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CHAPTER 1 INTRODUCTION 1.0 Overview Chapter one presents the insight as well as the background to the study which aimed at improving students’ performance in problems involving addition and subtraction of fractions. It also includes statement of the problem, purpose of the study, objectives of the study, research questions and significance of the research, limitations and delimitation of the study and organization of the study. 1.1 Background to the Study Mathematics in general is an essential subject for scientific and technological development of any nation. It is part of life without which man cannot function (Nabie, 2002). This suggests that no nation can grow scientifically and technologically above her mathematics status; an indication that mathematics is indispensable for science. Mathematics is the means of sharpening the individual’s mind, shaping his reasoning ability and developing his personality, hence its immense contribution to the general and basic education of the people of the world (Asiedu-Addo & Yidana, 2004). The entire world has become digital and this digital innovation is irreversible. It has become a world culture, which is progressing at a terrific speed for good. Any person, community or country that resists or refuses to join these forces of progress will be left behind and abandoned to languish in ignorance, penury and backwardness forever (Talabi, 2003). 1 With this understanding, the Ghana School Curriculum made mathematics a compulsory subject in her Basic, Second cycle and Tertiary levels of education apparently to be at par with the developing if not the developed countries (CRDD, 2004; CRDD, 2007 & Dogbe, Morrison and Speed, 1995). A thorough looked at the intended curriculum (syllabus) of schools in Ghana revealed that the study of Fractions is one aspect of mathematics that run through all levels of education of which the Colleges of Education are no exception. At the primary level for instance, Fractions can be found in Unit 2.8 for P2, Unit 3.4 and 3.11 for P3, Unit 4.6 and 4.9 for P4, Unit 5.11 for P5 and Unit 6.2 and 6.2.7 for P6 (CRDD, 2004) with the increasing levels of scope underplay. In the Junior High schools, Fractions can be found in Units 1.2, 1.13 and 1.14 for JHS1, Units 2.5, 2.11 and Unit 2.14 for JHS2 (CRDD, 2007). It is noted that the knowledge of Fractions is essential in learning other concepts like Money and Taxes, Handling of Data and Probability, Algebraic expressions, Volumes and Areas, Geometry and Trigonometry, Measurement, Calculus just to mention but a few. Doctors, nurses, and pharmacists need to figure out Fractions to determine the proper dosages of medicines for unusually large or small patients. The patient may have to take half sachet of medicine with half a glass of water two times daily. You need Fractions to bend a straight line to turn it into a curve. Students can plan their menu and have a balanced diet over a stipulated period of time with the idea of Fractions because some amount of carbohydrates may be taken with some amount of protein. Mother Earth is made up of Fractions whilst all financial operations work on Fractions in terms of percentages in their business transactions. 2 Having a solid foundation of basic Fraction concepts will make the advanced concepts easier to learn. This is a clear indication that the importance of Teaching and Learning of Fractions in our schools cannot be underestimated if students are to develop good conceptual knowledge in Mathematics which will eventually enable them to perform well in other subject areas. Most miserably however, teachers continue to skip this important area in Mathematics without teaching the concept of Fractions. Teachers who attempt teaching also teach without using materials that will enhance good understanding or enable pupils to grab the concept from the onset. There are topics that some teachers find difficult to teach. They call such topics challenging topics (INSET Project, 2007). Some of such topics are Operations on Fractions, Measurement of Area, Capacity, Volume and Time, Investigation with Numbers, Shape and Space and Collecting and Handling of Data. Teachers claim that the topics require subject teachers or specialists to teach them. However, with adequate preparation, teaching of these topics should not be problematic. The challenging topics are seen to be abstract in nature because they are not seen in real life situations. Besides, there are no Teaching and Learning Materials and relevant curriculum materials to support teachers to teach such topics. Most importantly, some teachers do not use appropriate methodology to enable students participate fully in the lesson (INSET Project, 2007). In line with the above development, the educational climate in institutions responsible for training the human resource needed for the nation for which the Colleges of Education in Ghana are of crucial importance, is essentially driven by an overriding interest in preparing students with the necessary Pedagogical Content Knowledge (PCK) 3 to become effective and efficient teachers at the Basic schools. Notwithstanding the government of Ghana efforts to upgrade the Pedagogical Content Knowledge of Mathematics teachers, the subject has not undergone much change in terms of how it is presented to students. These reflect consistently in low achievement levels in Mathematics among students especially at the basic schools. In this regards, the Colleges of Education formerly known as Teacher Training Colleges have undergone transformations in many areas particularly Certification and curriculum to uplift the standard of Content and Methodology so that trainees can handle these topics efficiently and effectively at the basic schools even before they graduate as professional teachers. The Mathematics Curriculum for Methods and Content were earlier integrated for Certificate ‘A’ until 1999, but in 2005, the curriculum was changed for the diploma programme. Between 1999 and 2005 though the Colleges were not awarding diploma, the curriculum for Methods and Contents were not integrated. The Mathematics Content courses for first years are now Number & Basic Algebra and Geometry & Trigonometry for first and second semesters respectively whilst the Mathematics Content courses for second years are Statistics & Probability and Further Algebra for first and second semesters correspondingly and simultaneously with the Methods of Teaching Primary School Mathematics and Methods of Teaching Junior High School Mathematics. Mathematics is hence assigned three (3) and six (6) credit hours for Arts and Mathematics & Science students respectively. In spite of this, over the years students’ performance in Mathematics especially at the basic level has not been impressive despite the huge resources and much attention paid to the study of Mathematics. The deplorable conditions of students can be observed 4 clearly from the summary of students’ performance as expressed by Anamuah-Mensah, Mereku and Asabere-Ameyaw (2004) report on results from the Junior Secondary School two (JSS2) students participation in Trends in International Mathematics and Science Study (TIMSS) in 2003, that Ghana’s overall performance in Mathematics was very poor. This performance placed: Ghana at the 45th position out of the 46 participating countries on the overall Mathematics achievement results table. The range of scores from 130 to 430 shows how diverse the JSS2 students were in their Mathematics abilities. The mean percentage correct answer on all Mathematics test items for each participating Ghanaian student was 15 and only 9% and 2% of the students reached the low and intermediate international benchmarks respectively. Ghana ranked 46th on the international benchmark for Mathematics. The students’ strong content areas in Mathematics were in Number and Data whilst the weakest areas were in Algebra, Measurement and Geometry. In almost all the content areas, the boys achieved significantly higher scores than the girls. A further analysis on the results by Anamuah-Mensah and Mereku (2005) indicated that The mean percentage of students making correct responses in Algebra, Measurement and Geometry were 13.6%, 17.3% and 13.4% respectively. For Number and Data, the mean percentage making correct responses were 22.6% and 27% respectively. The Ghanaian students found the constructed response items more difficult than the multiple-choice items. 5 The mean percentage of students who were able to provide the correct responses to the multiple-choice items was 21.6% while that observed for the constructed response items was 12.1%. These performances not withstanding; Ghana’s performances in TIMSS 2007 were better than that of 2003. According to Anamuah-Mensah, Mereku and Ghartey (2008), Ghana’s Junior High School two (JHS2) students’ performances in Mathematics, though improved significantly since TIMSS 2003, remains among the lowest in Africa and the world. In Mathematics, Ghana’s score of 309 was among the lowest and was statistically significantly lower than the TIMSS scale score average of 500. This poor performance place Ghana second from the bottom on the overall Mathematics results table doing slightly better than only Qatar. Ghana’s score in Mathematics was lower than those obtained by all the participating African countries. But the country’s performance level at TIMSS improved from that of 2003. In Mathematics, the 2007 score of 309 was significantly higher than the 2003 score of 276, a 33 point increase. My experiences as a teacher also revealed that, generally, Mathematics has been one of the subjects that most students fear to learn. As to why it is so feared, some students said it is simply difficult and they don’t like it. Others said the way it was taught them that made it difficult to understand. However, in my view, these situations could be attributed to several factors. Such factors include the abstractness of Mathematical concepts, the way the concepts are presented to the students and poor foundation among others. Today, many Mathematics teachers barely use materials in teaching; no matter the level they 6 teach, and it appears they lack the necessary pedagogical skills needed to be able to teach the subject with competency for pupils to grasp the concept from the onset. Like most branches of Mathematics therefore, Number and Algebra are the most important areas where ‘Fractions’; the topic which poses problems to both students and teachers alike can be found. The word ‘Fraction’ according to Downes and Paling (1965) is taken from the Latin word ‘Frangere’; meaning “to break”. Fractions are a wellidentify area of difficulty for many children even adults (Pamela, 1984); and as a result, Practicing Teachers of Dambai College of Education are no exception. It is important to note that Fractions form an integral part of the Mathematics curriculum of every level of the educational system. Most importantly, as implementers of the Government Policy on Education, the Colleges of Educations’ Mathematics Curriculum lays emphasis on the Pedagogical Content Know-how to give and enable the Teacher Trainees acquire the requisite skills to be able to teach this all important topic at the Basic level of education. This is evidently spelt out in the objectives of the Mathematics syllabus for Colleges of Education as: 1. To extend the students own Mathematical ability to a level significantly beyond that which he or she is likely to teach Mathematics in schools. 2. To give students an understanding of the Mathematical content and processes contained in the Kindergarten, Primary and Junior High School syllabuses. 3. To provide professional skills and understanding relating to Methods of Teaching which is appropriate for Basic Education. 