Advertisement

IDENTIFICATION AND ASSESSMENT OF TAIWANESE CHILDREN’S CONCEPTIONS OF LEARNING MATHEMATICS

  • Mei-Shiu ChiuEmail author
Article

Abstract

The aim of the present study was to identify children’s conceptions of learning mathematics and to assess the identified conceptions. Children’s conceptions are identified by interviewing 73 grade 5 students in Taiwan. The interviews are analyzed using qualitative data analysis methods, which results in a structure of 5 major conceptions, each having 2 subconceptions: constructivist (interest and understanding), interpretivist (liberty and innovation), objectivist (academic goal and perseverance), nativist (confidence and anxiety (reverse)), and pragmatist (vocational goal and application). The conceptions are assessed with a self-developed questionnaire, titled “the Conception of Learning Mathematics Questionnaire” (CLMQ), which is administered to 513 grade 5 students in Taiwan and examined with a reliability measure, confirmatory factor analysis, and correlations with 2 criteria: mathematics achievement and approaches to learning mathematics. The results show that the CLMQ has desirable internal consistency reliability and construct validity. The conceptions are also sensibly in relation to the 2 criteria, suggesting that the CLMQ is a valid measure for evaluating the quality of children’s learning mathematics in relation to teaching contexts.

Key words

approaches to learning conceptions/beliefs of learning mathematics learning 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Biggs, J. (1993). What do inventories of students’ learning processes really measure? A theoretical review and clarification. The British Journal of Educational Psychology, 63, 3–19.CrossRefGoogle Scholar
  2. Biggs, J. (2001). Enhancing learning: A matter of style or approach? In R. J. Sternberg & L. Zhang (Eds.), Perspectives on thinking, learning, and cognitive styles (pp. 73–102). Mahwah: Lawrence Erlbaum.Google Scholar
  3. Biggs, J., Kember, D. & Leung, D. Y. P. (2001). The revised two-factor Study Process Questionnaire: R-SPQ-2F. The British Journal of Educational Psychology, 71, 133–149.CrossRefGoogle Scholar
  4. Bollen, K. A. & Long, J. S. (1993). Testing structural equation models. Newbury Park: Sage.Google Scholar
  5. Booth, P., Luckett, P. & Mladenovic, R. (1999). The quality of learning in accounting education: The impact of approaches to learning on academic performance. Accounting Education, 8, 277–300.CrossRefGoogle Scholar
  6. Boyle, E. A., Duffy, T. & Dunleavy, K. (2003). Learning styles and academic outcome: The validity and utility of Vermunt’s inventory of learning styles in a British higher education setting. The British Journal of Educational Psychology, 73, 267–290.CrossRefGoogle Scholar
  7. Browne, M. W. & Cudeck, R. (1993). Alternative ways of assessing model fit. In K. A. Bollen & J. S. Long (Eds.), Testing structural equation models (pp. 136–162). Newbury Park: Sage.Google Scholar
  8. Burnett, P. C., Pillay, H. & Dart, B. C. (2003). The influences of conceptions of learning and learner self-concept on high school students’ approaches to learning. School Psychology International, 24, 54–66.CrossRefGoogle Scholar
  9. Burton, L. (2004). “Confidence is everything”—Perspectives of teachers and students on learning mathematics. Journal of Mathematics Teacher Education, 7, 357–381.CrossRefGoogle Scholar
  10. Burton, L. J., Taylor, J. A., Doswling, D. G. & Lawrence, J. (2009). Learning approaches, personality and concepts of knowledge of first-year students: Mature-age versus school leaver. Studies in Learning, Evaluation, Innovation and Development, 6, 65–81.Google Scholar
  11. Cano, F. (2005). Epistemological beliefs and approaches to learning: Their change through secondary school and their influence on academic performance. The British Journal of Educational Psychology, 75, 203–221.CrossRefGoogle Scholar
  12. Cano, F. & Cardelle-Elawar, M. (2004). An integrated analysis of secondary school students’ conceptions and beliefs about learning. European Journal of Psychology of Education, 19, 167–187.CrossRefGoogle Scholar
  13. Charmaz, K. (2000). Grounded theory: Objectivist and constructivist methods. In N. K. Denzin & Y. S. Lincoln (Eds.), Handbook of qualitative research (2nd ed., pp. 509–535). Thousand Oaks: Sage.Google Scholar
  14. Chiu, M.-S. (2009). Affective, cognitive, and social factors in reducing gender differences in measurement and algebra achievements. In M. Tzekaki, M. Kaldrimidou, & C. Sakonidis (Eds.), Proceedings of the 33rd Conference of the International Group for the Psychology of Mathematics Education, 2, pp. 321–328.Google Scholar
  15. Chiu, M.-S. & Whitebread, D. (2011). Taiwanese teachers’ implementation of a new ‘constructivist mathematics curriculum’: How cognitive and affective issues are addressed. International Journal of Educational Development, 31, 196–206.CrossRefGoogle Scholar
  16. Clute, P. S. (1984). Mathematics anxiety, instructional method, and achievement in a survey course in college mathematics. Journal for Research in Mathematics Education, 15, 50–58.CrossRefGoogle Scholar
  17. Crawford, K., Gordon, S., Nicholas, J. & Prosser, M. (1994). Conceptions of mathematics and how it is learned: The perspectives of students entering university. Learning and Instruction, 4, 331–345.CrossRefGoogle Scholar
  18. Crawford, K., Gordon, S., Nicholas, J. & Prosser, M. (1998). Qualitatively different experiences of learning mathematics at university. Learning and Instruction, 8, 455–468.CrossRefGoogle Scholar
  19. Dahl, T. I., Bals, M. & Turi, A. L. (2005). Are students’ beliefs about knowledge and learning associated with their reported use of learning strategies? The British Journal of Educational Psychology, 75, 257–273.CrossRefGoogle Scholar
  20. Dart, B. C., Burnett, P. C., Purdie, N., Boulton-Lewis, G., Campbell, J. & Smith, D. (2000). Students' conceptions of learning, the classroom environment, and approaches to learning. The Journal of Educational Research, 93, 262–270.CrossRefGoogle Scholar
  21. Diseth, A. & Martinsen, O. (2003). Approaches to learning, cognitive styles, and motives as predictors of academic achievement. Educational Psychology, 23, 195–207.CrossRefGoogle Scholar
  22. Driscoll, M. P. (2000). Psychology of learning for instruction (2nd ed.). Needham Height: Allyn & Bacon.Google Scholar
  23. Du Toit, M. & Du Toit, S. H. C. (2001). Interactive LISREL: User’s guide. Lincolnwood: Scientific Software International, Inc.Google Scholar
  24. Duarte, A. M. (2007). Conceptions of learning and approaches to learning in Portuguese students. Higher Education, 54, 781–794.CrossRefGoogle Scholar
  25. Eklund-Myrskog, G. (1998). Students’ conceptions of learning in different educational contexts. Higher Education, 35, 299–316.CrossRefGoogle Scholar
  26. Entwistle, N., McCune, V. & Walker, P. (2001). Conceptions, styles, and approaches within higher education: Analytic abstractions and everyday experience. In R. J. Sternberg & L. Zhang (Eds.), Perspectives on thinking, learning and cognitive styles (pp. 103–136). Mahwah: Lawrence Erlbaum.Google Scholar
  27. Entwistle, N. J. & Peterson, E. R. (2004). Conceptions of learning and knowledge in higher education: Relationships with study behavior and influences of learning environments. International Journal of Educational Research, 41, 407–428.CrossRefGoogle Scholar
  28. Grootenboer, P. & Hemmings, B. (2007). Mathematics performance and the role played by affective and background factors. Mathematics Education Research Journal, 19(3), 3–20.CrossRefGoogle Scholar
  29. Hair, J. F., Jr., Anderson, R. E., Tatham, R. L. & Black, W. C. (1998). Multivariate data analysis (5th ed.). Upper Saddle River: Prentice Hall.Google Scholar
  30. Hair, J. F., Jr., Black, W. C., Babin, B. J., Anderson, R. E. & Tatham, R. L. (2006). Multivariate data analysis (6th ed.). Upper Saddle River: Prentice Hall.Google Scholar
  31. Ho, H., Senturk, D., Lam, A. G., Zimmer, J. M., Hong, S., Okamoto, Y., Chiu, S., Nakazawa, Y. & Wang, C. (2000). The affective and cognitive dimensions of mathematics anxiety: A cross-national study. Journal for Research in Mathematics Education, 31, 362–379.CrossRefGoogle Scholar
  32. Joreskog, K. G. & Sorbom, D. (2001). LISREL 8: User’s reference guide. Lincolnwood: Scientific Software International.Google Scholar
  33. Joreskog, K. G. & Sorbom, D. (2005). LISREL 8.72 [computer software]. Lincolnwood: Scientific Software International, Inc.Google Scholar
  34. Juter, K. (2005). Students’ attitudes to mathematics and performance in limits of functions. Mathematics Education Research Journal, 17(2), 91–110.CrossRefGoogle Scholar
  35. Kember, D., Biggs, B. & Leung, D. Y. P. (2004). Examining the multidimensionality of approaches to learning through the development of a revised version of the Learning Process Questionnaire. The British Journal of Educational Psychology, 74, 261–280.CrossRefGoogle Scholar
  36. Kember, D., Charlesworth, M., Davies, H., McKay, J. & Stott, V. (1997). Evaluating the effectiveness of educational innovations: Using the study process questionnaire to show that meaningful learning occurs. Studies in Educational Evaluation, 23, 141–157.CrossRefGoogle Scholar
  37. Kember, D., Jamieson, Q. W., Pomfret, M. & Wong, E. T. T. (1995). Learning approaches, study time and academic performance. Higher Education, 29, 329–343.CrossRefGoogle Scholar
  38. Kember, D., Wong, A. & Leung, D. Y. P. (1999). Reconsidering the dimensions of approaches to learning. The British Journal of Educational Psychology, 69, 323–343.CrossRefGoogle Scholar
  39. Klatter, E. B., Lodewijks, H. G. L. C. & Aarnoutse, C. A. J. (2001). Learning conceptions of young students in the final year of primary education. Learning and Instruction, 11, 485–516.CrossRefGoogle Scholar
  40. Kloosterman, P. (2002). Beliefs about mathematics and mathematics learning in the secondary school: Measurement and implications for motivation. In G. C. Leder, E. Pehkonen & G. Torner (Eds.), Beliefs: A hidden variable in mathematics education? (pp. 247–269). Dordrecht: Kluwer.Google Scholar
  41. Kloosterman, P. & Stage, F. K. (1992). Measuring beliefs about mathematical problem solving. School Science and Mathematics, 92, 109–115.CrossRefGoogle Scholar
  42. Leder, G. C. & Forgasz, H. J. (2002). Measuring mathematical beliefs and their impact on the learning of mathematics: A new approach. In G. C. Leder, E. Pehkonen & G. Torner (Eds.), Beliefs: A hidden variable in mathematics education? (pp. 95–113). Dordrecht: Kluwer.Google Scholar
  43. Lester, F. K., Jr. (2002). Implications of research on students’ beliefs for classroom practice. In G. C. Leder, E. Pehkonen & G. Torner (Eds.), Beliefs: A hidden variable in mathematics education? (pp. 345–353). Dordrecht: Kluwer.Google Scholar
  44. Leung, D. Y. P., Ginns, P. & Kember, D. (2008). Examining the cultural specificity of approaches to learning in universities in Hong Kong and Sydney. Journal of Cross-Cultural Psychology, 39, 251–266.CrossRefGoogle Scholar
  45. Liem, G. A. D. & Bernardo, A. B. I. (2010). Epistemological beliefs and theory of planned behavior: Examining beliefs about knowledge and knowing as distal predictors of Indonesian tertiary students’ intention to study. The Asia-Pacific Education Researcher, 19, 127–142.CrossRefGoogle Scholar
  46. Liu, P.-H. (2010). Are beliefs believable? An investigation of college students’ epistemological beliefs and behavior in mathematics. Journal of Mathematical Behavior, 29, 86–98.CrossRefGoogle Scholar
  47. Lokan, J. & Greenwood, L. (2000). Mathematics achievement at lower secondary level in Australia. Studies in Educational Evaluation, 26, 9–26.CrossRefGoogle Scholar
  48. Malmivuori, M.-L. (2006). Affect and self-regulation. Educational Studies in Mathematics, 63, 149–164.CrossRefGoogle Scholar
  49. Marsh, H. W. & Hau, K. T. (2004). Explaining paradoxical relations between academic self-concepts and achievements: Cross-cultural generalizability of the internal/external frame of reference predictions across 26 countries. Journal of Educational Psychology, 96, 56–67.CrossRefGoogle Scholar
  50. Marshall, D., Summer, M. & Woolnough, B. (1999). Students’ conceptions of learning in an engineering context. Higher Education, 38, 291–309.CrossRefGoogle Scholar
  51. Marton, F. (1981). Phenomenography: Describing conceptions of the world around us. Instructional Science, 10, 177–200.CrossRefGoogle Scholar
  52. Marton, F. (1983). Beyond individual differences. Educational Psychology, 3, 189–303.CrossRefGoogle Scholar
  53. Marton, F., Dall'Alba, G. & Beaty, E. (1993). Conceptions of learning. International Journal of Educational Research, 19, 277–299.Google Scholar
  54. Marton, F. & Saljo, R. (1976). On qualitative differences in learning I: Outcome and process. The British Journal of Educational Psychology, 46, 4–11.CrossRefGoogle Scholar
  55. Marton, F., Watkins, D. & Tang, C. (1997). Discontinuities and continuities in the experience of learning: An interview study of high-school students in Hong Kong. Learning and Instruction, 7, 21–48.CrossRefGoogle Scholar
  56. Mason, L. & Scrivani, L. (2004). Enhancing students’ mathematical beliefs: An intervention study. Learning and Instruction, 14, 153–176.CrossRefGoogle Scholar
  57. McLean, M. (2001). Can we relate conceptions of learning to student academic achievement? Teaching in Higher Education, 6, 399–413.CrossRefGoogle Scholar
  58. McLeod, D. B. (1994). Research on affect and mathematics learning in the JRME: 1970 to the present. Journal for Research in Mathematics Education, 25, 637–647.CrossRefGoogle Scholar
  59. Meece, J. L., Wigfield, A. & Eccles, J. S. (1990). Predictors of mathematics anxiety and its influence on young adolescents’ course enrollment intentions and performance in mathematics. Journal of Educational Psychology, 82, 60–70.CrossRefGoogle Scholar
  60. Meyer, M. R. & Koehler, M. S. (1990). Internal influences on gender differences in mathematics. In E. Fennema & G. C. Leder (Eds.), Mathematics and gender (pp. 60–95). New York: Columbia University, Teachers College.Google Scholar
  61. Miles, M. B. & Huberman, A. M. (1994). Qualitative data analysis: an expanded sourcebook (2nd ed.). Thousand Oaks: Sage.Google Scholar
  62. Ministry of Education in Taiwan (1993). The curriculum standard for primary schools (in Chinese).Google Scholar
  63. Murphy, K. R. & Davidshofer, C. O. (2005). Psychological testing: Principles and applications (6th ed.). Upper Saddle River: Pearson.Google Scholar
  64. Newble, D. I. & Hejka, E. J. (1991). Approaches to learning of medical students and practising physicians: Some empirical evidence and its implications for medical education. Educational Psychology, 11, 333–342.CrossRefGoogle Scholar
  65. Onwuegbuzie, A. J. & Johnson, R. B. (2006). The validity issue in mixed research. Research in the Schools, 13, 48–63.Google Scholar
  66. Op’t Eynde, P., De Corte, E. & Verschaffel, L. (2002). Framing students’ mathematics-related beliefs: A quest for conceptual clarity and a comprehensive categorization. In G. C. Leder, E. Pehkonen & G. Torner (Eds.), Beliefs: A hidden variable in mathematics education? (pp. 13–37). Dordrecht: Kluwer.Google Scholar
  67. Op’t Eynde, P., De Corte, E. & Verschaffel, L. (2006). Epistemic dimensions of students’ mathematics-related belief systems. International Journal of Educational Research, 45, 57–70.CrossRefGoogle Scholar
  68. Österholm, M. (2009). Theories of epistemological beliefs and communication: A unifying attempt. In M. Tzekaki, M. Kaldrimidou & C. Sakonidis (Eds.), Proceedings of the 33rd conference of the international group for the psychology of mathematics education, 4 (pp. 257–264). Thessaloniki, Greece: PME.Google Scholar
  69. Pietsch, J., Walker, R. & Chapman, E. (2003). The relationship among self-concept, self-efficacy and performance in mathematics during secondary school. Journal of Educational Psychology, 95, 589–603.CrossRefGoogle Scholar
  70. Pintrich, P. R. & De Groot, E. V. (1990). Motivational and self-regulated learning components of classroom academic performance. Journal of Educational Psychology, 82, 33–40.CrossRefGoogle Scholar
  71. Presmeg, N. (2002). Beliefs about the nature of mathematics in the bridging of everyday and school mathematical practices. In G. C. Leder, E. Pehkonen & G. Torner (Eds.), Beliefs: A hidden variable in mathematics education? (pp. 293–312). Dordrecht: Kluwer.Google Scholar
  72. Purdie, N. & Hattie, J. (2002). Assessing students’ conceptions of learning. Australian Journal of Educational & Developmental Psychology, 2, 17–32.Google Scholar
  73. Saljo, R. (1979). Learning in the learner’s perspective 1: Some commonsense conceptions. Gothenburg: Institute of Education, University of Gothenburg.Google Scholar
