From Intuitive Spatial Measurement to Understanding of Units

  • Eliza L. CongdonEmail author
  • Marina Vasilyeva
  • Kelly S. Mix
  • Susan C. Levine
Part of the Research in Mathematics Education book series (RME)


The current chapter outlines children’s transition from an intuitive understanding of spatial extent in infancy and toddlerhood to a more formal understanding of measurement units in school settings. In doing so, the chapter reveals that children’s early competence in intuitive spatial thinking does not translate directly into success with standardized measurement units without appropriate scaffolding and support. Findings from cognitive science and education research are integrated to identify (a) the nature of children’s difficulties with measurement units, (b) some effective instructional techniques involving spatial visualization, and (c) suggestions for how instruction could be further modified to address children’s specific conceptual difficulties with standardized measurement units. The chapter ends by suggesting that the most effective instruction may be that which directly harnesses the power of children’s early intuitive reasoning as those children navigate the transition into a deeper conceptual understanding of standardized units of measure.


Mathematical development Spatial thinking Spatial visualization Units Linear units of measure Ruler measurement understanding Spatial extent Area Angle Misconceptions Manipulatives Gestures Children Infants Cognitive development Education Instruction Mathematics learning Procedural understanding of measurement Conceptual understanding of measurement 


  1. Baillargeon, R. (1987). Young infants’ reasoning about the physical and spatial properties of a hidden object. Cognitive Development, 2, 179–200.Google Scholar
  2. Baillargeon, R., & Graber, M. (1987). Where’s the rabbit? 5.5-month-old infants’ representation of the height of a hidden object. Cognitive Development, 2, 375–392.Google Scholar
  3. Baillargeon, R., Needham, A., & DeVos, J. (1992). The development of young infants’ intuitions about support. Early Development and Parenting, 1, 69–78.Google Scholar
  4. Barrantes, M., & Blanco, L. J. (2006). A study of prospective teachers’ conceptions of teaching and learning school geometry. Journal of Mathematics Teacher Education, 9, 411–436.Google Scholar
  5. Barrett, J., Clements, D., Sarama, J., Cullen, C., McCool, J., Witkowski, C., & Klanderman, D. (2012). Evaluating and improving a learning trajectory for linear measurement in elementary grades 2 and 3: A longitudinal study. Mathematical Thinking and Learning, 14(1), 28–54.Google Scholar
  6. Barth, H., & Paladino, A. M. (2011). The development of numerical estimation: Evidence against a representational shift. Developmental Science, 14, 125–135.Google Scholar
  7. Bragg, P., & Outhred, L. (2004). A measure of rulers—The importance of units in a measure. Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education (pp. 159–166). Bergen, Norway.Google Scholar
  8. Brooks, N., Pogue, A., & Barner, D. (2011). Piecing together numerical language: Children’s use of default units in early counting and quantification. Developmental Science, 14, 44–57. Scholar
  9. Boyer, T. W., Levine, S. C., & Huttenlocher, J. (2008). Development of proportional reasoning: Where young children go wrong. Developmental Psychology, 44(5), 1478.Google Scholar
  10. Bryant, P. E., & Kopytynska, H. (1976). Spontaneous measurement by young children. Nature, 260(5554), 773–773.Google Scholar
  11. Carpenter, T., Lindquist, M., Brown, C., Kouba, V., Edward, A., & Swafford, J. (1988). Results of the fourth NAEP assessment of mathematics: Trends and conclusions. Arithmetic Teacher, 36, 38–41.Google Scholar
  12. Clements, D. H. (1999). Teaching length measurement: Research challenges. School Science and Mathematics, 99(1), 5–11.Google Scholar
