# New opportunities in geometry education at the primary school

- 2.8k Downloads
- 11 Citations

## Abstract

This paper outlines the new opportunities that that will be changing the landscape of geometry education at the primary school level. These include: the research on spatial reasoning and its connection to school mathematics in general and school geometry in particular; the function of drawing in the construction of geometric meaning; the role of digital technologies; the importance of transformational geometry in the curriculum (including symmetry as well as the isometries); and, the possibility of extending primary school geometry from its typical emphasis on vocabulary (naming and sorting shapes by properties) to working on the composing/decomposing, classifying, comparing and mentally manipulating both two- and three-dimensional figures. We discuss these opportunities in the context of historical developments in the nature and relevance of school geometry. The aim is to motivate and connect the set of papers in this special issue.

### Keywords

Geometry Primary school Technology Drawing Transformations### References

- Ambrose, R., & Kenehan, G. (2009). Children’s Evolving Understanding of Polyhedra in the Classroom.
*Mathematical Thinking and Learning,**11*(3), 158–176.CrossRefGoogle Scholar - Bartolini Bussi, M. G., & Mariotti, M. A. (2008). Semiotic mediation in the mathematics classroom: Artifacts and signs after a Vygotskian perspective. In L. English, M. Bartolini Bussi, G. Jones, R. Lesh, & D. Tirosh (Eds.),
*Handbook of international research in mathematics education, second revised edition*(pp. 746–783). Mahwah: Lawrence Erlbaum.Google Scholar - Bartolini-Bussi, M. & Baccaglini-Frank, A. (2015). Geometry in early years: sowing seeds for a mathematical definition of squares and rectangles.
*ZDM Mathematics Education, 47*(3). doi:10.1007/s11858-014-0636-5 (this issue). - Battista, M.T. (2008). Development of shapemakers geometry microworld. In Blume, G.W., Heid, M.K. (Eds.),
*Research on Technology and the Teaching and Learning of Mathematics Vol. 2: Cases and Perspectives*(pp. 131-156). Information Age Publishing.Google Scholar - Battista, M. T. (2007). The development of geometric and spatial thinking. In F. Lester (Ed.),
*Second handbook of research on mathematics teaching and learning*(pp. 843–908). Reston: National Council of Teachers of Mathematics.Google Scholar - Brochard, R., Dufour, A., & Despres, O. (2004). Effect of musical expertise on visuospatial abilities: evidence from reaction times and mental imagery.
*Brain and Cognition,**54*(2), 103–109.CrossRefGoogle Scholar - Bruce, C. & Hawes, Z. (2015). The role of 2D and 3D mental rotations in mathematics for young children: what is it? Why does it matter? And what can we do about it?
*ZDM Mathematics Education, 47*(3). doi:10.1007/s11858-014-0637-4 (this issue). - Bruce, C., McPherson, R., Sabbati, M., & Flynn, T. (2011). Revealing significant learning moments with interactive whiteboards in mathematics.
*Journal of Educational Computing Research,**45*(4), 433–454.CrossRefGoogle Scholar - Bruce, C. D., Moss, J., Sinclair, N., Whiteley, W., Okamoto, Y., McGarvey, L., & Davis, B. (2013). Early years spatial reasoning: learning, teaching, and research implications. In
*Presented at the NCTM research presession: Linking research and practice.*Denver, CO.Google Scholar - Bruner, J. (1969).
*On knowing: essays for the left hand*. Athaneum: New York.Google Scholar - Bryant, P., & Watson, A. (2009).
*Key understandings in mathematics learning: Understanding space and its representation in mathematics*. Nuffield Foundation. http://foundation.bootle.biz/sites/default/files/P5.pdf. Accessed 34 April 2014. - Châtelet, G. (2000/1993).
*Les enjeux du mobile*. Paris: Seuil. (Engl. transl., by R. Shore & M. Zagha:*Figuring space: Philosophy, Mathematics and Physics*). Dordrecht: Kluwer Academy Press.Google Scholar - Clements, D. H., & Battista, M. T. (1989). Learning of geometric concepts in a Logo environment.
*Journal of Research in Mathematics Education,**20*, 450–467.CrossRefGoogle Scholar - Clements, D. H., & Battista, M. T. (1992). Geometry and spatial reasoning. In D. A. Grouws (Ed.),
*Handbook of research on mathematics teaching and learning*(pp. 420–464). New York: Macmillan.Google Scholar - Clements, D., Battista, A., Sarama, J., & Swaminathan, S. (1996). Development of turn and turn measurement concepts in a computer-based instructional unit.
*Educational Studies in Mathematics,**30*, 313–337.CrossRefGoogle Scholar - Clements, D. H., & Sarama, J. (2011). Early childhood teacher education: the case of geometry.
