CHAPTER 5: CONCLUSIONS AND IMPLICATIONS
5.4 LIMITATIONS
The study was limited to in-service secondary mathematics teachers in the John Taolo Gaetsewe district of the Northern Cape province. These teachers were participating in the RUMEP programme offered by Rhodes University, where the researcher was employed as a lecturer at the time of the study. The study material investigated in this study were compiled by the researcher and other RUMEP staff to train teachers. The researcher therefore interpreted the findings of the study as a RUMEP employee but also as a researcher.
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In view of these potentially conflicting concerns it was a challenge for the researcher to criticise the teachers’ comments as well as the material used in the role of researcher. However, the researcher ‘bracketed’ himself as researcher as best as he could and was aware of this possible limitation throughout the study. The researcher also cross-checked the findings of the study with other professionals who are knowledgeable in problem solving. Notwithstanding the limitations, the data generated for this study highlighted important areas to contribute to a better understanding of the experiences of BEd (in-service) secondary mathematics teachers facilitating mathematics and using (or not using) problem solving.
5.5 CONCLUSION
This study has shown how in-service secondary mathematics teachers can experience the benefits and challenges of using a problem-solving approach to their teaching. In spite of the limitations of this case study, the researcher was able to note some small changes in the attitude of teachers towards a problem-solving approach. If this positive attitude can be developed and extended to the mathematics classroom, mathematics may not be the ‘dreaded’ subject it is perceived to be at the moment.
However, it must also be noted that the use of a problem-solving approach by teachers is a long-term investment and cannot be achieved overnight. It may take a gradual approach to convince teachers that their present, traditional methods are less relevant and effective in relation to the needs of modern societies. To convince the majority of teachers of such a view, opportunities may be created where they successfully experience the actual methods used to enhance problem solving. Teachers also need to challenge and critically reflect on their own teaching methods more frequently.
The researcher finally contends that series of regular workshops, contact teaching blocks and classroom support visits for BEd in-service teachers, based on constructivist learning models and a problem-solving approach in mathematics teaching, may be a way to improve the quality and results of school mathematics in future. This may only materialise if universities, teachers and school authorities can synergistically muster their efforts and resources to achieve such an outcome.
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REFERENCE LIST
Anthony, G. & Walshaw, M. 2009. Characteristics of effective teaching of mathematics: A view from the West. Journal of Mathematics Education, 2(2):147–164.
Arjan, P. 1998. An evaluation of the proposed new curricula for schools in relation to Kuhn’s conception of paradigms and paradigm shifts. South African Journal of Higher
Education, 12(1):20–26.
Artzt, A.F., Armour-Thomas, E. & Curcio, F.R. 2008. Becoming a reflective mathematics
teacher: A guide for observations and self-assessment. Second edition. New York:
Lawrence Erlbaum.
Ary, D., Jacobs, L.C., Razavieh, A. & Sorensen, C. 2006. Introduction to research in
education. Seventh edition. Canada: Thomson Wadsworth.
Baggaley, J. 2008. Where did distance education go wrong? Distance Education, 29(1):39– 51.
Barr, R.B. & Tagg, J. 1995. From teaching to learning: A new paradigm for undergraduate education. Change, November/December:13–25.
Beldarrain, Y. 2006. Distance education trends: Integrating new technologies to foster student interaction and collaboration. Distance Education, 27(2):139–153.
Bobis, J., Anderson, J., Martin, A. & Way, J. 2011. A model for mathematics instruction to
enhance student motivation and engagement. Motivation and disposition: Pathways to learning mathematics. 73rd Yearbook. Reston: National Council of Teachers of Mathematics.
Boud, D., Keogh, R. & Walker, D. (Eds.). 1994. Reflection: Turning experience into
learning. New York: Nicholas.
Brahier, D.J. 2011. Motivation and disposition: The hidden curriculum. Pathways to
Learning Mathematics, (2011):1 – 8.
Brodie, K., Lelliott, A. & Davis, H. 2001. Forms and substance in learner-centred teaching: Teachers’ take-up from an in-service programme in South Africa. Teaching and
Teacher Education, 18(2002):541–559.
Brousseau, G. 1997. Theory of didactical situations in Mathematics. Didatique des
Mathematiques, 1970–1990. Netherlands: Kluwer Academic.
Bryan, C. & Clegg, K. 2006. Innovative assessment in higher education. London: Routledge. Burton, L. 1989. Thinking things through: Problem solving in mathematics. Oxford: Basil
115
Blackwell.
