# Student teachers’ mathematics attitudes, authentic investigations and use of metacognitive tools

- 852 Downloads
- 4 Citations

## Abstract

Based on findings from a semester-long study, this article examines the development of Samoan prospective teachers’ mathematical understandings and mathematics attitudes when investigating authentic contexts and applying working mathematically processes, mental computations and problem-solving strategies to find solutions of problems. The prospective teachers had enrolled for the *second* time (having failed their first attempt), in the first-year mathematics methods course of a 2-year Diploma of Education (Primary) programme. The group also included those enrolled in the Diploma of Education (Early Childhood and Special Needs) programmes, who recognizing their own limited understanding of mathematics would ordinarily shy away from opportunities for improvement. Given the negative mathematical and learning experiences, this group was ideal to engage in innovative and creative approaches that would make mathematics learning more meaningful and contextual in a Samoan environment. Only data from the attitudinal questionnaires and interviews are presented in this article. Main findings have implications for teaching and learning mathematics.

## Keywords

Attitudinal change Authentic investigation Metacognitive tools (concept maps and vee diagrams) Working mathematically Problem-solving strategies## Notes

### Acknowledgments

This study was funded by a grant from the National University of Samoa Research Fund.

## References

- Adams, R. J., & Khoo, S. T. (1996).
*QUEST: Interactive item analysis*. Melbourne: Australian Council for Education and Research.Google Scholar - Afamasaga-Fuata’i, K. (1998).
*Learning to solve mathematics problems through concept mapping and vee mapping*. National University of Samoa: Apia.Google Scholar - Afamasaga-Fuata’i, K. (2005). Students’ conceptual understanding and critical thinking? A case for concept maps and vee diagrams in mathematics problem solving. In M. Coupland, J, Anderson, & T. Spencer (Eds.),
*Making mathematics vital. Proceedings of the twentieth biennial conference of the Australian association of mathematics teachers*(AAMT) (pp. 43–52). January 17–21, 2005. University of Technology, Sydney, Australia.Google Scholar - Afamasaga-Fuata’i, K. (2008). Vee diagrams as a problem solving tool: Promoting critical thinking and synthesis of concepts and applications in mathematics. Published on AARE’s website http://www.aare.edu.au/07pap/code07.htm/afa07202.pdf.
- Afamasaga-Fuata’i, K. (2009). Enhancing undergraduate mathematics learning using concept maps and vee diagrams. In K. Afamasaga-Fuata’i (Ed.),
*Concept mapping in mathematics: Research to practice*(pp. 237–257). Norwell, MA: Springer.CrossRefGoogle Scholar - Afamasaga-Fuata’i, K. (2011a). Monitoring teacher trainees’ mathematical competence in an accelerated teacher education program.
*Educational Measurement and Evaluation Review, 2*, 35–61. http://pemea.club.officelive.com/vol2_emereview.aspx. - Afamasaga-Fuata’i, K. (2011b). Students’ attitudes and problem solving with vee diagrams.
*The Assessment Handbook*,*5*, 1–20. http://pemea.club.officelive.com/Documents/A1_V5_AH.pdf. - Afamasaga-Fuata’i, K., & Lauano, A. (2011). Students’ strategies and errors with items on limits of functions.
*PRISMCS Journal of the Faculty of Science,**3*(1), 52–79.Google Scholar - Afamasaga-Fuata’i, K., Meyer, P., & Falo, N. (2007). Primary student teachers’ diagnosed mathematical competence in semester one of their Studies. In J. Watson & K. Beswick (Eds.),
*Mathematics: Essential research, essential practice. Proceedings of the 30th Annual conference of the mathematics education research group of Australasia*(Vol. 1, pp. 83–92), University of Tasmania, Australia, MERGA. http://www.merga.net.au/documents/RP22007.pdf. - Afamasaga-Fuata’i, K., Meyer, P., & Falo, N. (2008). Assessing primary preservice teachers’ mathematical competence. In M. Goos, R. Brown, & K. Makar (Eds.),
