Conceptual Limitations in Curricular Presentations of Area Measurement: One Nation’s Challenges

References

• Ball, D. L., & Cohen, D. K. (1996). Reform by the book: What is—or might be—the role of curriculum materials in teacher learning and instructional reform? Educational Researcher, 25, 6–14.
   Google Scholar
• Baroody, A. J., Feil, Y., & Johnson, A. R. (2007). An alternative reconceptualization of procedural and conceptual knowledge. Journal for Research in Mathematics Education, 38, 115–131.
   Web of Science ®Google Scholar
• Battista, M. T. (2012). Cognition-based assessment & teaching of geometric measurement: Building on students’ reasoning. Portsmouth, NH: Heinemann.
   Google Scholar
• Battista, M. T., Clements, D. H., Arnoff, J., Battista, K., & Borrow, C. V. A. (1998). Students’ spatial structuring of 2D arrays of squares. Journal for Research in Mathematics Education, 29, 503–532. doi:10.2307/749731
   Web of Science ®Google Scholar
• Baturo, A., & Nason, R. (1996). Student teachers’ subject matter knowledge within the domain of area measurement. Educational Studies in Mathematics, 31, 235–268. doi:10.1007/BF00376322
   Google Scholar
• Berenson, S., Van Der Valk, T., Oldham, E., Runesson, U., Moreira, C. Q., & Broekman, H. (1997). An international study to investigate prospective teachers’ content knowledge of the area concept. European Journal of Teacher Education, 20, 137–150. doi:10.1080/0261976970200203
   Google Scholar
• Blume, G. W., Galindo, E., & Walcott, C. (2007). Performance in measurement and geometry from the viewpoint of principles and standards for school mathematics. In P. Kloosterman & F. K. Lester (Eds.), 2003 Mathematics assessment of the national assessment of educational progress (pp. 95–138). Reston, VA: National Council of Teachers of Mathematics.
   Google Scholar
• Bonotto, C. (2003). About students’ understanding and learning of the concept of surface area. In D. H. Clements & G. Bright (Eds.), Learning and teaching measurement: 2003 Yearbook (pp. 157–167). Reston, VA: National Council of Teachers of Mathematics.
   Google Scholar
• Brown, M., & Edelson, D. (2003). Teaching as design: Can we better understand the ways in which teachers use materials so we can better design materials to support their changes in practice? (Design brief). Evanston, IL: Center for Learning Technologies in Urban Schools.
   Google Scholar
• Chappell, M. F., & Thompson, D. R. (1999). Perimeter or area? Which measure is it? Mathematics Teaching in the Middle School, 5, 20–23.
   Google Scholar
• Charalambous, C. Y., Delaney, C., Hsu, H., & Mesa, V. (2010). A comparative analysis of the addition and subtraction of fractions in textbooks from three countries. Mathematical Thinking and Learning, 12, 117–151. doi:10.1080/10986060903460070
   Web of Science ®Google Scholar
• Charles, R., Crown, W., & Fennell, F. (2008). Mathematics, Michigan edition. Glenview, IL: Pearson Education Inc.
   Google Scholar
• Choppin, J. (2011). Learned adaptations: Teachers’ understanding and use of curriculum resources. Journal of Mathematics Teacher Education, 14, 331–353. doi:10.1007/s10857-011-9170-3
   Google Scholar
• Clements, D. H., & Sarama, J. (2009). Teaching and learning early math: The learning trajectories approach. New York, NY: Routledge.
   Google Scholar
• Confrey, J. (2012, November). Articulating a learning sciences foundation for learning trajectories in the CCSSM. Proceedings of the 34th Annual Meeting of the North American Chapter of the International Group for Psychology in Mathematics Education (PME-NA), 1–22. Kalamazoo, MI.
   Google Scholar
• Crosby, A. W. (1997). The measure of reality: Quantification and western society, 1250–1600. Cambridge, UK: Cambridge University Press.
   Google Scholar
• Ding, M., & Li, X. (2010). A comparative analysis of the distributive property in US and Chinese elementary mathematics textbooks. Cognition & Instruction, 28, 46–180. doi:10.1080/07370001003638553
   Web of Science ®Google Scholar
• diSessa, A. A. (1993). Toward an epistemology of physics. Cognition & Instruction, 10, 105–225. doi:10.1080/07370008.1985.9649008
   Web of Science ®Google Scholar
• diSessa, A. A. (1996). What do “just plain folks” know about physics? In D. R. Olson, & N. Torrance (Eds.), The handbook of education and human development: New models of learning, teaching, and schooling (pp. 709–730). Oxford, UK: Blackwell Publishers, Ltd.