Buschman, L. 2004. Teaching problem solving in mathematics. National Council of Teachers
of Mathematics, 10(6):302–309.
Chadwick, B.A., Bahr, H.M. & Albrecht, S.L. 1984. Social science research methods. New Jersey: Prentice-Hall.
Chambers, D.L. 1993. Integrating assessment and instruction. In N.L. Webb & A.F. Coxford (eds.). Assessment in the mathematics classroom. Reston, VA: The National Council of Teachers of Mathematics, 1993 Yearbook, 17 – 25.
Chappuis, S. & Stiggins R.J. 2002.: Classroom assessment for learning. Emotional
leadership, 60(1):40–43.
CHE (Council on Higher Education). 2004. Enhancing the contribution of distance higher
education in South Africa, Report of an investigation led by the South African
institute for distance education. Pretoria. September 2004.
Chisholm, L. 2005. The making of South Africa’s National Curriculum Statement. Journal of
Curriculum Studies, 37(2):193–208.
Clements, D.H. & Battista, M.T. 1990. Constructive learning and teaching. Arithmetic
Teacher, September.
Cockcroft, W.H. 1982. Mathematics count. Report of the Committee of Inquiry into the Teaching of Mathematics in Schools under the chairmanship of W.H. Cockcroft.
Retrieved from:
http://www.educationengland.org.uk/documents/cockcroft/cockcroft1982.html [12 September 2013].
Cohen, L., Manion, L. & Morrison, K. 2000. Research methods in education. Fifth edition. London: Routledge Falmer.
Cohen, L., Manion, L. & Morrison, K. 2007. Research methods in education. Sixth edition. London: Routledge.
Collopy, P. 2003. Curriculum materials as a professional development tool: How a
mathematics textbook affected two teachers’ learning. The Elementary School
Journal, 103(3):287–311.
Copes, L. & Shager, N.K. 2006. Phasing problem-based teaching into a traditional educational environment, in H.L. Schoen, & R.I. Charles, (eds.). Teaching
mathematics through problem solving: Grades 6 – 12. United States of America:
116
Creswell, J.W. 1994. Research design: Qualitative and quantitative approaches. Thousand Oaks, CA: Sage.
Creswell, J.W. 1998. Qualitative inquiry and research design: Choosing among five
traditions. Thousand Oaks, CA: Sage.
D’Ambrosio, B.S. 2006. Teaching mathematics through problem solving: A historical
perspective. In H.L. Schoen & R.I. Charles (eds.). Teaching mathematics through
problem solving: Grades 6–12. United States of America: Library of Congress, 39–
52.
David, M. & Sutton, C.D. 2004. Social research: The basics. New Delhi: Sage.
De Vos, A.S., Strydom, H., Fouché, C.B. & Delport, C.S.L. 2002. Research at grass roots:
For the social sciences and human service professions. Third edition. Pretoria: Van
Schaik.
DHET (Department of Higher Education and Training). 2011. Integrated strategic framework
for teacher education and development in South Africa. Retrieved from:
http://www.education.gov.za/LinkClick.aspx?fileticket=lXfDtQxRz3M%3D& [23 November 2012].
DoBE (Department of Basic Education). 2009. Report of the task team for the review of the
implementation of the National Curriculum Statement. Pretoria: Government Printing
Works.
DoBE (Department of Basic Education). 2010. Education statistics in South Africa 2009.
Retrieved from:
http://www.dhet.gov.za/LinkClick.aspx?fileticket=RfDvlKUWRdE%3d&tabid=1063 &mid=2158 [23 November 2012].
DoBE (Department of Basic Education). 2011. Curriculum and Assessment Policy Statement
Grades 10–12: Mathematics. Pretoria: Government Printing Works.
Driver, R. & Oldham, V. 1986. A constructivist approach to curriculum development.
Studies in Science Education, 13:105–122.
Ellington, H. & Race, P. 1994. Producing teaching materials: A handbook for teachers and
trainers. Second edition. New Jersey: Kogan Page.
Feuerstein, M. 1986. Partners in evaluation: Evaluating development and community
programs participants. London: Macmillan.
117 Theory into Practice, 40(2):102–109.
Freebody, P. 2004. Qualitative research in education: Interaction and practice. London: Sage.
Gavalcova, T. 2008. On strategies contributing to active learning. Teaching Mathematics and
its Applications, 27(3):116–122.
Geyser, H. 2009. Learning from assessment. In S. Gravett & H. Geyser (eds.). Teaching and
learning in higher education. Pretoria: Van Schaik, 90–111.
