The following is a growing list of references of research studies on students’ algebraic thinking. If you have any reference that you can add, please email us at algebrathinking@gmail.com.

Variables & Expressions

Analysis of Change: Graphing

Algebraic Relations: Equations & Inequalities

Functions & Patterns

Modeling & Word Problems

## Variables & Expressions

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Ainley, J., Bills, L., & Wilson, K. (2004). Constructing Meanings and Utilities within Algebraic Tasks. Paper presented at the 28th Conference of the International Group for the Psychology of Mathematics Education.

Ainley, J., Wilson, K., & Bills, L. (2003). Generalising the Context and Generalising the Calculation: International Group for the Psychology of Mathematics Education.

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Asquith, P., Stephens, A. C., Knuth, E. J., & Alibali, M. W. (2007). Middle School Mathematics Teachers' Knowledge of Students' Understanding of Core Algebraic Concepts: Equal Sign and Variable. Mathematical Thinking and Learning: An International Journal, 9(3), 249-272.

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Battista, M. T., & Van Auken Borrow, C. (1998). Using spreadsheets to promote algebraic thinking. Teaching Children Mathematics, 4(8), 470-478.

Bazzini, L. (1999). On the construction and interpretation of symbolic expressions. Paper presented at the Proceedings of the First Conference of the European Society for Research in Mathematics Education.

Bazzini, L., Boero, P. & Garuti, R. (2001). Moving symbols around or developing understanding: The case of algebraic expressions. Paper presented at the 25th Conference of the International Group for the Psychology of Mathematics Education, Utrecht, Netherlands.

Becker, J. R., & Rivera, F. (2005). Generalization Strategies of Beginning High School Algebra Students. Paper presented at the 29th Conference of the International Group for the Psychology of Mathematics Education, Melbourne, Australia.http://search.ebscohost.com/login.aspx?direct=true&db=eric&AN=ED496917&site=ehost-live

Bergsten, C. (1999). From sense to symbol sense. Paper presented at the Proceedings of the First Conference of the European Society for Research in Mathematics Education.

Bills, L., Wilson, K. & Ainley, J. (2006). Making Links Between Arithmetic and Algebraic Thinking. Research in Mathematic Education, 7, 67-82.

Bills, L., Ainley, J., & Wilson, K. (2003). Particular and General in Early Symbolic Manipulation. Paper presented at the 27th Conference of the International Group for the Psychology of Mathematics Education, Honolulu, Hawaii.

Blanton, M. L. K., J.J. (2002). Deveoping Elementary Teachers' Algebra "Eyes and Ears": Understanding Characteristics of Professional Development that Promote Generative and Self-Sustaining Change in Teacher Practice AERA. Dartmouth: University of Massachusetts Dartmouth.

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Booth, L. R. (1989). Seeing the Pattern: Approaches to Algebra. The Austrailian Mathematics Teacher, 45(3), 12.

Boulton-Lewis, G. M., Cooper, T.J., Atweh, B., Pillay, H., & Wilss, L. (2000, July 23-27). Readiness for Algebra. Paper presented at the 24th annual Psychology of Mathematics Education (PME) Conference, Hiroshima, Japan.

Breiteig, T., & Grevholm, B. (2006). The Transition from Arithmetic to Algebra: To Reason, Explain, Argue, Generalize and Justify. Paper presented at the Proceedings of the 30th Conference of The International Group for the Psychology of Mathematics Education.

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Britt, M., & Irwin, K. (2005). Algebraic Thinking in the Numeracy Project: Year One of a Three- Year Study. Paper presented at the 28th annual conference of the Mathematics Education Research Group of Australasia.

Britt, M., & Irwin, K. (2008). Algebraic thinking with and without algebraic representation: a three-year longitudinal study. ZDM, 40(1), 39-53.

Brown, S. A., & Mehilos, M. (2010). Using Tables to Bridge Arithmetic and Algebra.Mathematics Teaching in the Middle School, 15(9), 532-538.

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Cai, J. K., E. (2005). The development of students' algebraic thinking in earlier grades from curricular, instructional and learning perspectives. ZDM, 37(1), 1-4.

Calder, N., Brown, T., Hanley, U., & Darby, S. (2006). Forming Conjectures within a Spreadsheet Environment. Mathematics Education Research Journal, 18(3), 100-116.