7 On the basis of the above, Practicing Teachers’ inability to teach Fractions using Concrete Materials thereby enhancing pupils’ understanding of concepts better raised a lot of concerns in the researcher’s mind. It is evident that most of the practicing teacher trainees on teaching practice failed to teach this topic and a few who tried to teach the topic also fumbled with the teaching. This came to light whenever the researcher went out to supervise the students on their teaching practice. The most worrying aspect is that, these practicing teachers’ are the implementers of the Basic school Mathematics curriculum and as such need much more attention. More importantly, one of the weaknesses of the 2002, 2004 and 2008 Basic Education Certificate Examination (BECE) results listed in the Chief Examiner’s Report included candidates’ inability to answer questions that involved Fractions. In view of this, the teaching procedures that are not in contravention of the famous Chinese proverb that; ‘I hear and I forget, I see and I remember, I do and I understand’ must be used. The distinction between deepen approaches and surface approaches to learning is particularly useful for teachers who want to understand the pupils’ learning and create learning environments which encourage pupils to achieve desired learning outcomes. The fact that, the use of manipulation and representation is strongly advocated by many authors such as Martin (1994) and Apronti, Afful, Ibrahim, et al (2004), the selection of Cuisenaire rods as effective Teaching and Learning Materials (TLM) was necessitated since it is a major material outlined in the students’ Course to be abreast with its use and to enable pupils’ apply the three domains of learning-Cognitive (head), Affective (heart) and Psychomotor (hand) in learning a mathematical concept. This entails the use of concrete, physical, observable and touchable objects like Cuisenaire Rods to teach abstract concepts. 8 It is upon these bases that the researcher found it most expedient to use Cuisenaire rods as an intervention to curbing the problem which has pervaded the educational systems in Ghana. 1.2 Statement of the Problem Mathematics is a subject that has to do with understanding of concepts, mastery of skills and application of these concepts in various fields of life. In the researcher’s supervisory work as a Mathematics tutor in Dambai College of Education, it was discovered that second year students of Dambai College of Education could not teach the addition and subtraction of unlike Fractions using concrete materials. As a result of their inability to teach this topic, pupils could not solve problems involving addition and subtraction of Fractions since the concept was not well formed in their mind. As evidence, the Chief Examiner’s Report of 2005, 2006 and 2007 for Methods of Teaching Primary School Mathematics stated students’ difficulties in answering questions on Fractions using Cuisenaire rods as concrete material and urged tutors to pay particular attention to the teaching of Fractions using concrete materials. 1.3 Purpose of the Study The main reason for carrying research in this area is to enable Practicing Teachers of Dambai College of Education improve upon their performances in teaching addition and subtraction of Fractions with unlike denominators by the use of Cuisenaire rods. 9 1.4 Objectives of the Study According to Cooney, Davis and Henderson (1975), objectives should be stated in terms of observable student’s behaviour. The researcher’s objectives in this regard were as follows: i. Find out the effect of Cuisenaire rods on students’ performance in solving problems involving Fractions. ii. Determine the effect of Cuisenaire rods on students’ performance in teaching problems involving Fractions. iii. Determine the influence of Cuisenaire rods on students’ perception in teaching problems involving Fractions. 1.5 Research Questions It is expected that majority of students in any learning situation will do well in a test on a given topic if it is planned and taught well. Improving the performances of second year students of Dambai College of Education in teaching the addition and subtraction of fractions with unlike denominators raised the following questions in the researcher’s mind. 1. What effect has the use of Cuisenaire rods on students’ performance in teaching problems involving Fractions? 2. To what extent would the use of Cuisenaire rods as concrete materials sustain and motivate students’ interest in learning of Fractions? 3. Is there any difference in students mean score performance by the use of Cuisenaire Rods in solving problems involving Fractions? 10 1.6 Research Hypotheses The hypothesis designed to guide and direct the study is: Null hypothesis 𝐻0 : There is no significant difference in scores between the mean pre-test scores and the mean post- test scores of students. Alternative hypothesis 𝐻𝑎 : There is significant difference in scores between the mean pre-test scores and mean post- test scores of students. 1.7 Significance of the Study Improvements of teaching methods, strategies and techniques have been the concern of many Mathematics teachers and educators since time immemorial. These desires have motivated mathematics teachers to carry out research work in various aspects of the subject that interest them. These in effects serve as a guide to teaching and learning of the subject. Since students do not only performed the various activities involved but showed interest and asked questions, they stand a better position to explain the concept anytime they are called to do so. The findings of the study when implemented will help Mathematics teachers not only to be able to teach well but also identify the usefulness of using Cuisenaire rods as teaching and learning materials in teaching fractions. The teachers will develop less difficulty in teaching whilst the students will develop interest and be more courageous in solving problems involving fractions in general. The results of this study would also serve as a guide for teachers to vary their approach and methodology to enable students understands the concept of fractions. It would also serve as resource material for all stake holders and others who would like to research further into this area of national interest. 11 1.8 Limitations of the Study Research of this nature will not have ended without any restraints or drawbacks to its successful completion. However, a few of such limitations that impede the smooth running are enumerated below: The study would have been more representative if all the two hundred (200) second year students of Dambai College of Education were covered. This is because; the villages where the mentees (practicing students) practiced teaching are distance apart from the location of the college; couple with transportation problems. This might affect the results of the study. There were also different unplanned programmes that distorted the organised time set for the intervention. As a result of this, the researcher whacked time and money to these various stations without achieving the purpose for his travels. The conclusion will therefore be limited by these factors and as such generalizations cannot cover all teacher trainees in Ghana. 1.9 Delimitation of the Study Fractions are broad areas in Mathematics with so many aspects. However, the study was restricted to the Addition and Subtraction of fractions with unlike denominators. The study was also delimited to the use of Concrete Materials (Cuisenaire Rods) to improving the teaching and learning of fractions. It was also confined to only Second Year Students of Dambai College of Education in the Volta Region of Ghana; though this problem might exist in other Colleges of Education. 12 1.10 Organizational plan of the Study The study was organised in five chapters. Chapter 1 talked about the background to the study, statement of the problem, purpose of the study, objectives of the study, research questions, significance of the study, delimitation, limitation and organizational plan. The relevant literature review was presented in chapter 2 whilst chapter 3 talked about the methodology. Chapter 4 talked about data presentation, analysis and discussion of result whereas chapter 5 consisted of summary of key findings, implications for practice, conclusion, recommendations and areas for future research. 1.11 Definition of Terms For the purpose of the study, the following definitions are implied for the terms below. Concept: - The idea or knowledge we hold about something. Cuisenaire Rods: - They are versatile Teaching and Learning Materials (TLMs) which are used to teach the concepts of fractions, addition and subtraction of whole numbers whose sum does not exceed 10. The rods are made up of 10 different colours which are associated with numerals. Thus, 1 – white, 2 – red, 3 – light green (green), 4 – purple, 5 – yellow, 6 – dark green, 7 – black, 8 – brown, 9 – blue, 10 – orange. The numerals associated with the rods shows the number of white rods that can fit exactly when laid side-by side with that rod. Like Fraction: - They are fractions with the same denominators Unlike Fractions: - Fractions with different denominators. 13 CHAPTER 2 RELATED LITERATURE REVIEW 2.0 Overview The review of the related literature was focused on the theoretical framework underlying the research and related works on the study. This was based on the following themes: Theoretical framework Nature of mathematics Cuisenaire rods and mathematics teaching Meaning and concept of Fractions Equivalent Fractions Addition and Subtraction of Fractions The part of language in teaching mathematics 2.1 Theoretical Framework The theoretical framework for the study is based on the Shulman’s (1986) three knowledge domains in teaching; grounded with the Constructivists views of teaching and learning. These domains of Shulman (1986) encompass: Subject Matter Content Knowledge (SMCK) Pedagogical Content Knowledge (PCK) Curricular Knowledge (CK) To teach all students according to today’s standards, teachers need to understand subject matter deeply and flexibly so they can help students create useful cognitive maps, relate one idea to another, and address misconceptions. Teachers need to see how ideas connect 14 across fields and to everyday life. This kind of understanding provides a foundation for Pedagogical Content Knowledge that enables teachers to make ideas accessible to students (Shulman, 1987). Teaching is not a matter of knowing something. It is far more than mere transmitting of concepts and ideas to learners. It involves bringing out the accumulated ideas and experiences that students come to class with and working on those ideas and experiences together with the students by way of refining, reorganizing, co-constructing and repairing these ideas and experiences into meaningful and compressible form for students to assimilate (Shulman, 2000). This forms the foundation on which teaching mathematics through problem solving leans on. According to Shulman (2000), teaching is about making the internal and external capabilities of an individual and can only be achieved if teachers engage students in the classroom discourse. It is only when students are engaged in an interactive classroom environment that their ideas, conceptions and experiences are made bare to the teacher to put them on truck. 