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Reasoning Dynamically About the Area of a Rectangle: The Case of Lora and Isaac

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Abstract

This article reports on an exploratory study that engaged students in dynamic experiences of generating the area of a rectangle as a sweep of a line segment over a distance. A case study from a design experiment with one pair of third-grade students is presented to initiate a discussion around the forms of reasoning that students may exhibit as a result of their engagement with these dynamic motion tasks and the characteristics of the design that supported these particular forms of reasoning. The findings of this study show that engaging students in dynamic experiences of area may help them develop a conceptual understanding of the area of a rectangle as a continuous structure that can dynamically change based on the two linear measures that generate it: the length of the line segment swept and the distance of the sweep. These experiences can also help students from an early age develop a flexible understanding of a unit and reason covariationally about the continuous change of the quantities in measurement.

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Notes

  1. Note that she said “swipes” instead of sweeps, because that term was used during the experiment; however, since then it was decided that ‘sweep’ is a better term to describe the action above.

References

  • Barab, S., & Squire, K. (2004). Design-based research: Putting a stake in the ground. The Journal of the Learning Sciences, 13(1), 1–14.

    Google Scholar 

  • Barrett, J., & Clements, D. (2003). Quantifying path length: Fourth-grade children’s developing abstractions for linear measurement. Cognition and Instruction, 21(4), 475–520.

    Google Scholar 

  • Barrett, J., Clements, D., Sarama, J., Miller, A., Cullen, C., Van Dine, D., Newburgh, K., Vukovich, M., Eames, C., & Klanderman, D. (2017). Integration of results: A revised learning trajectory for area measurement. In J. Barrett, D. Clements, & J. Sarama (Eds.), Children’s measurement: A longitudinal study of children’s knowledge and learning of length, area, and volume (pp. 129–148). Reston, VA: National Council of Teachers of Mathematics.

    Google Scholar 

  • Barrett, J., Cullen, C., Miller, A., Eames, C., Kara, M., & Klanderman, D. (2017). Area in the middle and later elementary grades. In J. Barrett, D. Clements, & J. Sarama (Eds.), Children’s measurement: A longitudinal study of children’s knowledge and learning of length, area, and volume (pp. 105–127). Reston, VA: National Council of Teachers of Mathematics.

    Google Scholar 

  • Battista, M. (2004). Applying cognition-based assessment to elementary school students’ development of understanding of area and volume measurement. Mathematical Thinking and Learning, 6(2), 185–204.

    Google Scholar 

  • Battista, M. (2008). Development of the shape makers geometry microworld: Design principles and research. In G. Blume & K. Heid (Eds.), Research on technology and the teaching and learning of mathematics: Vol. 2 cases and perspectives (pp. 131–156). Charlotte, NC: Information Age Publishing.

    Google Scholar 

  • Battista, M., Clements, D., Arnoff, J., Battista, K., & Borrow, C. (1998). Students’ spatial structuring of 2D arrays of squares. Journal for Research in Mathematics Education, 29(5), 503–532.

    Google Scholar 

  • Baturo, A., & Nason, R. (1996). Student teachers’ subject matter knowledge within the domain of area measurement. Educational Studies in Mathematics, 31(3), 235–268.

    Google Scholar 

  • Brady, C. & Lehrer, R. (2020). Sweeping area across physical and virtual environments. Digital Experiences in Mathematics Education, 6(3).

  • Brown, A. (1992). Design experiments: Theoretical and methodological challenges in creating complex interventions in classroom settings. The Journal of the Learning Sciences, 2(2), 141–178.

    Google Scholar 

  • Clements, D., & Sarama, J. (2009). Learning and teaching early math: The learning trajectories approach. New York, NY: Routledge.

    Google Scholar 

  • Clements, D., & Stephan, M. (2004). Measurement in pre-K to grade 2 mathematics. In D. Clements, J. Sarama, & A. di Biase (Eds.), Engaging young children in mathematics: Standards for early childhood mathematics education (pp. 299–317). Mahwah, NJ: Lawrence Erlbaum Associates.

    Google Scholar 

  • Clements, D., Sarama, J., Van Dine, D., Barrett, J., Cullen, C., Hudyma, A., Dolgin, R., Cullen, A., & Eames, C. (2018). Evaluation of three interventions teaching area measurement as spatial structuring to young children. The Journal of Mathematical Behavior, 50, 23–41.

    Google Scholar 

  • Cobb, P., & Gravemeijer, K. (2008). Experimenting to support and understand learning processes. In A. Kelly, R. Lesh, & J. Baek (Eds.), Handbook of design research methods in education: Innovations in science, technology, engineering, and mathematics learning and teaching (pp. 68–95). New York, NY: Routledge.

