The Rational Number Project

The Rational Number Project is an
ongoing research project investigating
student learning and teacher enhancement.


The Rational Number Project has been funded by the
National Science Foundation
since 1979.

(will host and maintain this web site)


 
 

 

Bart, W., Post, T., Behr, M., Lesh, R. (1994). A diagnostic analysis of a proportional reasoning test item: An introduction to the properties of a semi-dense item. Focus on Learning Problems in Mathematics, 16(3), 1-11.

Behr, M. & Harel, G. (1990). Understanding the Multiplicative Structure. In G. Booker, P. Cobb, & T.N. de Merldicutti (Eds.) Proceedings of the PME XIV Conference Volume III (pp. 27-34). Mexico: Consejo Nacional de Ciencia y Technologia, Gobierno del Estado de Morelos.

Behr, M., Harel, G., Post, T., & Lesh, R. (1994). Units of quantity: A conceptual basis common to additive and multiplicative structures. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 123-180). Albany, NY: SUNY Press.

Behr, M., Harel, G., Post, T., & Lesh, R. (1993). Rational Numbers: Toward a Semantic Analysis - Emphasis on the Operator Construct. In T. Carpenter, E. Fennema & T. Romberg (Eds.), Rational Numbers: An Integration of Research (pp. 13-47). Hillsdale, NJ: Lawrence Erlbaum Associates.

Behr, M., Harel, G., Post, T., & Lesh, R. (1992). Rational number, ratio and proportion. In D. Grouws (Ed.), Handbook of Research on Mathematics Teaching and Learning (pp. 296-333). NY: Macmillan Publishing.

Behr, M., Harel, G., Post, T. & Lesh, R. (1991). The Operator Construct of Rational Number. In F. Furinghetti (Ed.) Proceedings of PME XV Conference (pp. 120-127). Assisi, Italy: PME.

Behr, M., Harel, G., Post, T., & Lesh, R. (1987). Theoretical analysis: Structure and hierarchy, missing value proportion problems. In J. Bergeron, N. Herscovics, & C. Kieran (Eds.), Proceedings of the Eleventh International Conference, Psychology of Mathematics Education PMR - XI Volume II (pp. 269-274). Montreal, Canada: PME.

Behr, M., Khoury, H., Harel, G., Post, T., Lesh, R., (1997) Conceptual Units Analysis of Preservice Elementary School Teachers' Strategies on a Rational-Number-as-Operator Task. Journal of Mathematics Education, 28(1), 48-69.

Behr, M., Lesh, R., Post, T., & Silver E. (1983). Rational Number Concepts. In R. Lesh & M. Landau (Eds.), Acquisition of Mathematics Concepts and Processes, (pp. 91-125). New York: Academic Press.

Behr, M. & Post, T. (1992). Teaching rational number and decimal concepts. In T. Post (Ed.), Teaching mathematics in grades K-8: Research-based methods (2nd ed.) (pp. 201-248). Boston: Allyn and Bacon.

Behr, M., & Post, T. (1988). Teaching Rational Number and Decimal Concepts. In T. Post, (Ed.), Teaching Mathematics in Grades K-8: Research Based Methods (pp. 190-231). Newton, MA: Allyn & Bacon, Inc.

Behr, M., & Post, T. (1986). Estimation and Children's Concept of Rational Number Size. In H. Schoen & M. Zweng (Eds.) Estimation and Mental Computation: 1986 NCTM Yearbook (pp. 103-111). Reston, VA: National Council of Teachers of Mathematics.

Behr, M., & Post, T. (1981). The Effect of Visual Perceptual Distractors on Children's Logical-Mathematical Thinking in Rational Number Situations. In T. Post & M. Roberts (Eds.), Proceedings of the Third Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 8-16). Minneapolis: University of Minnesota.

Behr, M., Post, T., & Lesh R. (1981, July). Construct Analyses, Manipulative Aids, Representational Systems and the Learning of Rational Numbers. In Proceedings of the Fifth Conference of the International Group for the Psychology of Mathematics Education. (pp. 203-209). Grenoble, France: PME.

Behr, M., Post, T., Silver, E., & Mierkiewicz, D. (1980, August). Theoretical Foundations for Instructional Research on Rational Numbers. In R. Karplus (Ed.) Proceedings of Fourth Annual Conference of International Group for Psychology of Mathematics Education (pp. 60-67). Berkeley, CA: Lawrence Hall of Science.