   Google Scholar
• diSessa, A. A., & Wagner, J. F. (2005). What coordination has to say about transfer. In J. Mestre (Ed.), Transfer of learning from a modern multi-disciplinary perspective (pp. 121–154). Greenwich, CT: Information Age.
   Google Scholar
• Floden, R. E. (2002). The measurement of opportunity to learn. In A. C. Porter & A. Gamoran (Eds.), Methodological advances in cross-national surveys of educational assessment (pp. 231–266). Washington, DC: National Academy Press.
   Google Scholar
• Greer, B. (1992). Multiplication and division as models of situations. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 276–295). New York, NY: MacMillan.
   Google Scholar
• Grouws, D. A., Smith, M. S., & Sztajn, P. (2004). The preparation and teaching practices of United States mathematics teachers: Grades 4 and 8. In P. Kloosterman & F. K. Lester (Eds.), Results and interpretations of the 1990 through 2000 mathematics assessments of the national assessment of educational progress (pp. 221–267). Reston, VA: National Council of Teachers of Mathematics.
   Google Scholar
• Heraud, B. (1987). Conceptions of area units by 8–9 year old children. In J. C. Bergeron, N. Herscovics, & C. Kieran (Eds.), Proceedings of the 11th International Conference of the Psychology of Mathematics Education (Vol. 3, pp. 299–304). Paris, France: CNRS-Paris.
   Google Scholar
• Herbel-Eisenmann, B. A., & Otten, S. (2011). Mapping mathematics in classroom discourse. Journal for Research in Mathematics Education, 42, 451–485. doi:10.5951/jresematheduc.42.5.0451
   Web of Science ®Google Scholar
• Hiebert, J. & Lefvre, P. (1986). Conceptual and procedural knowledge in mathematics: An introductory analysis. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 1–27). Hillsdale, NJ: Lawrence Erlbaum.
   Google Scholar
• Hino, K. (2002). Acquiring new use of multiplication through classroom teaching: An exploratory study. Journal of Mathematical Behavior, 20, 477–502. doi:10.1016/S0732-3123(02)00085-8
   Google Scholar
• Joram, E., Subrahmanyam, K., & Gelman, R. (1998). Measurement estimation: Learning to map the route from number to quantity and back. Review of Educational Research, 68, 413–449. doi:10.3102/00346543068004413
   Web of Science ®Google Scholar
• Kamii, C., & Kysh, J. (2006). The difficulty of “length x width”: Is a square the unit of measurement? Journal of Mathematical Behavior, 25, 105–115. doi:10.1016/j.jmathb.2006.02.001
   Google Scholar
• Kaur, B. (2014). Enactment of school mathematics curriculum in Singapore: Whither research! ZDM, 46, 829–836. doi:10.1007/s11858-014-0619-6
   Google Scholar
• Kobiela, M., Lehrer, R., & Pfaff, E. (2010, April). Students’ developing conceptions of area via partitioning and sweeping. Paper presented at the Annual Meeting of the American Education Research Association, Denver, Colorado.
   Google Scholar
• Kordaki, M., & Potari, D. (2002). The effect of area measurement tools on student strategies: The role of a computer microworld. International Journal for Computers for Mathematics Learning, 7, 65–100. doi:10.1023/A:1016051411284
   Google Scholar
• Kospentaris, G., Spyrou, P., & Lappas, D. (2011). Exploring students’ strategies in area conservation geometry tasks. Educational Studies in Mathematics, 77, 105–127. doi:10.1007/s10649-011-9303-8
   Web of Science ®Google Scholar
• Larson, N. (2004). Saxon Math. Austin, TX: Saxon Publishers.
   Google Scholar
• Lehrer, R. (2003). Developing understanding of measurement. In J. Kilpatrick, W. G. Martin, & D. Schifter (Eds.), A research companion to principles and standards for school mathematics (pp. 179–192). Reston, VA: National Council of Teachers of Mathematics.
