# Developing a network of and for geometric reasoning

## Abstract

In this article, we develop a theoretical model for restructuring mathematical tasks, usually considered advanced, with a network of spatial visual representations designed to support geometric reasoning for learners of disparate ages, stages, strengths, and preparation. Through our geometric reworking of the well-known “open box problem”, we sought to enrich learners’ conceptual networks for optimisation and rate of change, and to explore these concepts vertically across curricula for a variety of grades. We analyse a network of physical, geometric spatial visual representations that can support new inferences and key understandings, and that scaffold these advanced concepts so that they could be meaningfully addressed by learners of various ages, from elementary to university, and with diverse mathematical backgrounds.

### References

- Anghileri, J. (2006). Scaffolding practices that enhance mathematics learning.
*Journal of Mathematics Teacher Education,**9*(1), 33–52.CrossRefGoogle Scholar - Arcavi, A. (2003). The role of visual representations in the learning of mathematics.
*Educational Studies in Mathematics,**52*(3), 215–241.CrossRefGoogle Scholar - Battista, M. (2003). Understanding students’ thinking about area and volume measurement. In D. H. Clements & G. Bright (Eds.), Learning and teaching measurement (pp. 122–142). Reston: National Council of Teachers of Mathematics.Google Scholar
- Berry, J. S., & Nyman, M. A. (2003). Promoting students’ graphical understanding of the calculus.
*The Journal of Mathematical Behavior,**22*(4), 479–495.CrossRefGoogle Scholar - Bryant, P. (2008). Paper 5: understanding spaces and its representation in mathematics. In Nunez, T., Bryant, P. & Watson, A. (Eds.), Key understanding in mathematics learning: a report to the Nuffield Foundation.Google Scholar
- Cohen, E. G. (1994). Restructuring the classroom: conditions for productive small groups.
*Review of Educational Research,**64*(1), 1–35.CrossRefGoogle Scholar - Cohen, D. K., & Ball, D. L. (1999). Instruction, capacity, and improvement. CPRE research report series RR-43. Philadelphia: Consortium for Policy Research in Education, University of Pennsylvania.Google Scholar
- Coulson, S., & Oakley, T. (2005). Blending and coded meaning: literal and figurative meaning in cognitive semantics.
*Journal of Pragmatics,**37*(10), 1510–1536.CrossRefGoogle Scholar - Cuoco, A.A., & Goldenberg, E.P. (1997). Dynamic geometry as a bridge from Euclidean geometry to analysis. In King, J., &Schattschneider, D. (Eds.), Geometry turned on: dynamic software in learning, teaching, and research (No. 41). (pp. 33–44). Providence: The Mathematical Association of America (MAA).Google Scholar
- Dorko, A., & Speer, N. (2014). Calculus students’ understanding of volume. In Proceedings of the 17th annual conference on research in undergraduate mathematics education.Google Scholar
- Fauconnier, G. (1997).
*Mappings in thought and language*. Cambridge: Cambridge University Press.CrossRefGoogle Scholar - Grootenboer, P. (2009). Rich mathematical tasks in the maths in the Kimberley (MITK) Project. In R. Hunter, B. Bicknell, & T. Burgess (Eds.),
*Crossing divides: proceedings of the 32nd annual conference of the Mathematics Education Research Group of Australasia*(Vol. 1). Palmerston North: MERGA.Google Scholar - Herbert, S., & Pierce, R. (2008). An ‘emergent model’ for rate of change.
*International Journal of Computers for Mathematical Learning,**13*(3), 231–249.CrossRefGoogle Scholar - Jackiw, N. (1991).
*The geometer’s sketchpad computer software*. Emeryville: Key Curriculum Press.Google Scholar - Koedinger, K. R. (1992).
*Emergent properties and structural constraints: Advantages of diagrammatic representations for reasoning and learning*. Palo Alto: Paper presented at the AAAI spring symposium series: reasoning with diagrammatic reasoning.Google Scholar - Martin, T. (2000). Calculus students’ ability to solve geometric related-rates problems.
*Mathematics Education Research Journal,**12*(2), 74–91.CrossRefGoogle Scholar - Mason, J., & Pimm, D. (1984). Generic examples: seeing the general in the particular.
*Educational Studies in Mathematics,**15*(3), 277–289.CrossRefGoogle Scholar - Moreno-Armella, L. (2014). An essential tension in mathematics education.
*ZDM*—*The International Journal on Mathematics Education*,*46*(4), 621–633.Google Scholar - Moss, J., Hawes, Z., Naqvi, S., & Caswell, B. (2015). Adapting Japanese Lesson Study to enhance the teaching and learning of geometry and spatial reasoning in early years classrooms: a case study.
*ZDM*–*The International Journal on Mathematics Education*,*47*(3) (this issue).Google Scholar - National Research Council. (2006).
*Learning to think spatially: GIS as a support system in the K-12 curriculum*. Washington, D.C.: National Academies Press.Google Scholar - Natsheh, I., &Karsenty, R. (2014). Exploring the potential role of visual reasoning tasks among inexperienced solvers.
*ZDM*—*The International Journal on Mathematics Education*,*46*(1), 109–122.Google Scholar - Newcombe, N. S. (2006). A plea for spatial literacy.
