An analysis of the essential understandings in elementary geometry and a comparison to the common core standards with teaching implications

Helen Crompton 1, Sarah Ferguson 2 *
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1 Old Dominion University, Norfolk, VA, USA
2 University of Cincinnati, Batavia, OH, USA
* Corresponding Author
EUR J SCI MATH ED, Volume 12, Issue 2, pp. 258-275. https://doi.org/10.30935/scimath/14361
Published Online: 14 March 2024, Published: 01 April 2024
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ABSTRACT

Geometry and spatial reasoning form the foundations of learning in mathematics. However, geometry is a subject often ignored by curriculum writers and teachers until high school, leading to students lacking in critical skills in geometric reasoning. As the United States moves into a new curriculum epoch, heralding the commencement of the national common core standards (CCS), one could question if CCS in geometry align with the essential understandings children need to be successful geometric thinkers. This paper begins with an examination of the essential understandings of geometric reasoning leading to an interpretation and critique of the elementary geometry CCS. Finally, the instructional implications are discussed, considering the common core progression through what we know about how children learn geometry.

CITATION

Crompton, H., & Ferguson, S. (2024). An analysis of the essential understandings in elementary geometry and a comparison to the common core standards with teaching implications. European Journal of Science and Mathematics Education, 12(2), 258-275. https://doi.org/10.30935/scimath/14361

REFERENCES

  • Alghadari, F., & Noor, N. A. (2021). Conceptual technique for comparison figures by geometric thinking in analysis level. Journal for the Mathematics Education and Teaching Practices, 2(1), 1-8.
  • Ball, D. L., & Cohen, D. K. (1999). Instruction, capacity, and improvement. Consortium for Policy Research in Education.
  • Battista, M. T. (2001a). A research-based perspective on teaching school geometry. In J. Brophy (Ed.), Advances in research on teaching: Subject-specific instructional methods and activities (pp. 145-185). JAI Press. https://doi.org/10.1016/S1479-3687(01)80026-2
  • Battista, M. T. (2001b). Shape makers: A computer environment that engenders students’ construction of geometric ideas and reasoning. Computers in Schools, 17(1), 105-120. https://doi.org/10.1300/J025v17n01_09
  • Battista, M. T. (2002). Learning geometry in a dynamic computer environment. Teaching Children Mathematics, 8(6), 333-339. https://doi.org/10.5951/TCM.8.6.0333
  • Battista, M. T. (2007). The development of geometric and spatial thinking. In F. K. Lester Jr. (Ed.), Second handbook of research on mathematics teaching and learning (pp. 843-908). National Council of Teachers of Mathematics.
  • Battista, M. T. (2008). Development of the shape makers geometry microworld: Design principles and research. In G. W. Blume, & K. Heid (Eds.), Research on technology and the teaching and learning of mathematics: Cases and perspectives (pp. 131-156). Information Age Publishing.
  • Battista, M. T. (2009). Highlights of research on learning school geometry. In T. V. Craine, & R. Rubenstein (Eds.), Understanding geometry for a changing world (pp. 91-108). National Council of Teachers of Mathematics.
  • Battista, M. T., & Clements, D. H. (1988). A case for a Logo-based elementary schools geometry curriculum. Arithmetic Teacher, 36, 11-17. https://doi.org/10.5951/AT.36.3.0011
  • Battista, M. T., & Clements, D. H. (1995). Geometry and proof. Mathematics Teacher, 88(1), 48-54. https://doi.org/10.5951/MT.88.1.0048
  • Battista, M. T., & Clements, D. H. (1996). Students’ understanding of three-dimensional rectangular arrays of cubes. Journal for Research in Mathematics Education, 27(3), 258-292. https://doi.org/10.2307/749365
  • Beaton, A. E., Mullis, I. V. S., Martin, M. O., Gonzalez, E. J., Kelly, D. L., & Smith, T. A. (1996). Mathematics achievement in the middle school years: IEA’s third international mathematics and science study (TIMSS). Boston College.
  • Burger, W. F., & Shaughnessy, M. (1986). Characterizing the van Hiele levels of development in geometry. Journal for Research in Mathematics Education, 17(1), 31-48. https://doi.org/10.2307/749317
  • Carpenter, T. P., Corbitt, M. K., Kepner, H. S., Lindquist, M. M., & Reys, R. E. (1980). National assessment. In E. Fennema (Ed.), Mathematics education research: Implications for the 80s (pp. 22-38). Association for Supervision and Curriculum Development.
