Ambrose, R., & Kenehan, G. (2009). Children’s Evolving Understanding of Polyhedra in the Classroom. Mathematical Thinking and Learning,
(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.
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
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.
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,
(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
Bruce, C., McPherson, R., Sabbati, M., & Flynn, T. (2011). Revealing significant learning moments with interactive whiteboards in mathematics. Journal of Educational Computing Research,
(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.
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.
Clements, D. H., & Battista, M. T. (1989). Learning of geometric concepts in a Logo environment. Journal of Research in Mathematics Education,
, 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,
, 313–337.CrossRefGoogle Scholar
Clements, D. H., & Sarama, J. (2011). Early childhood teacher education: the case of geometry. Journal of Mathematics Teacher Education,
(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,
(3), 373–382.CrossRefGoogle Scholar
de Freitas, E., & Sinclair, N. (2012). Diagram, gesture, agency: theorizing embodiment in the mathematics classroom. Educational Studies in Mathematics,
(1–2), 133–152.Google Scholar
Delgado, A. R., & Prieto, G. (2004). Cognitive mediators and sex-related differences in mathematics. Intelligence,
(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,
, 1–19.CrossRefGoogle Scholar
Dieudonné, J. (1981). The universal domination of geometry. The Two-Year Cllege Mathematics Journal,
(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,
, 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,
, 184–206.CrossRefGoogle Scholar
Forsythe, S. K. (2011). Developing perceptions of symmetry in a Dynamic Geometry environment. Research in Mathematics Education,
(2), 225–226.CrossRefGoogle Scholar
Freudenthal, H. (1971). Geometry between the devil and the deep sea. Educational Studies in Mathematics,
, 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.
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), 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,
(3), 210–215.CrossRefGoogle Scholar
Gutiérrez, A. (1992). Exploring the links between van Hiele levels and 3-dimensional geometry. Structural Topology,
, 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
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,
(1), 61–76.Google Scholar
Highfield, K. (2009). Mapping, Measurement and Robotics. Reflections, Journal of the Mathematical Association of New South Wales,
(1), 52–55.Google Scholar
Highfield, K. (2010). Robotic toys as a catalyst for mathematical problem solving. Australian Primary Mathematics Classroom,
, 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.
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?
. 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,
, 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,
(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
Kell, H., Lubinski, D., Benbow, C., & Stieger, J. (2013). Who Rises to the Top? Early Indicators. Psychological Science,
, 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
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,
, 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).
Mamolo, A., Ruttenberg-Rozen, R. & Whitelely, W. (2015). Developing a network of and for geometric reasoning. ZDM Mathematics Education, 47
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,
, 8–24.CrossRefGoogle Scholar
Mitchelmore, M. C. (1998). Young Students’ Concepts of Turning and Angle. Cognition and Instruction,
(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
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
Page, E. I. (1959). Haptic perception: a consideration of one of the investigations of Piaget and Inhelder. Educational Review,
, 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,
, 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,
(1), 1–2.CrossRefGoogle Scholar
Sarama, J., & Clements, D. (2002). Building blocks for young children’s mathematical development. Journal of Educational Computing Research,
(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.
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,
(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,
, 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.
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).
Spelke, E. S., Gilmore, C. K., & McCarthy, S. (2011). Kindergarten children’s sensitivity to geometry in maps. Developmental Science,
(4), 809–821.CrossRefGoogle Scholar
Spencer, H. (1876). Inventional geometry. American Book Company.
Stipek, D. (2013). Mathematics in early childhood education: revolution or evolution? Early Education and Development,
, 431–435. doi:10.1080/10409289.2013.777285
Tahta, D. (1980). About geometry. For the Learning of Mathematics,
(1), 2–9.Google 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).
Tsamir, P., Tirosh, D., & Levenson, E. (2008). Intuitive nonexamples: the case of triangles. Educational Studies in Mathematics,
, 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
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
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).
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,
, 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,
, 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.