Behr, M., Reiss, M., Harel, G., Post, T., & Lesh, R. (1986, July). Qualitative Proportional Reasoning: Description of Tasks and Development of Cognitive Structures. In Proceedings of the Tenth International Conference for the Psychology of Mathematics Education (PME-10) (pp. 235-240). London, England.

Behr, M., Wachsmuth, I., & Post, T. (1988). Rational Number Learning Aids: Transfer From Continuous Models To Discrete Models. Focus on Learning Problems in Mathematics, 10(4), 1-17.

Behr, M., Wachsmuth, I., & Post, T. (1985, March). Construct a Sum: A Measure of Children's Understanding of Fraction Size. Journal for Research in Mathematics Education, 16(2), 120-131. A condensed earlier version appeared as On Children's Quantitative Concept of Rational Number: Construct and Estimate the Sum. In J. Bergeron & N. Herscovics (Eds.), Proceedings of the North American Chapter of the International Group for the Psychology of Mathematics Education Volume II (pp. 272-79). Montreal, Canada: September 1983.

Behr, M., Wachsmuth, I., & Post, T. (1984, August). Tasks to Assess Children's Perception of the Size of a Fraction. In A. Bell, B. Low & J. Kilpatrick (Eds.), Theory, Research and Practice in Mathematical Education (pp. 179-18). Fifth International Congress on Mathematical Education, South Australia: Shell Centre for Mathematics Education.

Behr, M., Wachsmuth, I., Post T., & Lesh R. (1984, November). Order and Equivalence of Rational Numbers: A Clinical Teaching Experiment. Journal for Research in Mathematics Education, 15(5), 323-341.

Bezuk, N., & Cramer, K. (1989). Teaching About Fractions: What, When, and How? In P. Trafton (Ed.), National Council of Teachers of Mathematics 1989 Yearbook: New Directions For Elementary School Mathematics (pp. 156-167). Reston, VA: National Council of Teachers of Mathematics.

Bright, G., Behr, M., Post, T., & Wachsmuth, I. (1988, May). Identifying fractions on number lines. Journal for Research in Mathematics Education., 19(3), 215-232.

Conner, G., Harel, G., & Behr, M. (1988). The effect of structural variables on the level of difficulty of missing value proportion problems. In M. Behr, C. Lacampagne, & M. Wheeler (Eds.), Proceedings of the Ninth Annual Conference of PME-NA (pp. 65-71). DeKalb, IL: PME.

Cramer, K., Behr, M., Post T., Lesh, R., (2013) Rational Number Project: Initial Fraction Ideas - Abridged for Grade Three
Abridged edition authors: Kathleen Cramer, Terry Wyberg, Susan Ahrendt, Debbie Monson, Christina Miller.
Abridged from
Cramer, K., Behr, M., Post T., Lesh, R., (2009) Rational Number Project: Initial Fraction Ideas.
Originally published in 1997 as Rational Number Project: Fraction Lessons for the Middle Grades - Level 1, Kendall/Hunt Publishing Co., Dubuque Iowa.

Cramer, K., Behr, M., Post T., Lesh, R., (2009) Rational Number Project: Initial Fraction Ideas.
Originally published in 1997 as Rational Number Project: Fraction Lessons for the Middle Grades - Level 1, Kendall/Hunt Publishing Co., Dubuque Iowa.

Cramer, K., Wyberg, T., & Leavitt, S. (2009). Fraction Operations and Initial Decimal Ideas. [Companion module to RNP: Fraction Lessons for the Middle Grades]

Cramer, K., & Wyberg, T. (2007). When getting the right answers is not always enough. In M. Strutchens & W. G. Martin (Eds). The learning of mathematics :2007 National Council of Teachers of Mathematics Yearbook (pp. 205 – 220). Reston, VA: NCTM.

Cramer, K. (2003) Using a translation model for curriculum development and classroom instruction. In Lesh, R., Doerr, H. (Eds.) Beyond Constructivism. Models and Modeling Perspectives on Mathematics Problem Solving, Learning, and Teaching. Lawrence Erlbaum Associates, Mahwah, New Jersey.

Cramer, Kathleen (2003). Mathematics for Elementary and Middle School Teachers: Functions and Proportionality Course. Unpublished curriculum.

Cramer, K., (2001) Using Models to Build Middle-Grade Students' Understanding of Functions. Mathematics Teaching in the Middle School. 6 (5), 310-318.

