Diagnóstico Nutricional
4.2 Análisis, Interpretación y Discusión de Resultados
Although there are many videos on the internet that help address different topics in mathematics, most of the videos are not informed by any theoretical framework. Research in integrating ICT should try to upload videos informed by various theoretical frameworks, for example, constructivism, and socio cultural or situated learning theories, to accommodate different beliefs held by teachers.
In this investigation, findings reveal that when both the traditional method of teaching and the use of computers were used to address learner errors and misconceptions, both groups of learners improved. With the learners having used technology showing a better improvement that the group whose teacher used traditional methods. However, the findings reveal many advantages associated with computer use, but the computer being a tool, it has its possibilities and constrains. Further research should document the possibilities of using computers in the mathematics classroom as well as the constraints so that teacher can intelligently incorporate computer technology, where possible, for effective learning and teaching
5.5Conclusion
This chapter concluded this investigation by answering the research questions in the study with respect to the data that was collected in Chapter 4 by linking it to the theoretical frameworks that informed this study. The findings revealed that there are more benefits for both the teacher and the learner in using ICT in addressing the errors that learners have in learning functions.
124 REFERENCES
Abdullah, S.A.S., & Saleh, F. (2005). Difficulties in comprehending the concept of functions: implication to instruction. CosMED International Conference, SEAMEO RECSAM. Gelugor Pulau Pinang Malaysia
Arcavi, A. (1995). Teaching and learning algebra: Past, present and future. Journal of Mathematical Behavior, 14(1), 145-162.
Barrera, R. R., Medina, M. P., & Robayna, M. C. (2004). Cognitive abilities and errors of students in secondary school in algebraic language processes. In D. E. McDougall & J. A. Ross (Eds.), Proceedings of the Twenty-sixth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Vol. 1,(pp. 253-260). Ontario Institute for studies in Education, University of Toronto, Canada.
Ben-Zeev, T. (1996). When Erroneous Mathematical Thinking is just as “Correct”: the oxymoron of rational errors. In R. J. Sternberg & T. Ben-Zeev (Eds.), The Nature of Mathematical Thinking (pp.12-21). Mahwah, NJ: Lawrence Erlbaum.
Bless, C. &Highson-Smith, C. (2000). Fundamentals of social research methods: An African perspective. Cape Town: Juta Education Pty Ltd.
Booth, L. R. (1984). Algebra: Children’s strategies and errors. Windsor: NEFR-Nelson.
Booth, L. R. (1988). Children’s difficulties in beginning algebra. In A. F. Coxford & A.P. Shulte (Eds.), The ideas of algebra, K-12 (pp. 20-32). Reston, VA: NCTM.
Booth, S. (2008). Researching learning in networked learning – phenomenography and variation theory as empirical and theoretical approaches. Proceedings of the 6th International Conference of Networked Learning. ICNW.
Brown, J. S., & Burton, R. R. (1978). Diagnostic Models for Procedural Bugs in Basic Mathematical Skills. Cognitive science, 2(2), 155-192.
Bruner, J. (1966). Some elements of discovery. In Shulman, L. S. and Keislar, E.R. (Eds) Learning by Discovery: a critical appraisal. Chicago: Rand McNally, 101-114. Bryman, A. (2001). Social Research Methods. Oxford University Press, London.
Bryman, A. (2008). Social Research Method (3rd Ed.). Oxford University Press, New York. Burns, R. B. (2000). Introduction to Research Methods . London: Sage Publications.
125
Cave, R. C. (1995). Graphing, bit by bit. The Mathematics Teacher, 88(5), 372-373.
Chik, P. P. M., & Lo, M. L. (2004). Simultaneity and the enacted object of learning, In Marton & A. B. M. Tsui (Eds.), Classroom discourse and the space of learning, (pp.89 – 110). Mahwah, NJ: Lawrence Erlbaum Associates, Inc.
Cobb, P., Yackel, F., & Wood, T. (1992). A constructive alternative to the representational view of mind in mathematics education. Journal for Research in Mathematics Education, 23, 2- 33.
