 It involves watching and listening to the subjects and recording what he sees rather than asking questions. The researcher adopted the participant type of observation so that he could get information which may not be accessible anyway. But it may lose its relevance if the researcher becomes emotionally involved and for that matter the researcher may lose its objectivity, this is the major weakness attached to observation. In course of teaching, the researcher observed that pupils were having some difficulties in the addition and subtraction of six and seven digit numbers on the part of the basic six pupils of Offinso State A primary. The pupils were having those difficulties because thorough observation brought to light the fact that in previous years the teachers failed to incorporate TLM’s in their lessons. Pupils’ dexterity in place value concept was also very low and it is also a domineering factor of the pupils’ difficulty. Observation Interview involves verbal questioning by the researcher to the respondents either in face to face situation or by phone. The researcher adopted the structured form of interview to collect relevant data to the study. Interview gives high response rate and also gives the opportunity to observe the non verbal behaviour of the respondents but it is very costly and time consuming. An interview guide was prepared by the researcher for both teachers and students and their responses were recorded (refer to appendix A and B). Tests 24 The researcher used tests to diagnose the extent of the problem i.e. pretest and also to determine the effectiveness of his intervention procedures i.e post-test. Each test consists of 5 questions which a correct answer is scored 2 marks and 0 for a wrong answer. (refer to appendix C and D). Data Collection Procedure The data collected followed a weekly planned research activity. The researcher carried out the research in three periods thus sixty minutes each for four weeks. The first week was used for pre-intervention and the second and third weeks were also used for the intervention and the fourth week was used for post-intervention. Pre-intervention Week one (1) During the first week observation was made by the researcher during one mathematics lesson. Pupils difficulty in addition and subtraction of six (6) and seven digit (7) numbers was revealed because it was the topic slated to be taught that very week. Pupils couldn’t solve simple problems involving addition and subtraction of six and seven digit numbers because their understanding of the place value concept was very low. These were some of the ways they approached the questions given to them. 645871 879604 + 459610 - 754821 109151481 125223 25 A pre-test was administered during the second mathematics lesson of the week. The researcher conducted the pre-test to find out the actual problem faced by the pupils. The pre-test consisted of 5 questions from the pupils’ text book. The test elapsed for 45 minutes. Refer to appendix C During the third mathematics lesson for the week, the researcher conducted an interview using an interview guide. All the respondents selected from the target population were interviewed. The researcher conducted the interview to find out why the pupils were having difficulties in the addition and subtraction of six and seven digit numbers. The researcher again, used free periods during the first week to interview some teachers using an interview guide prepared for the teachers. Ten teachers were selected and interviewed .the researcher wanted to know everything about the pupils’ difficulty that is why he tried gathering data on the pupils problem on the first week. Intervention The researcher conducted the intervention using two weeks i.e. the second and third week. The intervention was conducted in reference to pupils’ inability to add and subtract six and seven digit numbers as revealed by their response during the pre-test and the interview held on the first week. Week two (2) The researcher used the abacus during the whole week. Three days lesson was conducted during the week. 26 In the first lesson, the researcher introduced the abacus to the pupils. The researcher explained to the pupils that an abacus is a frame containing rods with small balls that slide alongside the rods. It is used as a tool for counting. The pupils were asked to later describe the abacus after they have been grouped to critically examine the abacus. Most of the pupils were able to describe the abacus as a wooden frame with small balls used for counting. Below is the picture of an abacus. Figure 1. The second day’s lesson was used by the researcher to demonstrate an example by using three seven spiked abacuses to