*Navigating currents and charting directions*.*Proceedings of the 31st annual conference of the mathematics education research group of Australasia*(Vol. 1, pp. 43–49). University of Queensland, Australia, MERGA. http://www.merga.net.au/documents/RP12008.pdf. - Afamasaga-Fuata’i, K, Meyer, P., & Falo, N. (2010). Mathematically speaking: Where are we now? Where are we going? Snapshots of some secondary and post-secondary students’ mathematics performance.
*Measina a Samoa*(Vol. 4, pp. 125–152). Measina a Samoa IV conference, 15–17 December 2008. NUS, Apia.Google Scholar - Afamasaga-Fuata’i, K., Meyer, P., Falo, N., & Sufia, P. (2007). Future teachers’ developing numeracy and mathematical competence as assessed by two diagnostic tests. Published on AARE’s website http://www.aare.edu.au/06pap/afa06011.pdf.
- Allport, G. W. (1935). Attitudes. In C. Murchison (Ed.),
*A handbook of social psychology*(pp. 798–844). Worcester, MA: Clark University Press.Google Scholar - Andrich, D. (2004). Controversy and the Rasch model: A characteristic or incompatible paradigm?
*Medical Care,**42*, 1–16.CrossRefGoogle Scholar - Australian Association of Mathematics Teachers (AAMT). (2007).
*AAMT standards for excellence in teaching mathematics in Australian schools.*Retrieved on October 6, 2007, from http://www.aamt.edu.au/standards. - Ausubel, D. P. (2000).
*The acquisition and retention of knowledge: A cognitive view*. Dordrecht: Kluwer Academic.CrossRefGoogle Scholar - Ball, D. L. (1990). The mathematical understandings that prospective teachers bring to teacher education.
*Elementary School Journal,**90*, 449–466.CrossRefGoogle Scholar - Ball, D. L., & Bass, H. (2000). Interweaving content and pedagogy in teaching and learning to teach: Knowing and using mathematics. In J. Boaler (Ed.),
*Multiple perspectives on the teaching and learning of mathematics*(pp. 83–104). Westport, CT: Ablex.Google Scholar - Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special?
*Journal of Teacher Education,**59*, 389–407.CrossRefGoogle Scholar - Baroody, A., & Coslick, R. T. (1998).
*Fostering children’s mathematical power: An investigative approach in K-8 mathematics instruction*. Mahwah, NJ: Lawrence Erlbaum.Google Scholar - Bond, T. G., & Fox, C. M. (2001).
*Applying the Rasch model: Fundamental measurement in the human sciences*. Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar - Bragg, L. (2007). Students’ conflicting attitudes towards games as a vehicle for learning mathematics: a methodological dilemma.
*Mathematics Education Research Journal,**19*(1), 29–44.CrossRefGoogle Scholar - Broun, C., Carpenter, T. P., Kouba, V. L., Lindqist, M. M., Silver, E. A., & Swafford, J. O. (1988). Secondary school results of the fourth NAEP mathematics assessment: algebra, geometry, mathematical methods and attitudes.
*Mathematics Teacher,**81*, 337–347.Google Scholar - Cobb, P. (1986). Contexts, goals, beliefs and learning mathematics.
*Journal for the Learning of Mathematics,**6*(2), 2–9.Google Scholar - Cockcroft, W. H. (1986).
*Mathematics counts*. London: Her Majesty’s Stationery Office.Google Scholar - Cohen, J. (1977).
*Statistical power analysis for behavioral sciences*(revised ed.). New York: Academic Press.Google Scholar - Dewey, J. (1938).
*Experience and Education*. Indianapolis, IN: Kappa Delta Pi.Google Scholar - Diezmann, C. M., Watters, J. J & English, L. D. (2001). Implementing mathematical investigations with young children. In
*Proceedings 24th annual conference of the mathematics education research group of Australasia*(pp. 170–177), Sydney.Google Scholar - Dossey, J. A., Mullis, I. V. S., Lindquist, M. M., & Chambers, D. L. (1988).
*The mathematics report card: Trends and achievement based on the 1986 national assessment*. Princeton: Educational Testing Service.Google Scholar - Eagly, A. H., & Chaiken, S. (1993).
*The psychology of attitudes*. Fort Worth, TX: Harcourt Brace Jovanovich.Google Scholar - Eleftherios, K., & Theodosios, Z. (2007). Students’ Beliefs and Attitudes about Studying and Learning Mathematics. In J.-H. Woo, H. Lew, K.-S. Park, D. Seo (Eds.),
