The Center for Algebraic Thinking

 

Algebra thinking References

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

Ainley, J. (1996). Purposeful Contexts for Formal Notation in a Spreadsheet Environment. Journal of Mathematical Behavior, 15(4), 405-422.

Ainley, J. (1999). Doing algebra type stuff: Emergent algebra in the primary school. Paper presented at the 23rd Conference of the International Group for the Psychology of Mathematics Education, Haifa, Israel.

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.

Arcavi, A. (1994). Symbol Sense: Informal Sense-Making in Formal Mathematics. For the Learning of Mathematics, 14(3), 24-35.

Arzarello, F., Bazzini, L., & Chiappini, G. (1994). The process of naming in algebraic problem solving. Paper presented at the 18th Annual Conference of the International Group for the Psychology of Mathematics Education (PME), Lisbon, Portugal.

Arzarello, F. (1998). The role of natural language in prealgebraic and algebraic thinking. In H. Steinbring, Barolini Bussi, M., & Sierpinska, A. (Ed.), Language and Communication in the Mathematics Classroom (pp. 1-8): National Council of Teachers of Mathematics.

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.

Balachef, N. (1984). French research activities in didactics of mathematics, some key words and related references. Theory of Mathematics Education, Occasional paper 54(Institut fur Didaktich der Mathmatik (IDM), Universitat, Bielefeld.).

Ball, L., Stacey, K., & Pierce, R. (2001). Assessing algebraic expectation. Paper presented at the 24th Annual Mathematics Education Research Group of Australasia Conference, Sydney, Australia.

Baroudi, Z. (2006). Easing students' transition to algebra. Austrailian Mathematics Teacher, 62(2), 28-33.

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.

Booker, G. (1987). Conceptual obstacles to the development of algebraic thinking. Paper presented at the 11th Conference of the International Group for the Psychology of Mathematics Education, Montreal, Quebec.

Booth, L. R. (1988). Children's difficulties in beginning algebra. The ideas of algebra, K-12 (1988 Yearbook) (pp. 20-32). Reston, VA: National Council of Teachers of Mathematics.

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.

Britain., N. C. C. f. G. (1992). Algebra: Some Common Misconceptions: National Curriculum Council for Great Britain.

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.

Burton, M. B. (1988). A Linguistic Basis for Student Difficulties with Algebra. For the Learning of Mathematics, 8(1), 2-7.

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.

Carpenter, T. F., M. (2001). Developing algebraic reasoning in the elementary school: Generalization and proof. In K. S. H. Chick, J. Vincent, & J. Vincent (Ed.), Proceedings ot the 12th ICMI Study Conference: The future of the teaching and learning of algebra (pp. 155-162). Melbourne, Australia: The University of Melbourne.

Carpenter, T. F., Megan; & Levi, Linda. (2003). Thinking Mathematicaly: Integrating Arithmetic and Algebra in the Elementary School. Portsmouth, NH: Heinemann.

Carraher, D., & Schliemann, A. (2007). Early algebra and algebraic reasoning. In F. Lester (Ed.), Second Handbook of Research on Mathematics Teaching and Learning (pp. 669-705): Information Age Publishing.

Carraher, D., Schliemann, A., & Brizuela, B. (2001). Can young students operate on unknowns? Paper presented at the 25th Conference of the International Group for the Psychology of Mathematics Education., Utrecht, Netherlands.

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.

Cauzinille-Marmeche, E., Mathieu, J., & Resnick, L. B. . (1984 (April)). Children's understanding of algebraic and arithmetic expressions. Paper presented at the Annual meeting of the American Educational Research Association, New Orleans, LA.

Cerulli, M., & Mariotti, M. A. (2001). Arithmetic and algebra, continuity or cognitive break? The case of Francesca. Paper presented at the 25th Conference of the International Group for the Psychology of Mathematics Education., Utrecht, Netherlands.

Chaiklin, S., & Lesgold, S. . (1984). Prealgebra students' knowledge of algebraic tasks with arithmetic expressions. Paper presented at the Annual meeting of the American Education Research Association, New Orleans, LA.

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.

Chun-Yi, L., & Ming-Puu, C. (2008). Bridging the gap between mathematical conjecture and proof through computer-supported cognitive conflicts. Teaching Mathematics & its Applications, 27(1), 1-1.

Clement, J. (1989). Preconceptions.

Clement, J., Narode, R., & Rosnick, P. (1981). Intuitive misconceptions in algebra as a source of math anxiety. Focus on Learning Problems in Mathematics, 3(4), 36-45.

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.

Donovan, I. I. J. E. (2005). Using the Dynamic Power of Microsoft Excel to stand on the Shoulders of GIANTS. Mathematics Teacher, 99(5), 334-339.

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.

Even, R., & Tirosh, D. (2008). Teacher knowledge and understanding of students' mathematical learning and thinking. In L. English (Ed.), Handbook of International Research on Mathematics Education (pp. 202-222). New York: Routledge.

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.

Filloy, E., Rojano, T., & Solares, A. (2010). Problems Dealing With Unknown Quantities and Two Different Levels of Representing Unknowns. Journal for Research in Mathematics Education, 41(1), 52-80.

Filloy, E. R., T. . (1984). From an arithmetical to an algebraic thought: A clinical study with 12-13 years

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

for the Psychology of Mathematics Education, Madison: University of Wisconsin.

Fujii, T. (2003). Probing students' understanding of variables through cognitive conflict problems: Is the concept of variable so difficult for students to understand? Paper presented at the 27th International Group for the Psychology of Mathematics Education Conference, Honolulu, HI.

Fujii, T., & Stephens, M. (2008). Using number sentences to introduce the idea of variable. In C. Greenes & R. Rubenstein (Eds.), Algebra and Algebraic Thinking in School Mathematics(pp. 127-140). Reston, VA: National Council of Teachers of Mathematics.

Garangon, M., Kieran, C., & Boileau, A. (1990). Introducing algebra: A functional approach in a computer environment. Paper presented at the Psychology of Mathematics Education.

Gaudin, N. (2002). Conceptions de fonction et registres de representation: etude de cas au lycee (Conceptions of Function and Representation: Case Study at School). For the Learning of Mathematics, 22(2), 35-47.

Gay, A. S. (2008). Uncovering variables in the context of modeling activities. In C. Greenes & R. Rubenstein (Eds.), Algebra and Algebraic Thinking in School Mathematics (pp. 211-222). Reston, VA: National Council of Teachers of Mathematics.

Goodson-Espy, T. (1998). The roles of reification and reflective abstraction in the development of abstract thought: transitions from arithmetic to algebra. Educational Studies in Mathematics, 36(3), 219-245.

Goulding, M. S., J.; Olwyn, C. (2000). Developing Student Teachers' Understanding of Algebra and Proof. Mathematics Teaching, 172, 18-20.

Gray, E., Pinto, M., Pitta, D., & Tall, D. (1999). Knowledge construction and diverging thinking in elementary & advanced mathematics. Educational Studies in Mathematics, 38(1-3), 111-133.

