Students’ Reasoning With Reversible Multiplicative Relationships

10.2307/748998

Ball D. L., 2000, Multiple perspectives on mathematics teaching and learning, 83

Ball D. L., 2001, Handbook of research on teaching, 4, 433

Bednarz N., 1996, Approaches to algebra: Perspectives for research and teaching, 115, 10.1007/978-94-009-1732-3_8

Behr M. J., 1992, Handbook of research on mathematics teaching and learning, 296

Behr M. J., 1993, Rational numbers: An integration of research, 13

10.2307/748423

Beth E. W., 1966, Mathematical epistemology and psychology

10.1016/S0732-3123(02)00117-7

Biddlecomb, B. and Olive, J. 2000. “JavaBars [Computer software]”. Retrieved June 4, 2002 fromhttp://jwilson.coe.uga.edu/olive/welcome.html

10.2307/748434

10.2307/749664

10.1207/s15327833mtl0602_1

10.1007/BF00304566

Confrey J., 1994, The development of multiplicative reasoning in the learning of mathematics, 291

Confrey J., 2000, Handbook of research design in mathematics and science education, 231

Davis G., 1993, Journal of Mathematical Behavior, 12, 63

10.1007/s10649-005-9000-6

Filloy E., 1989, For the Learning of Mathematics, 9, 19

Freudenthal H., 1983, Didactical phenomenology of mathematical structures

Greer B., 1992, Handbook of research on mathematics teaching and learning, 276

Greer B., 1994, The development of multiplicative reasoning in the learning of mathematics, 61

Hackenberg A. J., 2005, Construction of algebraic reasoning and mathematical caring relations

10.1016/j.jmathb.2007.03.002

10.1016/j.jmathb.2009.04.004

10.1007/BF01284528

10.2307/749369

10.1037/10034-000

Kieran C., 1992, Handbook of research on mathematics teaching and learning, 390

Kieren T. E., 1980, Recent research on number learning, 125

Kieren T. E., 1988, Number concepts and operations in the middle grades, 162

Kieren T. E., 1995, Providing a foundation for teaching mathematics in the middle grades, 31

Knuth E. J., 2006, Journal for Research in Mathematics Education, 37, 297

10.2307/749599

Lamon S. J., 2007, Second handbook of research on mathematics teaching and learning, 629

Lehrer R., 2003, Research companion to the principles and standards for school mathematics, 179

Ma L., 1999, Knowing and teaching elementary mathematics, 10.4324/9781410602589

10.2307/749454

10.2307/749431

10.2307/749828

Norton A., 2008, Journal for Research in Mathematics Education, 39, 401, 10.5951/jresematheduc.39.4.0401

Olive J., 2002, Teaching Children Mathematics, 8, 356, 10.5951/TCM.8.6.0356

10.1016/S0732-3123(02)00086-X

Piaget J., 1967, Biology and knowledge

Piaget J., 1968, Structuralism

Piaget J., 1970, Genetic epistemology, 10.7312/piag91272

Piaget J., 1977, Studies in reflecting abstraction

10.2307/748675

10.1016/j.jmathb.2008.07.005

Rosnick P., 1980, Journal of Mathematical Behavior, 3, 3

10.2307/749292

Schwartz J. L., 1988, Number concepts and operations in the middle grades, 41

10.1007/BF01273663

Siebert D., 2002, Making sense of fractions, ratios, and proportions, 247

10.1007/s10857-008-9089-5

10.2307/749205

10.1207/s15327833mtl0804_1

10.2307/30034818

Smith J. P., 2008, Algebra in the early grades, 95

Steffe L. P., 1991, Epistemological foundations of mathematical experience, 26, 10.1007/978-1-4612-3178-3_3

10.1016/1041-6080(91)90004-K

10.1016/1041-6080(92)90005-Y

Steffe L. P., 1994, The development of multiplicative reasoning in the learning of mathematics, 3

10.1016/S0732-3123(02)00075-5

Steffe L. P., 2007, Problems in mathematics education

10.1007/978-1-4419-0591-8

Steffe L. P., 2000, Handbook of research design in mathematics and science education, 267

10.1007/BF01273660

Steffe L. P., 1983, Children's counting types: Philosophy, theory, and application

Streefland L., 1991, Fractions in realistic mathematics education, 10.1007/978-94-011-3168-1

10.1007/BF01273861

Thompson P. W., 1994, The development of multiplicative reasoning in the learning of mathematics, 179

Thompson P. W., 2010, Children's fractional knowledge

Thompson P. W., 2003, Research companion to the principles and standards for school mathematics, 95

Tillema E., 2010, For the Learning of Mathematics, 30, 2

Toluk Z., 1999, Dissertation Abstracts International, 60, 3944

Toluk Z., 2004, International Journal of Mathematics Teaching and Learning, 5

Tunc-Pekkan Z., 2008, Modeling grade eight students’ construction of fraction multiplying schemes and algebraic operations (Unpublished doctoral dissertation)

10.2307/749707

10.1016/j.jmathb.2003.12.006

10.2307/748704

von Glasersfeld E., 1982, Archives de Psychologie, 50, 191

10.4324/9780203454220

von Glasersfeld E., 1998, Computing anticipatory systems, 38