teach addition and subtraction of six and seven digit numbers. During the lesson, the researcher stressed on the collection of ones, tens, hundreds, thousand and millions in ascending order. The process of exchanging ones for tens, tens for hundreds, hundreds for thousands was demonstrated to the pupils. The researcher then demonstrated to pupils how the abacus is used for adding six and seven digit numbers . Figure 2 below shows how the researcher demonstrated an addition operation with the abacus. Example 5342656 +4483129 27 Figure 2 The last day’s lesson was also used by the researcher to teach subtraction of six and seven digit numbers using an abacus. The pupils were taught that the difference between two numbers is found by subtracting a smaller number (subtrahend) from a larger number (minuends). Example: 7 8 5 6, 9 7 4 - 2 4 3 4, 5 2 3 28 Figure 3 All the three days lessons taught in the week was based on a detailed lesson plan prepared by the researcher. Refer to appendix E Week three (3) During the third week of intervention the pupils were taken through addition and subtraction of six and seven digit numbers using the place value chart. The three days lessons used for the intervention lasted for sixty minutes and a detailed lesson plan was again prepared to guide the researcher in his intervention. In day one’s lesson, the researcher introduced the pupils to the place value chart where it was explained to pupils how it is used. The pupils were told that materials including cardboards, pencils, pens, felt pens, erasers and ruler are used in preparing the place value chart. 29 The researcher further explained that with the use of a ruler and felt pen the positions on the place value chart were marked and drawn from positions representing ones to millions. Example of the amount in the chart (i.e 5 milliinton,793 thousand,6 hundred and 12; five million, seven hundred and ninety three thousand, six hundred and twelve. Millions 1,000,000 Hundred Ten thousand thousand 100,000 5 7 Thousands Hundreds 10,000 1,000 9 3 100 6 Tens 10 1 1 During the second day’s lesson, the researcher used the place value chart to teach addition of six and seven digit numbers. The researcher involved the pupils in an addition game involving (0-20), before introducing them into the use of place value chart to add six and seven digit numbers. The game is played by two pupils at a time. One starts from zero and add 1,2 or 3 . The addition alternates between the players up to the one who gets 20 first and he or she is declared the winner. The game is aimed at helping pupils to develop problem-solving strategies. The pupils to solve the question below using the place value chart. Example: Add 4,967,547 and 3,387,686 30 Ones 2 Figure 5 Place value chart M 4 3 H-Th T-Th Th H T 9 7 5 4 7 7 6 8 6 6 3 8 O 8 13 15 15 12 13 13 8 3 5 5 2 3 3 During the last days lesson of the third week the researcher then guided the pupils to solve questions on subtraction of six and seven digit numbers using the place value chart. It was explained to pupils that subtraction of numbers can be illustrated by the removal of a number from a group of objects. Example : 733526 - 375863 The above question can be solved with the place value chart as shown in figure 6. 31 Figure 6. Place Value Chart H-th T-th Th H T O 7 3 3 5 2 6 3 7 5 8 6 3 3 5 7 6 6 3 _ Post Intervention The fourth week was used to administer a post-test to find out the effectiveness of the researchers intervention procedure. The post- test consisted of 5 questions which was scored 2 marks each. Forty minutes was used to conduct the test and twenty minutes was used for making corrections and marking of the test. Data Analysis Procedure The research findings were based on the analysis of the pre-test and post-test as well as answers the teachers and pupils provided to the interview guide .the findings were analysed using tables and descriptive statistics such as frequencies and percentages together with charts 32 CHAPTER FOUR RESULTS, FINDINGS AND DISCUSSIONS Introduction This chapter presents the results of the study and discussion of findings. The results were the outcome of pupils’ performance in the pre-test and posttest. It further deals with the analysis of findings made through interviews administered to some selected teachers and students. The researcher used tables and bar charts to analyse the pre-test and post-test findings and only tables for the findings from the interview. Summary 1 Table 1 shows the results of the pre-test scores Table 1 Marks Frequency Percentage ∑fx 0 6 24 0 2 9 36 18 4 8 32 32 6 2 8 16 8 0 0 0 10 0 0 0 Total ∑f=25 100 ∑fx=66 33 Mean=∑∑ = Mean= 2.64 The data is further explained with the bar