Investigating functional thinking in the elementary classroom: Foundations of early algebraic reasoning

Journal of Mathematical Behavior

Volumn 25, Issue 3, 2006, Pages 208-223

  Warren, Elizabeth A ^a,b   Cooper, Tom J ^b,c   Lamb, Janeen T ^a,b  

^a AUSTRALIAN CATHOLIC UNIVERSITY   (Australia)

^b Queensland University of Technology ^*  (Australia)

^c QUEENSLAND UNIVERSITY OF TECHNOLOGY   (Australia)

Author keywords

Algebra; Elementary; Function; Learning; Mathematics

Indexed keywords

EID: 33751318004     PISSN: 07323123     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.jmathb.2006.09.006     Document Type: Article

References (35)

1. Ball S. Participant observation. In: Keeves J.P. (Ed). Educational research, methodology, and measurement: An international handbook (1997), Pergamon, Flinders University of South Australia, Adelaide
2. Brizuela B. Mathematical development in young children: exploring (2004), Teachers' College Press, New York
3. Brizuela B., and Schliemann A. Ten-year old students solving linear equations. For the Learning of Mathematics 24 2 (2004) 33-40
4. Chazan D. Algebra for all students. Journal of Mathematical Behavior 15 4 (1996) 455-477
5. Cobb P., Jaworski B., and Presmeg N. Emergent and social cultural views on mathematical activity. In: Steffe L.P., Mesher P., Cobb P., Goldin G., and Greer B. (Eds). Theories of mathematical learning (1996), Lawrence Erlbaum Associates, Mahwah, NJ
6. Confrey J., and Lachance A. Transformative teaching experiments through conjecture-driven research design. In: Kelly A.E., and Lesh R.A. (Eds). Handbook of research design in mathematics and science education (2000), Lawrence Erlbaum, Mahwah, NJ 231-265
7. Cuoco A. Computational media to support the learning and use of functions. In: diSessa A., Hoyles C., and Noss R. (Eds). Computers and exploratory learning (1995), Springer 79-108
8. Davis R., and Maher C. How students think: The role of representations. In: English L.D. (Ed). Mathematical reasoning: Analogies, metaphors and images (1997), Lawrence Erlbaum Associates, London 93-115
9. Davis, G., & McGowen, M. (2002). Function machines and flexible algebraic thought. Paper presented at the PME26. In A. Cockburn & E. Nardi (Eds.), Proceedings of the 26th conference of the international group for the psychology of mathematics education.
10. Dougherty B. Voyaging from theory to practice in learning: measure up. In: Pateman N., Dougherty B., and Zilliox J. (Eds). Proceedings of the 2003 joint meeting of PME and PMENA 1 (2003), University of Hawaii, Honolulu, HI 17-23
11. Dreyfus T. Advanced mathematical thinking processes. In: Tall D. (Ed). Advanced mathematical thinking (1991), Kluwer Academic Publishers, Dordtrecht 25-41
12. Dubinsky E., and Harel G. The nature of the process conception of function. In: Dubinsky E., and Harel G. (Eds). The concept of function-aspects of epistemology and pedagogy 25 (1992), M.A.A. Notes 85-106
13. English L.D. Analogies, metaphors, and images: Vehicles for mathematical reasoning. In: English L.D. (Ed). Mathematical reasoning: Analogies, metaphors, and images (1997), Lawrence Erlbaum Associates Inc., Mahwah, NJ
14. Henningsen M., and Stein M. Mathematical tasks and student cognition: Classroom-based factors that support and inhibit high-level mathematical thinking and reasoning. Journal for Research in Mathematics Education 28 5 (1997) 524-549
15. Kaput J., and Blanton M. Algebrafying the elementary mathematics experience. In: Chick H., Stacey K., Vincent J., and Vincent J. (Eds). The twelfth ICMI study, on the future of the teaching and learning of algebra 1 (2001), University of Melbourne, Melbourne, Australia 344-352
16. Kress G. Literacy in the new media age (2003), Routledge, London
17. Kress G., and van Leeuwen T. Multimodal discourse (2001), Arnold, London