Cramer, K., Behr, M., & Bezuk, N. (1989, October). Proportional Relationships and Unit Rates. Mathematics Teacher, 82 (7), 537-544.

Cramer, K., Henry, A., (2002) Using Manipulative Models to Build Number Sense for Addition of Fractions. National Council of Teachers of Mathematics 2002 Yearbook: Making Sense of Fractions, Ratios, and Proportions (pp. 41-48). Reston, VA: National Council of Teachers of Mathematics.

Cramer, K. & Lesh, R. (1988). Rational number knowledge of preservice elementary education teachers. In M. Behr (Ed.), Proceedings of the 10th Annual Meeting of the North American Chapter of the International Group for Psychology of Mathematics Education (pp. 425-431). DeKalb, Il.: PME.

Cramer, Kathleen; Monson, Debra; Whitney, Stephanie; Leavitt, Seth; Wyberg, Terry. (2010, February). Dividing Fractions and Problem Solving. Mathematics Teaching in the Middle School v15 n6 p338-346 Feb 2010.

Cramer, Kathleen A.; Monson, Debra S.; Wyberg, Terry; Leavitt, Seth; Whitney, Stephanie B. (2009, September). Models for initial decimal ideas. Teaching Children Mathematics, v16 n2 p106-117 Sept 2009.

Cramer, K., & Post, T. (1995). Facilitating children's development of rational number knowledge. In D. Owens, M. Reed, and G. Millsaps (Eds.), Proceedings of the Seventeenth Annual Meeting of PME-NA. (pp. 377-382). Columbus, OH: PME.

Cramer, K. & Post, T. (1993, May). Connecting Research To Teaching Proportional Reasoning. Mathematics Teacher, 86(5), 404-407.

Cramer, K. & Post, T. (1993, February). Making connections: A Case for Proportionality. Arithmetic Teacher, 60(6), 342-346.

Cramer, K., Post, T., & Behr, M. (1989, September). Interpreting Proportional Relationships. Mathematics Teacher, 82 (6), 445-452.

Cramer, K., Post, T., & Behr, M. (1989, January). Cognitive Restructuring Ability, Teacher Guidance and Perceptual Distracter Tasks: An Aptitude Treatment Interaction Study. Journal for Research in Mathematics Education, 20(1), 103-110.

Cramer, K., Post, T., & Currier, S. (1993). Learning and Teaching Ratio and Proportion: Research Implications. In D. Owens (Ed.), Research Ideas For the Classroom (pp. 159-178). NY: Macmillan Publishing Company.

Cramer, K. A, Post, T. R., del Mas, R. C. (2002) Initial Fraction Learning by Fourth- and Fifth-Grade Students: A Comparison of the Effects of Using Commercial Curricula With the Effects of Using the Rational Number Project Curriculum. Journal for Research in Mathematics Education. 33 (2) 111-144.

Cramer, K., Wyberg, T., & Leavitt, S. (2009). Fraction Operations and Initial Decimal Ideas. [Companion module to RNP: Fraction Lessons for the Middle Grades]

Cramer, K., Wyberg, T., & Leavitt, S. (2009). The Role of Representations in Fraction Addition and Subtraction. Mathematics Teaching in the Middle School. 13 (8), 490-496.

Cramer, K., & Wyberg, T. (2009). Efficacy of different concrete models for teaching the part-whole construct for fractions. Mathematical Thinking and Learning, 11 (4), 226-258.

Harel, G., & Behr, M. (1995). Teachers' solutions for multiplicative problems. Hiroshima Journal of Mathematics Education, 3, 31-51.

Harel, G. & Behr, M. (1990). The Construct Theory of Rational Numbers: Toward a Semantic Analysis. In G. Booker, P. Cobb, & T.N. de Merldicutti (Eds.) Proceedings of the PME XIV Conference (pp. 3-10). Mexico: Consejo Nacional de Ciencia y Technologia, Gobierno del Estado de Morelos.

Harel, G., & Behr, M. (1989). Structure and Hierarchy of Missing Value Proportion Problems and Their Representations. Journal of Mathematical Behavior, 8(1), 77-119.

Harel, G., Behr, M., Post, T. & Lesh, R. (1994). Invariance of ratio: The case of children's anticipatory scheme of constancy of taste. Journal for Research in Mathematics Education, 25(4), 324-345.

