Project References
Algebra/Algebraic Thinking
- Blanton, M. L. (2008). Algebra and the elementary classroom: Transforming thinking, transforming practice. Portsmouth, NH: Heinemann.
- Brown, L., & Drouhard, J. P. (2004). Responses to "The core of algebra". In K. Stacey, H. Chick & M. Kendal (Eds.), The future of the teaching and learning of algebra: The 12th ICMI study (pp. 35-40). Boston: Kluwer.
- Cai, J., & Knuth, E. J. (2005). The development of students' algebraic thinking in earlier grades from curricular, instructional and learning perspectives. ZDM, 37(1), 1-4.
- Cai, J., Lew, H. C., Morris, A. K., Moyer, J. C., Ng, S. F., & Schmittau, J. (2005). The development of students' algebraic thinking in earlier grades. ZDM, 37(1), 5-15.
- Carraher, D. W., & Schliemann, A. D. (2007). Early algebra and algebraic reasoning. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (Vol. 2, pp. 669-705). Charlotte, NC: Information Age.
- Carraher, D. W., Schliemann, A. D., & Schwartz, J. L. (2008). Early algebra is not the same as algebra early. In Algebra in the early grades (pp. 235-272). New York: Lawrence Erlbaum.
- Chazan, D. (2000). Beyond formulas in mathematics and teaching: Dynamics of the
high school algebra classroom. New York: Teachers College Press.
- Dougherty, B. (2008). Measure up: A quantitative view of early algebra. In J. J. Kaput, D. W. Carraher & M. Blanton (Eds.), Algebra in the early grades (pp. 389-412). New York: Lawrence Erlbaum.
- Driscoll, M. (1999). Fostering algebraic thinking: A guide for teachers, grades 6-10. Portsmouth, NH: Heinemann.
- Greenes, C., & Findell, C. (1999). Developing students' algebraic reasoning abilities. In L. V. Stiff (Ed.), Developing mathematical reasoning in grades K-12. Reston, VA: National Council of Teachers of Mathematics.
- Harel, G. Symbolic reasoning and transformational reasoning and their effect on algebraic reasoning. Unpublished manuscript.
- Healy, L., & Hoyles, C. (2000). A study of proof conceptions in algebra. Journal for Research in Mathematics Education, 31(4), 396-428.
- Kaput, J. J. (1998). Transforming algebra from an engine of inequity to an engine of mathematical power by "algebrafying" the K-12 curriculum. In S. Fennel (Ed.), The nature and role of algebra in the K-14 curriculum: Proceedings of a National Symposium (pp. 25-26). Washington, DC: National Research Council, National Academy Press.
- Kaput, J. J. (1999). Teaching and learning a new algebra. In E. Fennema & T. A.
Romberg (Eds.), Mathematics classrooms that promote understanding (pp. 133-155). Mahwah, NJ: Lawrence Erlbaum. - Kaput, J. J. (2008). What is algebra? What is algebraic reasoning? In J. J. Kaput, D. W. Carraher & M. L. Blanton (Eds.), Algebra in the early grades (pp. 5-17). New York: Lawrence Erlbaum.
- Kaput, J. J., Blanton, M. L., & Moreno, L. (2008). Algebra from a symbolization point of view. In J. J. Kaput, D. W. Carraher & M. L. Blanton (Eds.), Algebra in the early grades (pp. 19-55). New York: Lawrence Erlbaum.
- Kieran, C. (1992). The learning and teaching of school algebra. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 390-419). New York: Macmillan.
- Kieran, C. (1996). The changing face of school algebra: Invited Lecture, ICME-8 Congress, Spain.
- Kieran, C. (2004). The core of algebra: Reflections on its main activities. In K. Stacey, H. Chick & M. Kendal (Eds.), The future of the teaching and learning of algebra: The 12th ICMI study (pp. 21-33). Boston: Kluwer.
- Kieran, C. (2006). Research on the learning and teaching of algebra. In A. Gutierrez & P. Boero (Eds.), Handbook of research on the psychology of mathematics education (pp. 11-49). The Netherlands: Sense.
- Kieran, C. (2007). Learning and teaching algebra at the middle school through college levels. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 707-762). Charlotte, NC: Information Age.
- 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.