Cohen, L. &Manion, L. (1980). Research methods in education. London: Routledge.
Cohen, L., Manion, L., & Morrison, K. (2007). Research methods in education. Routledge, New York.
Cole, M. (1996). Cultural psychology: A once and future discipline. London: The Belknap Press of Harvard University Press.
Collins, A. (1991). The role of computer technology in restructuring schools. Phi Delta Kappan, 73(1), 28-36.
Confrey, J. (1987). 'Misconceptions' across subject matters; Science, mathematics, and programming. In Novack, J.D (Ed.), Proceedings of the Second International Seminar: Misconceptions and Educational Strategies in Science and mathematics, Vol 1 (pp.80- 106). Rotterdam: Sense Publishers
Confrey, J. (1990). A review of the research on student conceptions in mathematics, science, and programming. Review of Research in Education, 16, 3-56.
Confrey, J. (1995). A Theory of Intellectual Development, Part II. For the Learning of Mathematics: An International Journal of Mathematics Education 15, (1), 38-48.
Creswell, J. W. (2012). Educational research: Planning, conducting, and evaluating qualitative and quantitative research. (4thed.). Boston: Pearson.
Creswell, J. W., & Clark, P. (2011). Conducting and designing mixed methods research, (2nded.). Thousand Oaks, CA: Sage Publications, Inc.
Cuoco, A. A. & Goldenberg, E. P. (1996). A role for technology in mathematics education. Journal of Education, 178(2), 15-31.
Davis, R. B. (1984). Learning mathematics: The cognitive science approach to mathematics education. Norwood, N.J.: AblexPubling Corporation.
126
De Vos, A. S (Ed.). (2002). Research at grass roots: For the social sciences and human service professions. (2nded). Pretoria: Van Schaik publishers.
Department of Education. (2003). National curriculum statement Grades 10−12 (General): Mathematics. Pretoria: Department of Education.
Department of Basic Education. (2011). The Curriculum and Assessment Policy Statement for mathematics grades 10-12. Pretoria: Department of Education.
Department of Basic Education. (2011b). Report on the National Senior CertificateExamination 2011. Technical Report. Pretoria: Government Printers.
Department of Basic Education (2011c). Report on the qualitative analysis of ANA 2011 results. Pretoria: Government Printers.
Department of Basic Education. (2011d). Strategic Plan 2011-2014. Pretoria: Government Printers.
Donnelly P. (2002). A Study of Higher Order Thinking in the Early Years Classroom Through Doing Philosophy. Irish Educational Studies, 20, 278-295.
Dreiling, K. M. (2007). Graphing calculator use by high school mathematics teachers of Western Kansas. PhD Dissertation. Manhattan, Kansas: Kansas State University.
Dreyfus, T., & Eisenberg, T. (1986). On visual versus analytical thinking in mathematics, Proceedings of Psychology of Mathematics Education 10, 152–158.
Driver, R., & Easley, J. (1978). Pupils and paradigms: a review of literature related to concept development in adolescent science students. Studies in Science Education, 5, 61- 84. Dubinsky, E., & Harel, G. (1992). The concept of function, aspects of epistemology and
pedagogy. MAA Notes 25, 85-106.
Elia, I. &Spyrou, P. (2006). How students conceive function: A trairchic conceptual- semiotic model of the understanding of a complex concept. The Montana Mathematics Enthusiast, 3(2), 256-272.
Ely, R. (2010). Nonstandard student conceptions about infinitesimal and infinite numbers. Journal for Research in Mathematics Education, 41(2), 117–146.
Ensor, P., & Galant, J. (2005). Knowledge and pedagogy: sociological research inmathematics education in South Africa. In. R. Vithal, J. Adler, & C. Keitel, (Eds.), Researching Mathematics Education in South Africa: Perspectives and Possibilities (281-306). Cape Town: HSRC.
127
Ernest, P. (1991). The Philosophy of Mathematics Education. London: Falmer Press.
Even, R. (1988). Pre-Service Teachers Conceptions of the Relationships Between Functions and Equations. PME XII, Hungary.