Harel, G., Behr, M., Post, T., & Lesh, R. (1994). The impact of number type on the solution of multiplication and division problems: Further considerations. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 365-388). Albany, NY: SUNY Press.

Harel, G., Behr, M., Post, T., & Lesh, R. (1992). The Blocks Task: Comparative Analyses of the Task With Other Proportion Tasks and Qualitative Reasoning Skills of Seventh Grade Children in Solving the Task. Cognition and Instruction, 9(1), 45-96.

Harel, G., Behr, M., Post, T. & Lesh, R. (1991). Variables Affecting Proportionality: Understanding of Physical Principles, Formation of Quantitative Relations, and Multiplicative Invariance. In F. Furinghetti (Ed.) Proceedings of PME XV Conference (pp. 125-133). Assisi, Italy: PME.

Harel, G., Behr, M., Post, T., & Lesh, R. (1987). Qualitative differences among seventh grade children in solving a non numerical proportional reasoning blocks task. In J. Bergeron, N. Herscovics, & C. Kieran (Eds.), Proceedings of the Eleventh International Conference, Psychology of Mathematics Education PMR - XI Volume II (pp. 282-288). Montreal, Canada: PME.

Harel, G., Post, T., & Behr, M. (1988). An assessment instrument to examine knowledge of multiplication and division concepts and its implementation with in-service teachers. In M. Behr, C. Lacampagne, & M. Wheeler (Eds.), Proceedings of the Ninth Annual Conference of PME-NA (pp. 411-417). DeKalb, IL: PME.

Harel, G., Post, T., & Behr, M. (1988, July). On the textual and semantic structures of mapping rule and multiplicative compare problems. In A. Borbas (Ed.) Proceedings of the XII International Congress, Psychology of Mathematics Education (PME) Volume II (pp. 372-379). Budapest: PME.

Heller , P., Post, T., & Behr, M. (1985, October). The Effect of Rate Type, Problem Setting and Rational Number Achievement on Seventh Grade Students Performance on Qualitative and Numerical Proportional Reasoning problems. In S. Damarin & M. Shelton (Eds.), Proceedings of the seventh General Meeting of the North American chapter of the International Group for the Psychology of Mathematics Education (pp. 113-122). Columbus, Ohio: PME.

Heller, P., Ahlgren, A., Post, T., Behr, M., & Lesh, R. (1989, March). Proportional Reasoning: The Effect of Two Context Variables, Rate Type and Problem Setting. Journal for Research in Science Teaching, 26(1), 205-220.

Heller, P., Post, T., Behr, M., & Lesh, R. (1990). Qualitative and Numerical Reasoning About Fractions and Rates by Seventh and Eighth Grade Students. Journal for Research in Mathematics Education, 21(5), 388-402.

Heller, P., Post, T., Behr, M., & Lesh, R. (unpublished). The effect of two context variables on qualitative and numerical reasoning about rates.

Lacampagne, C., Post, T., Harel, G., Behr, M. (1988, November). A model for the development of leadership and the assessment of mathematical and pedagogical knowledge of middle school teachers. In M. Behr, C. Lacampagne, & M. Wheeler (Eds.), Proceedings of the Ninth Annual conference of PME-NA (pp. 418-425). DeKalb, IL: PME.

Leavitt, Seth (2003). Creating the Web Pages for the Rational Number Project.
Unpublished M.Ed. paper.

Lesh, R., Behr, M., & Post, T. (1987). Rational Number Relations and Proportions. In C. Janiver (Ed.), Problems of Representations in the Teaching and Learning of Mathematics (pp. 41-58). Hillsdale, NJ: Lawrence Erlbaum.

Lesh, R., Cramer, K., Doerr, H., Post, T., Zawojewski, J., (2003) Model Development Sequences. In Lesh, R., Doerr, H. (Eds.) Beyond Constructivism. Models and Modeling Perspectives on Mathematics Problem Solving, Learning, and Teaching. Lawrence Erlbaum Associates, Mahwah, New Jersey.

Lesh, R., Hoover, M., Hole, B., Kelly, A., Post, T., (2000) Principles for Developing Thought-Revealing Activities for Students and Teachers. In A. Kelly, R. Lesh (Eds.), Research Design in Mathematics and Science Education. (pp. 591-646). Lawrence Erlbaum Associates, Mahwah, New Jersey.

Lesh, R., Hoover, M. & Kelly, A. (1992). Equity, Technology, and Teacher Development. In I. Wirszup & R. Streit (Eds.), Developments in School Mathematics Education Around the World: Volume 3 (pp. ). Reston, VA: National Council of Teachers of Mathematics.

