Early Algebraization pp 161-185 | Cite as

# Examining Students’ Algebraic Thinking in a Curricular Context: A Longitudinal Study

## Abstract

This chapter highlights findings from the LieCal Project, a longitudinal project in which we investigated the effects of a *Standards*-based middle school mathematics curriculum (CMP) on students’ algebraic development and compared them to the effects of other middle school mathematics curricula (non-CMP). We found that the CMP curriculum takes a functional approach to the teaching of algebra while non-CMP curricula take a structural approach. The teachers who used the CMP curriculum emphasized conceptual understanding more than did those who used the non-CMP curricula. On the other hand, the teachers who used non-CMP curricula emphasized procedural knowledge more than did those who used the CMP curriculum. When we examined the development of students’ algebraic thinking related to representing situations, equation solving, and making generalizations, we found that CMP students had a significantly higher growth rate on representing-situations tasks than did non-CMP students, but both CMP and non-CMP students had an almost identical growth in their ability to solve equations. We also found that CMP students demonstrated greater generalization abilities than did non-CMP students over the three middle school years.

## Keywords

Middle School Growth Curve Modeling Abstract Strategy Algebraic Thinking Traditional Curriculum## Preview

Unable to display preview. Download preview PDF.

## References

- Bailey, R., Day, R., Frey, P., Howard, A. C., Hutchens, D. T., McClain, K., Moore-Harris, B., Ott, J. M., Pelfrey, R., Price, J., Vielhaber, K., & Willard, T. (2006a).
*Mathematics: Applications and Concepts*(teacher wraparound edition), course 1. Columbus, OH: The McGraw-Hill Company. Google Scholar - Bailey, R., Day, R., Frey, P., Howard, A. C., Hutchens, D. T., McClain, K., Moore-Harris, B., Ott, J. M., Pelfrey, R., Price, J., Vielhaber, K., & Willard, T. (2006b).
*Mathematics: Applications and Concepts*(teacher wraparound edition), course 2. Columbus, OH: The McGraw-Hill Company. Google Scholar - Bailey, R., Day, R., Frey, P., Howard, A. C., Hutchens, D. T., McClain, K., Moore-Harris, B., Ott, J. M., Pelfrey, R., Price, J., Vielhaber, K., & Willard, T. (2006c).
*Mathematics: Applications and Concepts*(teacher wraparound edition), course 3. Columbus, OH: The McGraw-Hill Company. Google Scholar - Bednarz, N., Kieran, C., & Lee, L. (Eds.) (1996).
*Approaches to Algebra: Perspectives for Research and Teaching*. The Netherlands: Kluwer Academic Publishers. Google Scholar - Bruner, J. S. (1990/1998).
*Acts of Meaning*. Cambridge, MA: Harvard University Press. Google Scholar - Cai, J. (2004). Why do U.S. and Chinese students think differently in mathematical problem solving? Exploring the impact of early algebra learning and teachers’ beliefs.
*Journal of Mathematical Behavior*,*23*, 135–167. Google Scholar - Cai, J., & Knuth, E. (Eds.) (2005). Developing algebraic thinking: Multiple perspectives.
*Zentralblatt fuer Didaktik der Mathematik (International Journal on Mathematics Education), 37*(1). Special Issue. Google Scholar - Cai, J., Moyer, J. C., Wang, N., & Nie, B. (2009). Learning from classroom instruction in a curricular content: An analysis of instructional tasks. In S. L. Swars, D. W. Stinson, & S. Lemons-Smith (Eds.),
*Proceedings of the 31st Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education*(Vol. 5, pp. 692–699). Atlanta, GA: Georgia State University. Google Scholar - Cai, J., Wang, N., Moyer, J. C., & Nie, B. (April/May, 2010a).
*Impact of curriculum reform: Evidence of student learning outcomes in the United States*. Paper presented at the annual meeting of American Educational Research Association. April 30–May 4, Denver, Colorado. Google Scholar - Cai, J., Wang, N., Nie, B., Moyer, J. C., & Wang, C. (April/May, 2010b).