Gimenez, T. 1999. Reflective teaching and teacher education. Linguagem & Ensino, 2(2):129–143.
Grouws, D.A. 2006. The teacher’s role in teaching mathematics through problem solving, in H.L. Schoen, & R.I. Charles, (eds.). Teaching mathematics through problem solving:
Grades 6 – 12. United States of America: Library of Congress Cataloging-in-
Publication Data. 129 – 141.
Hatfield, M.M., Edwards, N.T., Bitter, G.G. & Morrow, J. 2000. Mathematics methods for
elementary and middle school teachers. Fourth edition. New York: Wiley.
Henning, E., Van Rensburg, W. & Smit, B. 2004. Finding your way in qualitative research. Pretoria: Van Schaik.
Hiebert, J. & Carpenter, T.P. 1992. Learning and teaching with understanding. In D.A. Grouws (ed.). Handbook of research on mathematics teaching and learning. New York: Macmillan, 65–95.
Hiebert, J. & Wearne, D. 2006. Developing understanding through problem solving, in H.L. Schoen, & R.I. Charles, (eds.). Teaching mathematics through problem solving:
Grades 6 – 12. United States of America: Library of Congress Cataloging-in-
Publication Data. 3 – 13.
Hitchcock, G. & Hughes, D. 1995. Research and the teacher: A qualitative introduction to
school-based research. Second edition. London: Routledge.
Hofstee, E. 2009. Constructing a good dissertation: A practical guide to finishing master’s,
MBA or PHD on schedule. Johannesburg: EPE.
Huba, M.E. & Freed, J.E. 2000. Learner-centred assessment on college campuses: Shifting
the focus from teaching to learning. Needham Heights, MA: Allyn & Bacon.
Huitt, W. 2003. Constructivism. Educational Psychology Interactive. Retrieved from: http//Chiron.valdosta.edu/whuitt/col/cogsys/construct.html [12 July 2013].
118
Jansen, J.D. 1998. Curriculum reform in South Africa: A critical analysis of outcomes‐based education. Cambridge Journal of Education, 28(3):321–331.
Jaques, D., Gibbs, G. & Rust, C. 1995. Designing and evaluating courses, educational
methods unit. London: Oxford Polytechnic.
Johnson, D.A. & Rising, G.R. 1967. Guidelines for teaching mathematics. Belmont, MA: Wadsworth.
Kahan, J.A. & Wynberg, T.R. 2006. Mathematics as sense making, in H.L. Schoen, & R.I. Charles, (eds.). Teaching mathematics through problem solving: Grades 6 – 12. United States of America: Library of Congress Cataloging-in-Publication Data. 15 – 25.
Kaur, A.S. 1996. Managing distance education. New Delhi: Deep & Deep.
Kilpatrick, J. 1992. A history of research in mathematics education. In D.A. Grouws (ed.).
Handbook of research on mathematics teaching and learning. New York: Macmillan,
3–38.
Kilpatrick, J., Swafford, J. & Findell, B. (Eds.). 2001. Adding it up: Helping children learn
mathematics. Washington, DC: National Academy Press.
Kline, M. 1978. Why the professor can’t teach. New York: St. Martin’s Press.
Koehler, M.S. & Grouws, D.A. 1992. Mathematics teaching practices and their effects. In D.A. Grouws (ed.). Handbook of research on mathematics teaching and learning. New York: Macmillan, 115–126.
Kombo, D.K. & Tromp, D.L.A. 2006. Proposal and thesis writing: An introduction. Nairobi: Paulines.
Kowal, S. & O’Connel, D.C. 2004. The transcription of conversations. In U. Flick, E. von Kardorff & I. Steinke (eds.). A companion to qualitative research. London: Sage, 248–252.
Kuhn, T.S. 1996. The structure of scientific revolutions. Third edition. Chicago, IL: University of Chicago Press.
Lambdin, D.V. 1993. The NCTM’s 1989 Evaluation Standards: Recycled ideas, whose time has come. In N.L. Webb & A.F. Coxford (eds.). Assessment in the mathematics
classroom. 1993 Yearbook, 7–16. Reston, VA: NCTM.
Lankshear, C. & Knobel, M. 2005. A handbook for teacher research: From design to
119
Larson, L.C. 1983. Problem solving through problems. New York: Springer-Verlag. Lau, P.N.K., Singh, P. & Hwa, T.Y. 2009. Constructing mathematics in an interactive
classroom context. Educational Studies in Mathematics, 72(3):307–324.