Fey, J. T. (1989). Technology and mathematics education: A survey of recent developments and important problems. Educational Studies in Mathematics, 20(3), 237-272.
Fey, J.T. (1989). Technology and mathematics education: A survey of recent developments and important problems. Educational Studies in Mathematics, 20(3), 237- 272.
Graham, C. R. (2011). Theoretical considerations for understanding technological pedagogical content knowledge (TPACK). Computers & Education, 57(3), 1953-1960.
Graham, T. M. and Thomas, O. J. (2000). Building a versatile understanding of algebraic variables with a graphic calculator. Educational Studies in Mathematics, 41(3), 265-282. Gray, A. (2007). Constructivist Teaching and Learning. SSTA Research Centre Report #97-07.
OnlineatResearchReports/Instruction/97-07.htm#A%20Classroom%20Example%20of% 20Constructivist%20Teaching (accessed 1 September 2016, from Saskatchewan School Boards Association website).
Green, M., Piel, J., & Flowers C. (2008). Reversing education majors’ arithmetic misconceptions with short-term instruction using manipulatives. North Carolina at Charlotte: Heldref Publications.
Guba, E. G., & Lincoln, Y. S. (1994). Competing paradigms in qualitative research. In N. K. Denzin, & Y.S. Lincoln (Eds.), Handbook of qualitative research (105-117). Thousand Oaks, CA: Sage.
Harper, D. (2010). Online Etymology Dictionary. Retrieved May, 12th ,2016, from http://dictionary.reference.com/cite.html?qh=geometry&ia=etymon.
Hart, K.M. (1981). Children's Understanding of Mathematics: 11-16. London: John Murray. Hatano, G. (1996). A conception of knowledge acquisition and its implications for mathematics
education.In P. Steffe, P. Nesher, P. Cobb, G. Goldin & B. Greer (Eds.), Theories of mathematical learning (pp. 197-217). New Jersey: Lawrence Erlbaum.
Haugland, S. W. & Wright, J. L. (1997). Young children and technology: A world of discovery. Massachusetts: Ally and Bacon.
Heid, M. K. (1988). Resequencing skills and concepts in applied calculus using the computer as a tool. Journal for Research in Mathematics Education, 19(3), 3-25.
128
Hiebert, J., &Lefebvre, P. (1986). Conceptual and procedural knowledge in mathematics. An introductory analysis. In J. Hiebert (Ed.), Conceptual and procedural knowledge: the case of mathematics (pp 1-27). Hillsdale, NJ: Lawrence Erlbaum.
Howie, S. (2001). Mathematics and science performance in grade 8 in South Africa 1998/99: TIMMS-R 1999 South Africa. Pretoria: Human Sciences Research Council.
Insook, C. (1999). Mathematical and Pedagogical Discussions of the Function Concept. Journal of the Korea Society of Mathematical Education Series D: Research in Mathematical Education, 3(1), 35-56.
Jansen, J. D. (1998). Curriculum reform in South Africa: A critical analysis of outcomes‐based education. Cambridge journal of education, 28(3), 321-331.
Janvier, C. (1987). Translation processes in mathematics education. In C.Janvier (Ed), Problems of representations in the teaching and learning of mathematics (27-32) Hillsdale, NJ: Lawrence Erlbaum Associates.
Jaworski, B. (1994). Investigating Mathematics Teaching. A Constructivist Enquiry. Washington, DC: The Falmer Press.
Johnson, R. B., & Onwuegbuzie, A. J. (2004). Mixed methods research: a research paradigm whose time has come. Educational researcher, 33(7), 14 - 26.
Klein, D. (1998, 2005). Epistemology. London: Routledge.
Knuth, E. J. (2000).Student understanding of the cartesian connection: An exploratory study. Journal for Research in Mathematics Education, 31(4), 500-508.
Koehler, M. J., & Mishra, P. (2009). What is technological pedagogical content knowledge. Contemporary issues in technology and teacher education, 9(1), 60-70.
Lakatos, I. (1976). Proofs and refutations, the logic of mathematical discovery. Cambridge: Cambridge University Press.