  13. Clements, D. H. (2003). Learning and teaching measurement (2003 yearbook). Reston, VA: National Council of Teachers of Mathematics.Google Scholar
  14. Clements, D. H., & Battista, M. T. (1989). Learning of geometric concepts in a Logo environment. Journal for Research in Mathematics Education, 20(5), 450–467.Google Scholar
  15. Clements, D. H., & Battista, M. T. (1992). Geometry and spatial reasoning. In D. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 420–464). New York, NY: Macmillan.Google Scholar
  16. Clements, D. H., & Bright, G. (2003). Learning and teaching measurement: 2003 yearbook. Reston, VA: National Council of Teachers of Mathematics.Google Scholar
  17. Clements, D. H., & Burns, B. A. (2000). Students’ development of strategies for turn and angle measure. Educational Studies in Mathematics, 41(1), 31–45.Google Scholar
  18. Clements, D. H., & McMillen, S. (1996). Rethinking “concrete” manipulatives. Teaching Children Mathematics, 2(5), 270–279.Google Scholar
  19. Clements, D. H., & Stephan, M. (2004). Measurement in pre-K to grade 2 mathematics. Engaging young children in mathematics: Standards for early childhood mathematics education (pp. 299–317).Google Scholar
  20. Cohen, L. B., & Younger, B. A. (1984). Infant perception of angular relations. Infant Behavior and Development, 7(1), 37–47.Google Scholar
  21. Congdon, E.L. & Levine, S.C. (2017, April). Making measurement mistakes: How actions and gestures can rectify common student misconceptions. In B. Kolvoord (Chair), Supporting Spatial Thinking to Enhance STEM Learning. Symposium conducted at the Annual Meeting of the American Educational Research Association, San Antonio, TX.Google Scholar
  22. Congdon, E. L., Kwon, M. K., & Levine, S. C. (2018). Learning to measure through action and gesture: Children’s prior knowledge matters. Cognition, 180, 182–190.Google Scholar
  23. Davis, C., & Uttal, D. H. (2007). Map use and development of spatial cognition. In J. Plumert & J. Spencer (Eds.), The emerging spatial mind (pp. 219–247). New York: Oxford University Press.Google Scholar
  24. Davydov, V. V. (1975). The psychological characteristics of the prenumerical period of mathematics instruction. In L. P. Steffe (Ed.), Soviet studies in the psychology of learning and teaching mathematics (Vol. 7, pp. 109–206). Chicago: The University of Chicago Press.Google Scholar
  25. Duffy, S., Huttenlocher, J., & Levine, S. (2005). It is all relative: How young children encode extent. Journal of Cognition and Development, 6(1), 51–63.Google Scholar
  26. Duffy, S., Huttenlocher, J., Levine, S. C., & Duffy, R. (2005). How infants encode spatial extent. Infancy, 8, 81–90.Google Scholar
  27. Gal’perin, P. Y., & Georgiev, L. S. (1969). The formation of elementary mathematical notions. In J. Kilpatrick & I. Wirszup (Eds.), Soviet studies in the psychology of learning and teaching mathematics. Chicago: The University of Chicago.Google Scholar
  28. Gao, F., Levine, S., & Huttenlocher, J. (2000). What do infants know about continuous quantity? Journal of Experimental Child Psychology, 77, 20–29.Google Scholar
  29. Gibson, D. J., Congdon, E. L., & Levine, S. C. (2015). The effects of word-learning biases on children’s concept of angle. Child Development, 86(1), 319–326.Google Scholar
  30. Hiebert, J. (1984). Why do some children have trouble learning measurement concepts? Arithmetic Teacher, 3(7), 19–24.Google Scholar
  31. Hollich, G., Golinkoff, R. M., & Hirsh-Pasek, K. (2007). Young children associate novel words with complex objects rather than salient parts. Developmental Psychology, 43, 1051–1061. Scholar
  32. Huang, H. M. E., & Witz, K. G. (2011). Developing children’s conceptual understanding of area measurement: A curriculum and teaching experiment. Learning and Instruction, 21(1), 1–13.Google Scholar
