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PROSPECTIVE TEACHERS’ KNOWLEDGE OF CONNECTIONS AMONG EXTERNAL REPRESENTATIONS IN THE CONTEXT OF PROPORTIONALITY

Year 2017, Volume: 2 Issue: 2, 69 - 94, 31.12.2017

Ramazan Avcu

Abstract

Mathematics not only consists of procedures, symbols or operations, but
also involves connections, representations, problem solving, reasoning and
proof, communication, and conceptual understanding. Learners should make
necessary connections among mathematical concepts and representations in order
to develop deep mathematical understandings. For this reason, this study aims
to explore prospective middle school mathematics teachers’ views about
mathematical connections and their knowledge of connections among external
representations in the context of proportionality. Multiple case study was used
as a research design and the participants were three prospective middle school
teachers enrolled in a teacher education program at a public university in the
inner region of Turkey. Data were collected through semi-structured interviews,
note taking, and written tasks. The
findings of the study suggested that prospective teachers viewed connections
mainly as a link between different mathematical topics or concepts, as a link
between mathematics and daily life, and as a tool for improving students’
understanding of mathematics. Moreover, prospective teachers were able to
translate easily among different representations. However, the translations
were carried out without conceptual understanding. Besides, participants had limited understanding of
proportionality in the case of graphical representations. 

Keywords

Prospective teachers mathematical connections external representations proportionality