Laridon, P. E.J.M., Brink, M.E., Burgess, A.G., Jawurek, A.M., Kitto, A.L., Myburgh, M.J., et al (2007). Classroom Mathematics Grade 12. Sandton : Heinemann Publishers. Leinhardt, G., Zaslavsky, O. & Stein, M. K. (1990). Functions, graphs and graphing: Tasks,
learning and teaching. Review of Educational Research, 60(1), 1-64.
Luneta, K. (2013). Teaching Elementary Mathematics. Learning to teach elementary mathematics through mentorship and professional development. Saarbrucken: LAP LAMBERT Academic Publishing GmbH & Co. KG.
129
Makonye, J. P., &Luneta, K. (2010). Learner misconceptions in elementary analysis: A case study of a grade 12 class in South Africa. Acta Didactica Napocensia, 3(3), 45-56. Makonye, J., & Hantibi, N. (2014). Exploration of Grade 9 learners’ errors on operations with
directed numbers. Mediterranean Journal of Social Sciences, 5(20), 1564.
Mann, R. (2000). The ADAGE approach to mathematics and the concept of function (Doctoral dissertation, University of Nebraska, Lincoln, 2000).
Mapp T. (2008). Understanding phenomenology: the lived experience. British Journal of Midwifery 16(5), 308-311.
Markovits Z., Eylon B., & Bruckheimer M. (1988). Difficulties students have with thefunction concept. In A. Coxford & A. Shulte (eds), The Ideas of Algebra, K12, 43-60. Reston, VA: N.C.T.M.
Markovits, Z., Eylon, B., & Bruckheimer, M. (1986). Functions today and yesterday. For the Learning of Mathematics, 6(2), 18-24.
Marton, F., & Pang, M. (2006). On some necessary conditions of learning. The Journal of the learning sciences. 15, 2, 193 - 220.
Marton, F., Runesson, U., &Tsui, A. (2004). The space of learning in Marton, F., &Tsui, A.(Eds.), Classroom discourse and the space of learning (pp 3-40). New Jersey: Lawrence Erlbaum Association.
Mason, J. (1996). Expressing generality and roots of algebra. In N. Bednarz, C. Kieran, & L. Lee (Eds.), Approaches to algebra: Perspectives for research and teaching (pp.65–86). Dordrecht: The Netherlands: Kluwer
Merriam, S. B. (1992). Case study research in education: A qualitative approach. San Francisco: Jossey-Bass Inc, Publishers.
Miles, M. B., &Huberman, A. M. (2004). Qualitative data analysis: A sourcebook of new methods. Thousand Oaks, CA: Sage Publications.
Mishra, P., & Koehler, M. (2006). Technological pedagogical content knowledge: A framework for teacher knowledge. The Teachers College Record, 108(6), 1017-1054.
Moschkovich, J., Schoenfeld, A. H., & Arcavi, A. (1993). Aspects of understanding: On multiple perspectives and representations of linear relations and connections among them. In T. A. Romberg, E. Fennema, & T. P. Carpenter (Eds.), Integrating research on the
130
graphicalrepresentation of functions (69-100). Hillsdale, NJ: Lawrence Erlbaum Associates.
Mun Ling, L., & Marton, F. (2011). Towards a science of the art of teaching: Using variation theory as a guiding principle of pedagogical design. International Journal for Lesson and Learning Studies, 1(1), 7-22.
National Council of Teachers of Mathematics. (1989). Principles and standards for school mathematics. Reston, VA: NCTM.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: NCTM.
National Research Council. (2002). Adding it up: Helping children learn mathematics. J.
Kilpatrick, J. Swafford, & B. Findell (Eds.). Mathematics Learning Study Committee, Centre for Education, Division of Behavioral and Social Sciences and Education. Washington, DC: National Academy Press
Nesher, P. (1987). Towards an instructional theory: The role of learners’ misconception for the learning, of mathematics. For the Learning of Mathematics, 7(3), 33-39.
Niess, M. L. (2005). Preparing teachers to teach science and mathematics with technology: Developing a technology pedagogical content knowledge. Teaching and teacher education, 21(5), 509-523.