*Proceedings of the 31*st*conference of the international group for the psychology of mathematics education PME 31 Seoul, Korea, July 8*–*13, 2007 Vol 3, Research Reports Han*-*Miy.*Google Scholar - Ercikan, K., McCreith, T., & Lapointe, V. (2005). Factors associated with mathematics achievement and participation in advanced mathematics courses: An examination of gender differences from an international perspective.
*School Science and Mathematics,**105*(1), 5–14.CrossRefGoogle Scholar - Ernest, P. (1999). Forms of knowledge in mathematics and mathematics education: Philosophical and rhetorical perspectives.
*Educational Studies in Mathematics,**38*, 67–83.CrossRefGoogle Scholar - Furinghetti, F., & Morselli, F. (2009). Every unsuccessful problem solver is unsuccessful in his or her own way. Affective and cognitive factors in proving.
*Educational Studies in Mathematics,**70*, 71–90.CrossRefGoogle Scholar - Garden, R. A. (1997).
*Mathematics and science performance in middle primary school: Results from the New Zealand’s participation in the Third International Mathematics and Science Study*. Wellington: Research and International Section, Ministry of Education.Google Scholar - Gowin, B. (1981).
*Educating*. Ithaca: Cornell University Press.Google Scholar - Hammond, P., & Vincent, J. (1998). Early mathematics from the keys to life angle. In J. Gough & J. Mousley (Eds.),
*Mathematics: Exploring all angles*(pp. 156–164). Melbourne: Mathematical Association of Victoria.Google Scholar - Hill, H. C., Ball, D. L., & Schilling, S. G. (2008). Unpacking pedagogical content knowledge: Conceptualizing and measuring teachers’ topic-specific knowledge of students.
*Journal for Research in Mathematics Education,**39*(4), 372–400.Google Scholar - Hill, H. C., Rowan, B., & Ball, D. L. (2005). Effects of teachers’ mathematical knowledge for teaching on student achievement.
*American Educational Research Journal,**42*(2), 371–406.CrossRefGoogle Scholar - Hill, H., Schilling, S., & Ball, D. L. (2004). Developing measures of teachers’ mathematics knowledge for teaching.
*The Elementary School Journal,**105*(1), 11–30.CrossRefGoogle Scholar - Hogg, M., & Vaughan, G. (2005).
*Social psychology*(4th ed.). London: Prentice-Hall.Google Scholar - Item Response Theory. (2012).
*Item response theory.*Retrieved from http://en.wikipedia.org/wiki/Item_response_theory. - Jaworski, B. (1986).
*An investigative approach to teaching and learning mathematics*. Milton Keynes: Open University Press.Google Scholar - Kilpatrick, J., Swafford, J., & Findell, B. (Eds.). (2001).
*Adding it up: Helping children learn mathematics*. Washington, DC: National Academy Press.Google Scholar - Kloosterman, P., & Gorman, J. (1990). Building motivation in the elementary mathematics classroom.
*School Science and Mathematics,**90*(5), 375–382.CrossRefGoogle Scholar - Lazarsfeld, P. F., & Henry, N. W. (1968).
*Latent structure analysis*. Boston: Houghton Mifflin.Google Scholar - Leder, G. (1987). Attitudes towards mathematics. In T. A. Romberg & D. M. Stewart (Eds.),
*The monitoring of school mathematics*(Vol. 2, pp. 261–277). Madison, WI: Wisconsin Centre for Education Research.Google Scholar - Lefton, L. A. (1997).
*Psychology*(6th ed.). Boston: Allyn & Bacon.Google Scholar - McLeod, D. B. (1992). Research on affect in mathematics education: A reconceptualization. In D. A. Grouws (Ed.),
*Handbook of research on mathematics teaching and learning*(pp. 575–596). New York: MacMillan.Google Scholar - McLeod, S. A. (2009).
*Attitudes.*Retrieved from http://www.simplypsychology.org/attitudes.html. - National Council of Teachers of Mathematics (NCTM). (2007).
*Executive summary. Principles and standards for school mathematics.*Retrieved on October 4, 2007 from http://standards.nctm.org/document/chapter3/index.htm. - New Zealand Ministry of Education. (2011). The mathematics standards. Available from http://nzcurriculum.tki.org.nz/National-Standards/Mathematics-standards.
- Nisbet, S., & Williams, A. (2009). Improving students’ attitudes to chance with games and activities.
*Australian Mathematics Teacher, 65*, 3. Retrieved from http://www.freepatentsonline.com/article/Australian-Mathematics-Teacher/210224583.html. - Piaget, J. (1972).