References

  • Adjiage, R., & Pluvinage, F. (2007). An experiment in teaching ratio and proportion. Educational Studies in Mathematics, 65(2), 149-175.
  • Adu-Gyamfi, K., Bossé, M. J., & Chandler, K. (2017). Student connections between algebraic and graphical polynomial representations in the context of a polynomial relation. International Journal of Science and Mathematics Education, 15(5), 915-938.
  • Akkuş, O., & Duatepe Paksu, A. (2006). Orantısal akıl yürütme becerisi testi ve teste yönelik dereceli puanlama anahtarı geliştirilmesi. Eurasian Journal of Educational Research, 25, 1-10.
  • Ayan, R., & Bostan, M. I. (2016). Middle school students’ reasoning in nonlinear proportional problems in geometry. International Journal of Science and Mathematics Education, Online first, 1-16.
  • Baykul, Y. (2009). İlköğretimde matematik öğretimi (6-8. sınıflar). Ankara: Pegem Akademi Publishing.
  • Chapman, O. (2006). Classroom practices for context of mathematics word problems. Educational Studies in Mathematics, 62, 211-230.
  • Christensen, L. B., Johnson, R. B., & Turner, L. A. (2013). Research methods: Design and analysis. Boston: Pearson.
  • Common Core State Standards Initiative (CCSSI). (2010). Common core state standards for mathematics. Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers.
  • Coulombe, W. N., & Berenson, S. B. (2001). Representations of patterns and functions: Tools for learning. In A. A. Cuoco & F. R. Curcio (Eds.), The roles of representation in school mathematics (2001 Yearbook) (pp. 166–172). Reston, VA: National Council of Teachers of Mathematics.
  • Creswell, J. W. (2008). Educational research: Planning, conducting, and evaluating quantitative and qualitative research (3rd ed). Upper Saddle Creek, NJ: Pearson Education.
  • Daley, G., & Valdes, R. (2006). Value added analysis and classroom observation as measures of teacher performance: A preliminary report (Publication No. 311). Los Angeles: Los Angeles Unified School District; Program Evaluation and Research Branch; Planning, Assessment and Research Division.
  • Davis, R. B., Young, S., & McLoughlin, P. (1982). The roles of "understanding" in the learning of mathematics (Report No. NSF/SED-82008). Urbana/Champaign: Curriculum Laboratory, University of Illinois. (ERIC Document Reproduction Service No. ED220279)
  • De Bock, D., Neyens, D., & Van Dooren, W. (2017). Students’ ability to connect function properties to different types of elementary functions: An empirical study on the role of external representations. International Journal of Science and Mathematics Education, 15(5), 939-955.
  • De Lange, J. (1996). Using and applying mathematics in education. In: A. J. Bishop, K. Clements, C. Keitel, J. Kilpatrick, & C. Laborde (Eds.), International handbook of mathematics education (pp. 49-97). Boston: Kluwer Academic Publishers.
  • Denzin, N. (1988). The research act (Rev. ed.). New York: McGraw-Hill.
  • Driscoll, M. (1999). Fostering algebraic thinking: A guide for teachers grades 6-10. Portsmouth, NH: Heinemann.
  • Eli, J. A., Mohr-Schroeder, M. J., & Lee, C. W. (2011). Exploring mathematical connections of prospective middle-grades teachers through card-sorting tasks. Mathematics Education Research Journal, 23(3), 297-319.
  • Even, R. (1998). Factors involved in linking representations of functions. Journal of Mathematical Behavior, 17(1), 105-121.
  • Fielding, N., & Fielding., J. (1986). Linking data. Newbury Park, CA: Sage.
  • Fuson, K. C., & Abrahamson, D. (2005). Understanding ratio and proportion as an example of the apprehending zone and conceptual-phase problem-solving models. In J. Campbell (Ed.), Handbook of mathematical cognition (pp. 213–234). New York: Psychology Press.
  • Gainsburg, J. (2008). Real-world connections in secondary mathematics teaching. Journal of Mathematics Teacher Education, 11(3), 199-219.
  • Glesne, C., & Peshkin, A. (1992). Becoming qualitative researchers: An introduction. White Plains, NY: Longman.
  • Goldin, G. A., & Kaput J. J. (1996). A joint perspective on the idea of representations in learning and doing mathematics. In S. P., Leslie & N. Pearla (Eds.), Theories of mathematical learning. Mahwah, NJ: Lawrence Erlbaum Associates.
  • Hiebert, J., & Carpenter, T. (1992). Learning and teaching with understanding. In D.A. Grouws (Ed), Handbook of research on mathematics teaching and learning, (pp. 65-97). Reston, VA: National Council of Teachers of Mathematics. Hiebert, J., & Lefevre, 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: Erlbaum.
  • Johnson, B. R., Seigler, R. S., & Alibali, M. W. (2001). Developing conceptual understanding and procedural skill in mathematics: An iterative process. Journal of Educational Psychology, 93(2), 346-362.