- Lins, R., & Kaput, J. (2004). The early development of algebraic reasoning: The current state of the field. In K. Stacey, H. Chick & M. Kendal (Eds.), The future of the teaching and learning of algebra: The 12th ICMI study (pp. 47-70). Boston: Kluwer.
- MacGregor, M., & Price, E. (1999). An exploration of aspects of language proficiency and algebra learning. Journal for Research in Mathematics Education, 30(4), 449-467.
- Mason, J. (2008). Making use of children's powers to produce algebraic thinking. In J. J. Kaput, D. W. Carraher & M. L. Blanton (Eds.), Algebra in the early grades (pp. 57-94). New York: Lawrence Erlbaum.
Meira, L. Students' early algebraic activity: Sense making and the production of meanings in mathematics. Unpublished manuscript.
- National Research Council. (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academy of Sciences.
- Nesher, P., & Teubal, E. (1975). Verbal cues as an interfering factor in verbal problem solving. Educational Studies in Mathematics, 6, 41-51.
- Schmittau, J. (2005). The development of algebraic thinking. ZDM, 37(1), 16-22.
- 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, 1-36.
- Sfard, A., & Linchevski, L. (1994). The gains and the pitfalls of reification--the case of algebra Educational Studies in Mathematics, 26, 191-228.
- Smith, J. P., & Thompson, P. W. (2008). Quantitative reasoning and the development of algebraic reasoning. In J. J. Kaput, D. W. Carraher & M. L. Blanton (Eds.), Algebra in the early grades (pp. 95-132). New York: Lawrence Erlbaum.
- Stylianides, A. J. (2007). Proof and proving in school mathematics. Journal for
Research in Mathematics Education, 38(3), 289-321.
- Wong, N.-Y. (2005). The positioning of algebraic topics in the Hong Kong elementary school mathematics curriculum. ZDM, 37(1), 23-33.
- Wu, H. (2001). How to prepare students for algebra. American Educator/American Federation of Teachers, 25(2), 10-17.
- Yackel, E. (1997). A foundation for algebraic reasoning in the early grades. Teaching Children Mathematics, 3(6), 276-280.
- Equations/Expressions/Equality/Inequality
- Alibali, M. W. (1999). How children change their minds: Strategy change can be gradual or abrupt. Developmental Psychology, 35(1), 127-145.
- Alibali, M. W., Brown, A. N., Stephens, A. C., Kao, Y., & Nathan, M. J. (in preparation).
Middle school students' conceptual understanding of equations: Evidence from writing story problems.
- Alibali, M. W., Knuth, E. J., Hattikudur, S., McNeil, N. M., & Stephens, A. C. (2007). A longitudinal examination of middle school students' understanding of the equal sign and equivalent equations. Mathematical Thinking and Learning, 9(3), 221- 247.
- Baroody, A. J., & Ginsburg, H. P. (1983). The effects of instruction on children's
understanding of the "equals" sign. Elementary School Journal, 84(2), 199-212.
- Behr, M. J., Erlwanger, S., & Nichols, E. (1980). How children view the equals sign. Mathematics Teaching, 92, 13-15.
- Carpenter, T. P., Levi, L., Franke, M. L., & Zeringue, J. K. (2005). Algebra in the elementary school: Developing relational thinking. ZDM, 37(1), 53-59.
- Chappell, M. F. (1997). Preparing students to enter the gate. Teaching Children
Mathematics, 3(6), 266-267.
- Clement, J. (1982). Algebra word problem solutions: Thought processes underlying a common misconception. Journal for Research in Mathematics Education, 13(1), 16-30.
- de Lima, R. N., & Tall, D. (2008). Procedural embodiment and magic in linear equations. Educational Studies in Mathematics, 67(3), 3-18.
- Falkner, K. P., Levi, L., & Carpenter, T. P. (1999). Children's understanding of equality:
A foundation for algebra. Teaching Children Mathematics, 6(4), 56-60.
- Filloy, E., & Rojano, T. (1989). Solving equations: The transition from arithmetic to algebra. For the Learning of Mathematics, 9(2), 19-25.