   Google Scholar
• Lehrer, R., Jaslow, L., & Curtis, C. (2003). Developing an understanding of measurement in the elementary grades. In D. H. Clements & G. Bright (Eds.), Learning and teaching measurement: 2003 yearbook (pp. 100–121). Reston, VA: National Council of Teachers of Mathematics.
   Google Scholar
• Lehrer, R., Jenkins, M., & Osana, H. (1998). Longitudinal study of children’s reasoning about space and geometry. In R. Lehrer & D. Chazan (Eds.), Designing learning environments for developing understanding of geometry and space (pp. 137–167). Mahwah, NJ: Lawrence Erlbaum.
   Google Scholar
• Lehrer, R., & Slovin, H. (2014). Developing essential understanding of geometry and measurement in grades 3–5. Reston, VA: National Council of Teachers of Mathematics.
   Google Scholar
• Li, X., Ding, M., Capraro, M. M., & Capraro, R. M. (2008). Sources of differences in children’s understandings of mathematical equality: Comparative analysis of teacher guides and student texts in China and the United States. Cognition & Instruction, 26, 195–217. doi:10.1080/07370000801980845
   Web of Science ®Google Scholar
• Maloney, A. P., Confrey, J., & Nguyen, K. H. (2014). Learning over time: Learning trajectories in mathematics education. Charlotte, NC: Information Age.
   Google Scholar
• Murphy, C. (2012). The role of subject knowledge in primary student prospective teachers’ approaches to teaching topic of area. Journal of Mathematics Teacher Education, 15, 187–206. doi:10.1007/s10857-011-9194-8
   Google Scholar
• National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author.
   Google Scholar
• National Research Council. (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academy Press.
   Google Scholar
• Nesher, P. (1988). Multiplicative school word problems: Theoretical approaches and empirical findings. In J. Hiebert & M. Behr (Eds.), Number concepts and operations in the middle grades (pp. 19–40). Reston, VA: National Council of Teachers of Mathematics; Hillsdale, NJ: Lawrence Erlbaum Associates.
   Google Scholar
• Nunes, T., Light, P., & Mason, J. (1993). Tools for thought: The measurement of length and area. Learning and Instruction, 3, 39–54. doi:10.1016/S0959-4752(09)80004-2
   Google Scholar
• Outhred, L. N., & Mitchelmore, M. C. (2000). Young children’s intuitive understanding of rectangular area measurement. Journal for Research in Mathematics Education, 31, 144–167. doi:10.2307/749749
   Web of Science ®Google Scholar
• Owens, K., & Outhred, L. (1998). Covering shapes with tiles: Primary students’ visualization and drawing. Mathematics Education Research Journal, 10, 28–41. doi:10.1007/BF03217056
   Google Scholar
• Piaget, J., Inhelder, B., & Szeminska, A. (1960). The child’s conception of geometry. New York, NY: Basic Books.
   Google Scholar
• Polikoff, M. S. (2015). How well aligned are textbooks to the common core standards in mathematics? American Educational Research Journal, 53, 1185–1211. doi:10.3102/0002831215584435
   Web of Science ®Google Scholar
• Remillard, J. T., Harris, B., & Agodini, R. (2014). The influence of curriculum material design on opportunities for student learning. ZDM, 46, 735–749. doi:10.1007/s11858-014-0585-z
   Google Scholar
• Remillard, J. T., & Heck, D. J. (2014). Conceptualizing the curriculum enactment process in mathematics education. ZDM, 46, 705–718. doi:10.1007/s11858-014-0600-4
   Google Scholar
• Reys, B. J., & Reys, R. E. (2006). The development and publication of elementary mathematics textbooks: Let the buyer beware! Phi Delta Kappan, 87, 377–383. doi:10.1177/003172170608700510
   Web of Science ®Google Scholar
• Schifter, D., & Szymaszek, J. (2003). Structuring a rectangle: Teachers write to learn about their students’ thinking. In D. Clements, & G. Bright (Eds.), Learning and Teaching Measurement: 2003 Yearbook (pp. 143–156). Reston, VA: National Council of Teachers of Mathematics.
   Google Scholar
• Schmidt, W. H., McKnight, C. C., & Raizen, S. A. (1997). A splintered vision: An investigation of U.S. science and mathematics education. Dordrecht, The Netherlands: Kluwer.