  33. Hunting, R. P., & Sharpley, C. F. (1988). Fraction knowledge in preschool children. Journal for Research in Mathematics Education, 19(2), 175–180.Google Scholar
  34. Huntley-Fenner, G. (2001). Why count stuff? Young preschoolers do not use number for measurement in continuous dimensions. Developmental Science, 4(4), 456–462.Google Scholar
  35. Huttenlocher, J., Duffy, S., & Levine, S. (2002). Infants and toddlers discriminate amount: Are they measuring? Psychological Science, 13, 244–249.Google Scholar
  36. Huttenlocher, J., Newcombe, N., & Sandberg, E. H. (1994). The coding of spatial location in young children. Cognitive Psychology, 27, 115–148.Google Scholar
  37. Huttenlocher, J., Newcombe, N., & Vasilyeva, M. (1999). Spatial scaling in young children. Psychological Science, 10(5), 393–398.Google Scholar
  38. Izard, V., O’Donnell, E., & Spelke, E. S. (2014). Reading angles in maps. Child Development, 85(1), 237–249.Google Scholar
  39. Izard, V., & Spelke, E. S. (2009). Development of sensitivity to geometry in visual forms. Human Evolution, 23(3), 213.Google Scholar
  40. Kamii, C. (2006). Measurement of length: How can we teach it better? Teaching Children Mathematics, 13(3), 154–158.Google Scholar
  41. Kawanaka, T., Stigler, J. W., & Hiebert, J. (1999). Studying mathematics classrooms in Germany, Japan and the United States: Lessons from the TIMSS videotape study. In G. Kaiser, E. Luna, & I. Huntley (Eds.), International comparisons in mathematics education (Vol. 11, p. 86). London, UK: Falmer.Google Scholar
  42. Kerslake, D. (1986). Fractions: Children’s strategies and errors. A Report of the Strategies and Errors in Secondary Mathematics Project. NFER-NELSON Publishing Company, Ltd., Windsor, England.Google Scholar
  43. Kotovsky, L., & Baillargeon, R. (1998). The development of calibration-based reasoning about collision events in young infants. Cognition, 67, 311–351.Google Scholar
  44. Kwon, M. K., Levine, S. C., Ratliff, K., & Snyder, C. (2011, January). The importance of alignable differences in teaching linear measurement. In: Proceedings of the Cognitive Science Society (Vol. 33, p. 1156).Google Scholar
  45. Kwon, M. K., Ping, R., Congdon, E. L., & Levine, S. C. (under revision). Overturning children’s misconceptions about ruler measurement units: The power of disconfirming evidence.Google Scholar
  46. Landau, B., Smith, L. B., & Jones, S. S. (1988). The importance of shape in early lexical learning. Cognitive Development, 3, 299–321. Scholar
  47. Lehrer, R. (2003). Developing understanding of measurement. In J. Kilpatrick, W. G. Martin, & D. Schifter (Eds.), A research companion to principles and standards for school mathematics (pp. 179–192). Reston, VA: National Council of Teachers of Mathematics.Google Scholar
  48. 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 (Vol. 1, pp. 137–167). Mahwah, NJ: Lawrence Erlbaum Associates Publishers.Google Scholar
  49. Liben, L. S., & Downs, R. M. (1993). Understanding person-space-map relations: Cartographic and developmental perspectives. Developmental Psychology, 29(4), 739.Google Scholar
  50. Liben, L. S., & Yekel, C. A. (1996). Preschoolers’ understanding of plan and oblique maps: The role of geometric and representational correspondence. Child Development, 67(6), 2780–2796.Google Scholar
  51. Lin, P.-J., & Tsai, W.-H. (2003). Fourth graders’ achievement of mathematics in TIMSS 2003 field test. (in Chinese). Science Education Monthly, 258, 2e20.Google Scholar
  52. Lindquist, M., & Kouba, V. (1989). Measurement. In M. Linduist (Ed.), Results from the fourth mathematics assessment of the National Assessment of Educational Progress (pp. 35–43). Reston, VA: National Council of Teachers of Mathematics.Google Scholar