- Fosnot, C. T., & Jacob, B. (2009). Young mathematicians at work: The role of contexts and models in the emergence of proof In D. A. Stylianou, M. L. Blanton & E. J.Knuth (Eds.), Teaching and Learning Proof Across the Grades (pp. 102-119). New York: Routledge.
- Hattikudur, S., & Alibali, M. W. (2010). Learning about the equal sign: Does comparing with inequality symbols help? Journal of Experimental Psychology, 107, 15-30.
- Herscovics, N., & Kieran, C. (1999). Constructing meaning for the concept of equation. In B. Moses (Ed.), Algebraic thinking, grades K-12: Readings from NCTM's school-based journals and other publications (pp. 181-188). 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, 59-78.
- Huntley, M. A., Marcus, R., Kahan, J., & Miller, J. L. (2007). Investigating high-school students' reasoning strategies when they solve linear equations. Journal of Mathematical Behavior, 26, 115-139.
- Johanning, D. I. (2004). Supporting the development of algebraic thinking in middle school: a closer look at students' informal strategies. Journal of Mathematical Behavior, 23, 371-388.
- Kaput, J., & Sims-Knight, J. E. (1983). Errors in translations to algebraic equations: roots and implications. Focus on Learning Problems in Mathematics, 5(3-4), 63- 78.
- Kieran, C. (1981). Concepts associated with the equality symbol. Educational Studies in Mathematics, 12(3), 317-326.
- Kieran, C., & Sfard, A. (1999). Seeing through symbols: The case of equivalent
expressions. Focus on Learning Problems in Mathematics, 21(1), 1-17.
- Knuth, E. J., Alibali, M. W., Hattikudur, S., McNeil, N. M., & Stephens, A. C. (2008). Equal sign understanding in the middle grades. Mathematics teaching in the middle school, 13(9), 514-519.
- Knuth, E. J., Stephens, A. C., McNeil, N. M., & Alibali, M. W. (2006). Does
understanding the equal sign matter? Evidence from solving equations. Journal for Research in Mathematics Education, 37(4), 297-312.
- Koedinger, K. R., Alibali, M. W., & Nathan, M. J. (under review). Trade-offs between grounded and abstract representations: Evidence from algebra problem solving.
- Koehler, J. (2002). Algebraic reasoning in the elementary grades: Developing an understanding of the equal sign as a relational symbol. Unpublished Master's Thesis, University of Wisconsin-Madison, Madison, WI.
- 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, 39-65.
- Linchevski, L., & Livneh, D. (1999). Structure sense: The relationship between algebraic and numerical contexts. Educational Studies in Mathematics, 40, 173-196.
- MacGregor, M., & Stacey, K. (1993). Cognitive models underlying students' formulation of simple linear equations. Journal for Research in Mathematics Education, 24(3), 217-232.
- McNeil, N. M., & Alibali, M. W. (2002). A strong schema can interfere with learning: The case of children's typical addition schema. In C. D. Schunn & W. Gray (Eds.), Proceedings of the twenty-fourth annual conference of the cognitive science society (pp. 661-666). Mahwah, NJ: Erlbaum.
- McNeil, N. M., & Alibali, M. W. (2004). You'll see what you mean: Students encode equations based on their knowledge of arithmetic. Cognitive Science, 28, 451-
466.
- McNeil, N. M., & Alibali, M. W. (2005). Knowledge change as a function of mathematics
experience: All contexts are not created equal. Journal of Cognition and Development, 6, 385-406.
- McNeil, N. M., & Alibali, M. W. (2005). Why won't you change your mind? Knowledge of operational patterns hinders learning and performance on equations. Child Development, 76(4), 883-899.
- McNeil, N. M., Grandau, L., Knuth, E., Alibali, M., Stephens, A., Hattikudur, S., et al. (2006). Middle-School Students' Understanding of the Equal Sign: The Books They Read Can't Help. Cognition and Instruction, 24(3), 367-385.
- Morris, A. K. (2003). The development of children's understanding of equality and inequality relationships in numerical symbolic contexts. Focus on Learning Problems in Mathematics, 25(2), 18-51.
- Olive, J., Izsak, A., & Blanton, M. (2002). Investigating and enhancing the development of algebraic reasoning in the early grades (K-8): The Early Algebra Working
Group. In D. S. Mewborn, P. Sztajn, D. Y. White, H. G. Wiegel, R. L. Bryant, & K. Nooney (Eds.), Proceedings of the twenty-fourth annual meeting of the international group for the psychology of mathematics education (Vol. 1, pp. 119-
120). Columbus, OH: ERIC.