chart below A bar chart showing the pre-test scores of pupils 10 frequency 8 6 frequenc 4 2 0 0 2 4 6 8 10 marks Figure 4.1 A bar graph showing the pre-test scores of pupils Data analysis of pre-test scores From table 1 and figure 4.1 i.e. the bar graph, it was observed that out of the twenty five students (25) selected for the study, six (6) of the representing 24% scored zero (0). Nine (9) students representing 36% scored two (2) marks, eight (8) pupils representing 32% scored four marks each and two pupils (2) representing 8% scored six (6) marks each. The pre-test results again produced a mean of 2.64. this shows that averagely their performance was very low and shows pupils disability in adding and subtracting six and seven digit numbers. 34 Summary 2 The table below shows pupils post-test scores Table 2 Marks Frequency Percentage ∑fx 0 0 0 0 2 0 0 0 4 0 0 0 6 8 32 48 8 10 40 80 10 7 28 70 Total ∑f=25 100 ∑fx=198 Mean=∑∑ = Mean= 7.92 The data is further explained by the bar graph below 35 A bar chart showing the post-test scores of pupils 12 frequency 10 8 6 frequenc 4 2 0 0 2 4 6 8 10 marks Figure 4.2. A bar graph showing the post-test results Data analysis of post-test results From table 2 and figure 4.2, it can be observed that out of the twenty-five (25) students, eight pupils (8) representing 32% scored six (6) marks, ten (10) students eight (8) marks representing 40 % and seven (7) students scored ten (10) marks representing 28 %. The post-test results also produced an average of 7.9 as compared to 2.64 of the pre-test scores. This implies that intervention in the form of abacus and place value chart very successful hence contributing to the massive improvement of pupils results. 36 Interview The data below are the results of the structured interview administered by the researcher. The researcher used an interview guide for both teachers and pupils (refer to Appendix A and B respectively). Though the responses were open ended but the researcher made efforts to categorise them. Table 3 Table 3 shows the responses teachers gave to the interview question one meant for teachers (refer to appendix A). Why do pupils perform poorly in mathematics? Responses Respondents Percentage Bad perception about mathematics 2 20 Lack of access to maths books 3 30 Pupils background 3 30 Poor teaching methods 2 20 Total 10 100 Table 3 The table above displays information from some selected teachers on their responses to the interview question, why do pupils perform poorly in mathematics? Out of the ten teachers selected to be interviewed, two (2) respondents representing 20% pointed out that the bad perception pupils have about mathematics contributes to their poor performance in mathematics. Three (3) respondents representing 30% also talked about pupil’s background being a domineering factor in their low performance in mathematics. Another 37 three (3) representing 30% also said lack of access to mathematics books contributes to their poor performance in mathematics. Lastly, two (2) teachers representing 20% also responded that poor teaching methods on the part of teachers also contribute to their abysmal performance in mathematics. Table 4 The table below shows teachers responses to the interview question two for teachers (refer to appendix A). What method is suitable for the teaching of mathematics? Responses Respondents Percentage Activity method 7 20 Demonstration 3 20 Total 10 100 Table 4 From table 4 seven respondents representing 70% pointed out that Activity method is the most suitable method for the teaching of mathematics. Three (3) others representing 30% also said demonstration is suitable for the teaching of mathematics Table 5 The table 5 below shows the responses teachers gave to the interview question three (refer to appendix A). 38 Why do teachers fail to use TLM’s? Responses Respondents Percentage Not necessary 1 10 Lack of support from the government 9 90 Total 10 100 Table 5 From table 5 only one respondent representing 10% said that it is not necessary to use TLM’s in lesson delivery, nine respondents representing 90% also said lack of support from the government has resulted to the failure of teachers to use TLM‘s. Table 6 This table shows the responses students gave to the first interview question for students (refer to appendix B). Do you have access to mathematics books both at home and in school? Responses Respondents Percentage Sometimes 5 20 Not at all 15 60 Very often 5 20 Total 25 100 Table 6 From table 6 five (5) students representing 20% said that they sometimes get access to mathematic books at home and in school. Fifteen (15) others representing 60% responded that they don’t get access to mathematics books. 