*The Chronicle of Higher Education,**52*(26), B20.Google Scholar - Newcombe, N. S. (2010). Picture this: increasing math and science learning by improving spatial thinking.
*American Educator*,*34*(2), 29–35 (43).Google Scholar - Ontario Association for Mathematics Education. (2005). Growing up mathematically (GUM) (online version). http://www.oame.on.ca/main/index1.php?lang=en&code=gum. Accessed 26 January 2011.
- Presmeg, N. C. (2006). Research on visualization in learning and teaching mathematics. In A. Gutiérrez & P. Boero (Eds.),
*Handbook of research on the psychology of mathematics education*(pp. 205–235). Rotterdam: Sense Publishers.Google Scholar - Presmeg, N.C. (2014). Contemplating visualization as an epistemological learning tool in mathematics.
*ZDM—The International Journal on Mathematics Education*,*46*(1), 151–157.Google Scholar - Silver, E. A. (1997). Fostering creativity through instruction rich in mathematical problem solving and problem posing.
*ZDM–The International Journal on Mathematics Education,**29*(3), 75–80.CrossRefGoogle Scholar - Simon, M. A. (2006). Key developmental understandings in mathematics: a direction for investigating and establishing learning goals.
*Mathematical Thinking and Learning,**8*(4), 359–371.CrossRefGoogle Scholar - Sinclair, N., & Bruce, C. (2015). New opportunities in geometry education at the primary school.
*ZDM*–*The International Journal on Mathematics Education*,*47*(3) (this issue).Google Scholar - Sinclair, N., de Freitas, E., & Ferrara, F. (2013). Virtual encounters: the murky and furtive world of mathematical inventiveness.
*ZDM*––*The International Journal on Mathematics Education*,*45*(2), 239–252.Google Scholar - Sinclair, M., Mamolo, A., & Whiteley, W. J. (2011). Designing spatial visual tasks for research: the case of the filling task.
*Educational Studies in Mathematics,**78*(2), 135–163.CrossRefGoogle Scholar - Skemp, R. R. (1976). Relational understanding and instrumental understanding.
*Mathematics Teaching,**77*, 20–26.Google Scholar - Stein, M. K., & Lane, S. (1996). Instructional tasks and the development of student capacity to think and reason: an analysis of the relationship between teaching and learning in a reform mathematics project.
*Educational Research and Evaluation,**2*(1), 50–80.CrossRefGoogle Scholar - Strong, R., Thomas, E., Perini, M., & Silver, H. (2004). Creating a differentiated mathematics classroom.
*Educational Leadership*, 73–78.Google Scholar - Sullivan, P., Clarke, D., & Clarke, B. (2013).
*Teaching with tasks for effective mathematics learning*. New York: Springer.CrossRefGoogle Scholar - Tall, D. O. (2007). Developing a theory of mathematical growth.
*ZDM—The International Journal on Mathematics Education,**39*(1–2), 145–154.CrossRefGoogle Scholar - Tchoshanov, M., Blake, S., & Duval, A. (2002). Preparing teachers for a new challenge: teaching Calculus concepts in middle grades. In Proceedings of the second international conference on the teaching of mathematics (at the undergraduate level), Hersonissos, Crete.Google Scholar
- Tomlinson, C. (1999).
*The differentiated classroom: responding to the needs of all learners*. Alexandria: ASCD.Google Scholar - Turner, M., & Fauconnier, G. (2002).
*The way we think: conceptual blending and the mind’s hidden complexities*. New York: Basic Books.Google Scholar - Uttal, D. H., Meadow, N. G., Tipton, E., Hand, L. L., Alden, A. R., Warren, C., & Newcombe, N. S. (2013). The malleability of spatial skills: a meta-analysis of training studies.
*Psychological Bulletin,**139*(2), 352–402.CrossRefGoogle Scholar - Vasilyeva, M., Ganley, C. M., Casey, B. M., Dulaney, A., Tillinger, M., & Anderson, K. (2013). How children determine the size of 3D structures: investigating factors influencing strategy choice.
*Cognition and Instruction,**31*(1), 29–61.CrossRefGoogle Scholar - Weigand, H. G. (2014). A discrete approach to the concept of derivative.
*ZDM*—*The International Journal on Mathematics Education*,*46*(4), 603–619.Google Scholar - Whiteley, W. (2007a).
*Visual reasoning: rates of change without derivatives*. Barrie: OAME Conference Georgian College.Google Scholar - Whiteley, W. (2007b). Rates of change without derivatives. Training camp for the new Ontario curriculum.Google Scholar
- Whiteley, W. (2011). Optimizing with geometric reasoning. Mathwiki. http://wiki.math.yorku.ca/index.php/Optimizing_with_Geometric_Reasoning#The_Popcorn_Box_Exploration. Accessed 1 March 2014.
- Whiteley, W. (2012). Mathematical modeling as conceptual blending: exploring an example within mathematics education. In Bockaravo, M., Danesi, M., &Núñez, R. (Eds.), Cognitive science and interdisciplinary approaches to mathematical cognition.Google Scholar
- Whiteley, W., & Mamolo, A. (2011). The Popcorn box activity and reasoning about optimization.
*Mathematics Teacher,**105*(6), 420–426.Google Scholar - Whiteley, W., & Mamolo, A. (2013). Optimizing through geometric reasoning supported by 3-D models: Visual representations of change. In C. Margolinas (Ed.),
*Task design in mathematics education*(pp. 129–140). Oxford: Proceedings of the ICMI 22 conference.Google Scholar - Wood, D., Bruner, J. S., & Ross, G. (1976). The role of tutoring in problem solving.
*Journal of Child Psychology and Psychiatry*, 17(2), 89–100.Google Scholar