- O'Rode, N. (2003). A theoretical framework for understanding students' conceptions of equivalence. Paper presented at the annual meeting of the American Educational
Research Association, Chicago, IL.
- Rittle-Johnson, B., & Alibali, M. W. (1999). Conceptual and procedural understanding: Does one lead to the other? Journal of Educational Psychology, 91(1), 175-189.
- Rittle-Johnson, B., & Star, J. R. (2007). Does comparing solution methods facilitate conceptual and procedural knowledge? An experimental study on learning to solve equations. Journal of Educational Psychology, 99(3), 561-574.
- Schliemann, A. D., Carraher, D. W., Pendexter, W., & Brizuela, B. M. (1998). Solving algebra problems before algebra instruction. Unpublished manuscript.
- Stacey, K., & MacGregor, M. (2000). Learning the algebraic methods of solving
problems. Journal of Mathematical Behavior, 18(2), 149-167.
- Steinberg, R. M., Sleeman, D. H., & Ktorza, D. (1990). Algebra students' knowledge of equivalence of equations. Journal for Research in Mathematics Education, 22(2), 112-121.
- Stephens, A. C. (2001). A study of students' translations from equations to word
problems. In R. Speiser, C. A. Mahar & C. N. Walter (Eds.), Proceedings of the twenty-third annual meeting of the North American chapter of the international group for the psychology of mathematics education (pp. 133-134). Columbus, OH: ERIC Clearinghouse for Science, Mathematics, and Environmental Education.
- Stephens, A. C. (2003). Another look at word problems. Mathematics Teacher, 96(1), 63-66.
- Swafford, J., & Langrall, C. (2000). Grade 6 students' preinstructional use of equations to describe and represent problem situations. Journal for Research in Mathematics Education, 31(1), 89-112.
- van Amerom, B. A. (2003). Focusing on informal strategies when linking arithmetic to early algebra. Educational Studies in Mathematics, 54, 63-75.
- Vlassis, J. (2002). The balance model: Hindrance or support for the solving of linear equations with one unknown. Educational Studies in Mathematics, 49, 341-359.
- Functional Thinking
- Anderson, N. C. (2008). Walk the line: Making sense of y = mx + b. In C. E. Greenes& R. Rubenstein (Eds.), Algebra and algebraic thinking in school mathematics (pp.
233-246). Reston, VA: National Council of Teachers of Mathematics.
- Bastable, V., & Schifter, D. (2008). Classroom stories: Examples of elementary students engaged in early algebra. In J. J. Kaput, D. W. Carraher & M. L. Blanton (Eds.), Algebra in the early grades (pp. 165-184). New York: Lawrence Erlbaum.
- Blanton, M. L., & Kaput, J. J. (2004). Elementary grades students' capacity for functional thinking. In M. J. Hoines & A. B. Fuglestad (Eds.), Proceedings of the 28th PME Internatiomal Conference (Vol. 2, pp. 135-142).
- Brenner, M. E., Mayer, R. E., Moseley, B., Brar, T., Duran, R., Reed, B. S., et al. (1997). Learning by understanding: The role of multiple representations in learning algebra. American Educational Research Journal, 34(4), 663-689.
- Brizuela, B. M. (2003). Relationships among different mathematical representations: The case of Jennifer, Nathan, and Jeffrey. Paper presented at the American Educational Research Association, Chicago, IL.
- Brizuela, B. M., & Earnest, D. (2008). Multiple notational systems and algebraic
understandings: The case of the "best deal" problem. In J. J. Kaput, D. W.
Carraher & M. Blanton (Eds.), Algebra in the early grades (pp. 273-301). New York: Lawrence Erlbaum.
- Brizuela, B. M., & Lara-Roth, S. (2002). Additive relations and function tables. Journal of Mathematical Behavior, 20, 309-319.
- Carraher, D. W., Schliemann, A. D., Brizuela, B. M., & Earnest, D. (2006). Arithmetic and algebra in early mathematics education. Journal for Research in Mathematics Education, 37(2), 87-115.