*The psychology of the child*. New York: Basic Books.Google Scholar - Pugalee, D. K. (1999). Constructing a model of mathematical literacy.
*The Clearing House,**73*(1), 19–22.CrossRefGoogle Scholar - Rasch Analysis. (2005).
*What is Rasch analysis?*Retrieved from http://www.rasch-analysis.com. - Rasch, G. (1980).
*Probabilistic models for some intelligence and attainment test*(expanded version). Chicago: The University of Chicago Press.Google Scholar - Reynolds, A. J., & Walberg, H. J. (1992). A process model of mathematics achievement and attitude.
*Journal for Research in Mathematics Education,**23*(4), 306–328.CrossRefGoogle Scholar - Romberg, T. A. (1994). Classroom instruction that fosters mathematical thinking and problem solving. In A. H. Schoenfeld (Ed.),
*Mathematical thinking and problem solving*(pp. 287–304). Hillsdale, NJ: Lawrence Erlbaum.Google Scholar - Ryan, J. & McCrae, B. (2005/2006). Assessing Pre-service teachers’ mathematics subject matter knowledge.
*Mathematics Teacher Education and Development, 7*, 72–89.Google Scholar - Samoa’s Ministry of Education, Sports & Culture (SMESC). (2008). Educational Statistical Digest 2008, Part 2, pp. 3–4. http://www.mesc.gov.ws/pdf/edu_stats_digest_2008.pdf.
- Samoan Ministry of Education, Sports and Culture (SMESC). (1995).
*Education strategies: 1995–2005*. Apia: Education Policy and Planning Development Project.Google Scholar - Samoan Ministry of Education, Sports and Culture (SMESC). (2003).
*Primary mathematics and early secondary syllabi*. Apia: MESC.Google Scholar - Samoan Ministry of Education, Sports and Culture (SMESC). (2013).
*Primary mathematics curriculum: Years 1–8*. Apia: MESC.Google Scholar - Schoenfeld, A. H. (1989). Explorations of students’ mathematical beliefs and behaviour.
*Journal for Research in Mathematics Education,**2*(4), 335–338.Google Scholar - Schoenfeld, A. H. (1991). What’s all the fuss about problem solving?
*Zentrallblatt Für Didaktik der Mathematik,**91*(1), 4–8.Google Scholar - Shulman, L. (1986). Those who understand: Knowledge growth in teachers.
*Educational Researcher,**15*(2), 4–14.CrossRefGoogle Scholar - Shulman, L. S. (1987). Knowledge and teaching: Foundations of the new reform.
*Harvard Educational Review,**57*, 1–22.Google Scholar - Smith, R. M. (1990). Theory and practice of fit.
*Rasch Measurement Transactions, 3*(4), 78. http://rasch.org/rmt/rmt34b.htm. - Stacey, K., & Steinle, V. (2006). A case of the inapplicability of the Rasch model: Mapping conceptual learning.
*Mathematics Education Research Journal,**18*(2), 77–92.CrossRefGoogle Scholar - Steen, L. A. (2001). Mathematics and numeracy: Two literacies, one language. Retrieved on June 30, 2005 from http://www.stolaf.edu/people/steen/Papers/numeracy.html.
- Steinberg, J. (2000). Frederic lord, who devised testing yardstick, dies at 87.
*New York Times*, February 10, 2000.Google Scholar - Taylor, L. (1992). Mathematical attitude development from a Vygotskian perspective.
*Mathematics Education Research Journal,**4*(3), 8–23.CrossRefGoogle Scholar - Vygotsky, L. (1978).
*Mind and society*. Cambridge, MA: Harvard University Press.Google Scholar - Weber, A. (1992).
*Social psychology*. New York City, NY: Harper Collins.Google Scholar - Wilkins, J. L. M., Zembylas, M., & Travers, K. J. (2002). Investigating correlates in mathematics and science literacy in the final year of secondary school. In D. F. Robitaille & A. E. Beaton (Eds.),
*Secondary analysis of the TIMMS data*(pp. 291–316). Boston: Kluwer.CrossRefGoogle Scholar - Wiseman, D. C. (1999).
*Research strategies for education*. Belmont: Wadsworth Publishing Company.Google Scholar - Woolfolk, A. & Margetts, K. (2007).
*Educational Psychology.*French’s Forest, NSW: Pearson Education.Google Scholar - Wright, B. D., & Linacre, J. M. (1994). Reasonable mean-square fit values.
*Rasch Measurement Transactions,**8*(3), 370.Google Scholar - Yu, C. H. (2006). A simple guide to the item response theory (IRT). Retrieved from http://www.creative-wisdom.com/computer/sas/IRT.pdf.
- Zan, R., Brown, L., Evans, J., & Hannula, M. (2006). Affect in mathematics education: An introduction.
*Educational Studies in Mathematics,**63*, 113–121.CrossRefGoogle Scholar