*Journal of Mathematics Teacher Education,**14*(2), 133–148.CrossRefGoogle Scholar - Craine, T. (2009).
*Understanding geometry for a changing world*. Reston: NCTM.Google Scholar - Cupchik, G. C., Phillips, K., & Hill, D. S. (2001). Shared processes in spatial rotation and musical permutation.
*Brain and Cognition,**46*(3), 373–382.CrossRefGoogle Scholar - de Freitas, E., & Sinclair, N. (2012). Diagram, gesture, agency: theorizing embodiment in the mathematics classroom.
*Educational Studies in Mathematics,**80*(1–2), 133–152.Google Scholar - Delgado, A. R., & Prieto, G. (2004). Cognitive mediators and sex-related differences in mathematics.
*Intelligence,**32*(1), 25–32.CrossRefGoogle Scholar - Devichi, C., & Munier, V. (2013). About the concept of angle in elementary school: misconceptions and teaching sequences.
*Journal of Mathematical Behavior,**32*, 1–19.CrossRefGoogle Scholar - Dieudonné, J. (1981). The universal domination of geometry.
*The Two-Year Cllege Mathematics Journal,**12*(4), 227–231.CrossRefGoogle Scholar - Erez, M., & Yerushalmy, M. (2006). ‘‘If you can turn a rectangle into a square, you can turn a square into a rectangle…’’ young students experience the dragging tool.
*International Journal of Computers for Mathematical Learning,**11*, 271–299.CrossRefGoogle Scholar - Farmer, G., Verdine, B., Lucca, K., Davies, T., Dempsey, R., Newcombe, N., et al. (2013).
*Putting the pieces together: Spatial skills at age 3 predict to spatial and math performance at age 5*. Seattle: SRCD poster presentation.Google Scholar - Fennema, E., & Tartre, L. A. (1985). The use of spatial visualisation in mathematics by girls and boys.
*Journal for Research in Mathematics Education,**16*, 184–206.CrossRefGoogle Scholar - Forsythe, S. K. (2011). Developing perceptions of symmetry in a Dynamic Geometry environment.
*Research in Mathematics Education,**13*(2), 225–226.CrossRefGoogle Scholar - Freudenthal, H. (1971). Geometry between the devil and the deep sea.
*Educational Studies in Mathematics,**3*, 413–435.CrossRefGoogle Scholar - Fuys, D., Geddes, D., Lovett, C. J., & Tischler, R. (1988). The van Hiele model of thinking in geometry among adolescents.
*Journal for Research in Mathematics Education*[monograph number 3]. Reston: NCTM.Google Scholar - Gibson, D., Congdon, E. & Levine, S. (2012). The effects of word learning biases on children’s understanding of angle. Published by the Spatial Intelligence Learning Center. http://bit.ly/1qOBlyu. Accessed December 2014.
- Gonzales, G., & Herbst, P. (2006). Competing arguments for the geometry curse: why were American high school students supported to study geometry in the twentieth century?
*International Journal for the History of Mathematics Education,**1*(1), 7–33.Google Scholar - Goodwin, K., & Highfield, K. (2013). A framework for examining technologies and early mathematics learning. In L. D. English & J. T. Mulligan (Eds.),
*Reconceptualising early mathematics learning*(pp. 205–226). New York: Springer.CrossRefGoogle Scholar - Guay, R. B., & McDaniel, E. D. (1977). The relationship between mathematics achievement and spatial abilities among elementary school children.
*Journal for Research in Mathematics Education,**8*(3), 210–215.CrossRefGoogle Scholar - Gutiérrez, A. (1992). Exploring the links between van Hiele levels and 3-dimensional geometry.
*Structural Topology,**18*, 31–48.Google Scholar - Hallowell, D., Okamoto, Y, Romo, L. & LaJoy, J. (2015). First-grader’s spatial-mathematical reasoning about plane and solid shapes and their representations.
*ZDM Mathematics Education, 47*(3). doi:10.1007/s11858-015-0664-9 (this issue). - Henderson, D. W., & Taimina, D. (2005).
*Experiencing geometry. Euclidean and non-Euclidean with history*. Ithaca: Cornell University.Google Scholar - Hershkowitz, R. (1989). Visualization in geometry—two sides of the coin.
*Focus on Learning Problems in Mathematics,**11*(1), 61–76.Google Scholar - Highfield, K. (2009). Mapping, Measurement and Robotics.
*Reflections, Journal of the Mathematical Association of New South Wales,**34*(1), 52–55.Google Scholar - Highfield, K. (2010). Robotic toys as a catalyst for mathematical problem solving.