- Grandau, L., & Stephens, A. C. (2006). Algebra and geometry. Mathematics teaching in the middle school, 11(7), 344-349.
- Izsak, A., & Findell, B. R. (2005). Adaptive interpretation: Building continuity between students' experiences solving problems in arithmetic and in algebra. ZDM, 37(1), 60-67.
- Kenney, P. A., & Silver, E. A. (1997). Probing the foundations of algebra: Grade-4 pattern items in NAEP. Teaching Children Mathematics, 3(6), 268-274.
- Knuth, E. J. (2000). Student understanding of the Cartesian connection: An exploratory study. Journal for Research in Mathematics Education, 31(4), 500-507.
- Lannin, J. K. (2005). Generalization and justification: The challenge of introducing algebraic reasoning through patterning activities. Mathematical Thinking and Learning, 7(3), 231-258.
- 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, 299-317.
- Leinhardt, G., Zaslavsky, O., & Stein, M. K. (1990). Functions, graphs, and graphing: Tasks, learning, and teaching. Review of Educational Research, 60(1), 1-64.
- Lobato, J., Ellis, A. B., & Munoz, R. (2003). How "focusing phenomena" in the
instructional environment support individual students' generalizations.
Mathematical Thinking and Learning, 5(1), 1-36.
- Martinez, M., & Brizuela, B. M. (2006). A third grader's way of thinking about linear function tables. Journal of Mathematical Behavior, 25, 285-298.
- Mesa, V. (2004). Characterizing practices associated with functions in middle school textbooks: An empirical approach. Educational Studies in Mathematics, 56, 255-
286.
- Moschkovich, J., Schoenfeld, A. H., & Arcavi, A. (1993). Aspects of understanding: On multiple perspectives and representations of linear relations and connections
among them. In T. A. Romberg, E. Fennema & T. P. Carpenter (Eds.), Integrating research on the graphical representation of functions (pp. 69-100). Hillsdale, NJ: Lawrence Erlbaum.
- Moss, J., Beatty, R., Barkin, S., & Shillolo, G. (2008). "What is your theory? What is your rule?" Fourth graders build an understanding of functions through patterns and
generalizing problems. In C. E. Greenes & R. Rubenstein (Eds.), Algebra and algebraic thinking in school mathematics (pp. 155-168). Reston, VA: National Council of Teachers of Mathematics.
- Nathan, M. J., Stephens, A. C., Masarik, K., Alibali, M. W., & Koedinger, K. R. (2002). Representational fluency in the middle school: A classroom study. In D. S. Mewborn, P. Sztajn, D. Y. White, H. G. Wiegel, R. L. Bryant, & K. Nooney (Eds.), Proceedings of the twenty-fourth annual meeting of the international group for the
psychology of mathematics education (pp. 463-472). Columbus, OH: ERIC.
- Radford, L. (2000). Signs and meanings in students' emergent algebraic thinking: A semiotic analysis. Educational Studies in Mathematics, 42, 237-268.
- Smith, E. (2008). Representational thinking as a framework for introducing functions in the elementary curriculum. In J. J. Kaput, D. W. Carraher & M. L. Blanton (Eds.),
Algebra in the early grades (pp. 133-160). New York: Lawrence Erlbaum.
- Speiser, B., & Walter, C. (1997). Performing algebra: Emergent discourse in a fifthgrade classroom. Journal of Mathematical Behavior, 16(1), 39-49.
- Stacey, K., & MacGregor, M. (2001). Curriculum reform and approaches to algebra. In
R. Sutherland, T. Rojano, A. Bell & R. Lins (Eds.), Perspectives on school algebra (pp. 141-153). Dordrecht, The Netherlands: Kluwer.
- Tierney, C., & Monk, S. (2008). Children's reasoning about change over time. In J. J. Kaput, D. W. Carraher & M. L. Blanton (Eds.), Algebra in the early grades (pp. 185-200). New York: Lawrence Erlbaum.
- Warren, E., & Cooper, T. (2008). Generalising the pattern rule for visual growth
patterns: Actions that support 8 year olds' thinking. Educational Studies in
Mathematics, 67, 171-185.