*Student learning and classroom instruction using Standards-based and traditional curricula: An analysis of instructional tasks*. Paper presented at the annual meeting of American Educational Research Association. April 30–May 4, Denver, Colorado. Google Scholar - Cai, J., Nie, B., & Moyer, J. C. (2010c). The teaching of equation solving: Approaches in
*Standards*-based and traditional curricula in the United States.*Pedagogies: An International Journal*,*5*, 170–186. Google Scholar - Carpenter, T. P., Franke, M. L., & Levi, L. (2003).
*Thinking Mathematically: Integrating Arithmetic and Algebra in Elementary School*. Potsmouth, NH: Heinemann. Google Scholar - Chazan, D. (2008). The shifting landscape of school algebra in the United States. In C. E. Greenes & R. Rubenstein (Eds.),
*Algebra and Algebraic Thinking in School Mathematics*(pp. 19–33). Reston, VA: NCTM. Google Scholar - Cocking, R. R., & Mestre, J. P. (Eds.) (1988).
*Linguistic and Cultural Influences on Learning Mathematics*. Hillsdale, NJ: Lawrence Erlbaum Associates. Google Scholar - Cuoco, A., Goldenburg, P., & Mark, J. (1996). Habits of mind: An organizing principle for mathematics curricula.
*Journal of Mathematical Behavior*,*15*, 375–402. CrossRefGoogle Scholar - Detterman, D. K. (1993).
*Individual Differences and Cognition*. Norwood, NJ: Ablex. Google Scholar - Doyle, W. (1988). Work in mathematics classes: The context of students’ thinking during instruction.
*Educational Psychologist*,*23*, 167–180. CrossRefGoogle Scholar - Gardner, H. (1991).
*The Unschooled Mind: How Children Think and How Schools Should Teach*. New York: Basic Books. Google Scholar - Hiebert, J., & Wearne, D. (1993). Instructional tasks, classroom discourse, and students’ learning in second-grade arithmetic.
*American Educational Research Journal*,*30*(2), 393–425. Google Scholar - Jones, G., Langrall, C., Thornton, C., & Nisbet, S. (2002). Elementary students’ access to powerful mathematical ideas. In L. D. English (Ed.),
*Handbook of International Research in Mathematics Education*(pp. 113–142). Mahwah, NJ: Erlbaum. Google Scholar - Kaput, J. (1999). Teaching and learning a new algebra. In E. Fennema & T. Romberg (Eds.),
*Mathematics Classrooms that Promote Understanding*(pp. 133–155). Mahwah, NJ: Erlbaum. Google Scholar - Katz, V. J. (Ed.) (2007).
*Algebra: Gateway to a Technological Future*. Washington: The Mathematical Association of America. Google Scholar - Kieran, C. (1996). The changing face of school algebra. In C. Alsina, J. M. Alvarez, B. Hodgson, C. Laborde, & A. Pérez (Eds.),
*Eighth International Conference on Mathematics Education. Selected Lectures*(pp. 271–290). Seville, Spain: S.A.E.M. Thales. Google Scholar - Kieran, C. (2007). Learning and teaching algebra at the middle school through college levels: Building meaning for symbols and their manipulation. In F. K. Lester (Ed.),
*Second Handbook of Research on Mathematics Teaching and Learning*(pp. 707–762). Charlotte, NC: Information Age Publishing; Reston, VA. National Council of Teachers of Mathematics. Google Scholar - Kieran, C., & Chalouh, L. (1993). Prealgebra: The transition from arithmetic to algebra. In D. T. Owens (Ed.),
*Research Ideas for the Classroom: Middle Grades Mathematics*(pp. 179–198). New York: Macmillan Publishing Co. Google Scholar - Lappan, G., Fey, J. T., Fitzgerald, W. M., Friel, S. N., & Phillips, E. D. (2002).
*Connected Mathematics*. Upper Saddle River, NJ: Prentice Hall. Google Scholar - Lave, J. (1988).
*Cognition in Practice: Mind, Mathematics, and Culture in Everyday Life*. Cambridge: Cambridge University Press. CrossRefGoogle Scholar - Lee, L. (1996). An initiation into algebraic culture through generalization activities. In N. Bednarz, C. Kieran, & L. Lee (Eds.),
*Approaches to Algebra: Perspectives for Research and Teaching*. The Netherlands: Kluwer Academic Publishers. Google Scholar - Mason, J. (1996). Expressing generality and roots of algebra. In N. Bednarz, C. Kieran, & L. Lee (Eds.),
*Approaches to Algebra: Perspectives for Research and Teaching*. The Netherlands: Kluwer Academic Publishers. Google Scholar - Mathematical Sciences Education Board (1998).