Olivier, A. (1989) Handling pupils’ misconceptions. Pythagoras. 21, 10 -18.
Olivier, A. (1992). Handling learners' misconceptions. In M. Moodley; R.A, Njisane& N. C. Presmeg (Eds), Mathematics education for in-service and pre-service teachers (pp.193- 209). Pietermaritzburg: Shuter & Shooter.
Pang, M. F. (2002). Making learning possible. The use of variation in the teaching of school economics. Unpublished doctoral dissertation, University of Hong Kong.
Piaget, J. (1968). Development and Learning. Journal of Research in Science Teaching, 40, S8 – S18.
Piaget, J. (1970). Genetic epistemology. New York: Columbia University Press. Piaget, J. (1972). The principles of genetic epistemology. New York: Basic Books.
Pillay, V. (2006). Mathematical knowledge for teaching functions. Paper presented at the 14th Annual meeting of the Southern African Association for Research in Mathematics, Science and Technology Education, University of Pretoria, South Africa.
131
Polya, G. (1973). How to Solve It. Saltburn: Anchor.
Posamentier, A. S. (1998). Tips for the mathematics teacher: Research-based strategiestohelp students learn. CA: Corwin Press, Inc.
Pugalee, D. K. (2001). Algebra for all: The role of technology and constructivism in an algebra course for at-risk students. Preventing School Failure, 45(4), 171-177.
Radatz, H. (1979). Error analysis in mathematics education. Journal for Research in Mathematics Education, 10, 163-172.
Reddy, V. (2006). Mathematics and science achievement at South African schools in TIMSS 2003. Cape Town: Human Sciences Research Council.
Riccomini, P. J. (2005). Identification and remediation of systematic error patterns in subtraction. Learning Disability Quarterly, 28(3), 233-242.
Runesson, U. (2005). Beyond discourse and interaction. Variation: a critical aspect for teaching and learning mathematics. Cambridge journal of education, 35(1), 69-87.
Runesson, U., & Marton, F. (2002). The object of learning and the space of variation. In Marton, F., & Morris, P. (Eds.). What matters? Discovering critical conditions of Chapter 2 (p. 19 - 37).
Runesson, U., Kullberg, A., & Maunula, T. (2006). Sensitivity to student learning: A possible way to enhance teachers’ and students’ learning? In O. Zaslavsky, & P. Sullivan, (Eds.), constructing knowledge for teaching secondary mathematics, Mathematics teacher education 6, (pp. 263 – 275).
Sajka, M. (2005). What subject matter knowledge about the concept of function should the teacher have? Piazza Armerina, Forum of Ideas, Poster Session July 23-29.
Schacter, J. (1999). The impact of education technology on student achievement: What the most current research has to say.
Schoenfeld, A. H. (1985). Mathematical problem solving. Orlando: Academic Press.
Schoenfeld, A. H., Smith, J. P., & Arcavi, A. (1993). Learning: The micro-genetic analysis of one students’ evolving understanding of a complex subject matter domain. In R. Glaser (ed.), Advances in instructional psychology. (Vol. 4, pp. 55 – 175). Hillsdale, NJ: Erlbaum.
132
Schoenfeld, A.H. (1987). What’s all the fuss about metacognition? In A.H. Schoenfeld (Ed.), Cognitive science and mathematics education. Hillsdale, NJ: Lawrence Erlbaum Associates.
Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22(1), 1- 36.
Sfard, A. (1997). Two conceptions of mathematical notions, operational and structural. In A. Borb (Ed.), Proceedings of the 11th Annual Conference of the International Group for the Psychology of Mathematics Education (pp. 162-169). Montreal, Canada.
Shuard, H., & Neill, H. (1986). Teaching calculus. Glasgow: Blackie & Son.
Shulman, L. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4-14.
Siegler, R. S. (1995). Cognitive variability. Developmental science, 10(1), 104-109.
Sierpinska, A. (1992). On understanding the notion of function. In G. Harel& E. Dubinsky (Eds.), The concept of function: Aspects of epistemology and pedagogy Vol 25 (pp.25– 58).New York: Mathematical Association of America.