- Warren, E. A., Cooper, T. J., & Lamb, J. T. (2006). Investigating functional thinking in the elementary classroom: Foundations of early algebraic reasoning. Journal of Mathematical Behavior, 25, 208-223.
- 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, 125-147.
- Zaslavsky, O., Sela, H., & Leron, U. (2002). Being sloppy about slope: The effect of changing the scale. Educational Studies in Mathematics, 49, 119-140.
- Generalized arithmetic
- Carpenter, T. P., Franke, M. L., & Levi, L. (2003). Thinking mathematically: Integrating arithmetic and algebra in the elementary school. Portsmouth, NH: Heinemann.
- Carpenter, T. P., & Levi, L. (2000). Developing conceptions of algebraic reasoning in the primary grades: National Center for Improving Student Learning and
Achievement in Mathematics and Science. University of Wisconsin-Madison.
- Carpenter, T. P., Levi, L., Berman, P., & Pligge, M. Developing algebraic reasoning in
the elementary school. Unpublished manuscript.
- Davis, R. B. (1985). ICME 5 report: Algebraic thinking in the early grades. Journal of Mathematical Behavior, 4, 195-208.
- Gallardo, A. (2002). The extension of the natural-number domain to the integers in the transition from arithmetic to algebra. Educational Studies in Mathematics, 49, 171-192.
- 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, 169-188.
- Koehler, J. (2004). Learning to think relationally: Thinking relationally to learn.
Unpublished Dissertation, University of Wisconsin-Madison, Madison, WI.
- Lee, L., & Wheeler, D. (1989). The arithmetic connection. Educational Studies in Mathematics, 20, 41-54.
- Peled, I., & Carraher, D. W. (2008). Signed numbers and algebraic thinking. In J. J. Kaput, D. W. Carraher & M. Blanton (Eds.), Algebra in the early grades (pp. 303-
328). New York: Lawrence Erlbaum.
- Picciotto, H. A proposal for new directions in early mathematics: Operation sense, toolbased
pedagogy, curricular breadth. Unpublished manuscript.
- Rachlin, S. (1995). Learning to see the wind. Unpublished manuscript.
- Schifter, D. (1999). Reasoning about operations: Early algebraic thinking in grades K-6. In L. V. Stiff (Ed.), Developing mathematical reasoning in grades K-12 (pp. 62- 81). Reston, VA: National Council of Teachers of Mathematics.
- Slavit, D. (1999). The role of operation sense in transitions from arithmetic to algebraic thought. Educational Studies in Mathematics, 37, 251-274.
- Zazkis, R., & Gadowsky, K. (2001). Attending to transparent features of opaque
representations of natural number. In A. Cuoco (Ed.), The roles of representation in school mathematics (pp. 44-52). Reston, VA: National Council of Teachers of Mathematics.
- Proportional reasoning
- Baroody, A. J., & Coslick, R. T. (1998). Fostering children's mathematical power: An investigative approach to K-8 mathematics instruction. Mahwah, NJ: Lawrence Erlbaum.
- Christou, C., & Phillippou, G. (2002). Mapping and development of intuitive proportional reasoning. Journal of Mathematical Behavior, 20, 321-336.
- Clark, F. B., & Kamii, C. (1996). Identification of multiplicative thinking in children in grades 1-5. Journal for Research in Mathematics Education, 27(1), 41-51.
- Clark, M. R., Berenson, S. B., & Cavey, L. O. (2003). A comparison of ratios and fractions and their roles as tools in proportional reasoning. Journal of
Mathematical Behavior, 22, 297-317.
- Ellis, A. B. (2007). Connections between generalizing and justifying: Students'
reasoning with linear relationships. Journal for Research in Mathematics
Education, 38(3), 194-229.
- Kaput, J. J., & West, M. M. (1994). Missing-value proportional reasoning problems: Factors affecting informal reasoning patterns. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp.
235-287). Albany: SUNY.
- Kenney, P. A., Lindquist, M. M., & Heffernan, C. L. (2002). Butterflies and caterpillars:
Multiplicative and proportional reasoning in the early grades. In B. Litwiller & G
Bright (Eds.), Making sense of fractions, ratios, and proportions (pp. 87-99)
Reston, VA: National Council of Teachers of Mathematics.