Gray, J. (2001). Symbols and Suggestions: Communication of Mathematics in Print. Mathematical Intelligencer, 23(2), 59.

Gray, S., Loud, B., & Sokolowski, C. (2005). Undergraduates' errors in using and interpreting variables: A comparative study. Paper presented at the Proceedings of the 27th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Virginia Tech.

Hallagan, J. (2004). A teacher's model of students' algebraic thinking about equivalent expressions. Paper presented at the 28th International Conference of the International Group for the Psychology of Mathematics Education. .

Hallagan, J. (2006). The Case of Bruce: A Teacher’s Model of his Students’ Algebraic Thinking About Equivalent Expressions. Mathematics Education Research Journal, 18(1), 103-123.

Harper, E. W., & University of, B. (1979). The child's interpretation of a numerical variable.

Hart, K. M., Johnson, D.C., Brown, M., Dickson, L., & Clarkson, R. . (1989). Children's Mathematical Frameworks 8-13: A Study of Classroom Teaching.: Windsor: NFER-NELSON. .

Haspekian, M. (2003). Between arithmetic and algebra: A space for the spreadsheet? Contribution to an instrumental approach. Paper presented at the Third Conference of the European Society for Research in Mathematics Education, Pisa: University of Pisa.

Haspekian, M. (2005). An "Instrumental Approach" to study the integration of a computer tool into mathematics teaching: The case of spreadsheets. International Journal of Computers for Mathematical Learning, 10, 109-141.

Hathaway, P., & Prasad, K. (1994). Fruit salad algebra--A Fiji experience. Journal of Educational Studies, 16(2), 90-102.

Heid, M. K. (1996). A technology-intensive functional approach to the emergence of algebraic thinking. In C. K. N. Bednarz, & L. Lee (Ed.), Approaches to algebra: Perspectives for research and teaching (pp. 239-255). Dordrecht, The Netherlands: Kluwer.

Herscovics, N. (1989). Cognitive obstacles encountered in the learning of algebra. In S. W. C. Kieran (Ed.), Research issues in the learning and teaching of algebra (Vol. 4, pp. 60-86). Reston, VA: National Council of Teachers of Mathematics.

Herscovics, N., & Linchevski, L. (1994). A Cognitive Gap between Arithmetic and Algebra. Educational Studies in Mathematics, 27(1), 59-78.

Irwin, K. C., & Britt, M. S. (2005). The Algebraic Nature of Students' Numerical Manipulation in the New Zealand Numeracy Project. Educational Studies in Mathematics, 58(2), 169-188.

Johanning, D. I. (2004). Supporting the development of algebraic thinking in middle school: a closer look at students' informal strategies. The Journal of Mathematical Behavior, 23(4), 371-388.

Kaput, J. (1987). Toward a theory of symbol use in mathematics. Problems of representation in the teaching and learning of mathematics. (pp. 159-196). Hillsdale, NJ: Lawrence Erlbaum Associates.

Kieran, C. (1989). The early learning of algebra: A structural perspective. In S. W. Kieran (Ed.), Research issues in the learning and teaching of algebra. : (pp. (33-56)). Reston, VA: National Council of Teachers of Mathematics.

Kieran, C., Battista, M., & Clements, D. (1991). Helping to make the transition to algebra. Arithmetic Teacher, 38(7), 49-51.

Kieran, C. (2007). Learning and teaching algebra at the middle school through college levels. In F. Lester (Ed.), Second Handbook of Research on Mathematics Teaching and Learning (pp. 707-762): Information Age Publishing.

Kieran, C., & Saldanha, L. (2005). Computer Algebra Systems (CAS) as a Tool for Coaxing the Emergence of Reasoning about Equivalence of Algebraic Expressions: International Group for the Psychology of Mathematics Education.

Kieran, C., & Sfard, A. (1999). Seeing through Symbols: The Case of Equivalent Expressions. Focus on Learning Problems in Mathematics, 21(1), 1-17.

Kinzel, M. T. (1999). Understanding algebraic notation from the students' perspective. Mathematics Teacher, 92(5), 436-441.

Kirshner, D. (2001). The structural algebra option revisited. In R. Sutherland (Ed.), Perspectives on School Algebra (pp. 83-98). Netherlands: Kluwer Academic Publishers.

Kirshner, D., & Awtry, T. (2004). Visual Salience of Algebraic Transformations. Journal for Research in Mathematics Education, 35(4), 224-257.

Knuth, E., Slaughter, M., Choppin, J. & Sutherland, J. (2002). Mapping the conceptual terrain of middle school students' competencies in justifying and proving. Paper presented at the 24th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education.

Knuth, E., Alibali, M., McNeil, N., Weinberg, A., & Stephens, A. (2005). Middle school students' understanding of core algebraic concepts: Equivalence & Variable1. ZDM, 37(1), 68-76.

Kuchemann, D. (1978). Children's understanding of numerical variables. Mathematics in school, 7(4), 23-25.

Kuchemann, D. (1981). Algebra. In K. M. Hart (Ed.), Children's Understanding of Mathematics (pp. 102-119): London: John Murray.

Landy, D., & Goldstone, R. L. (2007). How Abstract Is Symbolic Thought? Journal of Experimental Psychology: Learning, Memory, and Cognition, 33(4), 720-733.

Lannin, J. K. (2003). Developing algebraic reasoning through generalization. Mathematics Teaching in the Middle School, 8(7), 343.

Lannin, J. K. (2005). Generalization and Justification: The Challenge of Introducing Algebraic Reasoning Through Patterning Activities. Mathematical Thinking & Learning, 7(3), 231-258.

Lannin, J. K., Townsend, B. E., Armer, N., Green, S., & Schneider, J. (2008). Developing Meaning for Algebraic Symbols: Possibilities & Pitfalls. Mathematics Teaching in the Middle School, 13(8), 478-483.

Le Teuff, Y. (2005). Exploration on the potential of using spreadsheets to support children’s mathematical thinking. University of Warwick Secondary PGCE 2004-2005.

Lee, L., & Wheeler, D. (1987). Algebraic thinking in high school students: Their conceptions of generalization and justification (research report). Montreal: Concordia University, Department of Mathematics.

Lee, L., & Wheeler, D. (1989). The Arithmetic Connection. Educational Studies in Mathematics, 20(1), 41-54.

Li, X., & Li, Y. (2008). Research on Students' Misconceptions to Improve Teaching and Learning in School Mathematics and Science. School Science and Mathematics, 108(1), 4-7.

Linchevski, L. (1995). Algebra with Numbers and Arithmetic with Letters. The Journal of Mathematical Behavior, 14, 113-120.

Linchevski, L. (2001). Operating on the unknowns: what does it really mean? Paper presented at the 25th Conference of the International Group for the Psychology of Mathematics Education., Utrecht, Netherlands.

Linchevski, L., & Herscovics, N. (1996). Crossing the Cognitive Gap between Arithmetic and Algebra: Operating on the Unknown in the Context of Equations. Educational Studies in Mathematics, 30(1), 39-65.