  • Kamii, C., Kirkland, L., & Lewis, B. A. (2001). Representation and abstraction in young children’s numerical reasoning. In A. A. Cuoco, & F. R. Curcio (Eds.), The roles of representation in school mathematics (pp. 24-35). Reston: NCTM Publications.
  • Kaput, J. J. (1989). Linking representation in the symbol systems of algebra. Hillsdale, NJ: Earlbaum Associates. Karplus, R., Pulos, S., & Stage, E. K. (1983). Early adolescents’ proportional reasoning on “rate” problems. Educational Studies in Mathematics, 14, 219-234.
  • Klein, P. D. (2003). Rethinking the multiplicity of cognitive resources and curricular representations: Alternatives to ‘learning styles’ and ‘multiple intelligences’. Journal of Curriculum Studies, 35(1), 45-81.
  • Knuth, E. J. (2000). Student understanding of the Cartesian connection: An exploratory study. Journal for Research in Mathematics Education, 31(4), 500-507.
  • Lamon, S. J. (2006). Teaching fractions and ratios for understanding: Essential content knowledge and instructional strategies for teachers (2nd ed.). Mahwah, NJ: Erlbaum.
  • Lamon, S. J. (2007). Rational numbers and proportional reasoning: Toward a theoretical framework for research. In F. K. Lester, Jr. (Ed.), Second handbook of research on mathematics teaching and learning (pp. 629–668). NC: Information Age Publishing.
  • Lee, J. E. (2012). Prospective elementary teachers’ perceptions of real-life connections reflected in posing and evaluating story problems. Journal of Mathematics Teacher Education, 15(6), 429-452.
  • Lesh, R., Post, T., & Behr, M. (1987). Representations and translations among representations in mathematics learning and problem solving. In C. Janvier (Ed.), Problems of representation in the teaching and learning of mathematics (pp. 33-40). Hillsdale, NJ: Erlbaum.
  • Ma, L. (2010). Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and the United States. New York: Taylor & Francis.
  • Merriam, S. (2001). Qualitative research and case study applications in education. San Francisco: Jossey-Bass. Ministry of National Education (2017). Matematik dersi öğretim programı (ilkokul ve ortaokul 1–8.sınıflar) [Elementary and middle school mathematics curriculum: Grades 1-8]. Ankara: Directorate of State Books.
  • National Academy of Sciences (2003). Engaging schools: Fostering high school students’ motivation to learn. Washington: National Academy Press.
  • National Council of Teachers of Mathematics (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: NCTM Publications.
  • National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics. Reston, VA: NCTM Publications.
  • National Council of Teachers of Mathematics (2009). Guiding principles for mathematics curriculum and assessment. Reston, VA: NCTM Publications. National Research Council (1998). High school mathematics at work: Essays and examples for the education of all students. Washington: National Academy Press.
  • Orrill, C. H., & Kittleson, J. M. (2015). Tracing professional development to practice: Connection making and content knowledge in one teacher’s experience. Journal of Mathematics Teacher Education, 18(3), 273-297.
  • Patton, M. Q. (1987). How to use qualitative methods in evaluation. Newbury Park, CA: Sage.
  • Patton, M. Q. (1990). Qualitative evaluation and research methods (2nd ed). Newbury Park, CA: Sage.
  • Porzio, D. T. (1999). Effects of differing emphasis in the use of multiple representations and technology on students understanding of calculus concepts. Focus on Learning Problems in Mathematics, 21(3), 1-29.
  • Post, T. R., Behr, M. J., & Lesh, R. (1988). Proportionality and the development of prealgebra understandings. In A. Coxford & A. Shulte (Eds.), The ideas of algebra, K-12 (pp. 78–90). Reston, VA: National Council of Teachers of Mathematics.
  • Reed, M. K. (1995). Making mathematical connections in middle school. Columbus OH: ERIC Clearinghouse for Science Mathematics and Environmental Education.
  • Umay, A. (2003). Matematiksel muhakeme yeteneği. Hacettepe University Journal of Education, 24, 234-243.
  • Van de Walle, J. A., Karp, K. S., & Bay-Williams, J. M. (2016). Elementary and middle school mathematics: Teaching developmentally (9th ed.). New York, NY: Pearson Education.
  • Van den Heuvel-Panhuizen, M., & Wijers, M. (2005). Mathematics standards and curricula in the Netherlands. ZDM, 37(4), 287-307.
  • Van der Kooij, H., & Goddijn, A. (2011). Algebra in science and engineering. In P. Drijvers (Ed), Secondary algebra education (pp. 203-226). Rotterdam: SensePublishers.
  • Verschaffel, L., Greer, B., & De Corte, E. (2000). Making sense of word problems. Exton: Swets & Zeitlinger Publishers.
  • Yin, R. K. (2003). Case study research: Design and methods (3rd ed.). Thousand Oaks, California: Sage Publications.
  • Zhang, J. (1997). The nature of external representations in problem solving. Cognitive Science, 21(2), 179-217.