Skemp, R. R. (1976). Relational understanding and instrumental understanding. Mathematics Teaching, 77, 20-26.
Skemp, R. R. (1987). The psychology of learning mathematics. Hillsdale, NJ: LEA.
Smith, J. P., diSessa S. A., & Roschelle, J. (1993). Misconceptions reconceived: A constructivist analysis of knowledge in transition. The journal of learning sciences, 3(2), 115-163. Spector, J. M. (2004). Multiple uses of information and communication technology in education.
Curriculum, plans, and processes in instructional design: International perspectives, 271-287.
Usiskin, Z. (1988). Conceptions of school algebra and uses of variable. In A. F. Coxford & A. P. Shulte (Eds.), The ideas of algebra, K-12 (pp. 8-19). Reston, VA: NCTM.
Vinner, S., & Dreyfus, T. (1989). Images and definitions for the concept function. Journal for Research in Mathematics Education. 20(4), 356-366.
Von Glassersfeld, E. (1989). Cognition, Construction of Knowledge and Teaching. Kluver Academic Publishers, Netherlands.
133
Wang, L., Bruce, C., & Hughes, H. (2011). Sociocultural theories and their application in information literacy research and education. Australian Academic & Research Libraries, 42(4), 296-308.
Wenglinsky, H. (1998). Does it compute? The relationship between educational technology and student achievement in mathematics.
Wertsch, J. V. (1979). From social interaction to higher psychological processes. London: Human Development.
Wheatley, G. H & Shumway, R. (1992). The potential of calculators to transform elementary school mathematics. In J. T. Fey (Ed.), Calculators in Mathematics Education. Reston: The NCTM: 1-8.
Yerushalmy, M. & Gilead, S. (1997). Solving equations in a technological environment. The Mathematics Teacher, 90(2), 156-162.
Young, R & O’Shea, T. (1981). Errors in children’s subtraction. Cognitive Science, 5, 152-177. Zaslavsky, O. (1997). Conceptual obstacles in the learning of quadratic functions. Focus on
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PRETEST ON FUNCTIONS DATE: June 2016
TOTAL: 40 TIME: 1H30
APPENDIX 1
QUESTION ONE
Find the equations of the given curves
(5) (5)
QUESTION TWO
2.1 On the DIAGRAM SHEET, on the same system of axes, draw the graphs of f(x) = (x+3)2 and g(x) =(x+3) – 4.Clearly show the coordinates of the intercepts (y intercept and the x intercepts)
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2.2 state the domain of f and the range of g (1)
2.3 Explain what has happened to the graph of f(x) to obtain the graph g(x) =(x+3)2 – 4 (2)
QUESTION THREE
A parabola f(x) =ax2 + bx + c with turning point (-32; - 14 ) and a straight-line g(x) = -x +1 4
intersect at the point S (2; 12). The two graphs not drawn to scale are drawn below. A and B are the x intercepts of the parabola. K is the y intercept of f. RT is a straight line parallel to the y- axis.
3.1 Show that a = 1, b = 3 and c = 2. (5)
3.2 Calculate the distance between A and B. (4)
3.3 Determine the length of KM, the distance between the two y intercepts. (3)
3.4 For which values of x is f a decreasing function (2)
3.5 Determine for which values of x is f(x) ≥ g(x), where x≥0 (2)
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137 APPENDIX 2
POST TEST ON FUNCTIONS DATE:
TOTAL: 30 TIME: 1 HOUR
Write yours answers on the answer sheet provided
QUESTION ONE
Find the equations of the following curves
1.1 1.2
(5) (5)
QUESTION TWO
For the following quadratic, function: f(x) = -x2 – 5x -6 a) Find the x and y - intercepts of the function. (2) b) Find the domain and range of the function. (2)
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c) Sketch the graphs of the function. (4)
d) For what values of x are the functions increasing? And for what value of x are the functions decreasing? (2)
QUESTION THREE
Generate the equation of a parabola passing through these given points: