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

- 502 Downloads
- 10 Citations

## 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.

## References

- 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 - 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 - 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 - Bollen, K. A. & Long, J. S. (1993).
*Testing structural equation models*. Newbury Park: Sage.Google Scholar - 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 - 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 - 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 - 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 - Burton, L. (2004). “Confidence is everything”—Perspectives of teachers and students on learning mathematics.
*Journal of Mathematics Teacher Education, 7*, 357–381.CrossRefGoogle Scholar - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - Diseth, A. & Martinsen, O. (2003). Approaches to learning, cognitive styles, and motives as predictors of academic achievement.
*Educational Psychology, 23*, 195–207.CrossRefGoogle Scholar - Driscoll, M. P. (2000).
*Psychology of learning for instruction*(2nd ed.). Needham Height: Allyn & Bacon.Google Scholar - Du Toit, M. & Du Toit, S. H. C. (2001).
*Interactive LISREL: User’s guide*. Lincolnwood: Scientific Software International, Inc.Google Scholar - Duarte, A. M. (2007). Conceptions of learning and approaches to learning in Portuguese students.
*Higher Education, 54*, 781–794.CrossRefGoogle Scholar - Eklund-Myrskog, G. (1998). Students’ conceptions of learning in different educational contexts.
*Higher Education, 35*, 299–316.CrossRefGoogle Scholar - 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 - 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 - 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 - 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 - 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 - 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 - Joreskog, K. G. & Sorbom, D. (2001).
*LISREL 8: User’s reference guide*. Lincolnwood: Scientific Software International.Google Scholar - Joreskog, K. G. & Sorbom, D. (2005).
*LISREL 8.72*[computer software]. Lincolnwood: Scientific Software International, Inc.Google Scholar - Juter, K. (2005). Students’ attitudes to mathematics and performance in limits of functions.
*Mathematics Education Research Journal, 17*(2), 91–110.CrossRefGoogle Scholar - 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 - 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 - 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 - 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 - 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 - 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 - Kloosterman, P. & Stage, F. K. (1992). Measuring beliefs about mathematical problem solving.
*School Science and Mathematics, 92*, 109–115.CrossRefGoogle Scholar - 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 - 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 - 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 - 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 - 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 - Lokan, J. & Greenwood, L. (2000). Mathematics achievement at lower secondary level in Australia.
*Studies in Educational Evaluation, 26*, 9–26.CrossRefGoogle Scholar - Malmivuori, M.-L. (2006). Affect and self-regulation.
*Educational Studies in Mathematics, 63*, 149–164.CrossRefGoogle Scholar - 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 - Marshall, D., Summer, M. & Woolnough, B. (1999). Students’ conceptions of learning in an engineering context.
*Higher Education, 38*, 291–309.CrossRefGoogle Scholar - Marton, F. (1981). Phenomenography: Describing conceptions of the world around us.
*Instructional Science, 10*, 177–200.CrossRefGoogle Scholar - Marton, F. (1983). Beyond individual differences.
*Educational Psychology, 3*, 189–303.CrossRefGoogle Scholar - Marton, F., Dall'Alba, G. & Beaty, E. (1993). Conceptions of learning.
*International Journal of Educational Research, 19*, 277–299.Google Scholar - 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 - 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 - Mason, L. & Scrivani, L. (2004). Enhancing students’ mathematical beliefs: An intervention study.
*Learning and Instruction, 14*, 153–176.CrossRefGoogle Scholar - McLean, M. (2001). Can we relate conceptions of learning to student academic achievement?
*Teaching in Higher Education, 6*, 399–413.CrossRefGoogle Scholar - 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 - 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 - 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 - Miles, M. B. & Huberman, A. M. (1994).
*Qualitative data analysis: an expanded sourcebook*(2nd ed.). Thousand Oaks: Sage.Google Scholar - Ministry of Education in Taiwan (1993).
*The curriculum standard for primary schools*(in Chinese).Google Scholar - Murphy, K. R. & Davidshofer, C. O. (2005).
*Psychological testing: Principles and applications*(6th ed.). Upper Saddle River: Pearson.Google Scholar - 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 - Onwuegbuzie, A. J. & Johnson, R. B. (2006). The validity issue in mixed research.
*Research in the Schools, 13*, 48–63.Google Scholar - 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 - 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 - 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 - 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 - 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 - 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 - Purdie, N. & Hattie, J. (2002). Assessing students’ conceptions of learning.
*Australian Journal of Educational & Developmental Psychology, 2*, 17–32.Google Scholar - Saljo, R. (1979).
*Learning in the learner’s perspective 1: Some commonsense conceptions*. Gothenburg: Institute of Education, University of Gothenburg.Google Scholar - Saljo, R. (1981). Learning approach and outcome: Some empirical observations.
*Instructional Science, 10*, 47–65.CrossRefGoogle Scholar - Schoenfeld, A. H. (1989). Explorations of students’ mathematical beliefs and behavior.
*Journal of Research in Mathematics Education, 20*, 338–355.CrossRefGoogle Scholar - 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 - Schumacker, R. E. & Lomax, R. G. (1996).
*A beginner’s guide to structural equation modeling*. Mahwah: Erlbaum.Google Scholar - 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 - 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 - Strauss, A. & Corbin, J. (1990).
*Basics of qualitative research: Grounded theory procedures and techniques*. Newbury Park: Sage.Google Scholar - 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 - 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 - Trigwell, K. & Ashwin, P. (2006). An exploratory study of situated conceptions of learning and learning environments.
*Higher Education, 51*, 243–258.CrossRefGoogle Scholar - 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 - 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 - 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 - 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 - 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 - 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 - 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 - Zeegers, P. (2001). Approaches to learning in science: A longitudinal study.
*The British Journal of Educational Psychology, 71*, 115–132.CrossRefGoogle Scholar - 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 - 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