*Australian Primary Mathematics Classroom,**15*, 22–27.Google Scholar - Highfield, K., & Mulligan, J. (2007). The Role of Dynamic Interactive Technological Tools in Preschoolers’ Mathematical Patterning. In J. Watson & K. Beswick (Eds),
*Proceedings of the 30th annual conference of the Mathematics Education Research Group of Australasia*(pp. 372–381). MERGA.Google Scholar - Highfield, K., & Mulligan, J. T. (2008). Young children’s engagement with technological tools: The impact on mathematics education. Paper presented to the
*International Congress of Mathematical Education (ICME 11): Discussion Group 27: How is technology challenging us to re*-*think the fundamentals of mathematics education?*http://dg.icme11.org/tsg/show/28. Accessed 1 December 2008. - Highfield, K., & Mulligan, J. T. (2009). Young children’s embodied action in problem-solving tasks using robotic toys. In M. Tzekaki, M. Kaldrimidou, & H. Sakonidis (Eds.),
*Proceedings of the 33rd conference of the International Group for the Psychology of Mathematics Education*(Vol. 2, pp. 273–280). Thessaloniki: PME.Google Scholar - Jaime, A., & Gutiérrez, A. (1995). Guidelines for teaching plane isometries in secondary school.
*Mathematics Teacher,**88*, 591–597.Google Scholar - Jones, K. (2000). Providing a foundation for deductive reasoning: students’ interpretations when using dynamic geometry software and their evolving mathematical explanations.
*Educational Studies in Mathematics,**44*(1–3), 55–85.CrossRefGoogle Scholar - Kaur, H. (2015). Two aspects of young children’s thinking about different types of dynamic triangles: prototypicality and inclusion.
*ZDM Mathematics Education, 47*(3). doi:10.1007/s11858-014-0658-z (this issue). - Kell, H., Lubinski, D., Benbow, C., & Stieger, J. (2013). Who Rises to the Top? Early Indicators.
*Psychological Science,**24*, 648–659.CrossRefGoogle Scholar - Kotsopoulos, D., Cordy, M. & Langemeyer, M. (2015). Children’s understanding of large-scale mapping tasks: an analysis of talk, drawings, and gesture.
*ZDM Mathematics Education, 47*(3). doi:10.1007/s11858-014-0661-4 (this issue). - Lakoff, G., & Núñez, R. E. (2000).
*Where mathematics comes from: How the embodied mind brings mathematics into being*(1st ed.). New York: Basic Books.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: Lawrence Erlbaum Associates.Google Scholar - Lovell, K. (1959). A follow-up study on some aspects of the work of Piaget and Inhelder on the child’s conception of space.
*British Journal of Educational Psychology,**29*, 104–117.CrossRefGoogle Scholar - Mammana, C., & Villani, V. (Eds.) (1998).
*Perspective on the teaching of geometry for the 21st century: an ICMI study.*Dordrecht: Kluwer Academic Publishers (New ICMI Studies Series; No. 5).Google Scholar - Mamolo, A., Ruttenberg-Rozen, R. & Whitelely, W. (2015). Developing a network of and for geometric reasoning.
*ZDM Mathematics Education, 47*(3). doi:10.1007/s11858-014-0654-3 (this issue). - Martin, J. L. (1976). An analysis of some of Piaget’s topological tasks from a mathematical point of view.
*Journal for Research in Mathematics Education,**7*, 8–24.CrossRefGoogle Scholar - Mitchelmore, M. C. (1998). Young Students’ Concepts of Turning and Angle.
*Cognition and Instruction,**16*(3), 265–284.CrossRefGoogle 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 Mathematics Education, 15*(3). doi:10.1007/s11858-015-0679-2 (this issue). - Moyer, P. S., Niezgoda, D., & Stanley, J. (2005). Young children’s use of virtual manipulatives and other forms of mathematical representations. In W. Masalski & P. C. Elliott (Eds.),
*Technology-supported mathematics learning environments: 67th yearbook*(pp. 17–34). Reston: National Council of Teachers of Mathematics.Google Scholar - Ng, O. & Sinclair, N. (2015). Young children reasoning about symmetry in a dynamic geometry environment.
*ZDM Mathematics Education, 47*(3). doi:10.1007/s11858-014-0660-5 (this issue). - Page, E. I. (1959). Haptic perception: a consideration of one of the investigations of Piaget and Inhelder.
*Educational Review,**11*, 115–124.CrossRefGoogle Scholar - Pittalis, M., & Christou, C. (2010). Types of reasoning in 3D geometry thinking and their relation with spatial ability.