Year 2017, Volume: 2 Issue: 2, 69 - 94, 31.12.2017

Ramazan Avcu

Abstract

References

  • Adjiage, R., & Pluvinage, F. (2007). An experiment in teaching ratio and proportion. Educational Studies in Mathematics, 65(2), 149-175.
  • Adu-Gyamfi, K., Bossé, M. J., & Chandler, K. (2017). Student connections between algebraic and graphical polynomial representations in the context of a polynomial relation. International Journal of Science and Mathematics Education, 15(5), 915-938.
  • Akkuş, O., & Duatepe Paksu, A. (2006). Orantısal akıl yürütme becerisi testi ve teste yönelik dereceli puanlama anahtarı geliştirilmesi. Eurasian Journal of Educational Research, 25, 1-10.
  • Ayan, R., & Bostan, M. I. (2016). Middle school students’ reasoning in nonlinear proportional problems in geometry. International Journal of Science and Mathematics Education, Online first, 1-16.
  • Baykul, Y. (2009). İlköğretimde matematik öğretimi (6-8. sınıflar). Ankara: Pegem Akademi Publishing.
  • Chapman, O. (2006). Classroom practices for context of mathematics word problems. Educational Studies in Mathematics, 62, 211-230.
  • Christensen, L. B., Johnson, R. B., & Turner, L. A. (2013). Research methods: Design and analysis. Boston: Pearson.
  • Common Core State Standards Initiative (CCSSI). (2010). Common core state standards for mathematics. Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers.
  • Coulombe, W. N., & Berenson, S. B. (2001). Representations of patterns and functions: Tools for learning. In A. A. Cuoco & F. R. Curcio (Eds.), The roles of representation in school mathematics (2001 Yearbook) (pp. 166–172). Reston, VA: National Council of Teachers of Mathematics.
  • Creswell, J. W. (2008). Educational research: Planning, conducting, and evaluating quantitative and qualitative research (3rd ed). Upper Saddle Creek, NJ: Pearson Education.
  • Daley, G., & Valdes, R. (2006). Value added analysis and classroom observation as measures of teacher performance: A preliminary report (Publication No. 311). Los Angeles: Los Angeles Unified School District; Program Evaluation and Research Branch; Planning, Assessment and Research Division.
  • Davis, R. B., Young, S., & McLoughlin, P. (1982). The roles of "understanding" in the learning of mathematics (Report No. NSF/SED-82008). Urbana/Champaign: Curriculum Laboratory, University of Illinois. (ERIC Document Reproduction Service No. ED220279)
  • De Bock, D., Neyens, D., & Van Dooren, W. (2017). Students’ ability to connect function properties to different types of elementary functions: An empirical study on the role of external representations. International Journal of Science and Mathematics Education, 15(5), 939-955.
  • De Lange, J. (1996). Using and applying mathematics in education. In: A. J. Bishop, K. Clements, C. Keitel, J. Kilpatrick, & C. Laborde (Eds.), International handbook of mathematics education (pp. 49-97). Boston: Kluwer Academic Publishers.
  • Denzin, N. (1988). The research act (Rev. ed.). New York: McGraw-Hill.
  • Driscoll, M. (1999). Fostering algebraic thinking: A guide for teachers grades 6-10. Portsmouth, NH: Heinemann.
  • Eli, J. A., Mohr-Schroeder, M. J., & Lee, C. W. (2011). Exploring mathematical connections of prospective middle-grades teachers through card-sorting tasks. Mathematics Education Research Journal, 23(3), 297-319.
  • Even, R. (1998). Factors involved in linking representations of functions. Journal of Mathematical Behavior, 17(1), 105-121.
  • Fielding, N., & Fielding., J. (1986). Linking data. Newbury Park, CA: Sage.
  • Fuson, K. C., & Abrahamson, D. (2005). Understanding ratio and proportion as an example of the apprehending zone and conceptual-phase problem-solving models. In J. Campbell (Ed.), Handbook of mathematical cognition (pp. 213–234). New York: Psychology Press.
  • Gainsburg, J. (2008). Real-world connections in secondary mathematics teaching. Journal of Mathematics Teacher Education, 11(3), 199-219.
  • Glesne, C., & Peshkin, A. (1992). Becoming qualitative researchers: An introduction. White Plains, NY: Longman.
  • Goldin, G. A., & Kaput J. J. (1996). A joint perspective on the idea of representations in learning and doing mathematics. In S. P., Leslie & N. Pearla (Eds.), Theories of mathematical learning. Mahwah, NJ: Lawrence Erlbaum Associates.
  • Hiebert, J., & Carpenter, T. (1992). Learning and teaching with understanding. In D.A. Grouws (Ed), Handbook of research on mathematics teaching and learning, (pp. 65-97). Reston, VA: National Council of Teachers of Mathematics. Hiebert, J., & Lefevre, 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: Erlbaum.
  • Johnson, B. R., Seigler, R. S., & Alibali, M. W. (2001). Developing conceptual understanding and procedural skill in mathematics: An iterative process. Journal of Educational Psychology, 93(2), 346-362.
  • Kamii, C., Kirkland, L., & Lewis, B. A. (2001). Representation and abstraction in young children’s numerical reasoning. In A. A. Cuoco, & F. R. Curcio (Eds.), The roles of representation in school mathematics (pp. 24-35). Reston: NCTM Publications.