2.1.1 Subject Matter Content Knowledge Shulman (1986) defined Subject Matter Content Knowledge as the amount and organization of knowledge intrinsically in the mind of the teacher. He argues that teachers’ subject matter content knowledge should not be limited to knowledge of facts and procedures; but also an understanding of both the substantive and syntactic structures of the subject matter. The substantive structures are the various ways in which the basic concepts and principles of the discipline are organized to incorporate its facts. Teachers will therefore 15 be able to use appropriate materials to teach mathematics well only when they comprehend the network of fundamental concepts and principles of problem solving in holistic manner (Shulman, 1986). The syntactic structure of a discipline is the set of ways in which truth or falsehood, validity or invalidity are established (Shulman, 1986). The syntactic structure is used to establish the most appropriate claims about a particular phenomenon. Teachers’ knowledge must therefore go beyond mere definitions of accepted truths in the subject matter domain. In sum, to provide for effective teaching and learning of mathematics, mathematics teachers’ Content Knowledge of concepts cannot be underplayed. The question that arises is ‘how can an individual handle a subject matter competently if the content knowledge is weak? The researcher believes that teacher’s knowledge of mathematics is essential to their ability to teach effectively as Brophy (1991) cited in Asiedu-Addo and Yidana (2004) indicates “where (teachers) knowledge is more explicit, better connected and more integrated, they will tend to teach the subject more dynamically, represent it in more varied ways, encourage and respond fully to students comments and questions. Where their knowledge is limited, they will tend to depend on the text for content, de-emphasise interactive discourse in favour of seatwork assignments and in general portray the subject as a collection of static, factual knowledge.” This suggests that the teacher uses mainly non thought provocative questions, and often selects only what he/she thinks can teach. 16 2.1.2 Pedagogical Content Knowledge Pedagogical knowledge includes generic knowledge about how students learn, teaching approaches, methods of assessment and knowledge of different theories about learning (Harris, Mishra & Koehler, 2009; Shulman, 1986). This knowledge alone though necessary; is insufficient for teaching purposes. According to Shulman (1986), pedagogical content knowledge is knowledge about how to combine pedagogy and content effectively. It includes, knowing what approaches fit the content, knowing how elements of content can be arranged for better teaching. It also involves knowledge of teaching strategies that incorporate appropriate conceptual representations to address learner difficulties and misconceptions and foster meaningful understanding; and knowledge of what the students bring to the learning situation; knowledge that might be either facilitative or dysfunctional for the particular learning task at hand. Shulman (1986) lay emphasis on the pedagogical content knowledge as the combination of the most regular taught topics, the most useful forms of representations of those ideas, the most powerful analogies, examples, illustrations, explanations and demonstrations in the art of teaching. Pedagogical Content Knowledge also includes the ways of representing and formulating the subject matter that makes it comprehensible to students with diverse views and understandings. In teaching Mathematics through activity oriented base and problem solving techniques, teachers need to design and present the lesson using appropriate teaching learning materials (TLMs) that can enable the students construct their own knowledge of the concept. They need to know the pedagogical strategies and techniques most appropriate for reorganizing the understanding of learners who might appear before them as blank slates (Shulman, 2000); hence the knowledge of subject 17 matter in the training of a mathematics teacher in particular and the classroom teacher in general is as important as the methodology aspect of it, and that the course outline in the Teacher Training Institutions should be reviewed in a more pragmatic approach by encouraging students to appreciate the need for both methodology and content courses (Asiedu-Addo & Yidana, 2004). 2.1.3 Curriculum Knowledge The word ‘curriculum’ comes from a Latin root which originally meant ‘a course to be run’, that is, a course in the sense of ‘race-course’ (Mereku & Agbemaka, 2009). Curriculum has numerous definitions which can be slightly confusing; especially meeting it the first time. It refers to all the courses offered at a school; it is the prescribed course of studies which students must fulfil in order to pass a certain level of education. Curriculum is really more than just what is taught in the classroom. The term was once used to refer only to the content of educational provision. It was therefore barely distinguishable from terms like ‘syllabus’ or even timetable (Mereku, 2004). It is anything and everything that teaches a lesson planned or otherwise. Humans are born learning, thus the learned curriculum actually encompasses a combination of the hidden, null, written, political and societal etc. Since students learn all the times through exposure and modelled behaviours, it means that they learn important social and emotional lessons from everyone who inhabits the school. According to Tanner and Tanner (1975) cited in Mereku and Agbemaka (2009), “Curriculum is the planned and guided learning experiences and intended outcomes, formulated through the systematic reconstruction of knowledge and experience under the auspices of the school, for the learner’s continuous and wilful growth in personal-social competence” 18 This definition according to Mereku and Agbemaka (2009) highlights the fact that the curriculum must take into account not only established knowledge but also emergent knowledge. This is because curriculum while transmitting the cumulative tradition of knowledge also concerns with the systematic reconstruction of knowledge in relation to the life experience, growth and development of the learner. An interesting interpretation of the term ‘curriculum’ by Costa and Liebmann (1997) cited in Mereku and Agbemaka (2009) is given below: “Curriculum is the pulse of the school; it is the currency through which educators exchange thoughts and ideas with students and the school community. It is the passion that binds the organization together. Curriculum, in the broader sense, is everything that influences the learning of students both overtly and covertly, inside and outside the school.” Whereas, Young (1998) cited in Mereku and Agbemaka (2009) looks at the ‘curriculum’ as ‘socially organized knowledge’; and said “….. academic curricula are as much the products of people’s actions in history as any other form of social organization. They are not given, nor, in today’s language, do they represent an unchanging gold standard. They can therefore be transformed. The issue is one of purposes and the extent to which the existing curriculum represents a future society that we can endorse or a past society that we want to change”, From the definitions above, it is possible to state that a curriculum has the following characteristics: It comprises the experiences of children for which the school is responsible. It has content. It is planned. It is a series of courses to be taken by students. 19 In addition, a curriculum considers the learners and their interaction with each other, the teacher and the materials. The output and outcomes of a curriculum are evaluated. Bringing all these points together, the curriculum is viewed as a composite whole including the learner, the teacher, teaching and learning methodologies, anticipated and unanticipated experiences, outputs and outcomes possible within a learning institution (Mereku & Agbemaka, 2009). The Mathematics Curriculum therefore is represented by a full range of programmes designed for the teaching of mathematics topics at a given grade level. It covers a wide variety of instructional materials available in relation to the subject matter to be handled and the set of characteristics that guides the use of particular curriculum materials in particular circumstances (Shulman, 1986). Teachers need to think hard about students mathematical ideas analyze textbooks presentations and judge the relative value of two different representations in the face of a particular mathematical issue (Ball & Bass, 2000). Mathematics teachers need to have thorough understanding of the curricular resources available for instruction so as to make them available to students when teaching mathematics for students to make their own meaning of concepts. 2.1.4 Constructivist Idea of Learning Constructivism is a philosophy of learning founded on the premise that by reflecting on our experiences, we construct our own understanding of the world we live in. I believe in constructivism because I view an individual as a knowledge constructor. Constructivists believe that mathematics does not grow through a number of indubitable established theorems, but through the incessant improvement of guesses by speculation and criticism (Fletcher, 2005). Constructivism can be traced at least to the eighteenth 20 century and the work of the Neapolitan philosopher Giambattista Vico, who held that humans can only clearly understand what they have themselves constructed (http://www.sedl.org/scimath/compass/v01n03/2.html). Each of us generates our own “rules” and “mental model”, which we use to make sense of our experiences. Learning therefore, is simply the process of adjusting our mental models to accommodate new experiences. The constructivists claim that learning is an active process and that knowledge is constructed rather than innate, or passively absorbed. Knowledge is invented not discovered and that all knowledge is personal, distinctive and socially constructed. To the constructivist, learning is essentially a process of making sense of the world and requires meaningful, open-ended, challenging problems for the learner to solve. However, social constructivist thesis is that mathematics is a social construction, a cultural product, fallible like any other branch of mathematics. They claim that knowledge is not passively received but actively built up by the cognizing subject and the function of cognition is adaptive and serves the organization of the experiential world (vonGlasersfeld, 1989). Many others worked with these ideas, but the first major contemporaries to develop a clear idea of constructivism as applied to classrooms and childhood development were Jean Piaget and John Dewey http://www.sedl.org/scimath/compass/v01n03/2.html). For Dewey education depends on action. Knowledge and ideas emerged only from a situation in which learners had to draw them out of experiences that had meaning and importance to them (Dewey, 1966). These situations had to occur in a social context, such as a classroom, where students joined in manipulating materials and, thus, created a 21 community of learners who built their knowledge together. Piaget's constructivism is based on his view of the psychological development of children. In a short summation of his educational thoughts, Piaget called for teachers to understand the steps in the development of the child's mind (Piaget, 1973). The fundamental basis of learning, he believed, was discovery: "To understand is to discover, or reconstruct by rediscovery” (http://www.sedl.org/scimath/compass/v01n03/2.html); which suggest that a student who understands a concept can explain it a variety of ways anytime without following a rigidly procedure. The philosophical view on how we come to understand or know; is characterized by three (3) propositions: Understanding is in our interactions with the environment This is the core concept of constructivism because we cannot talk about what is learned separately from how it is learned as if a variety of experiences all lead to the same understanding. Rather, what we understand is a function of the content, the context, the activity of the learner, and perhaps most importantly, the goals of the learner. Understanding is an individual construction; as such cannot be shared but rather, we can test the degree to which our understandings are compatible. An implication of this proposition is not just within the individual but rather it is a part of the entire context (Gaffney and Anderson, 1991). Cognitive conflict or puzzlement is the stimulus for learning and determines the organization and nature of what is learned When we are learning in an environment, there is some stimulus or goal for learning. The learner has a purpose for being there. That goal is not only the stimulus for learning, but 22 it is a primary factor in determining what the learner attends to, what prior experience the learner brings to bear in constructing an understanding, and, basically, what understanding is eventually constructed. In Dewey’s terms, it is “problematic” that leads to and is the organizer for learning (Dewey, 1938; Rochelle, 1992), but for Piaget, it is the need for accommodation when current experience cannot be assimilated in existing schema (Piaget, 1977; vonGlaserfeld, 1989). Knowledge evolves through social negotiation and through the evaluation of the viability of individual understandings The social environment is critical to the development of our individual understanding as well as to the development of the body of propositions we call knowledge. At the individual level, other individuals are a primary mechanism for testing our understanding. Collaborative groups are important because we can test our own understanding and examine the understanding of others as a mechanism for enriching, interweaving and expanding our understanding of particular issues or phenomena. vonGlaserfeld (1989) noted that, other people are the greatest source of alternative views to challenge our current views and hence serve as the source of puzzlement that stimulates new learning. 