    Google Scholar 

  • Cobb, P., & Wheatley, G. (1988). Children’s initial understandings of ten. Focus on Learning Problems in Mathematics, 10(3), 1–28.

    Google Scholar 

  • Cobb, P., Confrey, J., DiSessa, A., Lehrer, R., & Schauble, L. (2003). Design experiments in educational research. Educational Researcher, 32(1), 9–13.

    Google Scholar 

  • Cobb, P., Stephan, M., McClain, K., & Gravemeijer, K. (2001). Participating in classroom mathematical practices. The Journal of the Learning Sciences, 10(1–2), 113–163.

    Google Scholar 

  • Confrey, J., & Smith, E. (1994). Exponential functions, rates of change, and the multiplicative unit. Educational Studies in Mathematics, 26(2/3), 135–164.

    Google Scholar 

  • Confrey, J., Nguyen, K., Lee, K., Panorkou, N., Corley, A., & Maloney, A. (2012). Turn-on common core math: Learning trajectories for the common Core state standards for mathematics. Raleigh, NC: North Carolina State University (www.turnonccmath.net).

    Google Scholar 

  • Cuoco, A., & Goldenberg, P. (1997). Dynamic geometry as a bridge from Euclidean geometry to analysis. In J. King & D. Schattschneider (Eds.), Geometry turned on! Dynamic software in learning, teaching, and research (pp. 33–46). Washington, DC: The Mathematical Association of America.

    Google Scholar 

  • de Villiers, M. (1999). Rethinking proof with the Geometer’s sketchpad. Emeryville, CA: Key Curriculum Press.

    Google Scholar 

  • Duval, R. (2005). Les conditions cognitives de l’apprentissage de la géométrie: Développement de la visualisation, différenciation des raisonnements et coordination de leursfonctionnements. Annales de Didactique et de Sciences Cognitives, 10, 5–53.

    Google Scholar 

  • Harel, G. (2013). Intellectual need. In K. Leatham (Ed.), Vital directions for mathematics education research (pp. 119–151). New York, NY: Springer.

    Google Scholar 

  • Hoyles, C. (2018). Transforming the mathematical practices of learners and teachers through digital technology. Research in Mathematics Education, 20(3), 209–228.

    Google Scholar 

  • Izsák, A. (2005). “You have to count the squares”: Applying knowledge in pieces to learning rectangular area. The Journal of the Learning Sciences, 14(3), 361–403.

    Google Scholar 

  • Jackiw, N. (1995). The Geometer’s sketchpad, v3.0. Emeryville, CA: Key Curriculum.

    Google Scholar 

  • Kamii, C., & Kysh, J. (2006). The difficulty of “length × width”: Is a square the unit of measurement? The Journal of Mathematical Behavior, 25(2), 105–115.

    Google Scholar 

  • Kobiela, M., & Lehrer, R. (2019). Supporting dynamic conceptions of area and its measure. Mathematical Thinking and Learning, 21(3), 178–206.

    Google Scholar 

  • Laborde, C. &Laborde, J.-M. (2011). Interactivity in dynamic mathematics environments: What does that mean? (http://atcm.mathandtech.org/EP2011/invited_papers/3272011_19113.pdf).

  • Laborde, C., Kynigos, C., Hollebrands, K., & Strasser, R. (2006). Teaching and learning geometry with technology. In A. Gutierrez & P. Boero (Eds.), Handbook of research on the psychology of mathematics education: Past, present, and future (pp. 275–304). Rotterdam, The Netherlands: Sense Publishers.

    Google Scholar 

  • Lakoff, G., & Núñez, R. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. New York, NY: Basic Books.

    Google Scholar 

  • Lehrer, R. (2003). Developing understanding of measurement. In J. Kilpatrick, W. 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., Slovin, H., & Dougherty, B. (2014). Developing essential understanding of geometry and measurement for teaching mathematics in grades 3–5. Reston, VA: National Council of Teachers of Mathematics.

    Google Scholar 

  • Leung, A. (2008). Dragging in a dynamic geometry environment through the lens of variation. International Journal of Computers for Mathematical Learning, 13(2), 135–157.

    Google Scholar 

  • Nemirovsky, R., & Ferrara, F. (2009). Mathematical imagination and embodied cognition. Educational Studies in Mathematics, 70(2), 159–174.

    Google Scholar 

  • Ng, O.-L., & Sinclair, N. (2015). “Area without numbers”: Using touchscreen dynamic geometry to reason about shape. Canadian Journal of Science, Mathematics and Technology Education, 15(1), 84–101.

