# Analyzing the Relation Between Spatial and Geometric Reasoning for Elementary and Middle School Students

• Michael T. Battista
• Leah M. Frazee
• Michael L. Winer
Chapter
Part of the Research in Mathematics Education book series (RME)

## Abstract

Numerous studies have found that spatial ability and mathematical ability are positively correlated. But specifying the exact nature of the relation between these types of reasoning has been elusive, with much research focused on understanding correlations between mathematical performance and specific spatial skills as measured by spatial tests. We attempt to deepen understanding of the relationship between spatial and mathematical reasoning by precisely describing the spatial processes involved in reasoning about specific topics in geometry. We focus on two major components of spatial reasoning. Spatial visualization involves mentally creating and manipulating images of objects in space, from fixed or changing perspectives on the objects, so that one can reason about the objects and actions on them, both when the objects are and are not visible. Property-based spatial analytic reasoning decomposes objects into their parts using geometric properties to specify how the parts or shapes are related, and, using these relationships, operates on the parts. Spatial analytic reasoning generally employs concepts such as measurement, congruence, parallelism, and isometries to conceptualize spatial relationships.

## Keywords

Mathematics Geometry Spatial visualization Analytic Reasoning Mental model Learning progression Structure Shape Isometries Reflections Rotations Measurement Length Angle Area Volume