Linchevski, L., & Livneh, D. (1999). Structure Sense: The Relationship between Algebraic and Numerical Contexts. Educational Studies in Mathematics, 40(2), 173-196.

Livneh, D. (2002). The competition between numbers and structure: why expressions with identical algebraic structures trigger different interpretations. Focus on Learning Problems in Mathematics, 24(2), 1-17.

Livneh, D. L., L. (2007). Algebrification of arithmetic: Developing algebraic structure sense in the context of arithmetic. Paper presented at the Psychology of Mathematics Education 31, Seoul, Korea.

Loveless, T. (2009). The Misplaced Math Student: Lost in Eighth-Grade Algebra. Brown Center Report on American Education (Vol. 8): Brookings.

MacGregor, M., & Stacey, K. (1993). Seeing a pattern and writing a rule. Paper presented at the Seventeenth International Conference for the Psychology of Mathematics Education, University of Tsukuba, Japan.

MacGregor, M., & Stacey, K. (1994). Progress in learning algebra: Temporary and persistent difficulties. Paper presented at the 17th Annual Mathematics Education Research Group of Australasia Conference.

MacGregor, M. E. (1986). A Fresh Look at Fruit Salad. The Austrailian Mathematics Teacher, 42(3), 9-11.

MacGregor, M. E. (1987). Adding X to Y. Australian Association of Mathematics Teachers, 43(4), 12-13.

MacGregor , M. E., & Stacey, K. (1992). A comparison of pattern-based and equation-solving approaches to algebra 362-371.

MacGregor, M. S., K. (1993). What is X? Australian Association of Mathematics Teachers, 49(4), 28-30.

MacGregor, M. S., K. (1997). Students' Understanding of Algebraic Notation: 11-15. Educational Studies in Mathematics, 33(1), 1-19.

Malara, N. I., R. (1999). The interweaving of arithmetic and algebra: Some questions about syntactic and structural aspects and their teaching and learning. Paper presented at the First Conference of the European Society for Research in Mathematics Education.

Malara, N. N., G. ArAl: A project for an early approach to algebraic thinking.

Malisani, E., & Spagnolo, F. (2009). From arithmetical thought to algebraic thought: The role of the “variable”. Educational Studies in Mathematics, 71(1), 19-41.

Martinez, J. G. R. (1988). Helping students understand factors and terms. Mathematics Teacher, 81, 747-751.

Martinez, J. G. R. (2002). Building conceptual bridges from arithmetic to algebra. Mathematics Teaching in the Middle School, 70(6), 326-331.

Mason, J. (1987). What do symbols represent? In C. Janvier (Ed.), Problems of representation in the teaching and learning of mathematics. (pp. 73-82). Hillsdale, NJ: Lawrence Erlbaum Associates.

Mason, J. (1996). Expressing generality and roots of algebra. In C. K. N. Bednarz, & L. Lee (Ed.), Approaches to algebra: Perspectives for research and teaching (pp. 65-86). Dordrecht, Netherlands: Kluwer.

Mason, J., & Pimm, D. (1984). Generic Examples: Seeing the General in the Particular. Educational Studies in Mathematics, 15(3), 277-289.

McArthur, D., Stasz, C., & Zmuidzinas, M. (1990). Tutoring Techniques in Algebra. Cognition and Instruction, 7(3), 197-244.

McIntyre, Z. (2005). An analysis of variable misconceptions before and after various collegiate level math courses.

McNeil, N. M., Weinberg, A., Hattikudur, S., Stephens, A. C., Asquith, P., Knuth, E. J., & Alibali, M. W. (2010). A Is for Apple: Mnemonic Symbols Hinder the Interpretation of Algebraic Expressions. Journal of Educational Psychology, 102(3), 625-634.

Moreli, L. (1992). A visual approach to algebra concepts. The Mathematics Teacher, 85(6), 434-437.

Morelli, L. (1992). A Visual Approach to Algebra Concepts. Mathematics Teacher, 85(6), 434-437.

Moseley, B., & Brenner, M. E. (2009). A Comparison of Curricular Effects on the Integration of Arithmetic and Algebraic Schemata in Pre-Algebra Students. Instructional Science: An International Journal of the Learning Sciences, 37(1), 1-20.

Nathan, M. J., & Kim, S. (2007). Pattern Generalization with Graphs and Words: A Cross-Sectional and Longitudinal Analysis of Middle School Students' Representational Fluency.Mathematical Thinking and Learning: An International Journal, 9(3), 193-219.

Nathan, M. J., & Koellner, K. (2007). A Framework for Understanding and Cultivating the Transition from Arithmetic to Algebraic Reasoning. Mathematical Thinking and Learning: An International Journal, 9(3), 179-192.

Neurath, R. A., & Stephens, L. J. (2006). The Effect of Using Microsoft Excel in a High School Algebra Class. International Journal of Mathematical Education in Science & Technology, 37(6), 721-726.

Nguyen, D. M. (2004). Persistent Student Misconceptions About Algebra and Symbolism: What is that X anyway? Report from the Teacher Quality Project supported by the THECB and TEA.

Nie, B., Cai, J., & Moyer, J. (2009). How a standards-based mathematics curriculum differs from a traditional curriculum: with a focus on intended treatments of the ideas of variable. ZDM, 41(6), 777-792.

Niess, M. L., Sadri, P., & Kwangho, L. (2008). Variables and Spreadsheets Connect with Real-World Problems. Mathematics Teaching in the Middle School, 13(7), 423-431.

Norton, S., & Irvin, J. (2007). A Concrete Approach to Teaching Symbolic Algebra. Paper presented at the The 30th annual conference of the Mathematics Education Research Group of Australasia.

Noss, R. (1986). Constructing a Conceptual Framework for Elementary Algebra through Logo Programming. Educational Studies in Mathematics, 17(4), 335-357.

Novotna, J., & Hoch, M. (2008). How Structure Sense for Algebraic Expressions or Equations Is Related to Structure Sense for Abstract Algebra. Mathematics Education Research Journal, 20(2), 93-104.

Orton, A. (2004). Pattern in the Teaching and Learning of Mathematics. London, UK: Continuum International Publishing Group.

Orton, A. O., J. (2004). Pattern and the approach to algebra. In A. Orton (Ed.), Pattern in the Teaching and Learning of Mathematics. (pp. 104-120). London, UK: Continuum International Publishing Group.

Pegg, J., & Redden, E. (1990). From Number Patterns to Algebra: The Important Link. Australian Mathematics Teacher, 46(2), 19-22.

Pegg, J., & Redden, E. (1990). Procedures for, and Experiences in, Introducing Algebra in New South Wales. Mathematics Teacher, 83(5), 386-391.

Philipp, R. A. (1992). The Many Uses of Algebraic Variables. Mathematics Teacher, 85(7), 557-561.

Pierce, R., & Stacey, K. (2001). A Framework for Algebraic Insight. Paper presented at the 24th Annual Mathematics Education Research Group of Australasia Conference, Sydney, Australia.