2.1.5 Classroom Implication of Constructivism to the Teacher In constructivism, teachers and pupils are viewed as active meaning makers who continually give contextually based meanings to each others' words and actions as they interact. From this perspective, mathematical structures are not perceived, intuited or taken in but are constructed by reflectively abstracting from and re-organising 23 Sensori-Motor and conceptual activity. Thus the mathematical structures that the teacher 'sees' are considered to be the product of his or her own conceptual activity and could be different from those of the pupils. (VonGlasserfeld, 1989). Consequently, the teacher cannot be said to be a transmitter of such structures nor can he or she build any structures for pupils. The teacher's role here is viewed as that of a facilitator in the learning process. Indeed if pupils are to be empowered and given greater control over their own lives, then as Fletcher (1997) points out, they should be encouraged to choose their own areas of study in mathematics and should also be encouraged to work in groups and generate mathematical problems. In the classroom, the teacher’s view of learning must point to a number of different teaching practices. It means encouraging students to use active techniques to create more knowledge and then to reflect on and talk about what they are doing and how their understanding is changing. The teacher makes sure he/she understands the students' pre-existing conceptions, and guides the activity to address them and then build on them. Constructivist teacher encourages students to constantly assess how the activity is helping them gain understanding. By questioning themselves and their strategies, students in the constructivist classroom ideally become "expert learners." This gives them ever-broadening tools to keep learning. With a wellplanned classroom environment, the students learn how to learn. When they continuously reflect on their experiences, students find their ideas gaining in complexity and power, and they develop increasingly strong abilities to integrate new information. The teacher's main role is to facilitate and encourage this learning and reflection processes. 24 2.2 The Nature of Mathematics Mathematics is not only a list of facts and techniques which children memorise but is made up of a number of processes which together form a way of thinking. These processes are; problem solving and investigation, generalizing, abstracting, specializing, classifying, conjecturing, communicating mathematically, justifying, forming and testing hypothesis, applying, comparing and ordering among others (Martin, 1994; Apronti, et al 2004). Views held on the nature of mathematics according to Mereku (2004) can be described in terms of the constituents or elements of knowledge embodied in the subject. The constituents of mathematical knowledge or things that have to be learned to possess mathematical knowledge are usually expressed by rules, definitions, methods and conventions. Ernest (1985) and Van Dormolen (1986) cited in Mereku (2004) referred to this constituent as ‘Objects’ and ‘Kernels’ respectively. These constituents have theoretical, communicative and methodological implications. Children find it difficult to understand abstract concepts. Besides, the difficulties intrinsic in mathematics itself are termed subject-based factors. They arise from the nature of mathematics, its symbolism and language. The mathematical concepts are very many; and are represented using mathematical symbols, which by their nature are very abstract right from the concept one; that even the mathematical concepts we teach in the primary one are far removed from reality (Nabie, 2002). Similarly, mathematical symbols are seldom experienced in real life situations that have meaning to children. If they experience them at all, they have no real value to them until they start with symbolic work in school; hence this abstract nature and structure of mathematics make abstraction, generalisation, deduction and recall of concepts and principles difficult for learners. 25 2.3 Cuisenaire Rods and Mathematics Teaching Cuisenaire rods are versatile manipulative materials for teaching concepts in mathematics and one of such very important concepts is Fractions. These rods were invented over 75 years ago by George Cuisenaire – a Belgian Mathematics teacher (Kurumeh & Achor, 2008). However the use of Cuisenaire rod is still prominent in the intended curriculum of Colleges of Education today due to the important role it plays in teaching mathematical concepts especially Fractions. These materials were invented to help students grasp abstract concepts in mathematics using coloured cardboards strips of varying lengths called Cuisenaire rods. The original pack of Cuisenaire rods consist of 74 rectangular rods in 10 different lengths and 10 different colours as shown below: W R e d G r e e n P u r p l e Y e l l o w D a r k g r e e n 26 B l a c k B r o w n B l u e O r a n g e Each colour corresponds to a different length. The content of the pack is thus: 22 white rods of 1cm each, 12 red rods of 2cm each, 10 light green rods of 3cm each, 6 purple rods of 4cm each, 4 yellow rods of 5cm each, 4 dark green rods of 6cm each, 4 black rods of 7cm each brown rods of 8cm each, 4 blue rods of 9cm each and 4 orange rods of 10cm each. These rods could be used as manipulative and symbolic concrete representations in teaching concepts in mathematics. Learners explore whole numbers, Fractions, measurements, ratio, area and perimeter etc using Cuisenaire rods (Thompson, 1994) and develop a link between ordinal and cardinal numbers and counting (Martin, 1994). The use of Cuisenaire rods’ approach is a hands-on and minds-on manipulative activity filed approach for teaching abstract concepts in mathematics and sciences. It is a valuable educational tool for modelling relationships between what is taught in school and their everyday life activities (Elia, Gagatsis & Demetrico, 2007) thereby enabling students to work independently and in groups on meaningful mathematics while the teacher provides individual attention to other students. Because Cuisenaire rods are ready-made tools, its approach minimizes preparation and set up time both for the teacher and the students. This approach helps to develop key skills such as classification, critical thinking, problem solving and logical mathematics and spatial reasoning (Rule & Hllagan, 2006). However, Taylor-Chapman (1967) said there are several advantages for colouring the rods. Some of these may be doubtful but seem beyond argument: 1. You can use the rods for sorting and matching by colour; eg ‘all yellows together’ or ‘match each green with a black’. 2. The children quickly learn to pick out any length very quickly. 27 3. We can call a rod by its colour and make it our unit. Thus if a brown is the unit, purple will be found to be half, white (natural) will be an eighth and if light-green is the unit, then dark-green is two; and natural is one-third. These are great advantages when learning about relationships. 2.4 Meaning and Concept of Fractions The word Fraction is taking from the Latin word ‘Frangere’ which means ‘to break’ (Downes & Paling, 1965; Apronti et al, 2004). This suggests that, a Fraction may be described as a part of a whole where the whole could be ‘a unit’ or a set of objects. In a related development, Martin, et al (1994) pointed out the importance to realise that the pairs of numbers ‘12’, ‘23’, etc and the phrases ‘one third’, ‘two fifths’ etc are not Fractions but merely symbols and words representing the concept of particular Fractions. They are of the view that; to understand what a Fraction is, then we must first look at how they arise. ‘A half’ is what we get when we share something equally into two parts. They noted that what ‘a half’ is depends upon what we started with. This suggests that ‘a half’ of Mr. A may not be the same as ‘a half’ of Mr. B. It is not possible therefore to show any single object and say ‘this is what a half is’. They illustrated these in a diagram as all representing a half. 28 From the above, Fractions are not objects but actions (Martin, et al 1994, Kusi-Appau, 1997) and that; we perform an action to halve something. It is only when we learn to represent these actions that these symbols can be treated as objects. However, Hilton and Pedersen (1983) said the words ‘a half’, ‘a quarter’, ‘a third’, and ‘three-quarter’ etc are used frequently in everyday speech and their meaning is clear to the reader; suggesting that one can say ‘a fourth’ instead of ‘a quarter’. The phrase ‘half’ can be used in subtly different ways. For instance, would you like a piece of cake? We may reply ‘please just a half’. The host or hostess may then cut the piece of cake into two equal pieces and give him one of those pieces. He has received ‘a half’ of the original piece of cake. On the other hand, a realtor showing us two possible lots for purchase may say ‘Lots A is more attractively situated but there is only half as much land as on lots B’. There is no suggestion that lots A has been created by cutting up lot B; the realtor only means that the amount of land on lot A is the same as the amount of land we would get by taking half of lot B. This means that when we say that 10 Ghana pesewa is a tenth of GH¢ 1.0, we certainly do not mean that 10 Gp is obtained by cutting or breaking GH¢ 1.0 into ten equal pieces and taking one of the pieces; rather, we mean that 10 Gp is worth a tenth of GH¢ 1.0, that we would require ten 10 Gp to purchase what we can purchase for GH¢ 1.0. Fractions, Decimals and Percentages are number ideas that are not whole numbers. These three concepts are closely related to each other for the fact that one can move from one domain to the other. In line with this, the teacher must help the child to see these relations and how to move from one form to the other (Kusi-Appau, 1997). In developing the concept of Fractions, the teacher must be able to use activities in real life 29 situations (Apronti et al 2004). According to Barnette and Ted (2000), Kusi-Appau (1997) and Apronti et al (2004); Fractions can be considered in three ways: i. Part-whole model (Sharing) ii. Part-group model iii. Ratio model Part-whole model (sharing) This is the case where children share a number of items like oranges, among themselves. In a situation where the number of items being shared is not enough for the children, it becomes necessary for them to cut or break the items up into bits and to share; and as they do this they make use of Fractions to denote part of a whole. Very often, teachers 1 1 2 make mistake of telling children that the statements like 2, 3, 3 etc which are symbols and words representing the concept of Fractions, are Fractions. When teaching Fractions in schools, the emphasis is often on situations where the