  • CCSSO/​NGA. (2010a). Common core standards for mathematics. Council in Chief State School Officers and the National Governors Association Center for Best Practices.
  • CCSSO/​NGA. (2010b). Frequently asked questions. Common Core State Standards Initiative. https://corestandards.org/
  • Chaplin, S. H., O’Connor, C., & Canavan-Anderson, N. (2009). Classroom discussions: Using math talk to help students learn. Math Solutions.
  • Clements, D. H. (1998). Geometric and spatial thinking in young children. National Science Foundation.
  • Clements, D. H. (2004). Geometric and spatial thinking in early childhood education. In D. H. Clements, J. Sarama, & A-M. DiBiase (Eds.), Engaging young children in mathematics: Standards for early childhood mathematics education (pp. 267-298). Lawrence Erlbaum Associates.
  • 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). Macmillan.
  • Clements, D. H., & McMillen, S. (1996). Rethinking “concrete” manipulatives. Teaching Children Mathematics, 2(5), 270-279. https://doi.org/10.5951/TCM.2.5.0270
  • Clements, D. H., & Sarama, J. (2007). Effects of preschool mathematics curriculum: Summative research on the building blocks project. Journal for Research in Mathematics Education, 38(2), 136-163. https://doi.org/10.2307/748360
  • Clements, D. H., & Sarama, J. (2009). Learning and teaching early math: The learning trajectories approach. Routledge. https://doi.org/10.4324/9780203883389
  • Clements, D. H., Battista, M. T., Sarama, J., & Swaminathan, S. (1997). Development of students’ spatial thinking in a unit on geometric motions and area. The Elementary School Journal, 98(2), 171-186. https://doi.org/10.1086/461890
  • Clements, D. H., Sarama, J., & Wilson, D. C. (2001). Composition of geometric figures. In M. van den Heuvel-Panhuizen (Ed.), Proceedings of the 21st Conference of the International Group for the Psychology of Mathematics Education (pp. 163-184). Freudenthal Institute.
  • Clements, D. H., Swaminathan, S., Hannibal, M. A. Z., & Sarama, J. (1999). Young children’s concepts of shape. Journal for Research in Mathematics Education, 30(2), 192-212. https://doi.org/10.2307/749610
  • Clements, D. H., Wilson, D. C., & Sarama, J. (2004). Young children’s composition of geometric figures: A learning trajectory. Mathematical Thinking and Learning, 6(2), 163-184. https://doi.org/10.1207/s15327833mtl0602_5
  • Corcoran, T., Mosher, F. A., & Rogat, A. (2009). Learning progressions in science: An evidence-based approach to reform. Consortium for Policy Research in Education. https://doi.org/10.12698/cpre.2009.rr63
  • Crowley, M. L. (1987). The van Hiele model of the development of geometric thought. In M. M. Lindquist, & A. P. Shulte (Eds.), Learning and teaching geometry, K-12. 1987 yearbook (pp. 1-16). National Council of Teachers of Mathematics.
  • Darke, I. (1982). A review of research related to the topological primacy thesis. Educational Studies in Mathematics, 13(2), 119-142. https://doi.org/10.1007/BF00460707
  • Daro, P., Mosher, F. A., & Corcoran, T. (2011). Learning trajectories in mathematics: A foundation for standards, curriculum, assessment, and instruction. Consortium for Policy Research in Education. https://doi.org/10.12698/cpre.2011.rr68
  • de Villiers, M. D. (1987). Research evidence on hierarchical thinking, teaching strategies and the van Hiele theory: Some critical comments. In RUMEUS (Ed.), Report no.10. Research Unit for Mathematics Education.
  • Downs, R. M., & Liben, L. S. (1988). Through and map darkly: Understanding maps as representations. The Genetic Epistemologist, 16, 11-18.