## References

1. Battista, M. T. (1990). Spatial visualization and gender differences in high school geometry. Journal for Research in Mathematics Education, 21(1), 47–60.
2. Battista, M. T. (1994). On Greeno’s environmental/model view of conceptual domains: A spatial/geometric perspective. Journal for Research in Mathematics Education, 25(1), 86–94.
3. Battista, M. T. (1999). Fifth graders’ enumeration of cubes in 3D arrays: Conceptual progress in an inquiry-based classroom. Journal for Research in Mathematics Education, 30(4), 417–448.
4. Battista, M. T. (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.
5. 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, VA: National Council of Teachers of Mathematics.Google Scholar
6. Battista, M. T. (2008). Development of the shape maker’s geometry microworld: Design, principles, and research. In G. W. Blume & M. 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
7. Battista, M. T. (2012a). Cognition based assessment and teaching of geometric measurement (length, area, and volume): Building on students’ reasoning. Portsmouth, NH: Heinemann.Google Scholar
8. Battista, M. T. (2017a). Reasoning and sense making in the elementary grades: 3–5. National Council of Teachers of Mathematics.Google Scholar
9. Battista, M. T. (2017b). Reasoning and sense making in the elementary grades: 6–8. National Council of Teachers of Mathematics.Google Scholar
10. Battista, M. T., & Berle-Carman, M. (1996). Containers and cubes. Palo Alto, CA: Dale Seymour Publications.Google Scholar
11. Battista, M. T., & Clements, D. H. (1996). Students’ understanding of three-dimensional rectangular arrays of cubes. Journal for Research in Mathematics Education, 27(3), 256–292.
12. Battista, M. T., Clements, D. H., Arnoff, J., Battista, K., & Borrow, C. V. A. (1998). Students’ spatial structuring of 2D arrays of squares. Journal for Research in Mathematics Education, 29(5), 503–532.
13. Battista, M. T., & Frazee, L. M. (2018). Students’ use of property knowledge and spatial visualization in reasoning about 2D rotations. In ICME TSG-12 Secondary Geometry Monograph. New York, NY: Springer Publishing.Google Scholar
14. Battista, M.T., Frazee, L.M., & Winer, M. L. (2017). Adolescents’ understanding of isometries in a dynamic geometry curriculum. Paper presented at National Council of Teachers of Mathematics (NCTM) Research Conference, San Antonio, TX.Google Scholar
15. Battista, M. T. (2012b). Cognition based assessment and teaching of geometric shapes: Building on students’ reasoning. Portsmouth, NH: Heinemann.Google Scholar
16. Calvin, W. H. (1996). How brains think. New York: Basic Books.Google Scholar
17. Clements, D. H., & Battista, M. T. (1992). Geometry and spatial reasoning. In D. A. Grouws (Ed.), The handbook on research in mathematics teaching and learning. New York, NY: Macmillan.Google Scholar
18. English, L. D., & Halford, G. S. (1995). Mathematics education: Models and processes. Mahwah, New Jersey: Lawrence Erlbaum Associates.Google Scholar
19. Gorgorió, N. (1998). Exploring the functionality of visual and non-visual strategies in solving rotation problems. Educational Studies in Mathematics, 35(3), 207–231.
20. Greeno, J. G. (1991). Number sense as situated knowing in a conceptual domain. Journal for Research in Mathematics Education, 22(3), 170–218.
21. Hegarty, M. (2004). Mechanical reasoning by mental simulation. Trends in Cognitive Sciences, 8(6), 280–285.
22. Hegarty, M. (2010). Components of spatial intelligence. Psychology of Learning and Motivation, 52, 265–297.
23. Johnson-Laird, P. N. (1983). Mental models: Towards a cognitive science of language, inference, and consciousness (No. 6). Cambridge, MA: Harvard University Press.Google Scholar
24. Johnson-Laird, P. N. (1998). Imagery, visualization, and thinking. In J. Hockberg (Ed.), Perception and cognition at century’s end (pp. 441–467). San Diego, CA: Academic Press.
25. Knauff, M., Fangmeier, T., Ruff, C. C., & Johnson-Laird, P. N. (2003). Reasoning, models, and images: behavioral measures and cortical activity. Journal of Cognitive Neuroscience, 15(4), 559–573.
26. Krantz, D. H., Luce, R. D., Suppes, P., & Tversky, A. (1971). Foundations of measurement, Vol I: Additive and polynomial representations. New York: Academic Press.Google Scholar
27. Markovits, H. (1993). The development of conditional reasoning: A Piagetian reformulation of mental models theory. Merrill-Palmer Quarterly, 39(1), 131–158.Google Scholar
28. Mix, K. S., & Cheng, Y. L. (2012). The relation between space and math: Developmental and educational implications. Advances in Child Development and Behavior, 42, 197–243.
29. Mix, K. S., Levine, S. C., Cheng, Y. L., Young, C., & Hambrick, D. Z. (2016). Separate but correlated: The latent structure of space and mathematics across development. Journal of Experimental Psychology: General, 145(9), 1206–1227.
31. Newcombe, N., & Shipley, T. (2015). Thinking about spatial thinking: New typology, new assessments. In J. Gero (Ed.), Studying visual and spatial reasoning for design creativity. Dordrecht: Springer.Google Scholar
32. Sarama, J., Clements, D. H., Swaminathan, S., McMillen, S., & González Gómez, R. M. (2003). Development of mathematical concepts of two-dimensional space in grid environments: An exploratory study. Cognition and Instruction, 21(3), 285–324.
33. Steffe, L. P., & Thompson, P. W. (2000). Teaching experiment methodology: Underlying principles and essential elements. In R. Lesh & A. E. Kelly (Eds.), Research design in mathematics and science education (pp. 267–307). Hillsdale, NJ: Erlbaum.Google Scholar
34. Uttal, D. H., Meadow, N. G., Tipton, E., Hand, L. L., Alden, A. R., Warren, C., & Newcombe, N. (2013). The malleability of spatial skills: A meta-analysis of training studies. Psychological Bulletin, 139(2), 352–402.
35. van Hiele, P. M. (1986). Structure and insight. Orlando: Academic.Google Scholar
36. Winer, M. L. (2010). Fifth graders’ reasoning on the enumeration of cube-packages in rectangular boxes in an inquiry-based classroom. (Master’s thesis). Retrieved from http://www.ohiolink.edu/etd/.

© Springer Nature Switzerland AG 2018

## Authors and Affiliations

• Michael T. Battista
• 1
Email author
• Leah M. Frazee
• 2
• Michael L. Winer
• 3
1. 1.Department of Teaching and LearningThe Ohio State UniversityColumbusUSA
2. 2.Department of Mathematical SciencesCentral Connecticut State UniversityNew BritainUSA
3. 3.Department of MathematicsEastern Washington UniversityCheneyUSA