Students were subjected to an experiment and their opinions sought to determine the value of practical work in teaching fractions and division by fractions. Data from these instruments demonstrated the value of hands-on work in improving students' understanding of division by fractions.
OVERVIEW OF THE STUDY
- MOTIVATION
- Limited visual representation of the fraction concept
- OBE requirements for a learner-centred approach
- RELEVANCE OF THE STUDY
- RESEARCH QUESTIONS
- AIMS OF THE STUDY
This could ensure the discovery and successful application of the division algorithm in solving fraction division problems. Despite this, one of the assumptions of this study points to the teachers' minimal use of practical work.
REVIEW OF LITERATURE
- INTRODUCTION
- ENRICHED AND DIVERSE UNDERSTANDING OF THE FRACTION CONCEPT
- UNDERSTANDING WHOLE NUMBER DIVISION
- UNDERSTANDING VARIOUS INTERPRETATIONS OF FRACTION DIVISION
- Multiplication and Division of fractions: a general challenge for learners
- PRACTICAL WORK: A USEFUL VEHICLE FOR UNDERSTANDING VARIOUS INTERPRETATIONS OF FRACTION DIVISION
- SUMMARY
Multiple presentations of fraction concepts (Witherspoon, 1993) and multiplication of fractions (Cramer & Bezuk, 1991) should be used. Practical work is placed at the center of a meaningful understanding of the division of fractions.
RESEARCH METHODOLOGY
- INTRODUCTION
- QUALITATIVE NATURE OF THE STUDY
- THE PARTICIPANTS
- SAMPLING
- RESEARCH INSTRUMENTS
- Interviews
- TIME FRAMEWORK
- ANALYSIS
The salient and basic features of a. experimental studies are still prominent in quasi-experimental research. a) the experimental and control groups, (b) the treatment to which the experimental group is exposed (practical activities), (c) the pre- and post-test that both groups undergo before and after the treatment to determine the difference made in the experimental group. The interviewees were from the experimental group who had experience with practical division of fractions. The interview was a group interview.
RESULTS AND ANALYSIS
INTRODUCTION
THEORY VERSUS PRACTICE
- Real practices against teachers' claims
All respondents indicated that they would definitely recommend the use of practical work in learning fractions. The teacher claimed that she had received pre- and in-service training for practical work in teaching fractions.
FACTORS BEHIND TEACHERS' VIEWS
- Understanding mathematical concepts
This section looks at the factors behind teachers' views on practical work in teaching fractions as discussed in section 4.2. These perceptions refer to teachers' perceptions about practical work and the teaching of fractions in relation to their practices (see research question I). External factors included: (1) large numbers in classes, (2) pressure to complete the syllabus and (3) training in practical work.
The respondent was the teacher from school B whose observed lessons did not include practical work. With the exception of one respondent, it can be safely concluded that all respondents had some training in the use of practical work in learning fractions. This should be a strong factor behind the participants' (in theory) favorable disposition towards practical work.
STRENGTH OF PRACTICAL WORK IN FRACTION DIVISION
- Worksheets based on Bottle-tops
- Pre-test and Post-test
- PRACTICAL WORK IN DIVISION OF FRACTIONS
- The introductory exercise
- Worksheets based on the Ruler - With remainder
- Worksheets based on Bottle-tops - Without remainder
- Worksheets on Bottle-tops - With remainder
- Conclusions from Worksheets
- Pre-test and Post-test
- Conclusions from Pre-test and Post-test
- RULER OR BOTTLE -TOPS? A QUESTION OF EXPEDIENCY
Along with student performance on Worksheet 4 (see Table 4.2), student performance on Worksheet 6 was compared with student performance on Worksheet 1 (see Table 4.1) to see if there was any improvement in students' passing ability fractions using a ruler. Performance on Worksheet 3 (see Table 4.9) was compared with student performance on Worksheet 4 (see Table 4.2), which required students to use a ruler to find solutions to division problems without remainders. The results from this worksheet were compared with those from Worksheet 8 (see Table 4.11) to determine students' ability to use bottle caps to find solutions to division problems with fractions without remainders.
Performance on Worksheet 10 (see Table 4.12) was compared with performance on Worksheet 11 (see Table 4.6) to determine whether there was an improvement in student performance on division-by-remainder problems. For this purpose, only parts of division tests are reported in this subsection. Table 4.13 shows the performance of the students of both schools in dividing points with fractions in the pre-test. This is supported by improvements in the performance of students from the experimental group (in school A) on division problems with fractions.
LEARNERS' VIEWS
- ATTITUDES TOWARDS PRACTICAL WORK
- USE OF RULER
- CHALLENGES IN PRACTICAL FRACTION DIVISION
Sabelo: (Probably using divide in the same context as before)' • because sir, when you need a half, you can divide in the top of the bottle. Interviewer: (Continues to list division methods students have been exposed to) Use of ruler and bottle tops. Can you give me a reason why it was easy.. to show fractions using bottle tops.
Mthandeni: Ngoba ama bottle stoppers uwadi ukuwawalisa kabili (because you can separate the bottle stoppers into two groups). Sir, because you can count better with bottle caps (He moves his hands to indicate moving from one position to another, while the other students laugh). Interviewer: (Confirming his understanding of the meaning behind Sihle's gestures) You can count better with bottle caps.
LIMITATIONS OF THE STUDY
The result is that students fail to find the right solution to problems involving everyone else. Teachers should take advantage of students' positive attitudes and. receptivity to practical work and use of practical activities in learning fractions and fraction division. . relationships with students, especially those from the first school, the first research location. This is suggested from the background that the researcher is a high school teacher who normally teaches mathematics to older children.