Le Grange, L.L. 2004. Research and transformation. Hons BEd programme. Stellenbosch: Stellenbosch University.
Le Roux, A & Le Roux, C. 2004. Evaluating Unisa course materials using a course evaluation instrument. Progressio, 26(1):8–22.
Lesh, R. & Doerr, H.M. 2003. Foundations of a models and modeling perspective on
mathematics teaching, learning, and problem solving. In R. Lesh & H.M. Doerr (eds.).
Beyond constructivism: Models and modeling perspectives on mathematics problem solving, learning and teaching. Mahwah, NJ: Lawrence Erlbaum, 3–33.
Long, M. 2000. The psychology of education. London: Routledge Falmer.
Louw, W.P. & Sonnekus, I.P. 2005. Seeking common ground in the learning development process. Progressio, 27(1/2):14–23.
Mahomed, H. 2004. Challenges in curriculum transformation in South Africa. Paper presented at the Fifth Annual Educationally Speaking Conference, 15–18 May, Boksburg.
Malan, S.P.T. 2000. The ‘new paradigm’ of outcomes-based education in perspective.
Journal of Family Ecology and Consumer Sciences, 28:22–28.
Manouchehri, A. 2007. Inquiry-discourse mathematics instruction. The Mathematics
Teacher, 101(4):290–300.
Marcus, R. & Fey, J.T. 2006. Selecting quality tasks for problem-based teaching. In H.L. Schoen & R.I. Charles (eds.). Teaching mathematics through problem solving:
Grades 6–12. United States of America: Library of Congress, 55–67.
McMillan, J.H. & Schumacher, S. 2001. Research in education: A conceptual introduction. Fifth edition. New York: Addison Wesley Longman.
Merriam, S.B. 2009. Qualitative research and case study: Applications in education. Second edition. San Francisco, CA: Jossey-Bass.
Miles, M.B & Huberman, A.M. 1994. An expanded source book: Qualitative data analysis. Second edition. London: Sage.
Morgan, C. & O’Reilly, M. 1999. Assessing open and distance learners. London: Kogan Page.
120
Motshekga, A. 2012. Statement during the announcement of 2011 National Senior Certificate
Grade 12 Examination results. Retrieved from:
http://www.education.gov.za/Newsroom/Speeches/tabid/298/ctl/Details/mid/1929/Ite mID/3280/Default.aspx [22 November 2012].
Motshekga, A. 2014a. National Senior Certificate examinations: School subjects report.
Retrieved from:
http://www.education.gov.za/LinkClick.aspx?fileticket=h39W%2FQRLpOE%3D&ta bid=36&mid=2514 [14 January 2014].
Motshekga, A. 2014b. Report on the annual national assessment of 2013: Grade 1 to 6 and
9. Retrieved from
http://www.education.gov.za/LinkClick.aspx?fileticket=Aiw7HW8ccic%3D&tabid=3 6 [14 January 2014].
Mouton, J. & Marais, H.C. 1996. Basic concepts in the methodology of the social sciences. Pretoria: HSRC Press.
Nachmias, C.F. & Nachmias, D. 1996. Research methods in the social sciences. Fifth edition. New York: St Martin’s Press.
NCTM (National Council of Teachers of Mathematics). 2000. Executive summary:
Principles and standards for school mathematics. Retrieved from: www.nctm.org [16
September 2012].
News Chanel Africa. 2013. Thousands of unqualified teachers are teaching South African
children. Retrieved from: http://www.enca.com/south-africa/thousands-unqualified-
teachers-sa-schools [18 September 2013].
Novotná, M., Hofmannová, J. & Petrová, R. 2008. Assessment approaches to teaching mathematics in English as a foreign language (Czech experience). International
Content and Language Integrated Learning Research Journal, 1(1):20–35.
Orton, A. & Frobisher, L. 2000. Introduction to education: Insights into teaching
mathematics. London: British Library.
Panaoura, A. 2012. Improving problem solving ability in mathematics by using a mathematical model: A computerized approach. Computers in Human Behaviour, 28:2291–2297.
Poggenpoel, M. 1998. Data analysis in qualitative research. In A.S. de Vos (ed.). Research
at grass roots: A primer for the caring professions. Pretoria: Van Schaik, 334–353.
121 Mathematics, 2(2/3):393–404.
Polya, G. 1988. How to solve it: A new aspect of mathematical method. Second edition. Priceton, New Jersey: Princeton University Press.
Schoenfeld, A.H. 1985. Mathematical problem solving. London: Academic Press.