   Google Scholar
• Schoenfeld, A. H. (1985). Mathematical problem solving. Orlando, FL: Academic Press.
   Google Scholar
• Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition, and sense-making. In D. A. Grouws (Ed.), Handbook of Research in Teaching and Learning Mathematics (pp. 334–370). New York, NY: Macmillan.
   Google Scholar
• 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, 472–494. doi:10.2307/749486
   Web of Science ®Google Scholar
• Smith, J. P., Males, L. M., Dietiker, L. C., Lee, K., & Mosier, A. (2013). Curricular treatments of length measurement in the United States: Do they address known learning challenges? Cognition & Instruction, 31, 388–433. doi:10.1080/07370008.2013.828728
   Web of Science ®Google Scholar
• Stavy, R., & Tirosh, D. (2000). How students (mis-)understand science and mathematics: Intuitive rules. New York, NY: Teachers College Press.
   Google Scholar
• Stein, M. K., Grover, B., & Henningsen, M. (1996). Building student capacity for mathematical thinking and reasoning: An analysis of mathematical tasks used in reform classrooms. American Educational Research Journal, 33, 455–488. doi:10.3102/00028312033002455
   Web of Science ®Google Scholar
• Stein, M. K., & Kim, G. (2009). The role of mathematics curriculum materials in large-scale urban reform: An analysis of demands and opportunities for teacher learning. In J. T. Remillard, B. A. Herbbel-Eisenmann, & G. M. Lloyd (Eds.), Mathematics teachers at work: Connecting curriculum materials and classroom instruction (pp. 37–55). New York, NY: Routledge.
   Google Scholar
• Stein, M. K., Remillard, J., & Smith, M. S. (2007). How curriculum influences student learning. In F. K. Lester Jr. (Ed.), Second Handbook of Research on Mathematics Teaching and Learning (pp. 319–369). Reston, VA: National Council of Teachers of Mathematics.
   Google Scholar
• Stephan, M., & Clements, D. (2003). Linear and area measurement in Prekindergarten to grade 2. In D. H. Clements & G. Bright (Eds.), Learning and teaching measurement: 2003 Yearbook (pp. 3–16). Reston, VA: National Council of Teachers of Mathematics.
   Google Scholar
• Sweller, J., & Cooper, G. A. (1985). The use of worked examples as a substitute for problem solving in learning algebra. Cognition and Instruction, 2, 59–89. doi:10.1207/s1532690xci0201_3
   Google Scholar
• The University of Chicago School Mathematics Project. (2007). Everyday mathematics (3rd ed.). Chicago, IL: Wright Group/McGraw Hill.
   Google Scholar
• Thompson, A. G., Philipp, R. A., Thompson, P. W., & Boyd, B. (1994). Calculational and conceptual orientations in teaching mathematics. In D. B. Aichele (Ed.), 1994 yearbook of the national council of teachers of mathematics (pp. 79–92). Reston, VA: National Council of Teachers of Mathematics.
   Google Scholar
• Thompson, T. D., & Preston, R. V. (2004). Measurement in the middle grades: Insights from NAEP and TIMSS. Mathematics Teaching in the Middle School, 9, 514–519.
   Google Scholar
• Vamvakoussi, X., & Vosniadou, S. (2010). How many decimals are there between two fractions? Secondary school students’ understanding of rational numbers and their notation. Cognition & Instruction, 28, 181–209. doi:10.1080/07370001003676603
   Web of Science ®Google Scholar
• Vosniadou, S., Vamvakoussi, X., & Skopeliti, I. (2008). The framework theory approach to conceptual change. In S. Vosniadou (Ed.), International handbook of research on conceptual change (pp. 3–34). Mahwah, NJ: Lawrence Erlbaum.
   Google Scholar
• Watson, A. (2010). Key understandings in learning mathematics. Scottish Mathematics Council Journal, 40, 14–16.
   Google Scholar
• Woodward, E., & Byrd, F. (1983). Area: Included topic, neglected concept. School Science and Mathematics, 83, 343–347.
   Google Scholar
• Zacharos, K. (2006). Prevailing educational practices for area measurement and students’ failure in measuring areas. The Journal of Mathematical Behavior, 25, 224–239. doi:10.1016/j.jmathb.2006.09.003
   Google Scholar