Pillay, H., Wilss, L., & Boulton-Lewis, G. (1997). Sequential development of algebra knowledge: A cogntive analysis. Mathematics Education Research Journal, 10(2), 87-102.

Pimm, D. (1995). Symbols and meanings in school mathematics. London: Routledge.

Ploger, D., & Klingler, L. (1997). Spreadsheets, patterns, and algebraic thinking. Teaching Children Mathematics, 3(6), 330.

Prasad, B. S. (1994). Four levels of cognitive functioning in algebra: An empirical verification of Kuchemann's hypothesis. Paper presented at the 17th Annual Mathematics Education Research Group of Australasia Conference.

Putnam, R. T. (1987). Mathematics knowledge for understanding and problem solving. International Journal of Educational Research, 11(6), 687-705.

Radford, L. (2000). Signs and Meanings in Students' Emergent Algebraic Thinking: A Semiotic Analysis. Educational Studies in Mathematics, 42(3), 237-268.

Radford, L., & Puig, L. (2007). Syntax and Meaning as Sensuous, Visual, Historical forms of Algebraic Thinking. Educational Studies in Mathematics, 66(2), 145-164.

Resnick, L. B., Cauzinille-Marmeche, E., & Mathieu, J. (1987). Understanding algebra. In J. A. Sloboda & D. Rogers (Eds.), Cognitive processes in mathematics (pp. 169-203). Oxford: Clarendon Press.

Rojano, T. Mathematics learning in the junior secondary school: Students' access to significant mathematical ideas., 143-163.

Rojano, T. (1996). Developing algebraic aspects of problem solving within a spreadsheet environment. Approaches to Algebra (pp. 137-145). Netherlands: Kluwer Academic Publishers.

Rojano, T. (2003). Mathematics learning in the middle school/junior secondary school: Student access to powerful mathematical ideas. In L. English (Ed.), Handbook of International Research in Mathematics Education (pp. 136-153): Routledge.

Rosnick, P. (1981). Some Misconceptions Concerning the Concept of Variable. Are You Careful About Defining Your Variables? Mathematics Teacher, 74(6), 418-420.

Rosnick, P. (1982). Students' Symbolization Processes in Algebra (pp. 1-29). Washington, D.C.: National Science Foundation.

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Algebraic Relations: Equations & Inequalities

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Functions & Patterns

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Hines, E., Klanderman, D. B., & Khoury, H. A. (2001). The tabular mode: not just another way to represent a function. School Science and Mathematics, 101(7), 362-371.

Hollar, J. C., & Norwood, K. (1999). The Effects of a Graphing-Approach Intermediate Algebra Curriculum on Students' Understanding of Function. Journal for Research in Mathematics Education, 30(2), 220-226.

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Huntzinger Billings, E. M. (2008). Exploring generalization through pictorial growth patterns. In C. Greenes & R. Rubenstein (Eds.), Algebra and Algebraic Thinking in School Mathematics(pp. 279-293). Reston, VA: National Council of Teachers of Mathematics.

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Kalchman, M. (1998). Developing children's intuitive understanding of linear and nonlinear functions in the middle grades. Paper presented at the Annual Meeting of the American Educational Research Association, San Diego.

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Katz, V., & Barton, B. (2007). Stages in the History of Algebra with Implications for Teaching. Educational Studies in Mathematics, 66(2), 185-201. doi: 10.1007/s10649-006-9023-7

Kieran, C. (1993). Functions, graphing, and technology: Integrating research on learning and instruction. In T. A. Romberg, E. Fennema & T. P. Carpenter (Eds.), Integrating research on the graphical representation of functions (pp. 189-237). Hillsdale, NJ: Lawrence Erlbaum Associates Publishers.

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Lannin, J. K., Barker, D. D., & Townsend, B. E. (2006). Recursive and explicit rules: How can we build student algebraic understanding? Journal of Mathematical Behavior, 25(4), 299-317.

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Lin, F.-L., Yang, K.-L., & Chen, C.-Y. (2004). The Features and Relationships of Reasoning, Proving and Understanding Proof in Number Patterns. International Journal of Science and Mathematics Education, 2(2), 227-256.

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Radford, L. (2000). Students' processes of symbolizing in algebra: A semiotic analysis of the production of signs in generalizing tasks. Paper presented at the the 24th Conference of the International Group for the Psychology of Mathematics Education, Hiroshima, Japan.

Radford, L. (2010). Layers of generality and types of generalization in pattern activities. PNA, 4(2), 37-62.

Radford, L., Bardini, C., & Sabena, C. (2007). Perceiving the General: The Multisemiotic Dimension of Students' Algebraic Activity. Journal for Research in Mathematics Education, 38(5), 507-530.

Redden, T. (1996). Patterns, Language and Algebra: A Longitudinal Study. Paper presented at the 19th Annual Mathematics Education Research Group of Australasia Conference.

Rho, K. (2000). A Case Study on the Changes of University Students' Function Concept in a Virtual Environment. Paper presented at the Annual Meeting of the American Educational Research Association New Orleans, LA. http://search.ebscohost.com/login.aspx?direct=true&db=eric&AN=ED440869&site=ehost-live

Richardson, K., Berenson, S., & Staley, K. Prospective elementary teachers use of representation to reason algebraically. The Journal of Mathematical Behavior, 28(2-3), 188-199.

Rivera, F., & Becker, J. (2006). Accounting for sixth graders' generalizaton strategies in algebra. Paper presented at the 28th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Merida, Mexico.

Rivera, F., & Becker, J. (2008). Middle school children’s cognitive perceptions of constructive and deconstructive generalizations involving linear figural patterns. ZDM, 40(1), 65-82.

Rojano, T. S., R. (1993). Towards an Algebraic Notion of Function: The Role of Spreadsheets. Collaboration between the Centro de Investigación y Estudios, Avanzados del IPN, Mexico and the Institute of Education, University of London, UK.

Ron, T. (1999). An Integrated Study of Children's Construction of Improper Fractions and the Teacher's Role in Promoting That Learning. Journal for Research in Mathematics Education, 30(4), 390-416.

Ronda, E. (2009). Growth Points in Students’ Developing Understanding of Function in Equation Form. Mathematics Education Research Journal, 21(1), 31-53.

Rossi Becker, J., & 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.

Sajka, M. (2003). A Secondary School Student's Understanding of the Concept of Function--A Case Study. Educational Studies in Mathematics, 53(3), 229-254.

Saunders, J. D., J. (1988). Relating functions to their graphs. In A. F. Coxford (Ed.), The ideas of algebra, K-12 (1988 Yearbook). Reston, VA: National Council of Teachers of Mathematics.

Schliemann, A. D., Carraher, D. W., & Brizuela, B. M. (2001). When tables become function tables. Paper presented at the 25th Conference of the International Group for the Psychology of Mathematics Education, Utrecht, The Netherlands.

Schwarz, B. B., & Hershkowitz, R. (1999). Prototypes: Brakes or Levers in Learning the Function Concept? The Role of Computer Tools. Journal for Research in Mathematics Education, 30(4), 362-389.