object can easily be cut, folded, split or coloured in equal parts. Although there is some need for this sort of activity, children should be exposed to a wide variety of situations, some where such folding or splitting strategies will not be successful. In experiencing a variety of situations where Fractions can be found, learners will have the opportunity to reflect and abstract critical relations in different contextual situations. In other words, children must see a whole in all its representational forms. To overcome such misconception, teachers must let children see that Fractions are not objects but are actions of dividing objects into equal parts and taking some parts. In developing a sound understanding of the part-whole concept of Fractions, it is necessary for teachers to present situations of fair sharing, where the child is expected to reason out the consequences of different actions 30 http://nrich.maths.org/2550. Children must therefore know how Fractions arise. Confusion among children is from what we take as a whole. Very often teachers use a unit object as a whole and therefore when children come to meet groups of objects, they become confused when we take a Fraction from it. Considering the figures below: (i) (ii) (iii) The child may see the part shaded in (i) as half. However, in other situation when you have an object of the shape in (ii) and divide it into two as in (iii), the child finds it difficult to understand that the shaded portion is half. This creates misconception and confusion in the child’s mind. To overcome this confusion, the teacher must help the child understand that the whole could be one unit, a group or part of a unit or anything we are taking part could be our whole (Kusi-Appau, 1997). Fractions taught as a part-whole concept, can ensure that children have a sound foundation for conceptualizing other concepts in Fractions. However, it must be noted that despite the wealth of possible examples, an approach to Fractions based solely on "part-whole" is too restricted yielding proper Fractions only. Therefore other concepts of Fractions need to be explored if children are to have a fuller and better understanding of rational numbers http://nrich.maths.org/2550. Part-group model In everyday activities of children, it is often becomes necessary for them to consider part of a set in relation to the major set. This is illustrated in the diagram below: 31 1 3 Part of a group Ratio model According to Apronti et al (2004) the ratio model shows the relationship between objects or quantities of the same kind. It is a way of comparing the objects and this ends up in the form of a Fraction; that is to say if there are 30 boys and 50 girls in a class then ratio of 30 3 boys to girls is 30: 50 = 50 = 5 (Kusi-Appau, 1997). Using Cuisenaire rods to compare the lengths of two rods side by side for instance, it takes 5 white rods to equal 1 yellow rod. Hence the length of the white rod is 1 5 of the length of the yellow rod. This ratio of the length of the white rod to yellow rod is 1:5 as illustrated diagrammatically as: W Yellow According to Pamela (1984), Fractions are well identified area of difficulty for many children and even adults. There are two obstacles to understanding of Fractions. 1. Fractions cannot be taught of as separate, independent entities. They have meaning only in relation to the whole to which they apply. To recognise Fraction of something, you need a concept of the whole. It is relatively easy to imagine the whole apple of which you have a quarter, but it is not easy to imagine the whole kilogram of which you have a quarter, or the whole hour of which a quarter has passed. 32 2. Complicated notations by which Fractions are symbolized. The numeral at the bottom of a Fraction (denominator) has an entirely different function from the numeral at the top (numerator). For instance, the denominator of the Fraction 2 3 tells us that the ‘whole’ has been divided into three equal shares. The numerator tells us that two of those shares are under consideration. The word ‘denominator’ means ‘the thing that names’. The denominator of the Fraction 2 3 gives the Fraction its name; ‘third’. The word ‘numerator’ means; ‘the thing that numbers’. Hence the numerator of the 2 3 tells us the number of thirds to be considered. The denominator and numerator for Fractions also make it possible to denote the same 2 Fraction in infinitely many ways. For instance 4 6 , , 8 , 10 14 , 6 9 12 15 21 3 is the same as 2 𝑒𝑡𝑐. is called equivalent Fractions of 3. This idea takes a long time to sink in, and can prove another obstacle to understanding. To overcome the first obstacle, we should always in the early stages refer to the whole to which any Fraction applies. We should not talk about a ‘quarter’ but ‘a quarter of an apple’, ‘a quarter of a metre’ or ‘a quarter of twelve’ etc. 2.5 Equivalent Fractions Apronti et al (2004) explained equivalent Fractions as Fractions of the same value but different names. They are Fractions which represent that same number but have different names (Kusi-Appau, 1997: 97). For instance 12, 24, 36, 4 8 are equivalent Fractions. According to Apronti et al (2004) equivalent Fractions can be introduced using: 33 i. Paper folding and shading ii. Fractional boards iii. Cuisenaire rods Paper Folding and Shading They illustrated ½ by folding vertically, a sheet of paper strip into two equal parts and shaded one part shown below: 1 2 To have its equivalence, they again fold the strip of paper horizontally thereby having four equal parts with two parts shaded. 2 4 By folding the same paper again you will have eight equal parts with four of the parts shaded 4 8 It is noted from the above that the portion shaded for the first time, no other part had been shaded again. It is the same portion that has been named differently. 34 Fractional Boards Using Fractional boards, Apronti et al (2004) to identical strips of cards or paper, fold one into two equal parts and another into four equal parts and so on as shown below: Whole 1 2 1 2 1 4 1 8 1 4 1 8 1 8 1 4 1 8 1 8 1 4 1 8 1 8 1 8 From the illustrations, 2 halves make 1 whole, 2 one-fourths make one-half, 4 one1 2 4 8 eighths make one-half and so on. This implies that 2 = 4 = 8 = 16 etc. Using Cuisenaire Rod According to Kusi-Appau, (1997) and Apronti et al (2004), one can choose any rod or set of rods like Orange and Dark green to be the whole. You can then make up as many rows using rods of one colour only as shown below. Orange Dark green Brown Purple Red Red W W W Brown Purple Red W W Red W W Purple Red W W Purple Red W W Red W W It can be seen from the diagram that: Two browns make the orange and dark green whole Two purples make one brown 35 Red W W W Four purples make two browns Two reds make one purple Four reds make one brown Eights reds make the orange and dark green whole Two whites make one red Four whites make one purple Eight whites make one brown sixteen whites make the orange and dark green whole These colour observation can then be turned into Fractional statements as a brown is onehalf of the orange and dark green whole, a purple is one-fourth of the orange and dark green whole, a red is one-eighth of the orange and dark green whole and a white is onesixteenth of the orange and dark green whole. Pupils can the progress from concrete objects to diagrams, to words and eventually to 3 4 symbols and deduce that 12 = 24 = 36 = 48 etc and 14 = 28 = 12 = 16 and so on. 2.6 Addition and Subtraction of Fractions Dolan (2000) observes that apart from whole-number computations, no topic in elementary mathematics curriculum demands more time than the study of Fractions. To him, for students to understand, the teaching about Fractions and their operations must be grounded in concrete models. A firm foundation for number sense involving Fractions and a deeper understanding of the algorithms for operations with must be developed before formal work with Fractions. 36 According to Owusu and Manu, (2007) before pupils are introduced to addition and subtraction of Fractions, they must be able to rename Fractions using their equivalence to confirm their readiness for operations on rational numbers. However, children often think that whenever two Fractions are added, the result is less than 1 (Owusu & Manu, 2007). This is because their exposure to addition of Fractions is always less than 1. This means that they need early exposure to problems where the sum is greater than 1 to erase such misconception. Teaching addition and subtraction of Fractions for better understanding, it is expected that we use concrete materials. However, usually the first step is to learn to add and subtract Fractions with the same denominator which is fairly straightforward and activities using concrete materials are easy to devise (Martin et al, 1994). To Apronti et al (2004), paper folding and shading, number line and Cuisenaire rods could be used to teach addition and subtraction of like Fractions. i. Using Paper folding and Shading 3 8 2 8 5 8 This shows that 3 8 2 5 +8=8 and 3 8 2 1 − 8 = 8 respectively. 37 1 8 ii. Using Number line 0 2 8 1 8 3 8 3 8 2 8 3 8 3 8 iii. 6 8 5 8 7 8 8 8 6 8 7 8 + 28 = 58 1 8 0 4 8 4 8 5 8 8 8 − 28 = 18 Using Cuisenaire rods Pupils pick the brown rod as the whole. W W W W W W W W Brown Pupils to take two whites and three whites to represent two-eighths and three-eighths respectively. 2 8 W 3 8 W W W W They have to join the white rods end-to-end and then compare to the whole as: 5 W W W W W 8 Brown 2 3 5 + = 8 8 8 For subtraction, the two whites are subtracted from the three whites to have one white. One white rod is then compared to the whole which is the brown rod as shown below. 1 8 W Brown 38 2.7 The Part of Language in Teaching Mathematics The language used by the teacher in teaching any topic is very important if he or she is to make positive impact on his or her learners. Language plays an important role in the teaching and learning process. Mathematics language should be carefully and accurately used from the beginning of the child’s learning experiences (William, 1986). Torbe (1982) cited in Mereku and Cofie (2008) pointed out that “without language, without the telling and listening, the reading and the writing which fills every school day, there could be no communication and no educational process; it is language which makes the whole educational process possible”. One of the main reasons why children experience difficulty in mathematics is in the understanding the nuance of mathematical language (Warrant, 2006) and since mathematics, as a language makes use of symbolic notation as such requires using and interpreting this symbolic notation and grasping the abstract ideas and concepts which underlie it (Mereku & Cofie, 2008). They laid emphasis on the fact that the child’s inability to use language in mathematics will not only hinder his understanding of the subject but will also prevent the teacher from having a deeper insight into the child’s grasp of mathematics. The appropriate use of mathematical language and symbols can also help children develop mathematical concepts. An understanding of the mathematical symbols and examples will enhance children’s mathematical ability (Nabie, 2002, 2009). This is more so if they are combined jointly manipulated and linked to their everyday life situations. In line with this, Skemp (1986) cited in Martin (1994) and Apronti et al (2004) proposed that: “concepts of a higher order than those a person already has cannot be communicated to that person by a definition. Only by arranging for the person to encounter a suitable collection of examples can such concept be communicated”. 39 “Since in mathematics examples are almost invariably other concepts, the concepts used in the examples must already be formed in the mind of the learner”. Beginning a lesson with children’s previous knowledge, of which the mother tongue forms a part, gives them a perpetual momentum to forge ahead in the learning process. New ideas are easily developed and understood if they are linked with already existing ones. Hence children mathematical concepts can easily be developed if incoming concepts are correctly linked with what the child knows already in an understandable language. 