39 Five students representing 20% also said they always get access to mathematics books. Table 7 This table shows the responses students gave to the interview question two meant for students (refer to appendix B). Are you given the opportunity to interact with the TLM’s teachers bring to the class? Responses Respondents Percentage Sometimes 1 4 Not at all 3 12 Very often 21 84 Total 25 100 Table 4.7 From table 7 it can be deduced that pupils are not always allowed to interact with the TLM’s teachers bring to class. Because one (1) student representing 4% said he always have access to the materials teachers bring to class for instruction, three other representing 12% also responded that they are sometimes made to interact with the materials teachers bring to class for instruction. Twenty-one (21) pupils representing 84% also said they don’t get the chance to interact with the instructional materials teachers bring to class. 40 Conclusion From the above tables, findings and discussions, it is noted that a whole lot of problems was associated to the inability of the basic six pupils of Offinso State A primary to add and subtract six and seven digit numbers. The researcher took into consideration all these problems and designed a suitable intervention in a form place value chart and abacus and used them. It is worth knowing that the study to a large extent successful and the post-test results of the basic six pupils of Offinso State A is also evident to the success of the intervention Hence, there have been remarkable improvements in the pupils’ performance. 41 CHAPTER FIVE SUMMARY, CONCLUSIONS AND RECOMMENDATIONS Introduction This chapter concludes the study by presenting a summary of the research findings, conclusions, recommendations and suggestions for further research Summary The research unveiled and examined the causes of the basic six pupils of Offinso State A primary difficulty in adding and subtracting six and seven digit numbers. After unveiling the causes, how to improve upon pupils’ performance was a key goal. Finally the study was to find out whether the abacus and place value chart used as intervention to teach the basic six pupils of Offinso State A primary school yielded any gains in pupils performance in the post- test. The study consist of twenty-five (25) pupils selected from the fifty (50) pupils in basic six of Offinso State A primary school. The instruments used were tests, observations and structured interview. Responses of the twenty-five (25) pupils and ten (10) ten teachers from the interview were analysed with tables and percentages The results of the pre-test showed that pupils indeed faced problem with addition and subtraction of six and seven digit numbers based on the scores they had. 42 However analysis of numerous data collected revealed the following as the causes of pupils’ difficulty. Firstly, it was found that the pupils were having bad perception about mathematics and this impeded their understanding of basic mathematical concepts Secondly, most of the pupils were not having access to mathematics books. This implied that they don’t have any textbook to dwell on for practice after lessons. Thirdly, it was revealed that pupils’ background was also expedient in their poor performance in mathematics. Most of the parents of the pupils were farmers since the school is situated at an area which has farming to be a predominant profession. Their interest on their wards education wasn’t encouraging; sometimes they even take their wards to farm during schooling days which made their wards to miss some mathematics lessons. Again, poor teaching methods on the part of some teachers were also noticed. Though most teachers accepted the fact that activity method was the most suitable method for the teaching of mathematics yet very few of them applied it. Lack of teaching aids for mathematics lesson also resulted to pupils’ abysmal performance in the pre-test and their performance before intervention as a whole. Most teachers attributed it to lack of support from the government. The very few teachers that made efforts to send teaching aids to class also denied the pupils from interacting with them. 43 After the causes were obtained a painstaking effort was made to curb the situation. Therefore the researcher designed an abacus and place value chart as intervention to curb the situation. The intervention procedures put in place brought light the following findings • Teaching and learning materials make lessons interesting and practical • Pupils understand concepts better when made to interact with TLM’s and this affirms Piaget’s assertion that children are tied to concrete materials • Activity oriented lessons ensures