*The Nature and Role of Algebra in the K-14 Curriculum: Proceedings of a National Symposium*. Washington, DC: National Research Council. Google Scholar - Mayer, R. E. (1987).
*Educational Psychology: A Cognitive Approach*. Boston: Little & Brown. Google Scholar - Moyer, J. C., Cai, J., Laughlin, C., & Wang, N. (2009). The effect of curriculum type on middle grades instruction. In S. L. Swars, D. W. Stinson, & S. Lemons-Smith (Eds.),
*Proceedings of the 31st Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education*(Vol. 5, pp. 201–209). Atlanta, GA: Georgia State University. Google Scholar - Moyer, J. C., Cai, J., Wang, N., & Nie, B. (April/May, 2010).
*Impact of curriculum reform: Evidence of change in classroom practice in the United States*. Paper presented at the annual meeting of American Educational Research Association, April 30–May 4, Denver, Colorado. Google Scholar - National Assessment of Educational Progress (2006).
*Mathematics Framework for the 2007 National Assessment of Educational Progress*. Washington, DC: National Assessment Governing Board. Google Scholar - National Council of Teachers of Mathematics (1989).
*Curriculum and Evaluation Standards for School Mathematics*. Reston, VA: Author. Google Scholar - National Council of Teachers of Mathematics (1991).
*Professional Standards for Teaching Mathematics*. Reston, VA: Author. Google Scholar - National Council of Teachers of Mathematics (2000).
*Principles and Standards for School Mathematics*. Reston, VA: Author. Google Scholar - National Research Council (1989).
*Everybody Counts: A Report to the Nation on the Future of Mathematics Education*. Washington, DC: National Academy Press. Google Scholar - National Research Council (2004).
*On Evaluating Curricular Effectiveness: Judging the Quality of K-12 Mathematics Evaluations*. J. Confrey & V. Stohl (Eds.). Washington, DC: National Academy Press. Google Scholar - Nie, B., Cai, J., & Moyer, J. C. (2009). How a Standards-based mathematics curriculum differs from a traditional curriculum: With a focus on intended treatments of the ideas of variable.
*Zentralblatt fuer Didaktik der Mathematik (International Journal on Mathematics Education)*,*41*(6), 777–792. Google Scholar - Orton, A., & Orton, J. (1999). Pattern and the approach to algebra. In A. Orton (Ed.),
*Pattern in the Teaching and Learning of Mathematics*(pp. 237–268). London: Cassell. Google Scholar - RAND Mathematics Study Panel (2003).
*Mathematical proficiency for all students: Toward a strategic research and development program in mathematics education*(MR-1643-OERI). Google Scholar - Raudenbush, S. W., & Bryk, A. S. (2002).
*Hierarchical Linear Models: Applications and Data Analysis Methods*(2nd ed.). Thousand Oaks, CA: Sage. Google Scholar - Resnick, L. B. (1987). Learning in school and out.
*Educational Researcher*,*16*, 13–19. Google Scholar - Rogoff, B., & Chavajay, P. (1995). What’s become of research on the cultural basis of cognitive development?
*American Psychologist*,*50*, 859–877. CrossRefGoogle Scholar - Senk, S. L., & Thompson, D. R. (Eds.) (2003).
*Standards-Based School Mathematics Curricula: What Are They? What Do Students Learn?*Mahwah, NJ: Lawrence Erlbaum Associates. Google Scholar - Sfard, A. (1995). The development of algebra: Confronting historical and psychological perspectives.
*Journal of Mathematical Behavior*,*14*, 15–39. CrossRefGoogle Scholar - Stein, M. K., Grover, B. W., & Henningsen, M. A. (1996). Building student capacity for mathematical thinking and reasoning: An analysis of mathematical tasks used in reform classrooms.
*American Educational Research Journal*,*33*, 455–488. Google Scholar - Wozniak, R. H., & Fischer, K. W. (1993).
*Development in Context: Acting and Thinking in Specific Environments*. Hillsdale, NJ: Erlbaum. Google Scholar - Zazkis, R., & Liljedahl, P. (2002). Arithmetic sequence as a bridge between conceptual fields.
*Canadian Journal of Mathematics, Science and Technology Education*,*2*(1), 93–120. Google Scholar