40 CHAPTER 3 METHODOLOGY 3.0 Overview This aspect of the research dealt with the methods and procedures used to obtain data for the research work. This was done under the following themes: 3.1 Research design Population and Sample Sampling Techniques Instrument Intervention Method of data collection Data analysis procedure Research Design The design for this study is an Action Research in the form of pre-test, intervention and post-test structured to examine how the performance of second year students in teaching fractions is improved using Cuisenaire rods at Dambai College of Education. Action research according to Cannae (2004) involves the application of scientific methods to solve classroom problems. It uses pre-test and post-test data from the teaching of two instructional units to identify student teacher controlled factors which promote or inhibit pupils’ academic achievement (Bill, 1986). Action research is attractive to educational researchers because it seeks to identify peculiar problem in the educational field especially in the classroom and suggest possible 41 rectification to the problem by offering suitable intervention and recommendations for use by other educators to also apply such intervention. Put simply by O’Brien, (1998), action research is “learning by doing” - a group of people identify a problem, do something to resolve it, see how successful their efforts were, and if not satisfied, try again http://www.web.net/~robrien/papers/arfinal.html. This design was chosen in order to find possible solution to the problem identified and that teachers will be able to have command on the teaching of fractions in Ghanaian schools. 3.2 Population and Sample According to Anamoah-Mensah et al (2004), the quality and integrity of any study depend on the validity and the efficiency of the samples used in the study. In this regard, the sample was carefully selected. Out of the target population of two hundred (200) Second Year Students of Dambai College of Education in the Krachi-East District of the Volta Region for which 165 are males and 35 females, a sample size of 50 students comprising 41 males and 9 females were selected for the study. 3.3 Sampling Techniques The sampling technique used was the stratified sampling alongside with the random sampling techniques to select the samples for the study. Stratified sampling according to Awanta and Asiedu-Addo (2008) is the process of selecting a sample in such a way that identified subgroups in the population are represented in the sample the same proportion that they exist in the population. The percentage representation of male and female students was calculated to be 82% and 18% respectively. Therefore to have a 42 good representation, these percentages were used on the sample size of 50 students. The sample was therefore selected by writing “YES” and “NO” on paper and fold for both the males and the female to pick at random. For the female group, 9 “YES” and 26 “NO” was written on paper for the 35 female to pick randomly. All those with the “YES” were selected for the study. The same process was repeated for the male students to have 41 of them on the study. 3.4 Research Instruments The instruments used for the collection of data were tests. The tests were used in two folds, that is, pre-test and post-test. 3.5 Validity and Reliability 3.5.1 Validity Validity of an instrument according to Taale and Ngman (2003) refers to whether the instrument truthfully does what it is constructed to do. In other words, when the instrument measures what it is intended to measure, then it is valid. To ensure the validity of the test items, the researcher consulted the curriculum for methodology and some prescribed mathematics textbooks for teacher trainees. The purpose was to gain insight into what learners were expected to learn in order to develop the instrument accordingly. The researcher made sure that the content of the test was based on what the research questions were set to find out. Thus,, only questions on teacher trainees Pedagogical Content Knowledge were asked. After constructing the test items, the researcher approached other tutors in the Mathematics Department to cross check the appropriateness of the test items. Durrheim 43 (1999) suggests that the researcher approach others in the academic community to check the appropriateness of his or her measurement tools. Colleague tutors responses indicated that the contents examined in this study reflected the prescribed Content for students’ Pedagogical Content Knowledge 3.5.2 Reliability Reliability on the other hand refers to how well the instrument provides a consistent set of results across similar test situation, time periods and examiners (Taale & Ngman-Wara, 2003). It means the degree of dependability of a measuring instrument. It is worth mentioning that it is possible to have an instrument which is reliable because the responses are consistent, but may be invalid because it fails to measure the concept it intends to measure (Fraenkel & Wallen, 2000). In this study, the split-half method was used to check the reliability of the instrument because it is a “more efficient way of testing reliability” and was less time consuming (Durrheim, 1999). The split-half method requires the construction of a single test consisting of a number of items. These items are then divided or split into two parallel halves (usually, making use of the even-odd item criterion). Students’ scores from these halves were then correlated using the Spearman-Brown formula used in reliability testing. The value of the coefficient was 0.72. This value indicates a good degree of reliability of the instrument as asserted by (Fraekel & Wallen, 2000). 3.6 Data Collection Procedure All the fifty (50) students sampled for the study responded to the pre-test administered to determine their previous thoughts on teaching Fractions using concrete 44 materials. The pre-test was conducted on 5th February, 2011. In the 2nd week of March 2011, the implementation for the intervention begun. After four (4) weeks of lessons and activities on the teaching and learning of Fractions, the students were again tested (Posttest) which involved similar concepts but different set of questions as compared to that of the pre-test to determine the amount of knowledge the students have gained from the intervention activities. The Pre-test and the Post-test were marked and the scores by the students are shown in appendices C and D. 3.7 Intervention The intervention the researcher employed in the study is the use of Cuisenaire rods as a Concrete Material to help teacher trainees in teaching and learning of fractions. The intervention spanned four (4) weeks and lessons were conducted three times a week for 60 minutes per meeting. Students were introduced to the concept of Fractions, comparing Fractions (equivalent fractions) and addition and subtraction of fractions using materials including Cuisenaire rods, paper folding, number lines and fractional boards. The researcher demonstrated the concepts using the materials in range of activities to help the teacher trainees overcome their difficulties in teaching the concept of Fractions. The lesson was taken out of their syllabuses and other sources of information that researcher deem vital to use. Different methods, techniques and strategies were employed to enable students’ involvement in the lesson by relating the object of learning to the needs of the learner, their involvement in the learning process become increasingly significant. The students were allowed to work in groups and in pairs as they manipulate these materials while the teacher serve as a facilitator providing help when needed and asking thought provoking questions to stimulate critical analytical and complex thinking 45 in order to help them construct their own meaning as the study focused on promoting constructivist approach of learning. 3.8 Intervention Implementation Students were taken through series of activities using concrete materials but concentrated on the use of Cuisenaire rods in solving and teaching problems involving Fractions. 3.8.1 Concept of Fractions The researcher demonstrated the concept of fractions by folding a sheet of paper equally for the students to see. The researcher discussed with students to identify one part as one-half because a whole has been divided into two halves. 1 2 After demonstration the students were asked to do likewise. Students were put in groups and the Cuisenaire rods were given to them. The researcher instructed them to choose a rod say (orange) and try to make up as many rows as they can using rods of one colour only as shown below. W W W W W W Yellow Red W W W W Yellow Red Red Red Red Orange From the diagram above, five red rods make an orange rod. In fraction statement a red is 1 one-fifth of the orange whole which is written symbolically as 5. In the same vein, two 46 yellow rods make an orange rod. In a fraction form, a yellow is one-half of the orange rod 1 1 written as 2 and a white is one-tenth written as 10. 3.8.2 Equivalent Fractions The idea of equivalence occurs and every opportunity should be taken during discussion with teacher trainees. The idea should grow out of the teacher trainees’ experience rather than being taught as a separate topic. It is helpful to draw the various ideas which they have acquired. Using paper folding, students were taken through the following activities. 3.8.3 Paper Folding and Shading The students were guided to fold vertically, a sheet of paper strip into two equal parts and shade one part to represent one-half as shown below: 1 2 The activities continued by guiding them again to fold the strip of paper horizontally thereby having four equal parts with two parts shaded. 2 4 47 By folding the same paper again you will have eight equal parts with four of the parts shaded 4 8 It was noted from the above that the portion shaded for the first time, no other part had been shaded again. It is the same portion that has been named differently. 3.8.4 Using Cuisenaire Rods The researcher guided the teacher trainees to choose any rod or set of rods to be the ‘whole’ for instance the orange and dark green rods joined end-to-end to make up as many rows as possible using rods of the same colour only making sure that each row is of the same length as the original ‘whole’ chosen and write down their observations in words. W W W Red Red W W W W Red Purple W W Red Red Purple W W W W Red Red Purple Brown W W W Red Purple Brown Orange Dark green From the diagram students were guided to identify the following: i. Two browns make the orange and dark green rods put end-to-end as ‘whole’. ii. Two purples make one brown. iii. Four purples make two browns. iv. Four purples make orange and dark green whole put end-to end. 48 v. Two reds make one purple. vi. Four reds make one brown. vii. Eight reds make the orange and dark green whole put end-to-end. viii. Two whites make one red. ix. Four whites make one purple. x. Eight whites make one brown. xi. Sixteen whites make the orange and dark green whole. etc These colour observations were then turned into fractional statements as 1. A brown is one-half of the orange and dark green rods put end-to-end. 2. A purple is one-fourth of the orange and dark green rods wholes put end-to-end. 3. A red is one-eighth of the orange and dark green whole. 4. A white is one-sixteenth of the orange and dark green whole. A critical look at the diagram shows a pattern as 1 2 = 2 4 4 = 8 = 8 16 and also 1 4 = 2 8 = 4 16 etc. Equivalent fractions are fractions of the same value but different names illustrated by the diagram above i.e. 1 2 = 2 4 = 4 8 = 8 16 1 2 etc and 4 = 8 = 4 16 etc. It was therefore deduce that when the top (numerator) and bottom (denominator) of a Fraction is multiply by the same counting number, the value of the fraction remains unchanged though with different names. 