Lesh, R., Kelly, A., (2000) Multitiered Teaching Experiments. In A. Kelly, R. Lesh (Eds.), Research Design in Mathematics and Science Education. (pp. 197-230). Lawrence Erlbaum Associates, Mahwah, New Jersey.

Lesh, R. & Lamon, S. (1992) Assessing Authentic Mathematical Performance. In R. Lesh & S. Lamon (Eds.), Assessments of Authentic Performance in School Mathematics (pp. 17-62). Washington, DC: American Association for the Advancement of Sciences Press.

Lesh, R. & Lamon, S. (1992). Introduction: Trends, Goals, and Priorities in Mathematics Assessment. In R. Lesh & S. Lamon (Eds.), Assessments of Authentic Performance in School Mathematics (pp. 3-16). Washington, DC: American Association for the Advancement of Sciences Press.

Lesh, R., Lamon, S., Gong, B. & Post, T. (1992) Using Learning Progress Maps to Improve Educational Decision Making. In R. Lesh & S. Lamon (Eds.), Assessments of Authentic Performance in School Mathematics (pp. 343-375 ). Washington, DC: American Association for the Advancement of Sciences Press.

Lesh, R., Lamon, S., Lester, F. & Behr, M. (1992) Future Directions for Mathematics Assessment. In R. Lesh & S. Lamon (Eds.), Assessments of Authentic Performance in School Mathematics (pp. 379-425). Washington, DC: American Association for the Advancement of Sciences Press.

Lesh, R., Landau, M. & Hamilton, E. (1983). Conceptual models in applied mathematical problem solving research. In R. Lesh & M. Landau (Eds.), Acquisition of Mathematics Concepts & Processes (pp. 263-343). NY: Academic Press.

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 & National Council of Teachers of Mathematics.

Lesh, R., Post, T., & Behr, M. (1987). Dienes revisited: Multiple embodiments in computer environments. In I. Wirsup & R. Streit (Eds.), Development in School Mathematics Education Around the World (pp. 647-680). Reston, VA: National Council of Teachers of Mathematics.

Lesh, R., Post, T., & Behr, M. (1987). Representations and Translations among Representations in Mathematics Learning and Problem Solving. In C. Janvier, (Ed.), Problems of Representations in the Teaching and Learning of Mathematics (pp. 33-40). Hillsdale, NJ: Lawrence Erlbaum.

Orton, R., Post, T., Behr, M., Cramer, K., Harel, G., & Lesh, R. (1995). Logical and psychological aspects of rational number pedagogical reasoning. Hiroshima Journal of Mathematics Education, 3, 63-75.

Post, T. (1989, September). One Point of View - Fractions and Other Rational Numbers. Arithmetic Teacher, 37(1), 3-28.

Post, T. (1988). Some notes on the nature of mathematics learning. In T. Post (Ed.), Teaching Mathematics in Grades K-8: Research Based Methods (pp. 1-19). Boston: Allyn & Bacon.

Post, T. (1981). The Role of Manipulative Materials in the Learning of Mathematical Concepts. In Selected Issues in Mathematics Education (pp. 109-131). Berkeley, CA: National Society for the Study of Education and National Council of Teachers of Mathematics, McCutchan Publishing Corporation.

Post, T. (1981, May). Fractions: Results and Implications from National Assessment. The Arithmetic Teacher, 28(9), 26-31.

Post, T. (1979). Making Time for the Basics: Some Thoughts on Viable Alternatives Within a Balanced Mathematics Program. In S. Sharron & R. Reys (Eds.), Applications in School Mathematics, 1979 Yearbook (pp. 352-356). Reston, VA: National Council of Teachers of Mathematics.

Post, T., Behr, M., & Lesh, R. (1988). Proportionality and the development of pre-algebra understandings. In A. Coxford & A. Shulte (Eds.) The Idea of Algebra K-12: Yearbook National Council of Teachers of Mathematics (pp. 78-90). Reston, VA: NCTM.

Post, T., Behr, M., & Lesh, R. (1986). Research-Based Observations About Children's Learning of Rational Number Concepts. Focus on Learning Problems in Mathematics, 8(1), 39-48.

Post, T., Behr, M., & Lesh, R. (1982, April). Interpretations of Rational Number Concepts. In L. Silvey & J. Smart (Eds.), Mathematics for Grades 5-9, 1982 NCTM Yearbook (pp. 59-72). Reston, Virginia: NCTM.