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Carraher, D. B., B.; & Schliemann, A. (2000). Bringing Out the Algebraic Character of Arithmetic: Instantiating Variables in Addition and Subtraction. Paper presented at the 24th annual Psychology of Mathematics Education (PME) Conference, Hiroshima, Japan.

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Chalouh, L., & Herscovics, N. (1988). Teaching algebraic expressions in a meaningful way. In A. F. Coxford (Ed.), The ideas of algebra, K-12 (1988 Yearbook) (pp. 33-42). Reston, VA: National Council of Teachers of Mathematics.

Chazan, D., & Yerushalmy, M. (2003). On appreciating the cognitive compexity of school algebra: Research on algebra learning and directions of curricular change. In J. Kilpatrick, G. Martin & D. Schifter (Eds.), A Research Companion to Principles and Standards for School Mathematics. (pp. 123-135). Reston, VA: NCTM.

Christou, K., Vosniadou, S., & Vamvakoussi, X. (2007). Students' interpretations of literal symbols in algebra. In S. Vaosniadou, Baltas, A., & Vamvakoussi, X. (Ed.),Reframing the conceptual change approach in learning and instruction. New York: Elsevier.

Christou, K., & Vosniadou, S. (2006). Students' Interpretation of the use of Literal Symbols in Algebra- a Conceptual Change Approach. Paper presented at the Proceedings of the 30th Conference of The International Group for the Psychology of Mathematics Education.

Christou, K. V., S. (2005). How students interpret literal symbols in algebra: A conceptual change approach. Paper presented at the Cognitive Science 2005 Conference, Stressa, Italy.

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Coady, C., & Pegg, J. (1993). An exploration of student responses to the more demanding Kuchemann test items. Paper presented at the 16th Annual Mathematics Education Research Group of Australasia Conference.

Collis, K. F. (1976). Levels of Thinking in Elementary Mathematics.

Cortes, A. V., G.; & Kavafian, N. . (1990). From arithmetic to algebra: Negotiating a jump in the learning process. Paper presented at the Psychology of Mathematics Education.

Crowley, L., Thomas, M., & Tall, D. (1994). Algebra, symbols, and translation of meaning. Paper presented at the 18th International Conference for the Psychology of Mathematics Education, Lisbon, Portugal.

Darley, J. (2009). Traveling from arithmetic. Mathematics Teaching in the Middle School, 14(8), 458-464.

Dettori, G., Garuti, R., & Lemut, E. (2001). From arithmetic to algebraic thinking by using a spreadsheet. In R. Sutherland (Ed.), Perspectives on School Algebra (pp. 191-208). Netherlands: Kluwer Academic Publishing.

Dindyal, J. (2004). Algebraic Thinking in Geometry at High School Level: Students’ Use of Variables and Unknowns. 183-190.

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Driscoll, M. (1999). Fostering algebraic thinking: A guide for teachers grades 6-10. Portsmouth, NH: Heinemann.

Driscoll, M. J., & Moyer, J. C. (2001). Using students' work as a lens on algebraic thinking. Mathematics Teaching in the Middle School, 6(5), 282-287.

Dugdale, S. (1998). A Spreadsheet Investigation of Sequences and Series for Middle Grades through Precalculus. Journal of Computers in Mathematics and Science Teaching, 17, 203-222.

Dunkels, A. (1989). What's the next number after G? Journal of Mathematical Behavior, 8, 15-20.

English, L., & Sharry, P. (1996). Analogic reasoning and the development of algebraic abstraction. Educational Studies in Mathematics, 30, 135-157.

English, L. D., & Warren, E. A. (1995). General Reasoning Processes and Elementary Algebraic Understanding: Implications for Initial Instruction. Focus on Learning Problems in Mathematics, 17(4), 1-19.

Ernest, P. (1990). The Meaning of Mathematical Expressions: Does Philosophy Shed Any Light on Psychology? The British Journal for the Philosophy of Science, 41(4), 443-460.

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Falle, J. (2005). Towards a Language-based Model of Students’ Early Algebraic Understandings: Some Preliminary Findings. Paper presented at the 28th annual conference of the Mathematics Education Research Group of Australasia. .

Falle, J. (2007). Students’ Tendency to Conjoin Terms: An Inhibition to their Development of Algebra. Paper presented at the 30th annual conference of the Mathematics Education Research Group of Australasia.

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old. Paper presented at the Sixth Annual Meeting of the North American Chapter of the International Group

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## Modeling & Word Problems/Problem Solving

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