the maximum participation of all pupils • There was a considerable improvement in pupils’ performance in the post-test. The post-test produced an average an average of 7.92 as compared to 2.64 of the pre-test Conclusion The research based on the totality of action research concludes that the use of TLM’s i.e. abacus and place value chart has helped the basic six pupils of Offinso State A primary school to overcome their difficulty in adding and subtracting six and seven digit numbers To add to the above strategies, teachers should adopt different approaches and techniques during lesson delivery. Therefore parents, teachers, 44 stakeholders and beneficiaries of education should help find appropriate means to help pupils improve in mathematics since it is an indispensible subject in the curriculum. Recommendations Despite efforts made by the researcher to achieve the purpose of the study, there is still the demand for recommendation. Firstly, the researcher recommends that child centred methods should be adopted and used by all teachers in the teaching of mathematics and other subjects in the curriculum. This will ensure effective participation of pupils in the lesson. Secondly, the Ministry of Education and Ghana Education Service should provide enough mathematics textbooks so that pupils will also have them as reference materials after lessons in the classroom. This will improve their practice and further improve their performance. Again, the Ministry of Education and other stakeholders should provide some basic teaching and learning aids used in teaching mathematics so as to help make lesson delivery easier for teachers and also aid easy understanding on the part of pupils. Moreover, parents should assist their wards at home and also buy mathematics textbook for them so as to improve their performance. 45 Finally, the Ministry of Education, school authorities as well as NGO’s should institute special packages for pupils who excel in mathematics as a means of motivation. It is hoped that it will raise the morale and interest of pupils in mathematics. Suggestion for further research In every human activity, there is no perfection since we are bound to make mistakes. It is based on this that the researcher wishes to suggest the following modifications for further researchers who will undertake similar research in order to enhance efficiency. The researcher suggests that other researchers should try different intervention procedures such as using multi base block, number tracks, bundles of sticks and loose ones to teach addition and subtraction of six and seven digit numbers Also, it is suggested that the researcher should look at multiplication and division of numbers to make teaching and learning of mathematics worthwhile 46 REFERENCE Asafo Adjei (2002); Teaching Basic Mathematics for Training Colleges (Methodology), USA; Global Journals Inc. Barmby P, Lynn B, Tony H & Higgins S (1990); Primary Mathematics Teaching for Understanding: maidenhead; Open Head press. Brunner J (1966) ; Towards the theory of Instruction; Cambridge mass ; Belknap press Harvard University. Collins CoBuild Advanced dictionary (2009); Henle cengage Learning. Coppley J.V (2010); our young Children and Mathematics; Virginia: National Association of Education of Young children Crowder & Wheeler (1968), Teaching Elementary School Mathematics: George overhead limited’ Chen J & Weiland L (2007), Helping young children learn Mathematics, (digital paper), Available on children exchange website: www.children exchange.com Encyclopedia Britanica (2008), 15 th edition; Encyclopedia Britanica Inc (official website). En.wikipedia,org/Action Research. http://wikieducator.org/ review of mathematics in Ghana. http://en.wikipedia.org/wiki/definition of mathematics. http://en.wikipedia.org/wiki/branches of mathematics. Kramer (1971), Teaching elementary school Mathematics, Ohio; Claire parkyns publication. 47 Kerry Dyke (2012), Action research The Manual; a journal to be a guide and Understanding of Action research (available online), www.mun.ca/ chapter 3 Kilpatrick, J., Swafford, J., Findell, B., & National Academy of Sciences National Research Council, W. N. (2001). Adding it up: Helping children learn mathematics; Washington DC :national academy press Lowe R (1993), Successful instructional diagram, London: Kopan page limited Liebeck P, (1981), How children learn Mathematics a guide to Parents and teachers Mereku K.D (1999), School Matthematics in Ghana 1960 -2000, a paper Delivered at the 6th biennial delegates conference of mathematical association of Ghana (MAG) Petty R (2001), Teaching generic skills, a journal of using assessment to raise achievement in mathematics, available online at www.geofpetty.com Renga R and Delta C (1993), Effects of Mathematics: London, pinguin books Skemp R (1971), Psychology of learning Mathematics, New York; pinguin books Swenson T (1973), Teaching Mathematics to children ,New York : Oxford university press Vygostsky L.S (1986), Thought and language , Cambridge MA; MIT press Zoltan D (1971), 4th edition, Building up Mathematics, London UK; Hutchinson educational limited 48 APPENDICES APPENDIX A Interview guide for teachers 1. Why do pupils perform poorly in mathematics? 