  • Duatepe-Paksu, A. (2009). Effects of drama-based geometry instruction on student achievement, attitudes, and thinking levels. The Journal of Educational Research, 102(4), 272-286. https://doi.org/10.3200/JOER.102.4.272-286
  • Fennema, E. H. (1972). The relative effectiveness of a symbolic and a concrete model in learning a selected mathematics principle. Journal of Research in Mathematics Education, 3, 233-238. https://doi.org/10.2307/748490
  • Fuys, D., Geddes, D., & Tischler, R. (1988). The van Hiele model of thinking in geometry among adolescents. Journal for Research in Mathematics Education Monograph, 3. https://doi.org/10.2307/749957
  • Gagatsis, A., & Patronis, T. (1990). Using geometrical models in a process of reflective thinking in learning and teaching mathematics. Educational Studies in Mathematics, 21, 29-54. https://doi.org/10.1007/BF00311014
  • Genkins, E. F. (1975). The concept of bilateral symmetry in young children. In M. F. Rosskopf (Ed.), Children’s mathematical concepts: Six Piagetian studies in mathematics education (pp. 5-43). Teachers College Press.
  • Gutiérrez, A. (1992). Exploring the links between van Hiele levels and 3-dimensional geometry. Structural Topology, 18, 31-48.
  • Gutiérrez, A., Jaime, A., & Fortuny, J. M. (1991). An alternative paradigm to evaluate the acquisition of the van Hiele levels. Journal for Research in Mathematics Education, 22(3), 237-251. https://doi.org/10.2307/749076
  • Hess, K. K. (2010). Learning progressions frameworks designed for use with the Common Core State Standards in mathematics K-12. National Alternate Assessment Center at the University of Kentucky and the National Center for the Improvement of Educational Assessment.
  • Hess, K. K. (2011). E-x-p-a-n-d-e-d learning progressions frameworks for K-12 mathematics. National Center and State Collaborative. http://www.nciea.org/publications/Math%20Expanded%20LPF%20k4_KH11.pdf
  • Hiebert, J., Carpenter, T. P., Fennema, E., Fuson, K., Wearne, D., Murray, H., Olivier, A., & Human, P. (1997). Making sense: Teaching and learning mathematics with understanding. Heinemann.
  • Hoffer, A. (1981). Geometry is more than proof. Mathematics Teacher, 74, 11-18. https://doi.org/10.5951/MT.74.1.0011
  • Jamie, A., & Gutiérrez, A. (1989). The learning of plane isometries from the viewpoint of the van Hiele model [Paper presentation]. The 13th PME Conference.
  • Johnson-Gentile, K., Clements, D. H., & Battista, M. T. (1994). The effects of computer and non-computer environment on students’ conceptualizations of geometric motions. Journal of Educational Computing Research, 11, 121-140. https://doi.org/10.2190/49EE-8PXL-YY8C-A923
  • Kamalodeen, V., Ramsawak-Jodha, N., Figaro-Henry, S., Jaggernauth, S., & Dedovets, Z. (2021). Designing gamification for geometry in elementary schools: Insights from the designers. Smart Learning Environments, 8. https://doi.org/10.1186/s40561-021-00181-8
  • Kamina, P., & Iyer, N. N. (2009). From concrete to abstract: Teaching for transfer of learning when using manipulatives. In Proceedings of the Northeastern Educational Research Association (pp. 1-9). NERA.
  • Karp, K., Caldwell, J., Zbiek, & Bay-Williams, J. (2011). Developing essential understanding of addition and subtraction for teaching mathematics in pre-k-grade 2. National Council of Teachers of Mathematics.
  • Laurendeau, M., & Pinard, A. (1970). The development of the concept of space in the child. International University Press.
  • Lehrer, R., & Chazan, D. (1998). Preface. In R. Lehrer, & D. Chazan (Eds.), Designing learning environments for developing understanding of geometry and space (pp. ix-xii). Psychology Press.
  • 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). Lawrence Erlbaum Associates.
  • Lewellen, H. (1992). Description of van Hiele levels of geometric development with respect to geometric motions [Unpublished master’s thesis]. Kent State University.
  • Lovell, K. (1959). A follow-up study of some aspects of the work of Piaget and Inhelder on the child’s conception of space. British Journal of Educational Psychology, 29, 104-117. https://doi.org/10.1111/j.2044-8279.1959.tb01484.x
  • Mansfield, H. M., & Scott, J. (1990). Young children solving spatial problems. In P. G. Booker, Cobb, & T. N. deMendicuti (Eds.), Proceedings of the 13th Conference of the International Group for the Psychology of Mathematics Education (pp. 275-282). International Group for the Psychology of Mathematics Education.