3.8.5 Comparing Fractions With the idea of equivalent fractions, students were also guided to compare both like fractions and unlike fractions using paper folding. I guided the students to compare firstly like fractions. For instance comparing 49 1 4 2 and 4. Students were guided to take two strips of paper of the same size and fold them such that each one is divided equally into four parts and shade the corresponding parts. Students were asked to compare the shaded parts by putting the strip of papers side by side as shown below. It was clear from the 2 1 diagram that 4 is greater than 4 since they all the same denominator. 𝟏 𝟒 2 4 Activities concerning unlike Fractions were also carried out using the paper folding. For instance using 2 3 1 and 2 , the students were asked to take two strips of paper with the same 2 size, fold first one into three equal parts and shade two of the equal parts as 3. The second 1 strip of paper is also folded into two equal parts and shade one part of it to have 2. The strips of papers were placed side by side for students to identify which of the shaded area 2 1 is larger as shown below. Students did see clearly that 3 is greater than 2. 2 3 𝟏 𝟐 50 3.8.6 Addition of like Fractions Cuisenaire rods were used extensively here to teach and develop algorithm of Fraction. Specific examples were used e.g. addition of 1 5 2 + 5. Students were guided to choose a rod (whole) that can be split into five exactly of other rods. Students were able to pick orange and yellow since orange can be split into five reds and yellow into five whites a rod of each represents one-fifth respectively. I did not restrict the students of which rods to work with, rather some used the orange rod whiles others used the yellow rods as shown below. W W W W W Yellow Students were guided to pick one and two of the white rods to represents respectively. Putting these rods end-to-end, we have W W 1 5 and 2 5 W Comparing the three rods put end-to-end with the whole (i.e. yellow rod), we have this diagram below. W W W Yellow Thus 1 5 + 2 5 = 3 5 . On the other hand students who picked orange as the whole were also guided to pick one red rod and two red rod to represent 1 5 and 2 5 respectively. They joined the rods and then compared to the whole which is the orange rod and got Red Red Red Orange 51 Red 3 5 . Red Red Red Red Orange Thus 1 5 + 2 5 = 3 5 . 1 2 In the same way, adding 3 and 3, students were guided to pick a rod that can be split into three equal rods. Students pick light green and dark green as the whole and worked with them as follows. Red Red Red Dark green 2 1 The red rods are each one-third so two red rods are taken as 3 whiles a red rod is taken as 3. Thus two reds and one red joined end-to-end and compared to the whole fit exactly onto 1 the whole as above. Thus 3 + 2 3 3 = 3 = 1. 3.8.7 Subtraction of like Fractions Students were guided to understand that the algorithm of subtraction is done in 3 the same way as in addition. In solving 5 − 1 5 for instance, students were again guided to pick rod or a train of rods that can be split into five of other rod. Students were able to pick yellow and orange rods. The yellow can be split into five whites whiles the orange rod can also be split into five red rods. It suggests that any of the rods could be worked with. Using the yellow rod as shown bellow, each of the white rods represent one-fifth. 3 Students were guided to take three whites rods and compare to the whole represent 5 and out of the 3 whites rods take one from it and the result is 2 whites. 52 W W W W W Yellow 3 5 3 1 Thus 5 − 3.8.8 5 W W W 1 5 Yellow W Yellow 2 = 5. Addition of unlike Fractions Through discussion, I explained to the understanding of students that Fractions with unlike denominators can be classified under the following categories: i. one denominator as a multiple of the other ii. a common factor iii. denominators being prime numbers There was a thorough discussion on fractions with one denominator as a multiple of the other the same before they add them. For instance 1 3 = 2 6 was obtained by multiplying the numerator and denominator by 2. In adding fractions with different denominators for instance 1 + 2 1 3 we listed down sets of Fractions that are equivalent to both shown below. 1 2 1 3 = = 2 4 2 6 = = 3 6 3 9 = = 4 8 4 8 = = 5 10 5 15 = = 6 12 6 18 7 = 14 = = 7 21 = 8 16 8 24 53 etc. etc. 1 2 and 1 3 as For the set of equivalent Fractions shown above, we pick those with the same name to 1 1 3 1 2 represent the original fractions 2 and 3. This shows that 6 is equivalent to 2 whiles6 is also 1 1 equivalent to3. This shows that 2 + 1 3 =6 + 3 2 6 5 = 6. 3 2 Using Cuisenaire rods as a concrete material to solve for example 5 + , it is clear that 10 10 is a multiple of 5 hence we need to choose a rod which is the whole such that the rod can be split into ten which is the orange rod. W W W Red W W Red W W Red W W Red W Red Orange 1 1 From the diagram one red represents 5 and one white represents 10 of the whole which is 3 2 the orange rod. Therefore in the question 5 + 3 Red whiles 3 red rods represents 5 2 , 2 white rods represent 10 10 Red W W Red It is clear that we cannot combine two different rods and have meaningful explanation 6 hence we change 3 red rods for 6 whites rods to have its equivalent as 10. W W W Red Now the two white rods 2 10 W W W Red W W 6 10 3 5 Red and six white rods W W W W W are joined end-to-end and then compared side-by-side to the whole which is the orange rod. 8 10 W W W W W W Orange Therefore 3 5 + 2 10 6 becomes 10 + 2 10 = 8 10 . 54 W W W 6 10 1 For the question 3 + 1 , you need a rod that can be split into 2 and 3 respectively and that 2 rod is dark green rod. 1 red rod represents 1 3 and 1 light green rod represents 1 2 of the whole below. W W W Red W Red Light green W W 1 3 Red Light green 1 2 Dark green Students were guided to exchange 1 light green rod for 3 whites and 1 red rod for 2 whites. i.e. 1 2 = Light green 3 6 W W W and 1 3 Putting the exchanged white rods end-to-end we have = Red W W 2 6 W Comparing the 5 white rods to the whole which is Dark green rod we have 1 Hence 3 + 1 2 = 2 6 + 3 6 = W W 5 6 W W . 5 . 6 Also there are situations in which one has to choose a set of rods as the whole. 2 For instance in questions like 3 + 1 , one need to choose a rod that could be split into 3 4 and 4. No single rod could be split in this way hence any of the following could be chosen as the whole Orange and Red Blue and Light Rod Brown and Purple Black and Yellow Dark green and Dark green 55 Any of the above combination could be used as a whole for the question and discussion was done for all for students to realize that they all arrive at the same answer. Using the orange and Red rods as the whole, I guided the students to put the two rods end-to-end and look for a rod that can go into the whole 3 and 4 respectively as shown below. Light green Light green Purple Light green Purple Light green 1 4 Purple 1 3 Orange Red From the above diagram 1 purple represents one-third (1/3) and 1 light green represent one-fourth (1/4). Purple 2 3 Purple 1 4 Light green Students now change the two purple rods for eight (8) white rods and one light green rod for three white rods, join end-to-end and compared to the whole. 11 12 W W W W W W W W W W W Orange It implies that 2 3 + 1 4 = 8 12 + 3 12 = Red 11 . 12 3.8.9 Subtraction of unlike Fractions The researcher guided the students to understand that subtraction and addition of Fractions follow the same procedure or algorithm. The only different thing you have to do is where you need to take away instead of addition. Using 1 2 1 − 8, students were guided to choose a rod which could be split into eight equal parts. Since 2 is a factor of 8 56 or 8 is a multiple of 2 the rod chosen can be split into 2 too. The appropriate rod is the brown rod. As shown in the diagram below, 1 purple rod represents 1 2 and 1 white rod 1 represents 8 of the whole (which is the brown rod). W W W W W W Purple W W Purple Brown 1 2 Purple 1 8 W Students change 1 purple for 4 white rods and compare with the whole and can now take away1 white rod from 4 white rods to have 3 white rods. Students now compare 3 white rods with the whole to have 3/8. 1 2 4 8 1 Hence 2 − 1 8 = Purple W 4 8 + W 1 8 W W = 4 8 W W - 1 8 W = 3 = 8. 57 W 3 8 W W W W CHAPTER 4 DATA PRESENTATION, ANALYSIS AND DISCUSSION 4.0 Overview This chapter dealt with the presentation of data, analysis of scores collected from the pre-test and the post-test and discussions based on the results of the study. The data collected was analyzed quantitatively and qualitatively. The results from the pre-test and post-test raw scores were analyzed using both the descriptive and inferential statistics employed on the Statistical Package of Social Sciences (SPSS). The descriptive statistic used to analyze the data projected the sample size, minimum and maximum scores, the mean scores and standard deviation for both the pre-test and post-test. The data was further analyzed using inferential statistics to project the p-values and t-values from the paired sample T-test. 4.1 Data Presentation and Analysis Table 4.1 shows the frequency distribution of the raw scores of the pre-test conducted for fifty (50) students. (See Appendix C). Table 4.1: Frequency distribution of Pre-test Scores by percentage Scores Frequency Percentage (%) 1 – 10 10 20 11 - 20 27 54 21 – 30 10 20 31 – 40 3 6 41 – 50 - 0 Total 50 100 58 Table 4.1 shows that 10 students representing 20% of the total number of students scored between 1 and 10 inclusive, 27 students representing 54% of the total number of students scored marks ranging from 11 to 20. Again, 10 students representing 20% of the total number of students involved in the study scored marks ranging from 21 to 30 whiles only 3 students representing 6% the number scored marks between 31 and 40 inclusive. No student scored marks ranging from 41 to 50. It is obvious from the marks that out of the 50 students who took the test, 42 students representing 84% obtained marks less than half of the total marks and only 8 students representing about 16% of the total number of students scored half or more of the total marks indicating poor performance of students in teaching problems involving addition and subtraction of Fractions. Table 4.2: Frequency distribution of post-test results for fifty (50) students by percentage Scores Frequency 1 – 10 - 0 11 - 20 2 4 21 – 30 22 44 31 – 40 19 38 41 – 50 7 14 Total 50 100 Percentage (%) From the post-test results in Table 4.2, it can be seen that no student obtained marks between 1 and 10 inclusive. Two (2) students representing 4% of the total number of students got marks from 11 to 20 whiles 22 students representing 44% of the total 59 number of students scored marks ranging from 21 to 30. Again, 19 students which represent 38% of the total number scored marks between 31 and 40 inclusive whiles 7 students representing 14% of the students’ total number scored marks from 41 to 50. It was realized from the post-test results that, 44 students representing 88% of the total number of students who took the test obtained half or more of the total marks. These improvements in students’ performance indicate the effect of the use of Cuisenaire Rods with questioning skills in teaching students on problems involving Fractions. However, 6 students representing 12% of the total number of students scored marks less than half of the total marks which indicate that some students still have little problems in solving Fraction related problems using concrete materials. Table 4.3: Descriptive Statistics of Pre-test and Post-test Scores N Minimum Maximum Mean Std. Deviation Pretest 50 5 37 16.94 7.175 Posttest 50 20 48 32.12 6.915 From table 4.3, the mean of pre-test score was 16.94 and that of post-test score was 32.12. Thus, the Gain score which is 32.12 − 16.94 is 15.18 when compared to the pre-test mean score of 16.94 show a significant improvement in students’ performance in solving problems involving Fractions by the use of Cuisenaire rods. A comparison of standard deviations of the pre-test score which was 7.175 and post-test score which was 6.915 revealed that the standard deviation of the post-test was less than that of the pre-test which indicates that the scores in the post-test were more spread around the mean mark which is 32.12 than it was in the pre-test scores. The minimum and maximum marks of pre-test scores and post-test scores are respectively 5, 37 and 20, 60 48. It is clear that both the minimum and the maximum marks of post-test scores are by far larger than that of the pre-test scores. Also, the range which is the difference in the maximum and minimum marks of pre-test and post-test is 32 and 28 respectively. It could be realised that, the range for pre-test is larger than the range for post-test which buttresses the interpretation for standard deviation; the fact that the data for the pre-test are less spread around the mean compared to that of the post-test. Testing of the Hypothesis Null hypothesis 𝐻0 : There is no significant difference in scores between the mean pre-test scores and the mean post- test scores of students. Alternative hypothesis 𝐻𝑎 : There is significant difference in scores between the mean pre-test scores and mean post- test scores of students. Table 4.4: Paired Sample Test for Pre-test and Post-test scores 𝑃𝑟𝑒𝑡𝑒𝑠𝑡 𝑵 𝑴𝒆𝒂𝒏 𝑺𝒕𝒅 𝑫𝒆𝒗𝒊𝒂𝒕𝒊𝒐𝒏 𝒕 𝒅𝒇 𝑷 − 𝒗𝒂𝒍𝒖𝒆 50 −15.180 8.395 −12.786 49 0.000 − 𝑃𝑜𝑠𝑡𝑡𝑒𝑠𝑡 From the Table 4.4 above, analyzing the data with paired sample test produced the P-value (0.000) which is less than the level of significance (0.05). Hence we reject the null hypothesis 𝐻0 and accept the alternative hypothesis 𝐻𝑎 (Asiedu-Addo et al 2004 ). Accordingly, it is concluded that there is a significant difference in mean scores between students in the Pre-test (𝜇 =16.94) and the post-test (𝜇 = 32.12). 