Post, T., Behr, M., Lesh, R., Wachsmuth, I. (1986, Spring). Selected Results from the Rational Number Project. In Proceedings of Ninth Psychology of Mathematics Education Conference the Netherlands (pp. 342-351). International Group for the Psychology of Mathematics Education, ANTWERP The Netherlands. This paper was reprinted in The Math Times Journal-Official Journal of the Minnesota Council of Teachers of Mathematics, Vol. 1, No. 1.

Post, T., & Cramer, K. (1989, March). Knowledge, Representation and Quantitative Thinking. In M. Reynolds (Ed.) Knowledge Base for the Beginning Teacher - Special publication of the AACTE (pp. 221-231). Oxford: Pergamon Press.

Post, T., & Cramer, K. (1987, October). Children's strategies when ordering rational numbers. Arithmetic Teacher, 35(2), 33-35.

Post, T., Cramer, K., Behr, M., Lesh, R., & Harel, G. (1993). Curriculum implications of Research on the Learning, Teaching, and Assessing of Rational Number Concepts. In T. Carpenter, E. F& Harel, G. (In press). Designing instructionally relevant assessment reports. In T. Carpenter & E. Fennema (Eds.), Research on the Learning, Teaching, and Assessing of Rational Number Concepts. Lawrence Erlbaum and Associates.

Post, T., Cramer, K., Harel, G., Kiernen, T., & Lesh, R. (1998) Research on rational number, ratio and proportionality. Proceedings of the Twentieth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, PME-NA XX Volume I (pp. 89-93). Raleigh, North Carolina.

Post, T., Harel, G., Behr, M. & Lesh, R. (1991). Intermediate Teachers' Knowledge of Rational Number Concepts. In E. Fennema, T. Carpenter, S. Lamon (Eds.), Integrating research on teaching and learning mathematics (pp. 177-198). NY: State University of NY Press.

Post, T., Harel, G., Behr, M., & Lesh, R. (1988). Intermediate teachers knowledge of rational number concepts. In Fennema, et al. (Eds.), Papers from First Wisconsin Symposium for Research on Teaching and Learning Mathematics (pp. 194-219). Madison, WI: Wisconsin Center for Education Research.

Post, T., & Reys, R. E. (1979). Abstraction Generalization and Design of Mathematical Experiences for Children. In K. Fuson & W. Geeslin (Eds.), Models for mathematics learning. (pp. 117-139). Columbus, OH: ERIC/SMEAC.

Post T., Wachsmuth I., Lesh R., & Behr M. (1985, January). Order and Equivalence of Rational Number: A Cognitive Analysis. Journal for Research in Mathematics Education, 16(1), 18-36.

Reiss, M., Behr, M., Lesh, R., & Post, T. (1988, July) The assessment of cognitive structures in proportional reasoning. In J. Bergeron, et al. (Eds.), Proceedings of the Eleventh International Conference, Psychology of Mathematics Education PMR - XI Volume II (pp. 310-316). Montreal, Canada: PME.

Reiss, M., Behr, M., Lesh, R., & Post, T. (1985, July). Cognitive Processes And Products in Proportional Reasoning. In L. Streefland (Ed.), Proceedings of the Ninth International Conference for the Psychology of Mathematics Education (pp. 352-356). Noordwijkerhout (Utrecht), Holland: PME.

Titus, J. (1995). The concept of fractional number among deaf and hard of hearing students. American Annals of the Deaf, 140(3), 255-263.

Wachsmuth I., Behr M., & Post T. (1983, July). Children's perception of fractions and ratios in grade 5. In R. Hershkowitz (Ed.), Proceedings of the International Group for the Psychology of Mathematics Education VII (pp. 164-169). Rehovot, Israel: Department of Science Teaching, The Weizmann Institute of Science.

Wachsmuth, I., Bright, G., Behr, M., & Post, T. (1986). Assessing fifth grade children's rational number knowledge in a non verbal application context: The darts game. Recherches en Didactique des Mathematiques., 7(3), 51-74.

Wyberg,Terry; Whitney, Stephanie R.; Cramer, Kathleen A.; Monson, Debra S.; & Leavitt, Seth. (2011, December) Unfolding Fraction Multiplication Mathematics Teaching in the Middle School. 17(5)5, 288-294.

 


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