  • Martin, T., Lukong, A., & Reaves, R. (2007). The role of manipulative in arithmetic and geometry tasks. Journal of Education and Human Development, 1(1).
  • Mason, M. (1998). The van Hiele levels of geometric understanding. McDougal Littell.
  • McNeil, N. M., & Jarvin, L. (2007). When theories don’t add up: Disentangling the manipulatives debate. Theory into Practice, 46(4), 309-316. https://doi.org/10.1080/00405840701593899
  • Milgram, J. (2010). Review of the final draft core standards. http:/​/​concernedabouteducation.posterous.com/​review-of-common-core-math-standards
  • Milgram, R. J., & Stotsky, S. (March 2010). Fair to middling: A national standards progress report. Pioneer Institute.
  • Mullis, I. V. S., Martin, M. O., Beaton, A. E., Gonzalez, E. J., Kelly, D. L., & Smith, T. A. (1997). Mathematics achievement in the primary school years: IEA’s third international mathematics and science study (TIMSS). Boston College.
  • Mullis, I. V. S., Martin, M. O., Beaton, A. E., Gonzalez, E. J., Kelly, D. L., & Smith, T. A. (1998). Mathematics and science achievement in the final year of secondary school: IEA’s third international mathematics and science study. Boston College.
  • NCTM. (2000). Principals and standards for school mathematics. National Council of Teachers of Mathematics.
  • NCTM. (2006). Curriculum focal points for prekindergarten through grade 8 mathematics: A quest for coherence. National Council of Teachers of Mathematics.
  • Nyikos, M., & Hashimoto, R. (1997). Constructivist theory applied to collaborative learning in teacher education: In search of ZPD. The Modern Language Journal, 81(4), 506-517. https://doi.org/10.1111/j.1540-4781.1997.tb05518.x
  • Olson, D. R. (1970). Cognitive development: The child’s acquisition of diagonality. Academic Press.
  • Page, E. I. (1959). Haptic perception: A consideration of one of the investigations of Piaget and Inhelder. Educational Review, 11, 115-124. https://doi.org/10.1080/0013191590110205
  • Peel, E. A. (1959). Experimental examination of some of Piaget’s schemata concerning children’s perception and thinking, and a discussion of their educational significance. British Journal of Educational Psychology, 29, 89-103. https://doi.org/10.1111/j.2044-8279.1959.tb01483.x
  • Pegg, J., & Davey, G. (1998). Interpreting student understanding in geometry: A synthesis of two models. In D. Chazan (Ed.), Designing learning environments for developing understanding of geometry and space (pp. 109-135). Lawrence Erlbaum Associates.
  • Perham, F. (1978). An investigation into the effect of instruction on the acquisition of transformation geometry concepts in first grade children and subsequent transfer to general spatial ability. In R. Lesh, & M. Mierkiewicz (Eds.), Concerning the development of spatial and geometric concepts (pp. 229-241). ERIC Clearinghouse for Science, Mathematics, and Environmental Education.
  • Piaget, J. (1955a). Language and thought of the child. New American Library.
  • Piaget, J. (1955b). The child’s construction of reality. Routledge and Kegan Paul. https://doi.org/10.1037/11168-000
  • Piaget, J. (1971). Biology and knowledge. Edinburgh University Press.
  • Piaget, J., & Inhelder, B. (1967). The child’s conception of space. W. W. Norton & Company.
  • Resnick, L. B., & Omanson, S. (1987). Learning to understand arithmetic. In R. Glaser (Ed.), Advances in instructional psychology (pp. 41-95). Erlbaum.
  • Richardson, K. (1999). Understanding geometry. Lummi Bay Publishing.
  • Richardson, K. (2002). Math time: The learning environment. Educational Enrichment, Inc.
  • Rosser, R. A., Ensing, S. S., Glider, P. J., & Lane, S. (1984). An information-processing analysis of children’s accuracy in predicting the appearance of rotated stimuli. Child Development, 55, 2204-2211. https://doi.org/10.2307/1129792
  • Rosser, R. A., Lane, S., & Mazzeo, J. (1988). Order of acquisition of related geometric competencies in young children. Child Study Journal, 18, 75-90.
  • Sales, C. (1994). A constructivist instructional project on developing geometric problem solving abilities using pattern blocks and tangrams with young children [Unpublished master’s thesis]. University of Northern Iowa.