Sfard, A. (1991). On the Dual Nature of Mathematical Conceptions: Reflections on Processes and Objects as Different Sides of the Same Coin. Educational Studies in Mathematics, 22(1), 1-36.

Sfard, A. (1994). Reification as a birth of a metaphor. For the Learning of Mathematics, 14(1), 44-55.

Sheehy, L. (1996). the History of the Function Concept in the Intended High School Curriculum over the Past Century: What Has changed and What Has Remained the Same. 1-17.

Shield, M. (2008). The Function Concept in Middle-Years Mathematics. Australian Mathematics Teacher, 64(2), 36-40.

Sierpinska, A. (1992). On Understanding the Notion of Function. The Concept of Function.

Singh, P. (2000). Understanding the concepts of proportion and ratio among grade nine students in Malaysia. International Journal of Mathematical Education in Science & Technology, 31(4), 579-599.

Slavit, D. (1994). The Effect of Graphing Calculations on Students' Conceptions of Function. Paper presented at the Annual Meeting of the American Educational Research Association, New Orleans, LA.

Slavit, D. (1998). Three women's understandings of algebra in a precalculus course integrated with the graphing calculator. The Journal of Mathematical Behavior, 17(3), 355-372.

Staats, S., & Batteen, C. (2010). Linguistic indexicality in algebra discussions. The Journal of Mathematical Behavior, 29(1), 41-56.

Stohl Drier, H. (2001). Teaching and Learning Mathematics with Interactive Spreadsheets. School Science & Mathematics, 101(4), 170-179.

Tabach, M., Hershkowitz, R., & Schwarz, B. (2001). The struggle towards algebraic generalization and its consolidation. Paper presented at the 25th Conference of The International Group for the Psychology of Mathematics Education. .

Thomas, L. (1971). The concept of function. In M. F. Rosskopf (Ed.), Children's Mathematical Concepts: Six Piagetian Studies in Mathematics Education (pp. 145-172). New York: Teachers College Press.

Thomas, M., Wilson, A., Corballis, M., Lim, V., & Yoon, C. (2010). Evidence from cognitive neuroscience for the role of graphical and algebraic representations in understanding function.ZDM, 42(6), 607-619.

Thompson, P. W. (1994). Students, functions, and the undergraduate curriculum. In A. H. S. E. Dubinsky, & J. J. Kaput (Ed.), Research in Collegiate Mathematics Education (Vol. 4, pp. 21-44). Providence, RI: American Mathematical Society.

Tourniaire, F., & Pulos, S. (1985). Proportional reasoning: A review of the literature. Educational Studies in Mathematics, 16(2), 181-204.

Trigueros, M., & Martínez-Planell, R. (2010). Geometrical representations in the learning of two-variable functions. Educational Studies in Mathematics, 73(1), 3-19.

Ursini, S., & Trigueros, M. (2004). How Do High School Students Interpret Parameters in Algebra? http://search.ebscohost.com/login.aspx?direct=true&db=eric&AN=ED489663&site=ehost-live

Vinner, S., & Dreyfus, T. (1989). Images and Definitions for the Concept of Function. Journal for Research in Mathematics Education, 20(4), 356-366.

Wells, P. J., & Maynard, T. (2004). Using Patterning Problems to Develop Proportional Reasoning. Yearbook, 2004, 21-30.

Williams, C. G. (1998). Using Concept Maps to Assess Conceptual Knowledge of Function. Journal for Research in Mathematics Education, 29(4), 414-421.

Willoughby, S. (2000). Functions from kindergarten through sixth grade. Algebraic Thinking, Grades K-12: Readings from NCTM's School-Based Journals and Other Publications. (pp. 197-201). Reston, VA: NCTM.

Wilson, M. R. (1994). One Preservice Secondary Teacher's Understanding of Function: The Impact of a Course Integrating Mathematical Content and Pedagogy. Journal for Research in Mathematics Education, 25(4), 346-370.

Yerushalmy, M. (1997). Designing Representations: Reasoning about Functions of Two Variables. Journal for Research in Mathematics Education, 28(4), 431-466.

Yerushalmy, M. (2000). Problem Solving Strategies and Mathematical Resources: A Longitudinal View on Problem Solving in a Function Based Approach to Algebra. Educational Studies in Mathematics, 43(2), 125-147.

Yerushalmy, M., & Schwartz, J. (1993). Seizing the opportunity to make algebra mathematically and pedagogically interesting. In T. A. Romberg, E. Fennema & T. P. Carpenter (Eds.),Integrating research on the graphical representation of functions (pp. 41-68). Hillsdale, NJ: Lawrence Erlbaum Associates Publishers.

Yerushalmy, M., & Shternberg, B. (2001). Charting a visual course to the concept of function. The role of representation in school mathematics: Yearbook of the National Council of Teachers of Mathematics (pp. 251-268). Reston, VA: NCTM.

Yerushalmy, M., Shternberg, B., & Gilead, S. (1999). Visualization as a Vehicle for Meaningful Problem Solving in Algebra. . Paper presented at the Conference of the Psychology of Mathematics Education.

Zaslavsky, O. (1997). Conceptual Obstacles in the Learning of Quadratic Functions. Focus on Learning Problems in Mathematics, 19 (Winter)(1), 20-44.

Zazkis, R., & Liljedahl, P. (2002). Generalization of Patterns: The Tension between Algebraic Thinking and Algebraic Notation. Educational Studies in Mathematics, 49(3), 379-402.

Zazkis, R., Liljedahl, P., & Chernoff, E. (2008). The role of examples in forming and refuting generalizations. ZDM, 40(1), 131-141. doi: 10.1007/s11858-007-0065-9

Zbiek, R. M. (1998). Prospective Teachers' Use of Computing Tools to Develop and Validate Functions as Mathematical Models. Journal for Research in Mathematics Education, 29(2), 184-201.


Modeling & Word Problems/Problem Solving

Aiken, L. (1971). Verbal factors and mathematics learning: A review of research. Journal for Research in Mathematics Education, 304-313.

Amit, M., & Fried, M. (2005). Multiple representations in 8th grade algebra lessons: Are learners really getting it? Paper presented at the 29th International Conference of the International Group for the Psychology of Mathematics Education. , Melbourne.

Archetti, A., Armiento, S., Basile, E., Cannizzaro, L., Crocini, P., & Saltarelli, L. (2000). The influence of the sequence of information in the solution of a word problem. Paper presented at the Psychology of Mathematics Education 24, Hiroshima, Japan.

Arzarello, F., Bazzini, L., & Chiappini, G. (1994). The process of naming in algebraic problem solving. Paper presented at the 18th Annual Conference of the International Group for the Psychology of Mathematics Education (PME), Lisbon, Portugal.

Asghari, A., & Tall, D. (2005). Students' experience of equivalence relations: A phenomenographic approach. Paper presented at the 29th International Conference of the International Group for the Psychology of Mathematics Education.