- Lamon, S. J. (1994). Ratio and proportion: Cognitive foundations in unitizing and norming. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 89-120). Albany: SUNY.
- Lamon, S. J. (1999). Teaching fractions and ratios for understanding. Mahwah, NJ: Lawrence Erlbaum.
- Lamon, S. J. (2007). Rational numbers and proportional reasoning. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (Vol. 1, pp. 629-667). Charlotte, NC: Information Age.
- Langrall, C., & Swafford, J. (2000). Three balloons for two dollars: Developing
proportional reasoning. Mathematics Teaching in the Middle School, 6(4), 254- 261.
- Lesh, R., Post, T., & Behr, M. (1988). Proportional reasoning. In J. Hiebert & M. Behr (Eds.), Number concepts and operations in the middle grades (pp. 93-118). Reston, VA: Lawrence Erlbaum and National Council of Teachers of Mathematics.
- Lo, J.-J., & Watanabe, T. (1997). Developing ratio and proportion schemes: A story of a fifth grader. Journal for Research in Mathematics Education, 28(2), 216-236.
- Lobato, J., & Thanheiser, E. (2002). Developing understanding of ratio-as-measure as a
foundation for slope. In B. Litwiller & G. Bright (Eds.), Making sense of fractions,
ratios, and proportions (pp. 162-175). Reston, VA: National Council of Teachers of Mathematics.
- Miller, J. L., & Fey, J. T. (2000). Proportional reasoning. Mathematics Teaching in the Middle School, 5(5), 310-313.
- Misailidou, C., & Williams, J. (2003). Diagnostic assessment of children's proportional
reasoning. Journal of Mathematical Behavior, 22, 335-368.
- Singh, P. (2000). Understanding the concepts of proportion and ratio constructed by two grade six students. Educational Studies in Mathematics, 43, 271-292.
- van Dooren, W., De Bock, D., Evers, M., & Verschaffel, L. (2009). Students' overuse of proportionality on missing-value problems: How numbers may change solutions.
Journal for Research in Mathematics Education, 40(2), 187-211.
- van Dooren, W., De Bock, D., Janssens, D., & Verschaffel, L. (2008). The linear
imperative: An inventory and conceptual analysis of students' overuse of linearity. Journal for Research in Mathematics Education, 39(3), 311-342.
- Weinberg, S. L. (2002). Proportional reasoning: One problem, many solutions! In B. Litwiller & G. W. Bright (Eds.), Making sense of fractions, ratios, and proportions (pp. 138-144). Reston, VA: National Council of Teachers of Mathematics.
- Variable
- Booth, L. R. (1988). Children's difficulties in beginning algebra. In A. Coxford & A. Schulte (Eds.), The ideas of algebra, K-12 (pp. 20-32). Reston, VA: The National Council of Teachers of Mathematics.
- English, L. D., & Warren, E. A. (1998). Introducing the variable through pattern
exploration. Mathematics Teacher, 91, 166-170.
- Kieran, C. (1991). Helping to make the transition to algebra. Mathematics Teacher, 84(3), 49-51.
- Knuth, E. J., Alibali, M. W., McNeil, N. M., Weinberg, A., & Stephens, A. C. (2005). Middle school students' understanding of core algebraic concepts: Equality and variable. Zentralblatt für Didaktik der Mathematik, 37(1), 68-76.
- Kuchemann, D. (1981). Algebra. In K. M. Hart (Ed.), Children's understanding of mathematics: 11-16 (pp. 102-119). London: John Murray.
- MacGregor, M., & Stacey, K. (1997). Students' understanding of algebraic notation: 11-15. Educational Studies in Mathematics, 33(1), 1-19.
- McNeil, N. M., Weinberg, A., Hattikudur, S., Stephens, A. C., Asquith, P., Knuth, E. J., et al. (In press). A is for Apple: Mnemonic Symbols Hinder the Interpretation of
Algebraic Expressions. Journal of Educational Psychology.
- Rosnick, P. (1981). Some misconceptions concerning the concept of variable.
Mathematics Teacher, 74(9), 418-420, 450.
- Stacey, K., & MacGregor, M. (1997). Ideas about symbolism that students bring to algebra. Mathematics Teacher, 90(2), 110-113.