  • Sarama, J., & Clements, D. H. (2004). Building blocks for early childhood mathematics. Early Childhood Research Quarterly, 19, 181-189. https://doi.org/10.1016/j.ecresq.2004.01.014
  • Sarama, J., & Clements, D. H. (2006). Mathematics, young students, and computer: Software, teaching strategies and professional development. The Mathematics Educator, 9(2), 112-134.
  • Sarama, J., & Clements, D. H. (2009). “Concrete” computer manipulatives in mathematics education. Child Development Perspectives, 3, 145-150. https://doi.org/10.1111/j.1750-8606.2009.00095.x
  • Sarama, J., Clements, D. H., & Vukelic, E. B. (1996). The role of a computer manipulative in fostering specific psychological/​mathematical processes. In D. Watkins, E. Jakubowski, & H. Biske (Eds.), Proceedings of the 18th Annual Meeting of the North America Chapter of the International Group for the Psychology of Mathematics Education (pp. 567-572). ERIC Clearinghouse for Science, Mathematics, and Environmental Education.
  • Schultz, K. A., & Austin, J. D. (1983). Directional effect in transformational tasks. Journal for Research in Mathematics Education, 14, 95-101. https://doi.org/10.2307/748577
  • Smith, E. E. (1995). Concepts and categorization. In D. N. Osherson, & L. R. Gleitman (Eds.), An invitation to cognitive science (pp. 3-34). MIT Press. https://doi.org/10.7551/mitpress/3966.003.0005
  • Soto-Johnson, H., Cribari, R. D., & Wheeler, A. (2009). The impact of written reflections in a geometry course for preservice teachers. Quaderni Di Ricerca in Didattica (Matematica) [Research Notebooks in Teaching (Mathematics)], 4(19), 65-73.
  • Stotsky, S., & Wurman, Z. (2010). Common core’s standards still don’t make the grade: Why Massachusetts and California must regain control over their academic destinies. Pioneer Institute.
  • Teppo, A. (1991). Van Hiele levels of geometric through revisited. Mathematics Teacher, 84, 210-221. https://doi.org/10.5951/MT.84.3.0210
  • Uttal, D., O’Doherty, K., Newland, R., Hand, L. L., & DeLoache, J. (2009). Dual representation and the linking of concrete and symbolic representations. Child Development Perspectives, 3, 156-159. https://doi.org/10.1111/j.1750-8606.2009.00097.x
  • Van de Walle, J. A., & Lovin, L. H. (2006). Teaching student-centered mathematics: Grades 3-5. Pearson.
  • van Hiele, P. M. (1984a). Summary of Pierre van Hiele’s dissertation entitled: The problem of insight in connection with school children’s insight into the subject-matter of geometry. In D. Geddes., D. Fuys, & R. Tischler (Eds.), English translation of selected writings of Dina van Hiele-Geldof and Pierre M. van Hiele (pp. 237-242). Brooklyn College.
  • van Hiele, P. M. (1984b). The child’s thought and geometry. In D. Geddes., D. Fuys, & R. Tischler (Eds.), English translation of selected writings of Dina van Hiele-Geldof and Pierre M. van Hiele (pp. 243-252). Brooklyn College.
  • van Hiele, P. M. (1999). Developing geometric thinking through activities that begin with play. Teaching Children Mathematics, 5, 310-316. https://doi.org/10.5951/TCM.5.6.0310
  • van Hiele-Geldof, D. (1984). The didactics of geometry in the lowest class of secondary school. In D. Fuys, D. Geddes, & R. Tischler (Eds.), English translation of selected writings of Dina van Hiele-Geldof and Pierre M. van Hiele (pp. 1-214). Brooklyn College.
  • Vygotsky, L. S. (1978). Mind in society: The development of higher psychological processes. Harvard University Press.
  • Watt, D. L., Clements, D., & Lehrer, R. (2002). Technology enhanced learning of geometry. http:/​/​www2.edc.org/telg/​research.html
  • Wiggins, G., & McTighe, J. (2005). Understanding by design. Association for Supervision and Curriculum Development.
  • Zbiek, R. M., Heid, M. K., Blume, G. W., & Dick, T. P. (2007). Research on technology in mathematics education: A perspective of constructs. In F. K. Lester Jr. (Ed.), Second handbook of research on mathematics in teaching and learning (pp. 1169-1208). National Council of Teachers of Mathematics.