*Educational Studies in Mathematics,**75*, 191–212.CrossRefGoogle Scholar - Rivera, F. D., Steinbring, H., & Arcavi, A. (2014). Visualisation as an epistemological learning tool: an introduction.
*ZDM—The International Journal on Mathematics Education,**46*(1), 1–2.CrossRefGoogle Scholar - Sarama, J., & Clements, D. (2002). Building blocks for young children’s mathematical development.
*Journal of Educational Computing Research,**27*(1&2), 93–110.CrossRefGoogle Scholar - Sfard, A. (2008).
*Thinking as communicating: Human development, the growth of discourses, and mathematizing*. New York: Cambridge University Press.CrossRefGoogle Scholar - Sinclair, N. (2008).
*The history of the geometry curriculum in the United States*. IAP—Information Age Publishing Inc.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.CrossRefGoogle Scholar - Sinclair, N., & Moss, J. (2012). The more it changes, the more it becomes the same: the development of the routine of shape identification in dynamic geometry environments.
*International Journal of Education Research,**51&52*, 28–44.CrossRefGoogle Scholar - Sinclair, N., Pimm, D., & Skelin, M. (2012).
*Developing Essential Understanding of Geometry for Teaching Mathematics in Grades 5*–*8*.*Essential Understanding Series*. Reston: National Council of Teachers of Mathematics.Google Scholar - Soury-Lavergne, S. & Maschietto, M. (2015). Intertwining of spatial and geometrical knowledge in problem solving with technology at primary school.
*ZDM Mathematics Education, 47*(3) (this issue).Google Scholar - Spelke, E. S., Gilmore, C. K., & McCarthy, S. (2011). Kindergarten children’s sensitivity to geometry in maps.
*Developmental Science,**14*(4), 809–821.CrossRefGoogle Scholar - Spencer, H. (1876).
*Inventional geometry*. American Book Company.Google Scholar - Stipek, D. (2013). Mathematics in early childhood education: revolution or evolution?
*Early Education and Development,**24*, 431–435. doi:10.1080/10409289.2013.777285.CrossRefGoogle Scholar - Thom, J., & McGarvey, L. (2015). The act and artifact of drawing(s): observing geometric thinking with, in, and through children’s drawings.
*ZDM Mathematics Education, 47*(3) (this issue).Google Scholar - Tsamir, P., Tirosh, D., & Levenson, E. (2008). Intuitive nonexamples: the case of triangles.
*Educational Studies in Mathematics,**69*, 81–95.CrossRefGoogle Scholar - Tsamir, P., Tirosh, D., Levenson, E., Barkai, R. & Tabach, M. (2015). Early-years teachers’ concept images and concept definitions: triangles, circles, and cylinders.
*ZDM Mathematics Education, 47*(3). doi:10.1007/s11858-014-0641-8 (this issue). - Usiskin, Z. (1987). Resolving the continuing dilemmas in school geometry. In M. M. Lindquist & A. P. Shulte (Eds.),
*Learning and Teaching Geometry, K-12: 1987 Yearbook*(pp. 17–31). Reston: National Council of Teachers of Mathematics.Google Scholar - Van den Heuvel-Panhuizen, M., & Buys, K. (Eds.). (2008).
*Young children learn measurement and geometry. A learning-teaching trajectory with intermediate attainment targets for the lower grades in primary school*. Rotterdam/Tapei: Sense Publishers.Google Scholar - Van den Heuvel-Panhuizen, M., Iliade, E., & Robitzsch, A. (2015). Kindergartners’ performance in two types of imaginary perspective-taking.
*ZDM Mathematics Education, 47*(3). doi:10.1007/s11858-015-0677-4 (this issue). - Van Hiele, P. M. (1985). The child’s thought and geometry. In D. Geddes & R. Tischler (Eds.),
*English translation of selected writings of Dina van Hiele*-*Geldof and Pierre M. van Hiele*(pp. 243–252). Brooklyn: Brooklyn College, School of Education (Original work published 1959).Google Scholar - Wai, J., Lubinski, D., & Benbow, C. P. (2009). Spatial ability for STEM domains: aligning over 50 years of cumulative psychological knowledge solidifies its importance.
*Journal of Educational Psychology,**101*, 817–835.CrossRefGoogle Scholar - Walcott, C., Mohr, D., & Kastberg, S. E. (2009). Making sense of shape: an analysis of children’s written responses.
*Journal of Mathematical Behavior,**28*, 30–40.CrossRefGoogle Scholar - Whiteley, W. (1999). The decline and rise of geometry in 20th century North America. In J. G. McLoughlin (Ed.),
*Canadian Mathematics Study Group Conference Proceedings*, (pp. 7–30). St John’s, NF: Memorial University of Newfoundland.Google Scholar