61 4.2 Discussions of Results Considering the pre-test scores, 84% of the students obtained marks less than the total mark. This is an indication that students had difficulty in working with fraction related problems. From the post-test results, 88% of the students scored half or more of the total mark. This is an improvement in students’ performance compared to their performance in the pre-test. The mean pre-test score of 16.94 and the mean post-test score of 32.12 with a gain score of 15.18 indicated that students’ performance was about twice better than when the intervention was not administered. In answering the research question, it is clear that after the intervention, the evidence gathered suggest that incorporating the intervention tool (Cuisenaire Rods) into mathematics classroom teaching of fractions improved the achievement scores of students. Also in the paired sample t-test, the P-value of (0.000) is far less than the significance level of (0.05) which means that the null hypothesis must be rejected in order to accept the alternative hypothesis indicating that there is significant difference in the mean scores between students in the pre-test and post-test scores. 4.3 Summary of Findings from Research Questions 4.3.1 Research Question 1 What effect has the use of Cuisenaire rods on students’ performance in teaching problems involving Fractions? In answering the first research question, the initial result (Pre-test results) from Table 4.1 and appendix C suggest that the overall performance of students in teaching was very poor which reflected in the performance of pupils as well. It was also realised from Table 4.1 and appendix C that about 84% of the students obtained marks less than 62 half of the total mark. This is a clear indication that students had difficulty in teaching Fraction. After the intervention, the evidence gathered from the results (Post-test results) suggested that incorporating the intervention tool (Cuisenaire Rods) into Mathematics classroom teaching improved the achievement scores of students, since about 88% of the students scored half or more marks of the total score. This is shown in the mean value of 32.12 over the mean value of 16.94 by the pre-test scores with a gain score of 15.18 indicating that students’ performance was twice better than when the intervention tool was not used. These evidences showed that the use of Cuisenaire rods on students have positive effect since students’ performance improved tremendously as a result of the use of the intervention tool. 4.3.2 Research Question 2 To what extent would the use of Cuisenaire rods as concrete materials sustain and motivate students’ interest in teaching and learning of Fractions? Table 4.3 shows descriptive statistics of pre-test and post-test scores. The minimum and maximum marks of pre-test and post-test scores are respectively 5, 37 and 20, 48. Comparing the standard deviations of pre-test scores which was 7.175 and posttest scores 6.915 (Table 4.3) revealed that the standard deviation of post-test was less than that of the pre-test indicating that the scores in the post-test (ie by the use of intervention tool) were more spread around the mean mark of 32.12 than it was in the pre-test indicating significant improvement in students performance. This improvement is as a result of the motivation derived from the use of the intervention tool (Cuisenaire Rods) thereby sustaining their interest in problems involving Fractions. 63 4.3.3 Research Question 3 Is there any difference in students mean score performance by the use of Cuisenaire Rods in solving problems involving Fractions? From Table 4.4, the Paired Sample Test analysis of the data yielded the P-value of (𝑃 = 0.000) which is less that the level of significance of 0.05. Hence, we reject the null hypothesis (𝐻𝑜 ) which states “there is no significant difference in students means score performance by the use of Cuisenaire rods in solving problems involving Fractions”, accordingly accept the alternative hypothesis(𝐻𝑎 ). We conclude from the results by this result that there is a significant difference in the mean scores of students in the pre-test (16.94) and the post-test (32.12). 64 CHAPTER 5 SUMMARY, CONCLUSION AND RECOMMENDATION 5.0 Overview This chapter summarises the research findings, conclusion and gives recommendation and suggestions for further research and for curriculum development. 5.1 Summary The research was conducted to improve second year students of Dambai College of Education in solving problems involving Fractions using Cuisenaire rods. The data collected from the pre-test scores and post-test scores were first subjected to descriptive statistics using Statistical Package for Social Sciences (SPSS). It was realised that before the intervention, only 16% of the students were able to solve problems involving fractions satisfactorily. But after teaching them by the use of Cuisenaire Rods through activities, about 88% of the students were able to solve problems involving fractions satisfactorily. The main research question the researcher asked was to find out whether “there is any difference in students’ performance in using Cuisenaire rods in solving problems involving Fractions” The process of intervention revealed that there is a significant difference in the mean achievement scores of students using the Cuisenaire rods to their mean achievement scores than when Cuisenaire rods were not used. The statistical difference showed that the intervention tool (Cuisenaire rods) used improved students knowledge in problems involving fractions. The students now develop more positive attitudes towards fractions and mathematics as a whole because they were excited as they could easily answer 65 thought provoking problems and reach conclusions once they can manipulate the materials. The findings have serious implications for mathematics teaching and learning. The question about teachers’ pedagogical and content knowledge requirements cannot be over-emphasised for effective classroom mathematics teaching and learning. This is because effective lesson presentation requires expert execution of a set of decisions and actions in the pre-instructional, interactive and post instructional phases of teaching that depend on the knowledge base of the teacher. At the pre-instructional phase, decisions about what content to include in lesson presentation and organising the content in a logical and meaningful manner require extensive content knowledge base with a repertoire of pedagogical strategies; thereby enabling the students to construct ideas and make meaning of on their own. The teachers’ pedagogical content knowledge cannot exclude the issue of language since it is only through language of a kind that any form of teaching can be possible. 5.2 Conclusion It is evident from the findings of the study that using Cuisenaire Rods improved immensely on students’ achievement in Mathematics and Fractions in particular. There was a common feeling of confidence among students using the Cuisenaire rods. There is therefore growing evidence in the research conducted by Kurumeh and Achor (2008) who have found Cuisenaire rods effective in the teaching and learning of Fractions and other topics in Mathematics. 66 Since Cuisenaire rods are materials recommended in the curriculum of the Colleges of Education, a deep insight into its use is required so that students can follow the step by step procedure in its use to teach the topics in Mathematics at the Basic schools. 5.3 Recommendation The importance of Mathematics cannot be over-emphasised. From the study, it was useful in helping students through the use of Cuisenaire rods develop a meaningful understanding in problems involving Fractions. Based on the results gathered from this study, the researcher has this recommendations and suggestion to make. Since Cuisenaire rod is one important material that students are required to be abreast with its usage in teaching mathematics concepts, the researcher recommends that tutors teaching mathematics in the Colleges of Education must be familiar with its usage so that they can incorporate these materials in their teaching processes. This study as well as the study conducted by Kurumeh and Achor (2008) in Nigeria suggests that there is significantly positive effect of using Cuisenaire rods in teaching mathematics. It is therefore recommended that more emphasis should be laid on its use especially at the Colleges of Education. 67 REFERENCES Anamuah-Mensah, J., Mereku, D. K. & Asabere-Ameyaw, A. (2004). 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Educational Studies in mathematics, vol. 62 No.2, 169-189. William, G. C. (1986). Primary Mathematics Today. Hong Kong: Longman Group Ltd. 74 APPENDICES APPENDIX A PRE-TEST ITEMS 1 1 1. Show, step by step, how you would use concrete materials to solve 4 + 2 = ? 2. Describe one way in which you would guide pupils in primary class 4 to 3 6 4 8 determine for themselves that and are equivalent Fractions, using concrete materials. 2 1 3. Describe an activity you would use to guide pupils in primary class 4 to find 3 + 6 3 4. Describe an activity you would use to guide pupils in primary class 4 to find 5 − 2 . 10 5. Describe briefly how you would use a named concrete material to introduce the 3 Fraction 8 to pupils in primary class 3. 75 APPENDIX B POST-TEST ITEMS 1 1. Describe an activity you would use to guide pupils in primary class 4 to find 3 + 1 . 5 7 3 2. How will you explain to an upper primary pupil that 8 is greater than 4 using concrete material. 1 1 2 3. A primary class four (4) pupil does the following 2 + 3 = 5 i. State the child’s mistakes. ii. Describe how you would use concrete materials to help the pupil to overcome his/her problem. 2 1 4. Show, step by step, how you would use concrete materials to solve 3 − 4. 3 1 5. Describe an activity you would use to guide pupils in primary class 4 to find 5 − 4 76 APPENDIX C PRE-TEST SCORES 12 23 11 9 13 16 18 12 19 20 11 16 21 14 17 18 16 14 19 17 25 18 31 28 33 10 15 8 21 9 19 7 37 23 25 12 18 10 14 15 11 20 15 9 26 8 30 5 21 8 77 APPENDIX D POST-TEST SCORES 26 27 26 29 30 31 30 36 48 29 38 27 31 46 25 43 41 32 40 20 46 34 43 36 37 22 36 20 33 38 30 31 32 25 29 33 30 28 26 32 23 24 33 27 24 30 44 36 39 30 78