    Google Scholar 

  • Norton, A. (2016). (Ir)reversability in mathematics. In M. Wood, E. Turner, M. Civil, & J. Eli (Eds.), Proceedings of the 38th annual meeting of the north American chapter of the International Group for the Psychology of mathematics education (pp. 1596–1603). Tucson, AZ: The University of Arizona.

    Google Scholar 

  • Outhred, L., & Mitchelmore, M. (2000). Young children’s intuitive understanding of rectangular area measurement. Journal for Research in Mathematics Education, 31(2), 144–167.

    Google Scholar 

  • Panorkou, N., & Pratt, D. (2016). Using google sketch up to research students’ experiences of dimension in geometry. Digital Experiences in Mathematics Education, 2(3), 199–227.

    Google Scholar 

  • Papert, S. (1980). Mindstorms: Children, computers, and powerful ideas. New York, NY: Basic Books.

    Google Scholar 

  • Piaget, J., Inhelder, B., & Szeminska, A. (1960). The child’s conception of geometry. London, UK: Routledge & Kegan Paul.

    Google Scholar 

  • Reynolds, A., & Wheatley, G. (1996). Elementary students’ construction and coordination of units in an area setting. Journal for Research in Mathematics Education, 27(5), 564–581.

    Google Scholar 

  • Roschelle, J., Kaput, J., & Stroup, W. (2000). SimCalc: Accelerating students’ engagement with the mathematics of change. In M. Jacobsen & R. Kozma (Eds.), Innovations in science and mathematics education: Advanced designs for technologies of learning (pp. 47–75). Hillsdale, NJ: Lawrence Erlbaum Associates.

    Google Scholar 

  • Saldanha, L., & Thompson, P. (1998). Re-thinking covariation from a quantitative perspective: Simultaneous continuous variation. In S. Berensen & W. Coulombe (Eds.), Proceedings of the 20th annual meeting of the north American chapter of the International Group for the Psychology of mathematics education (Vol. 1, pp. 298–304). Raleigh, NC: North Carolina State University.

    Google Scholar 

  • Simon, M. (2018). An emerging methodology for studying mathematics concept learning and instructional design. The Journal of Mathematical Behavior, 52, 113–121.

    Google Scholar 

  • Simon, M., & Blume, G. (1994). Building and understanding multiplicative relationships: A study of prospective elementary teachers. Journal for Research in Mathematics Education, 25(5), 472–494.

    Google Scholar 

  • Sinclair, N., Pimm, D., & Skelin, M. (2012). Developing essential understanding of geometry for teaching mathematics in grades 6–8. Reston, VA: National Council of Teachers of Mathematics.

    Google Scholar 

  • Steffe, L. (1983). Children’s algorithms as schemes. Educational Studies in Mathematics, 14(2), 109–125.

    Google Scholar 

  • Steffe, L. (1992). Schemes of action and operation involving composite units. Learning and Individual Differences, 4(3), 259–309.

    Google Scholar 

  • Thompson, P. (2000). What is required to understand fractal dimension? The Mathematics Educator, 10(2), 33–35.

    Google Scholar 

  • Thompson, P., & Saldanha, L. (2003). Fractions and multiplicative reasoning. In J. Kilpatrick, G. Martin, & D. Schifter (Eds.), A research companion to principles and standards for school mathematics (pp. 95–113). Reston, VA: National Council of Teachers of Mathematics.

    Google Scholar 

  • van den Heuvel-Panhuizen, M., & Drijvers, P. (2014). Realistic mathematics education. In S. Lerman (Ed.), Encyclopedia of mathematics education (pp. 521–525). Dordrecht, The Netherlands: Springer.

    Google Scholar 

  • Wheatley, G., & Reynolds, A. (1996). The construction of abstract units in geometric and numeric settings. Educational Studies in Mathematics, 30(1), 67–83.

    Google Scholar 

  • Wildi, T. (1991). Units and conversion charts: The metrification handbook for engineers and scientists. Los Alamitos, CA: IEEE Computer Society Press.

    Google Scholar 

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Acknowledgments

This research was supported by the Spencer Foundation under grant#201600101. The views expressed do not necessarily reflect official positions of the Foundation. The author would like to thank Dr. Debasmita Basu, Erell Germia, Toni York, Dr. Steven Greenstein and the anonymous reviewers for their valuable feedback to earlier drafts of this manuscript.

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  1. Department of Mathematics, Montclair State University, 1 Normal Avenue, Montclair, NJ, 07030, USA

    Nicole Panorkou

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Panorkou, N. Reasoning Dynamically About the Area of a Rectangle: The Case of Lora and Isaac. Digit Exp Math Educ 6, 257–292 (2020). https://doi.org/10.1007/s40751-020-00074-4

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