Baker, R., Corbett, A., & Koedinger, K. (2002). Distinct errors arising from a single misconception. Paper presented at the Twenty-Forth Annual Conference of the Cognitive Science Society.

Balacheff, N. (1986). Cognitive versus Situational Analysis of Problem-Solving Behaviors. For the Learning of Mathematics : An International Journal of Mathematics Education, 6(3), 10-12.

Bayat, S., & Tarmizi, R. A. (2010). Assessing Cognitive and Metacognitive Strategies during Algebra Problem Solving Among University Students. Procedia - Social and Behavioral Sciences, 8, 403-410.

Bednarz, N., Radford, L., Janvier, B., & Lepage, A. (1992). Arithmetical and algebraic thinking in problem solving. Paper presented at the the 16th International Conference for the Psychology of Mathematics Education, Durham, NH.

Bell, A., Swan, M., & Taylor, G. (1981). Choice of operations in verbal problems with decimal numbers. Educational Studies in Mathematics, 12, 399-420.

Berdugo, G. C. (2002). Understanding and Solving Algebra Word-Problems in a Second Language AERA. New Orleans: McGilll University.

Bernardo, A. B. I., & Okagaki, L. (1994). Roles of Symbolic Knowledge and Problem-Information Context in Solving Word Problems. Journal of Educational Psychology, 86(2), 212-220.

Blessing, S., & Ross, B. (1996). Content effects in problem categorization and problem solving. Journal of Experimental Psychology, 22(3), 792-810.

Boero, P. (2001). Transformation and anticipation as key processes in algebraic problem solving. In R. Sutherland (Ed.), Perspectives on School Algebra (pp. 99-119). Netherlands: Kluwer.

Booth, J. K., K. (2009). Facilitating the Diagrammatic Advantage for Algebraic Word Problems Paper presented at the Annual meeting of the American Educational Research Association, San Diego, CA.

Booth, J. L., Koedinger, K.R., & Siegler, R.S. (2007). The effect of prior conceptual knowledge on procedural performance and learning in algebra. Poster presented at Paper presented at the The 29th annual meeting of the Cognitive Science Society, Nashville, TN.

Brenner, M., Herman, S., Ho, H.-Z., & Zimmer, J. (1999). Cross-national comparison of representational competence. Journal for Research in Mathematics Education, 30(5), 541-557.

Brenner, M. E., Mayer, R. E., Moseley, B., Brar, T., Duran, R., Reed, B. S., & Webb, D. (1997). Learning by Understanding: The Role of Multiple Representations in Learning Algebra.American Educational Research Journal, 34(4), 663-689.

Burton, M. B. (1988). A Linguistic Basis for Student Difficulties with Algebra. For the Learning of Mathematics, 8(1), 2-7.

Campbell, K. J., Collis, K. F., & Watson, J. M. (1995). Visual Processing during Mathematical Problem Solving. Educational Studies in Mathematics, 28(2), 177-194.

Chinnappan, M. (2010). Cognitive load and modelling of an algebra problem. Mathematics Education Research Journal, 22(2), 8-23.

Cifarelli, V. (1991). Conceptual Structures in Mathematical Problem Solving. Paper presented at the the Annual Meeting of the American Educational Research Association.http://search.ebscohost.com/login.aspx?direct=true&db=eric&AN=ED338642&site=ehost-live

Clement, J., Lochhead, J., & Monk, G. (1981). Translation difficulties in learning mathematics. American Mathematical Monthly, 88, 286-290.

Clement, J. (1982). Algebra Word Problem Solutions: Thought Processes Underlying a Common Misconception. Journal for Research in Mathematics Education, 13(1), 16-30.

Clement, J., Narode, R., & Rosnick, P. (1981). Intuitive misconceptions in algebra as a source of math anxiety. Focus on Learning Problems in Mathematics, 3(4), 36-45.

Cohen, E., & Kanim, S. E. (2005). Factors influencing the algebra ``reversal error''. American Journal of Physics, 73(11), 1072-1078.

Cohors-Fresenborg, E. (1993). Integrating algorithmic and axiomatic ways of thinking in mathematics lessons in secondary schools. Paper presented at the South East Asia Conference on Mathematics Education (SEACME-6) and the Seventh National Conference on Mathematics., Kampus Sukolilo, Surabaya.

Collis, K. F., Romberg, T. A., & Australian Council for Educational Research, H. (1992). Collis-Romberg Mathematical Problem Solving Profiles.

Collis, K. F., Watson, J., & Campbell, J. (1993). Cognitive Functioning in Mathematical Problem Solving during Early Adolescence. Mathematics Education Research Journal, 5(2), 107-123.

Cooper, M. (1986). The Dependence of Multiplicative Reversal on Equation Format. Journal of Mathematical Behavior, 5(2), 115-120.

Curcio, F. R. (1987). Comprehension of Mathematical Relationships Expressed in Graphs. Journal for Research in Mathematics Education, 18(5), 382-393.

De Bock, D., Van Dooren, W., Janssens, D., & Verschaffel, L. (2002). Improper Use of Linear Reasoning: An In-Depth Study of the Nature and the Irresistibility of Secondary School Students' Errors. Educational Studies in Mathematics, 50(3), 311-334.

De Bock, D., Van Dooren, W., Verschaffel, L., & Janssens, D. (2001). Secondary school pupils' improper proportional reasoning: An in-depth study of the nature and persistence of pupils' errors. Paper presented at the 25th International Conference of the International Group for the Psychology of Mathematics Education.

De Bock, D., Verschaffel, L., & Janssens, D. (1998). The predominance of the linear model in secondary school students' solutions of word problems involving length and area of similar plane figures. Educational Studies in Mathematics, 35(1), 65-83.

Elia, I., & Philippou, G. (2004). The functions of pictures in problem solving. Paper presented at the 28th Conference of The International Group for the Psychology of Mathematics Education.

English, L., & Warren, E. (1994). The interaction between general reasoning processes and achievement in algebra and novel problem solving. Paper presented at the 17th Annual Mathematics Education Research Group of Australasia Conference.

Fan, L., & Zhu, Y. (2007). Representation of problem-solving procedures: A comparative look at China, Singapore, and US mathematics textbooks. Educational Studies in Mathematics, 66(1), 61-75.

Fernandez, C., Llinares, S., Dooren, W., Van Dooren, W., De Bock, Dirk, & Verschaffel, L. (2009). Effect of the Number Structure and the Quantity Nature on Secondary School Students' Proportional Reasoning. Paper presented at the Proceedings of the 33rd Conference of The International Group for the Psychology of Mathematics Education.

Filloy, E., & Rojano, T. (2001). Algebraic syntax and word problems solution: First steps. Paper presented at the 25th Conference of The International Group for the Psychology of Mathematics Education.

Filloy, E., Rojano, T., & Solares, A. (2004). Arithmetic/Algebraic Problem-Solving and the Representation of Two Unknown Quantities. Paper presented at the 28th Conference of The International Group for the Psychology of Mathematics Education. http://search.ebscohost.com/login.aspx?direct=true&db=eric&AN=ED489744&site=ehost-live

Fisher, K., Borchert, K., & Bassok, M. (2011). Following the standard form: Effects of equation format on algebraic modeling. Memory & Cognition, 39(3), 502-515.