  53. Lourenco, S. F., & Huttenlocher, J. (2008). The representation of geometric cues in infancy. Infancy, 13(2), 103–127.Google Scholar
  54. Markman, E. M., & Hutchinson, J. E. (1984). Children’s sensitivity to constraints on word meaning: Taxonomic versus thematic relations. Cognitive Psychology, 16, 1–27. Scholar
  55. Martin, W. G., & Strutchens, M. E. (2000). Geometry and measurement. Results from the seventh mathematics assessment of the National Assessment of Educational Progress (pp. 193–234).Google Scholar
  56. Miller, K. F. (1989). Measurement as a tool for thought: The role of measuring procedures in children’s understanding of quantitative invariance. Developmental Psychology, 25, 589–600.Google Scholar
  57. Mitchelmore, M. C., & White, P. (2000). Development of angle concepts by progressive abstraction and generalisation. Educational Studies in Mathematics, 41, 209–238. Scholar
  58. Mix, K. S., Levine, S. C., & Newcombe, N. S. (2016). Development of quantitative thinking across correlated dimensions. In A. Henik (Ed.), Continuous issues in numerical cognition: How many or how much (pp. 1–33). San Diego: Academic.Google Scholar
  59. Mix, K. S., & Paik, J. H. (2008). Do Korean fraction names promote part-whole reasoning? Journal of Cognitive Development, 9, 145–170.Google Scholar
  60. Möhring, W., Frick, A., Newcombe, N., & Levine, S. C. (2015). Spatial proportional reasoning is associated with formal knowledge about fractions. Journal of Cognition and Development, 17(1), 67–84.Google Scholar
  61. Mullis, I., Martin, M., Gonzalez, E., & Chrostowski, S. (2004). TIMSS 2003 international mathematics report: Findings from IEA’s trends in international mathematics and science study at the fourth and eighth grades. Chestnut Hill, MA: Boston College. National Center for Educational Statistics.Google Scholar
  62. National Center for Educational Statistics. (2009). NAEP Questions. Retrieved on September 3, 2017, from
  63. Newcombe, N., Huttenlocher, J., & Leamonth, A. (2000). Infants’ coding of location in continuous space. Infant Behavior and Development, 22, 483–510.Google Scholar
  64. Newcombe, N. S., Levine, S. C., & Mix, K. S. (2015). Thinking about quantity: The intertwined development of spatial and numerical cognition. Wiley Interdisciplinary Reviews: Cognitive Science, 6(6), 491–505.Google Scholar
  65. Newcombe, N. S., Sluzenski, J., & Huttenlocher, J. (2005). Pre-existing knowledge versus on-line learning: What do infants really know about spatial location? Psychological Science, 16, 222–227. Scholar
  66. Nunes, T., & Bryant, P. (1996). Children doing mathematics. Wiley-Blackwell.Google Scholar
  67. Piaget, J., Inhelder, B., & Szeminska, A. (1960). The child’s conception of geometry. New York: Basic Books.Google Scholar
  68. Ramscar, M., Dye, M., Popick, H. M., & O’Donnell-McCarthy, F. (2011). The enigma of number: Why children find the meanings of even small number words hard to learn and how we can help them do better. PLoS One, 6(7), e22501.Google Scholar
  69. Rescorla, R. A., & Wagner, A. R. (1972). A theory of Pavlovian conditioning: Variations in the effectiveness of reinforcement and nonreinforcement. In A. H. Black & W. F. Prokasy (Eds.), Classical conditioning II: Current research and theory (Vol. 2, pp. 64–99). New York: Appleton Century Crofts.Google Scholar
  70. Shipley, E. F., & Shepperson, B. (1990). Countable entities: Developmental changes. Cognition, 34, 109–136.Google Scholar
  71. Shusterman, A., Ah Lee, S., & Spelke, E. S. (2008). Young children’s spontaneous use of geometry in maps. Developmental Science, 11(2), F1–F7.Google Scholar
  72. Slater, A., Mattock, A., Brown, E., & Bremner, J. G. (1991). Form perception at birth: Cohen and Younger (1984) revisited. Journal of Experimental Child Psychology, 51, 395–406.Google Scholar