18. Kruteskii V.A. The psychology of mathematical abilities in schoolchildren (1976), University of Chicago Press, Chicago
19. Malara N., and Navarra G. ArAl Project: Arithmetic pathways towards favouring pre-algebraic thinking (2003), Pitagora Editrice, Bologna, Italy
20. McGowen, M., DeMarios, P., & Tall, D. (2000). Using the function machine as a cognitive root. In M.L. Fernandez (Ed.), Proceedings of the 22nd annual meeting of the North American chapter of the international group for the psychology of mathematical education Vol. 1, pp. 255-261).
21. National Councils of Teachers of Mathematics. Principles and standards for school mathematics (2000), Author, Reston, VA
22. Neria D., and Amit M. Students' preference of non-algebraic representations in mathematical communication. In: Hoines M.J., and Fuglestad A.B. (Eds). Proceedings of the 28th Conference of the International group for the psychology of mathematics education 3 (2004), Bergen University College, Bergen 409-416
23. Nisbet S., and Warren E. Primary school teachers' beliefs relating to mathematics, teaching and assessing mathematics and factors that influence these beliefs. Mathematics teacher education and development 2 (2000) 34-47
24. Presmeg N. Generalization using imagery in mathematics. In: English L.D. (Ed). Mathematical reasoning: analogies, metaphors, and images (1997), Lawrence Erlbaum Associates Inc., Mahwah, NJ
25. Redden T. Patterns language and algebra: A longitudinal study. In: Clarkson P. (Ed). Technology in mathematics education (Proceedings of the 19th annual conference of the Mathematics Education Research Group) (1996), MERGA, Rotorua 469-476
26. Rogoff B. Apprenticeship in thinking: Cognitive development in a social context (1990), Oxford University Press, New York
27. Smith, E. (2003) Representational thinking as a framework for introducing functions in the elementary curriculum. Retrieved August 25, 2006, from Web site: http://www.simcalc.unassd.edu/earlyalgebra/EAdownloads/Smith.pdf.
28. Stacey K., and MacGregor M. The effect of different approaches to algebra on students' perceptions of functional relationships. Mathematics Education Research Journal 7 (1995) 69-85
29. Steffe L.P., von Glaserfeld E., Richards J., and Cobb P. Children's counting types: Philosophy theory and application (1983), Praeger Publishers, New York
30. Tall, D., McGowen, M., & DeMarios, P. (2000). The function machine as a cognitive root for building a rich concept image of the function concept. In M. L. Fernandez (Ed.), Proceedings of the 22nd annual meeting of the North American chapter of the international group for the psychology of mathematical education (Vol. 1, pp. 247-254).
31. Warren, E. (1996). Interactions between instructional approaches, students' reasoning processes, and their understanding of elementary algebra. Unpublished Ph.D. thesis, QUT.
32. Warren, E. (2000). Visualisation and the development of early understanding in algebra. In: T. Nakahara, M. Koyama (Eds.), Proceedings of the 24th conference of the international group for the psychology of mathematics education (Vol. 4). Hiroshima, Japan.
33. Warren, E. (2005). Patterns supporting the development of early algebraic thinking. Paper presented at the MERGA 28-2005 Building Connections: Research, Theory and Practice, Melbourne.
34. Warren, E., & Cooper, T. J. (2001). Theory and practice: Developing an algebra syllabus for Years P-7. In H. Chick, K. Stacey, J. Vincent, & J. Vincent (Eds.), Proceedings of the 12th ICMI study conference: the future of the teaching and learning of Algebra (pp. 641-648).
35. Warren E., and Cooper T. Introducing functional thinking in year 2: A case study of early algebraic thinking. Contemporary Issues in Early Childhood 6 2 (2005) 150-162