# 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*Symbolic reasoning and transformational reasoning and their effect on algebraic reasoning.*Unpublished manuscript.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*(3), 289-321.

Research in Mathematics Education, 38

- 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*(6), 266-267.

Mathematics, 3

- 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*67, 171-185.

Mathematics,

- 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*22, 297-317.

Mathematical Behavior,

- Ellis, A. B. (2007). Connections between generalizing and justifying: Students'

reasoning with linear relationships.*Journal for Research in Mathematics*(3), 194-229.

Education, 38

- 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

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