  74. Saljo, R. (1981). Learning approach and outcome: Some empirical observations. Instructional Science, 10, 47–65.CrossRefGoogle Scholar
  75. Schoenfeld, A. H. (1989). Explorations of students’ mathematical beliefs and behavior. Journal of Research in Mathematics Education, 20, 338–355.CrossRefGoogle Scholar
  76. Schommer-Aikins, M., Duell, O. K. & Hutter, R. (2005). Epistemological belief, mathematical problem-solving, and academic performance of middle school students. Elementary School Journal, 105, 289–304.CrossRefGoogle Scholar
  77. Schumacker, R. E. & Lomax, R. G. (1996). A beginner’s guide to structural equation modeling. Mahwah: Erlbaum.Google Scholar
  78. Seegers, G. & Boekaerts, M. (1996). Gender-related differences in self-referenced cognitions in relation to mathematics. Journal for Research in Mathematics Education, 27, 215–240.CrossRefGoogle Scholar
  79. Seegers, G., van Putten, C. M. & de Brabander, C. J. (2002). Goal orientation, perceived task outcome and task demands in mathematics tasks: Effects on students’ attitude in actual task settings. The British Journal of Educational Psychology, 72, 365–384.CrossRefGoogle Scholar
  80. Strauss, A. & Corbin, J. (1990). Basics of qualitative research: Grounded theory procedures and techniques. Newbury Park: Sage.Google Scholar
  81. Strauss, A. & Corbin, J. (1998). Grounded theory methodology: An overview. In N. K. Denzin & Y. S. Lincoln (Eds.), Strategies of qualitative inquiry (pp. 158–183). Thousand Oaks: Sage.Google Scholar
  82. Sullivan, P., Tobias, S. & McDonough, A. (2006). Perhaps the decision of some students not to engage in learning mathematics in school is deliberate. Educational Studies in Mathematics, 62, 81–99.CrossRefGoogle Scholar
  83. Trigwell, K. & Ashwin, P. (2006). An exploratory study of situated conceptions of learning and learning environments. Higher Education, 51, 243–258.CrossRefGoogle Scholar
  84. Tsai, C.-C. (2004). Conceptions of learning science among high school students in Taiwan: A phenomenographic analysis. International Journal of Science Education, 26, 1733–1750.CrossRefGoogle Scholar
  85. Tsai, C.-C. & Kuo, P.-C. (2008). Cram school students’ conceptions of learning and learning science in Taiwan. International Journal of Science Education, 30, 353–375.CrossRefGoogle Scholar
  86. Turner, J. C., Thorpe, P. K. & Meyer, D. K. (1998). Students’ reports of motivation and negative affect: A theoretical and empirical analysis. Journal of Educational Psychology, 90, 758–771.CrossRefGoogle Scholar
  87. Wheeler, D. L. & Montgomery, D. (2009). Community college students’ views on learning mathematics in terms of their epistemological beliefs: A Q method study. Educational Studies in Mathematics, 72, 289–306.CrossRefGoogle Scholar
  88. Whitebread, D. & Chiu, M. S. (2004). Patterns of children’s emotional responses to mathematical problem-solving. Research in Mathematics Education, 6, 129–153.CrossRefGoogle Scholar
  89. Winkler, J. D., Kanouse, D. E. & Ware, J. E. (1982). Controlling for acquiescence response set in scale development. The Journal of Applied Psychology, 67, 555–561.CrossRefGoogle Scholar
  90. Yackel, E. & Rasmussen, C. (2002). Beliefs and norms in the mathematics classroom. In G. C. Leder, E. Pehkonen & G. Torner (Eds.), Beliefs: A hidden variable in mathematics education? (pp. 313–330). Dordrecht: Kluwer.Google Scholar
  91. Zeegers, P. (2001). Approaches to learning in science: A longitudinal study. The British Journal of Educational Psychology, 71, 115–132.CrossRefGoogle Scholar
  92. Zhu, C., Valcke, M. & Schellens, T. (2008). The relationship between epistemological beliefs, learning conceptions, and approaches to study: A cross-cultural structural model? Asia Pacific Journal of Education, 28, 411–423.CrossRefGoogle Scholar
  93. Zimmerman, B. J. (1995). Self-efficacy and educational development. In A. Bandura (Ed.), Self-efficacy in changing societies (pp. 202–231). Cambridge: Cambridge University Press.CrossRefGoogle Scholar

Copyright information

© National Science Council, Taiwan 2011

Authors and Affiliations

  1. 1.Department of EducationNational Chengchi UniversityTaipeiRepublic of China

Personalised recommendations