Through continued interaction with Year 7 students and the associated accumulated experience, it is possible that the above relationships could have improved by the time the study was conducted at School B. It is possible that such developments have tipped the balance in favor of school B. as far as student performance at the two schools is concerned, making a comparative analysis of performance at the two schools less conclusive. The principles of transferability and generalization cannot be guaranteed for conclusions drawn from data generated by such a small sample.
SUMMARY
Due to the small number of teachers who answered the questionnaires, it was extremely difficult to generalize that the conclusions reached will also apply in other similar situations. On the other hand, the practical work proved to be effective in helping to improve students' understanding and competence on the concepts of fractions and division by fractions. The students' answers to the interview questions confirmed their positive inclination towards practical work in dividing fractions.
They particularly embraced bottles which they claimed made it easy for them to work with fractions as they are easy to move around. This is because of the unique relationships the researcher had with learners in each of the schools where the study was conducted. The limited number of participants who participated in the study also made the application of principles of transferability and generalization impossible.
CHAPTERS
DISCUSSION OF DATA FINDINGS
- INTRODUCTION
- TEACHERS' PERCEPTIONS ON PRACTICAL FRACTION DIVISION Teachers' responses to questionnaires and practices in observed lessons confirmed
- Teachers' difficulties in constructing practical fraction division activities
- The relevance of practical fraction division to OBE
- FACTORS BEHIND TEACHERS' PERCEPTIONS
- Teachers' beliefs
- PRACTICAL WORK AND CONCEPTUAL DEVELOPMENT
- Whole numbers in fraction division
- LEARNERS' VIEWS OF PRACTICAL FRACTION DIVISION
- Ungrounded teaching of the algorithm
- SUMMARY
One of the factors behind this overemphasis of the subregional perspective of the fraction concept is the over-. The success of practical work to improve learners' understanding of conceptual processes involved in fraction division is documented under the following sub-headings: (l) whole numbers in fraction division, (2) concrete experience in fraction division, (3) the problem of writing the rest, (4) overemphasis of the partial region fraction perspective, and (5) learners' successful application of the algorithm. The views of learners on practical fraction division are documented under the following subheadings: (1) preference for concrete experience, (2) the value of the subset perspective, (3) learners' challenges, and (4) unfounded teaching of the algorithm.
Therefore, the subgroup interpretation has proven to be a strong alternative to the partial region perspective of the fraction concept. It has proven to be a useful option that teachers should seriously consider to improve and expand students' understanding of the concepts of fractions and division by fractions. The successful use of the fraction division algorithm by students from School B also disproved claims in the literature that the algorithm is difficult for students to use.
IMPLICATIONS AND RECOMMENDATIONS
- INTRODUCTION
- IMPLICATIONS FOR TEACHER TRAINING
- In-service training
- TEACHING IMPLICATIONS
- IMPLICATIONS FOR FURTHER RESEARCH The following themes for further research are suggested
The embracing attitude of teachers towards the relevance of practical fraction division for OBE is an encouraging starting point. School B's teacher's ideas about aspects of practical fraction division that should be covered in OBE workshops pretty much summarize all teachers' needs in this area. (see section 4.2.3). Such workshops should also delve teachers into deeper aspects of the concepts of fractions and fraction division (e.g. other fraction perspectives and fraction division situations). The positive results of the introduction of fraction division with whole number division make it necessary to continue using whole numbers to explain the meaning of fraction division situations.
Therefore, teaching about division of fractions remains dependent on understanding the division of integers and must continue to use integers as a starting point. Just as it was possible for students to meaningfully experience the measurement meaning of dividing fractions by using the subset perspective of the fraction, students should be helped to understand the shared/partitive and other meanings of division by using practical representations of fractions. This requires the commitment of teachers to seek and design effective strategies to help students understand partitive and other meanings of fraction division.
APPENDIX A
This section of the questionnaire aims to establish the level of your training in the use of practical work in the teaching of Mathematics in general, and fractions in particular. To answer questions, please put a cross in the appropriate box or provide a written answer where applicable. 1) Fractions provide enough opportunities to learn Mathematics through practical work. Have you ever received any form of training in teaching fractions through practical work?
Have you ever attended an in-service course on the use of practical work in teaching fractions?. To answer questions, please put a cross in the appropriate box or provide a written answer where applicable. This section is intended to inform the researcher about the challenges and needs of educators in the use of practical work for the teaching of fractions.
APPENDIXB
Now use your group of 12 bottle caps to complete the following tasks on dividing fractions. The remaining bottle caps do not form a complete group of 12 bottle 2. caps.
Finding ~ of 12 bottle-tops
APPENDIXC
GRADE 7
APPENDIXD TEST 2
Show the required fraction by shading the part or parts that represent that fraction
You can draw diagrams or use a ruler or bottle tip to find answers to these problems. If you decide to use bottle tops, use a set of 12 bottle caps. e) Write your solutions in the space provided. f) Show all your work.
APPENDIXE
APPENDIX F
In my investigation I have to work with seventh graders, one of whom is your child. Therefore, I ask for your permission to work with your child, in the company of other seventh grade students. You are assured that the real names of the participants will not be revealed after the study findings are published.
You are also assured that the findings will not be used for purposes other than those related to the objectives of the study. Your child's participation depends on your parental will and your child's own. His or her participation will be terminated appropriately if you and/or your child so desire. I also understand that real names will not be used in reporting findings, but that these will always be protected.