  • Kaput, J. J. (1989). Linking representation in the symbol systems of algebra. Hillsdale, NJ: Earlbaum Associates. Karplus, R., Pulos, S., & Stage, E. K. (1983). Early adolescents’ proportional reasoning on “rate” problems. Educational Studies in Mathematics, 14, 219-234.
  • Klein, P. D. (2003). Rethinking the multiplicity of cognitive resources and curricular representations: Alternatives to ‘learning styles’ and ‘multiple intelligences’. Journal of Curriculum Studies, 35(1), 45-81.
  • Knuth, E. J. (2000). Student understanding of the Cartesian connection: An exploratory study. Journal for Research in Mathematics Education, 31(4), 500-507.
  • Lamon, S. J. (2006). Teaching fractions and ratios for understanding: Essential content knowledge and instructional strategies for teachers (2nd ed.). Mahwah, NJ: Erlbaum.
  • Lamon, S. J. (2007). Rational numbers and proportional reasoning: Toward a theoretical framework for research. In F. K. Lester, Jr. (Ed.), Second handbook of research on mathematics teaching and learning (pp. 629–668). NC: Information Age Publishing.
  • Lee, J. E. (2012). Prospective elementary teachers’ perceptions of real-life connections reflected in posing and evaluating story problems. Journal of Mathematics Teacher Education, 15(6), 429-452.
  • Lesh, R., Post, T., & Behr, M. (1987). Representations and translations among representations in mathematics learning and problem solving. In C. Janvier (Ed.), Problems of representation in the teaching and learning of mathematics (pp. 33-40). Hillsdale, NJ: Erlbaum.
  • Ma, L. (2010). Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and the United States. New York: Taylor & Francis.
  • Merriam, S. (2001). Qualitative research and case study applications in education. San Francisco: Jossey-Bass. Ministry of National Education (2017). Matematik dersi öğretim programı (ilkokul ve ortaokul 1–8.sınıflar) [Elementary and middle school mathematics curriculum: Grades 1-8]. Ankara: Directorate of State Books.
  • National Academy of Sciences (2003). Engaging schools: Fostering high school students’ motivation to learn. Washington: National Academy Press.
  • National Council of Teachers of Mathematics (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: NCTM Publications.
  • National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics. Reston, VA: NCTM Publications.
  • National Council of Teachers of Mathematics (2009). Guiding principles for mathematics curriculum and assessment. Reston, VA: NCTM Publications. National Research Council (1998). High school mathematics at work: Essays and examples for the education of all students. Washington: National Academy Press.
  • Orrill, C. H., & Kittleson, J. M. (2015). Tracing professional development to practice: Connection making and content knowledge in one teacher’s experience. Journal of Mathematics Teacher Education, 18(3), 273-297.
  • Patton, M. Q. (1987). How to use qualitative methods in evaluation. Newbury Park, CA: Sage.
  • Patton, M. Q. (1990). Qualitative evaluation and research methods (2nd ed). Newbury Park, CA: Sage.
  • Porzio, D. T. (1999). Effects of differing emphasis in the use of multiple representations and technology on students understanding of calculus concepts. Focus on Learning Problems in Mathematics, 21(3), 1-29.
  • Post, T. R., Behr, M. J., & Lesh, R. (1988). Proportionality and the development of prealgebra understandings. In A. Coxford & A. Shulte (Eds.), The ideas of algebra, K-12 (pp. 78–90). Reston, VA: National Council of Teachers of Mathematics.
  • Reed, M. K. (1995). Making mathematical connections in middle school. Columbus OH: ERIC Clearinghouse for Science Mathematics and Environmental Education.
  • Umay, A. (2003). Matematiksel muhakeme yeteneği. Hacettepe University Journal of Education, 24, 234-243.
  • Van de Walle, J. A., Karp, K. S., & Bay-Williams, J. M. (2016). Elementary and middle school mathematics: Teaching developmentally (9th ed.). New York, NY: Pearson Education.
  • Van den Heuvel-Panhuizen, M., & Wijers, M. (2005). Mathematics standards and curricula in the Netherlands. ZDM, 37(4), 287-307.
  • Van der Kooij, H., & Goddijn, A. (2011). Algebra in science and engineering. In P. Drijvers (Ed), Secondary algebra education (pp. 203-226). Rotterdam: SensePublishers.
  • Verschaffel, L., Greer, B., & De Corte, E. (2000). Making sense of word problems. Exton: Swets & Zeitlinger Publishers.
  • Yin, R. K. (2003). Case study research: Design and methods (3rd ed.). Thousand Oaks, California: Sage Publications.
  • Zhang, J. (1997). The nature of external representations in problem solving. Cognitive Science, 21(2), 179-217.
There are 52 citations in total.

Details

Subjects Studies on Education
Journal Section Articles
Authors

Ramazan Avcu

Publication Date December 31, 2017
Submission Date December 10, 2017
Published in Issue Year 2017Volume: 2 Issue: 2

Cite

APA Avcu, R. (2017). PROSPECTIVE TEACHERS’ KNOWLEDGE OF CONNECTIONS AMONG EXTERNAL REPRESENTATIONS IN THE CONTEXT OF PROPORTIONALITY. Ihlara Eğitim Araştırmaları Dergisi, 2(2), 69-94.
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