Fisher, K. M. (1988). The students-and-professors problem revisited. Journal for Research in Mathematics Education, 19, 260-262.

Galbraith, P. L. (1986). The Use of Mathematical Strategies: Factors and Features Affecting Performance. Educational Studies in Mathematics, 17(4), 413-441.

Gay, A. S. (2008). Uncovering variables in the context of modeling activities. In C. Greenes & R. Rubenstein (Eds.), Algebra and Algebraic Thinking in School Mathematics (pp. 211-222). Reston, VA: National Council of Teachers of Mathematics.

Gilead, S., & Yerushalmy, M. (2001). Deep structures of algebra word problems: Is it approach (in)dependent? Paper presented at the 25th Conference of the International Group for the Psychology of Mathematics Education. , Melbourne.

Goldin, G. (2003). Perspectives on representation in mathematical learning and problem solving. In L. English (Ed.), Handbook of International Research in Mathematics Education (pp. 176-201): Routledge.

Gray, E. (1995). Research on the problem of translating natural language sentences into algebra. The Mathematics Educator, 6(2), 41-43.

Guillermo, R., & del Valle, R. (2004). The competent use of the analytic method in the solution of algebraic word problems. Paper presented at the 28th International Conference of the International Group for the Psychology of Mathematics Education.

Hall, R. K., D.; Wenger, E.; Truxaw, C. (1989). Exploring the Episodic Structure of Algebra Story Problem Solving. Cognition and Instruction, 6(3), 223-283.

Hallel, E., & Peled, I. (2001). Composing analogical word problems to promote structure analysis in solving algebra word problems. Paper presented at the 25th International Conference of the International Group for the Psychology of Mathematics Education. .

Herbel-Eisenmann, B., & Smith, J. (1999). Middle school students’ algebra learning: Understanding linear relationships in context. Paper presented at the Annual meeting of the American Educational Research Association.

Hershkovitz, S., & Nesher, P. (2001). Pathway between text and solution of word problems. Paper presented at the 25th Conference of the International Group for the Psychology of Mathematics Education.

Hinsley, D. A., Hayes, J. R., & Simon, H. A. . (1977). From words to equations: Meaning and representation in algebra word problems. In M. A. J. P. Carpenter (Ed.), Comprehension and cognition Hillsdale, NJ: Lawrence Erlbaum.

Hospesova, A., & Novotna, J. (2009). Intentionality and Word Problems in School Dialogue. Paper presented at the Proceedings of the 33rd Conference of The International Group for the Psychology of Mathematics Education.

Hubbard, R. (2004). An Investigation into the Modelling of Word Problems. Paper presented at the 27th annual conference of the Mathematics Education Research Group of Australasia.

Humberstone, J., & Reeve, R. A. (2008). Profiles of algebraic competence. Learning and Instruction, 18(4), 354-367.

Izsak, A. (2000). Inscribing the Winch: Mechanisms by Which Students Develop Knowledge Structures for Representing the Physical World with Algebra. The Journal of the Learning Sciences, 9(1), 31-74.

Izsak, A. (2003). "We Want a Statement That Is Always True": Criteria for Good Algebraic Representations and the Development of Modeling Knowledge. Journal for Research in Mathematics Education, 34(3), 191-227.

Izsak, A. (2004). Students' Coordination of Knowledge When Learning to Model Physical Situations. Cognition and Instruction, 22(1), 81-128.

Izsák, A. (2011). Representational Competence and Algebraic Modeling. In J. Cai & E. Knuth (Eds.), Early Algebraization (pp. 239-258): Springer Berlin Heidelberg.

Izsak, A., Caglayan, G., & Olive, J. (2009). Meta-Representation in an Algebra Classroom. Paper presented at the Proceedings of the 33rd Conference of The International Group for the Psychology of Mathematics Education.

Janvier, C. (1987). Translation Processes in Mathematics Education. 27-32.

Jiang, C., & Chua, B. (2010). Strategies for Solving Three Fraction-Related Word Problems on Speed: A Comparative Study between Chinese and Singaporean Students. International Journal of Science and Mathematics Education., 8(1), 73-96.

Johanning, D. I. (2004). Supporting the development of algebraic thinking in middle school: a closer look at students' informal strategies. The Journal of Mathematical Behavior, 23(4), 371-388.

Johari, A. (2003). Effects of Inductive Multimedia Programs in Mediating Word Problem Translation Misconceptions. Journal of Instructional Psychology, 30(1), 47-68.

Kaput, J. J., & Maxwell West, M. (1994). Missing-Value Proportional Reasoning Problems: Factors Affecting Informal Reasoning Patterns. 235-287.

Kaput, J. J., & Sims-Knight, J. (1983). Errors in translations to algebraic equations: Roots and implications. Focus of Learning Problems in Mathematics, 5(3), 63-78.

Khng, K. H., & Lee, K. (2009). Inhibiting interference from prior knowledge: Arithmetic intrusions in algebra word problem solving. Learning and Individual Differences, 19(2), 262-268.

Koedinger, K., & Nathan, M. J. (2004). The real story behind story problems: Effects of representations on quantitative reasoning. The Journal of the Learning Sciences, 13(2), 129-164.

Koedinger, K. R., Alibali, M. W., & Nathan, M. J. (1999). A developmental model of algebra problem solving: Trade-offs between grounded and abstract representations. Paper presented at the Annual Meeting of the American Educational Research Association, Montreal, Canada.

Koedinger, K. R., Alibali, M. W., & Nathan, M. J. (2008). Trade-Offs Between Grounded and Abstract Representations: Evidence From Algebra Problem Solving. Cognitive Science, 32, 366-397.

Laborde, C. (1990). Language and Mathematics. In P. N. J. Kilpatrick (Ed.), Mathematics and Cognition: A Research Synthesis by the International Group for the Psychology of Mathematics Education (pp. 53-69). Cambridge: Cambridge University Press.

Le Teuff, Y. (2005). Exploration on the potential of using spreadsheets to support children’s mathematical thinking. University of Warwick Secondary PGCE 2004-2005.

Lee, Y. (2002). Students' Understanding of Algebraic Word Problem Solving and Roles of Representation of Algebraic Word Problem Solving. In AERA (Ed.). New Orleans: University of Washington.

Lesh, R. (1987). The evolution of problem representations in the presence of powerful conceptual amplifiers. In C. Janvier (Ed.), Problems of representation in the teaching and learning of mathematics. (pp. 197-206). Hillsdale, NJ: Lawrence Erlbaum Associates.

Lesh, R., & ENGLISH, L. D. (2005). Trends in the Evolution of Models and Modeling Perspectives on Mathematical Learning and Problem Solving. Paper presented at the The 29th International Conference of the International Group for the Psychology of Mathematics Education.