2. What method is suitable for teaching mathematics? 3. Why do teachers fail to use TLM’s? 49 APPENDIX B Interview guide for pupils 1. Do you get access to mathematics books both at home and in school? 2. Are you given the opportunity to interact with the TLM’s teachers bring to the classroom? 50 APPENDIX C Pre-test items Solve the following questions 1. 243,936 2. +186,432 2,147,865 3. 3,466,609 + 1,489,347 _ 2,386,677 4. 4,149,903 5. 891,142 _ + 364,686 1,387,646 51 APPENDIX D Post-test items Solve the following questions using the abacus or place value chart 1. 879,604 2. 645,871 3. 4,103,827 _ 754,821 +459,610 +2,398.758 4. 6,786,785 5. 2,461,289 _ 247,898 +4,287,920 52 APPENDIX E LESSON PLAN SCHOOL: OFFINSO STATE A PRIMARY AVERAGE AGE: 11+ YRS SUBJECT: MATHEMATICS REFERENCE(s): MATHEMATICS SYLLABUS PG. TEACHER’S GUIDE PG. CLASS: BS 6 PUPILS’ TEXTBOOK 2, PG. WEEK ENDING: 29-02-2016 DAY/ TOPIC/SUB TOPIC OBJECTIVE (S) ASPECT RPK T L M ‘s / T L A ‘s CORE POINTS DURATION REMARKS DAY TOPIC OBJECTIVES TLM: Abacus Monday ADDITION AND SUBTRACTION OF SIX AND SEVEN DIGIT NUMBERS By the end of the lesson, the pupil will be able to: INTRODUCTION DATE EVALUATION i. Use abacus to add six and seven digit Introduce the lesson by asking pupils to add and subtract four and five digit numbers. E.g. 5415 65471 +3457 _45360 25-02-2016 53 Abacus is a frame containing rods with small balls that slide alongside the rods EXERCISE Solve the following: 1.243936 + 186423 2.2147865 +1489347 numbers SUB-TOPIC DURATION 60minutes ADDITION AND SUBTRACTION OF SIX AND SEVEN DIGIT NUMBERS USING THE ABACUS ACTIVITIES RPK Pupils can add and subtract four and five digit numbers 1. Introduce the abacus to pupils by guiding them to know that it consist of column and beads. The first column from right represents ones, second column represents tens, third column represents hundreds up to the seventh column which represents millions 3. 3466609 - 22386677 4.4149903 ii. Demonstrate to pupils how beads on each column is exchanged with the other. E.g, ten beads on the ones column is equivalent to one bead on the tens column iii. Guide pupils to solve 422345 + 344321 using the abacus by guiding them to first represent 422345 on the abacus iv. Ask pupils to represent 344321 on the abacus v. Ask pupils again to combine all the beads represented on each column and count them.. vi. Pupils will find out that 422345 + 344321 54 - 1387646 5.891142 +364686 766566 vii. Lead pupils to use the idea they used in addition of six and seven digit numbers using the subtraction operation viii. Solve more examples with pupils 55 REMARKS abacus to perform the APPENDIX F LESSON PLAN SCHOOL: OFFINSO STATE A PRIMARY SUBJECT: MATHEMATICS AVERAGE AGE: 11+ YRS REFERENCE(s): MATHEMATICS SYLLABUS PG. TEACHER’S GUIDE PG. CLASS: BS 6 PUPILS’ TEXTBOOK 2, PG. WEEK ENDING: 29-02-2016 DAY/ DURA TION TOPIC/SU B TOPIC OBJECT IVE (S) ASPECT RPK DAY TOPIC Monday ADDITIO N AND SUBTRAC TION OF SIX AND SEVEN DIGIT DATE OBJECT IVES By the end of the lesson, the pupil will be T L M ‘s / T L A ‘s CORE POINTS EVALUA TION REMARK S TLM: Place value chart Place value chart is made with a cardboard , felt pens and ruler EXERCIS E solve the following INTRODUCTION Revise pupils knowledge on the addition and subtraction of six and seven digit numbers using the abacus 56 1. 243936 + 186423 25-022016 NUMBER S able to: Place value chart M DURA TION 60minut es ix. Use place value chart to SUBadd six TOPIC and seven ADDITIO digit N AND numbers SUBTRAC TION OF SIX AND SEVEN DIGIT RPK NUMBER Pupils S USING THE can add PLACE and VALUE subtract CHART six and seven digit numbers using the abacus HTh T Th - H T O 2.2147865 + 1489347 T ACTIVITIES h 1. introduce the place value chart to pupils and demonstrate how it is designed with chalkboard illustrations 3. 3466609 - 22386677 x. Guide pupils to solve 4967547 + 3387686 using the place value chart 4.4149903 Place value chart - 1387646 M HTh 4 9 TTh Th H T O 6 7 5 4 7 8 6 3 8 7 6 8 13 15 15 12 13 13 8 3 5 5 2 3 3 3 5.891142 +364686 57 xi. Lead pupils to perform subtraction operation using the place value chart xii. Lead pupils to solve more examples. REMARK S 58