Schoenfeld, A.H. 1987. Cognitive science and mathematics education. Hillsdale: Lawrence Erlbaum.
Schroeder, T.L. & Lester, F.K. 1989. Developing understanding in mathematics via problem solving. In P.R. Trafton & A.P. Schulte (eds.). New directions in elementary school
mathematics. Reston, VA: NCTM, 31 – 42.
Silver, E.A. 1987. Foundations of cognitive theory and research for mathematics problem solving. In A.H. Schoenfeld (ed.). Cognitive science and mathematics education. Hillsdale, New Jersey: Lawrence Erlbaum, 33–60.
Simon, M.A., Saldanha, L., McClintock, E., Akar, G.K., Watanabe, T. & Zembat, I.O. 2010. A developmental approach studying students’ learning through their mathematical activity. Cognition and Instruction, 28(1):70–112.
Simon, M.A. & Schifter, D. 1993. Toward a constructivist perspective: The impact of a mathematics teacher in-service programme on students. Educational Studies in
Mathematics, 25:331–340.
Slimp, M. (Ed.). 2014. Trends in distance education—What college leaders should consider.
A white paper from the Instructional Technology Council, November.
Smith, B.O. & Ennis, R.H. 1961. Language and concepts. Chicago, IL: Rand McNally. Spector, J.M. Reflections: Reconsidering the notion of distance in distance education.
Distance Education, 30(1):157–161.
Stacy, K. 2005. The place of problem solving in contemporary mathematics curriculum documents. Journal of Mathematics Behaviour, 24:341–350.
Steinke, I. 2004. Quality criteria in qualitative research. In U. Flick, E. von Kardorff & I. Steike (eds.). A companion to qualitative research. London: Sage, 184–190.
Stigler, J.W. & Hiebert, J. 1998. Teaching is a cultural activity. American Educator, 22(4): 4 – 11.
Thanasoulas, D. 2002. Constructivist learning. Karen’s linguistic issues. Retrieved from:
http://www3.telus.net/linguisticsissues/constructivist.html [9 July 2013].
122
Journal Mathematics Teacher Education, 13:295–311.
Walker, D. 1994. Writing and reflection. In D. Boud, R. Keogh & D. Walker (eds.).
Reflection: Turning experience into learning. New York: Nicholas, 52 - 68.
Walshaw, M. & Anthony, G. 2008. The teacher’s role in classroom discourse: A review of recent research into mathematics classrooms. Review of Educational Research, 78(3):516–551.
Warwick, P. 2007. Reflective practice. Published class notes on the development of the notion of professional reflection. New York: Cambridge University.
Webb, N.L. 1992. Assessment of students’ knowledge of mathematics: Steps toward theory. In D.A. Grouws (ed.). Handbook of research on mathematics teaching and learning. New York: Macmillan, 661–709.
Webb, N.L. 1993. Assessment for the mathematics classroom. In N.L. Webb & A.F. Coxford (eds.). Assessment in the mathematics classroom. 1993 Yearbook, 1–6. Reston, VA: NCTM.
Wickelgren, W.A. 1995. How to solve mathematical problems. New York: Dover. Yin, R.K. 1993. Applications of case study research. Volume 34. London: Sage.
Yin, R.K. 2009. Case study research: Design and methods. Fourth edition. Los Angeles, CA: Sage.
Ziebarth, S.W. 2006. Classroom assessment issues related to teaching mathematics through problem solving. In H.L. Schoen & R.I. Charles (eds.). Teaching mathematics
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Appendix A: QUESTIONNAIRE
This questionnaire is aimed at identifying opportunities and challenges you are faced with regarding the facilitation of mathematics through a problem solving approach while participating in a B.Ed (in-service) degree offered by Rhodes University. Answer both Section A and Section B. You are requested in Section B to answer each question by marking either strongly agree, agree, disagree or strongly disagree. Please only choose ONE response per statement by marking the appropriate block with an “X”.
The questionnaire is completed anonymously and will take approximately 20 minutes of your time. Thank you kindly for your cooperation.
Section A: Teacher Information
Your age range Less than 30 years
31 – 40 years 41 – 50 years Over 50 years
Your highest educational level Teachers Diploma/ACE Bachelor’s degree in Education B. Ed Honours in Education Master’s degree in Education
Your current post level
One Two Three Four
Years in Mathematics teaching per Grade
(s)
Grade 9 Grade 10 Grade 11 Grade 12
Your current Mathematics teaching Grade (s)
Grade 9 Grade 10 Grade 11 Grade 12
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