  73. Slater, A., Mattock, A., Brown, E., Burnham, D., & Young, A. (1991). Visual processing of stimulus compounds in newborn infants. Perception, 20(1), 29–33.Google Scholar
  74. Smith, J. P., Males, L. M., Dietiker, L. C., Lee, K., & Mosier, A. (2013). Curricular treatments of length measurement in the United States: Do they address known learning challenges? Cognition and Instruction, 31(4), 388–433.Google Scholar
  75. Solomon, T. L., Vasilyeva, M., Huttenlocher, J., & Levine, S. C. (2015). Minding the gap: Children’s difficulty conceptualizing spatial intervals as linear measurement units. Developmental Psychology, 51(11), 1564.Google Scholar
  76. Sophian, C. (2007). The origins of mathematical knowledge in childhood. Lawrence Erlbaum Associates.Google Scholar
  77. Sophian, C., Garyantes, D., & Chang, C. (1997). When three is less than two: Early developments in children’s understanding of fractional quantities. Developmental Psychology, 33(5), 731.Google Scholar
  78. Spelke, E. S., Gilmore, C. K., & McCarthy, S. (2001). Kindergarten children’s sensitivity to geometry in maps. Developmental Science, 14, 809–821.Google Scholar
  79. Spelke, E., Lee, S. A., & Izard, V. (2010). Beyond core knowledge: Natural geometry. Cognitive Science, 34, 863–884. Scholar
  80. Strutchens, M. E., Harris, K. A., & Martin, W. G. (2001). Assessing geometry and measurement. Understanding using manipulatives. Mathematics Teaching in the Middle School, 6, 402–405.Google Scholar
  81. Strutchens, M. E., Martin, W. G., & Kenney, P. A. (2003). What students know about measurement: perspectives from the national assessments of educational progress. In D. H. Clements & G. Bright (Eds.), Learning and teaching measurement. 2003 Year book (pp. 195–207). Reston, VA: NCTM.Google Scholar
  82. Tipps, S., Johnson, A., & Kennedy, L. M. (2011). Guiding children’s learning of mathematics. Belmont, CA: Cengage Learning.Google Scholar
  83. TIMSS 2011 Assessment. Copyright © 2012 International Association for the Evaluation of Educational Achievement (IEA). Publisher: TIMSS & PIRLS International Study Center, Lynch School of Education, Boston College, Chestnut Hill, MA and International Association for the Evaluation of Educational Achievement (IEA), IEA Secretariat, Amsterdam, the Netherlands.Google Scholar
  84. Vasilyeva, M., Casey, B., Dearing, E., & Ganley, C. (2009). Measurement skills in low-income elementary school students: Exploring the nature of gender differences. Cognition and Instruction, 27, 401–428.Google Scholar
  85. Vasilyeva, M., Duffy, S., & Huttenlocher, J. (2007). Developmental changes in the use of absolute and relative information: The case of spatial extent. Journal of Cognition and Development, 8, 455–471.Google Scholar
  86. Vasilyeva, M., & Lourenco, S. F. (2012). Development of spatial cognition. Wiley Interdisciplinary Reviews: Cognitive Science, 3(3), 349–362.Google Scholar
  87. Wilson, P. S., & Rowland, R. (1993). Teaching measurement. In R. J. Jensen (Ed.), Research ideas for the classroom: Early childhood mathematics (Vol. 30, pp. 171–194). Old Tappan, NJ: Macmillan.Google Scholar
  88. Yuzawa, M., Bart, W. M., & Yuzawa, M. (2000). Development of the ability to judge relative areas: Role of the procedure of placing one object on another. Cognitive Development, 15(2), 135–152.Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Eliza L. Congdon
    • 1
    Email author
  • Marina Vasilyeva
    • 2
  • Kelly S. Mix
    • 3
  • Susan C. Levine
    • 4
  1. 1.Department of PsychologyBucknell UniversityLewisburgUSA
  2. 2.Lynch School of Education, Boston CollegeChestnut HillUSA
  3. 3.Department of Human Development and Quantitative MethodologyUniversity of MarylandCollege ParkUSA
  4. 4.Departments of Psychology, and Comparative Human Development and Committee on EducationUniversity of ChicagoChicagoUSA

Personalised recommendations