- Stephens, A. C. (2005). Developing students' understandings of variable. Mathematics teaching in the middle school, 11(2), 96-100.
- Weinberg, A. D., Stephens, A. C., McNeil, N. M., Krill, D. E., Knuth, E. J., & Alibali, M. W. (2004). Students' initial and developing conceptions of variable. Paper presented at the Annual meeting of the American Education Research Conference, San Diego, CA.
- Algebra teacher knowledge, development, and practice
- Asquith, P., Stephens, A., Knuth, E., & Alibali, M. (2007). Middle school mathematics teachers' knowledge of students' understanding of core algebraic concepts:
equal sign and variable. Mathematical Thinking and Learning, 9(3), 249-272.
- Blanton, M. L., & Kaput, J. J. (2003). Developing elementary teachers' "algebra eyes and ears". Teaching Children Mathematics, 10(2), 70-77.
- Blanton, M. L., & Kaput, J. J. (2005). Helping elementary teachers build mathematical
generality into curriculum and instruction. Zentralblatt für Didaktik der Mathematik, 37(1), 34-42.
- Blanton, M. L., & Kaput, J. J. (2005). Characterizing a classroom practice that promotes algebraic reasoning. Journal for Research in Mathematics Education, 36(5), 412-446.
- Borko, H., Frykholm, J., Pittman, M., Eiteljorg, E., Nelson, M., Jacobs, J., et al. (2005). Preparing teachers to foster algebraic thinking. ZDM, 37(1), 43-52.
- Doerr, H. M. (2004). Teachers' knowledge and the teaching of algebra. In K. Stacey, H. Chick & M. Kendal (Eds.), The future of the teaching and learning of algebra: The 12th ICMI study (pp. 267-290). Boston: Kluwer.
- Jacobs, V. R., Franke, M. L., Carpenter, T. P., Levi, L., & Battey, D. (2007). Professional development focused on children's algebraic reasoning in elementary school. Journal for Research in Mathematics Education, 38(3), 258-288.
- Nathan, M. J., & Koedinger, K. R. (2000). An Investigation of Teachers' Beliefs of Students' Algebra Development. Cognition and Instruction, 18(2), 209-237.
- Parker, M. (1999). Building on "building up": Proportional reasoning activities for future teachers. Mathematics Teaching in the Middle School, 4(5), 286-289.
- Simon, M. A., & Blume, G. W. (1994). Building and understanding multiplicative relationships: A study of prospective elementary teachers. Journal for Research in Mathematics Education, 25(5), 472-494.
- Stephens, A. C. (2006). Equivalence and relational thinking: Preservice elementary teachers' awareness of opportunities and misconceptions. Journal of Mathematics Teacher Education, 9, 249-278.
- Stephens, A. C. (2008). What "counts" as algebra in the eyes of preservice elementary teachers? Journal of Mathematical Behavior, 27, 33-47.
- Stephens, A. C., Grandau, L., Asquith, P., Knuth, E., & Alibali, M. W. (2004). Developing teachers' attention to students' algebraic thinking. Paper presented at the Annual Meeting of the American Education Research Association, San Diego, CA.
- Stump, S. L., & Bishop, J. (2002). Preservice elementary and middle school teachers' conceptions of algebra revealed through the use of exemplary curriculum
materials. In D. S. Mewborn, P. Sztajn, D. Y. White, H. G. Wiegel, R. L. Bryant,& K. Nooney (Eds.), Proceedings of the twenty-fourth annual meeting of the international group for the psychology of mathematics education (Vol. 4, pp. 1903-1914). Columbus, OH: ERIC.
- van Dooren, W., Verschaffel, L., & Onghena, P. (2002). The impact of preservice teachers' content knowledge on their evaluation of students' strategies for solving arithmetic and algebra word problems. Journal for Research in Mathematics Education, 33(5), 319-351.
- van Dooren, W., Verschaffel, L., & Onghena, P. (2003). Pre-service teachers' preferred
strategies for solving arithmetic and algebra word problems. Journal of Mathematics Teacher Education, 6, 27-52.
- Zazkis, R., & Liljedahl, P. (2002). Generalization of patterns: The tension between algebraic thinking and algebraic notation. Educational Studies in Mathematics, 49, 379-402.