Lesh, R., & Zawojewski, J. (2007). Problem solving and modeling Second Handbook of Research on Mathematics Teaching and Learning (pp. 763-801): Information Age Publishing.

Lochhead, J. M., J. . (1988). From words to algebra: Mending misconceptions. In A. F. Coxford (Ed.), The ideas of algebra, K-12 (1988 Yearbook). Reston, VA: National Council of Teachers of Mathematics.

Lopez-Real, F. (1995). How Important is the Reversal Error in Algebra? Paper presented at the 18th Annual Mathematics Education Research Group of Australasia Conference.

MacGregor, M. (1990). Writing in natural language helps students construct algebraic equations. Mathematics Education Research Journal, 2(2), 1-11.

MacGregor, M., & Stacey, K. (1995). Formulating Equations for Word Problems. Paper presented at the 18th Annual Mathematics Education Research Group of Australasia Conference.

MacGregor, M., & Stacey, K. (1996). Using Algebra to Solve Problems: Selecting, Symbolising, and Integrating Information. Paper presented at the 19th Annual Mathematics Education Research Group of Australasia Conference.

Mathews, S. M. (1997). The Effect of Using Two Variables When There Are Two Unknowns in Solving Algebraic Word Problems. Mathematics Education Research Journal, 9(2), 122-135.

Mayer, R. E. (1982). Memory for algebra story problems. Journal of Educational Psychology, 74(2), 199-216.

Mayfield, K. H., & Glenn, I. M. (2008). An Evaluation of Interventions to Facilitate Algebra Problem Solving. Journal of Behavioral Education, 17(3), 278-302.

Monk, S. (2007). Representation in school mathematics: Learning to graph and graphing to learn A Research Companion to Principles and Standards (pp. 250-261). Reston, VA: NCTM.

Naftaliev, E., & Yerushalmy, M. (2011). Solving algebra problems with interactive diagrams: Demonstration and construction of examples. The Journal of Mathematical Behavior, 30(1), 48-61.

Nathan, M. J., Kintsch, W., & Young, E. (1992). A Theory of Algebra-Word-Problem Comprehension and Its Implications for the Design of Learning Environments. Cognition & Instruction, 9(4), 329-389.

Nathan, M. J., & Koedinger, K. R. (2000). Teachers' and Researchers' Beliefs about the Development of Algebraic Reasoning. Journal for Research in Mathematics Education, 31(2), 168-190.

Nesher, P., Hershkovitz, S., & Novotna, J. (2003). Situation model, text base, and what else? Factors affecting problem solving. Educational Studies in Mathematics, 52(2), 151-176.

Niess, M. L., Sadri, P., & Kwangho, L. (2008). Variables and Spreadsheets Connect with Real-World Problems. Mathematics Teaching in the Middle School, 13(7), 423-431.

Nortvedt, G. (2008). Understanding word problems. Paper presented at the 32nd Conference of the International Group for the Psychology of Mathematics Education.

Olive, J., & Caglayan, G. (2008). Learners' Difficulties with Quantitative Units in Algebraic Word Problems and the Teacher's Interpretation of Those Difficulties. International Journal of Science and Mathematics Education, 6(2), 269-292.

Panasuk, R., & Beyranevand, M. (2010). Algebra students' ability to recognise multiple representations and achievement. International Journal for Mathematics Teaching and Learning, 1-21.

Patterson, A. C. (1997). Building Algebraic Expressions: A Physical Model. Mathematics Teaching in the Middle School, 2(4), 238-242.

Peled, Z., & Wittrock, M. C. (1990). Generated meanings in the comprehension of word problems in mathematics. Instructional Science, 19, 171-205.

Perkins, D. N., & Simmons, R. (1988). Patterns of Misunderstanding: An Integrative Model for Science, Math, and Programming. Review of Educational Research, 58(3), 303-326.

Pirie, S. (1998). Crossing the gulf between thought and symbol: Language as (slippery) stepping-stones. In H. Steinbring, M. Bussi, E. Bartolini & A. Sierpinska (Eds.), Language and Communication in the Mathematics Classroom (pp. 7-29). Reston, VA: NCTM.

Presmeg, N., & Nenduradu, R. (2005). An investigation of a preservice teacher's use of representations in solving algebraic problems involving exponential relationships. Paper presented at the 29th Conference of The International Group for the Psychology of Mathematics Education. , Melbourne.

Pugalee, D. K. (2004). A Comparison of Verbal and Written Descriptions of Students' Problem Solving Processes. Educational Studies in Mathematics, 55(1), 27-47.

Reed, S. (1999). Word Problems: Research and Curriculum Reform. . Mahwah, NJ: Lawrence Ehrlbaum Associates.

Reed, S. K. (1984). Estimating answers to algebra word problems. [Feature]. Journal of Experimental Psychology. Learning, Memory and Cognition, 10, 778-790.

Reed, S. K., Dempster, A., & Ettinger, M. (1985). Usefulness of Analogous Solutions for Solving Algebra Word Problems. Journal of Experimental Psychology: Learning, Memory, and Cognition, 11(1), 106-125.

Reed, S. K., & Ettinger, M. (1987). Usefulness of Tables for Solving Word Problems. Cognition & Instruction, 4(1), 43-59.

Roehler, M. (2004). Middle school students' intuitive techniques for solving algebraic word problems., University of Arizona.

Rojano, T. (1996). Developing algebraic aspects of problem solving within a spreadsheet environment. Approaches to Algebra (pp. 137-145). Netherlands: Kluwer Academic Publishers.

Ron, P. (1997). Translating Word Problems: Language Issues in the Spanish/English Bilingual Mathematics Classroom. Paper presented at the Annual Meeting of American Educational Research Association, Chicago, IL. .

Rosnick, P. (1982). Students' Symbolization Processes in Algebra (pp. 1-29). Washington, D.C.: National Science Foundation.

Rosnick, P. C., J. (1980). Learning without understanding: The effect of tutoring strategies on algebra misconceptions. Journal of Mathematical Behavior, 3(1), 3-27.

Schoen, H. (1988). Teaching elementary algebra with a word problem focus. In A. F. Coxford (Ed.), The ideas of algebra, K-12 (1988 Yearbook). Reston, VA: National Council of Teachers of Mathematics.

Sebrechts, M. M., Enright, M., Bennett, R. E., & Martin, K. (1996). Using Algebra Word Problems to Assess Quantitative Ability: Attributes, Strategies, and Errors. Cognition and Instruction, 14(3), 285-343.

Simon, M. S., V. . (1988). Developing algebraic representation using diagrams. In A. F. Coxford (Ed.), The ideas of algebra, K-12 (1988 Yearbook). Reston, VA: National Council of Teachers of Mathematics.

Slovin, H. (1990 (April)). A study of the effect of an in-service course in teaching algebra through a problem -solving process. Paper presented at the Research presession of the annual meeting of the National Council of Teachers of Mathematics, Salt Lake City, UT.

Stacey, K., & MacGregor , M. E. (2000). Learning the Algebraic Method of Solving Problems. Journal of Mathematical Behavior, 18(2), 149-167.

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