Tethering towards number: synthesizing cognitive variability and stage-oriented development in children’s arithmetic thinking

  • Jonathan Norris ThomasEmail author
  • Shelly Sheats Harkness


Differing research worldviews have typically resulted in interpretations at odds with one another. Yet, leveraging distinct perspectives can lead to novel interpretations and theoretical construction. Via an empirically grounded research commentary, we describe the value of such activity through the lens of previously reported findings. This synthesis of research from dissimilar scholarly traditions is one example of how paradigms in related but sometimes disconnected fields were used to provide a more comprehensive model of foundational numeracy development. While critique and skepticism may be valuable scholarly tools, we argue that such practices should be balanced with openness and belief towards ideas from worldviews different than our own. This balance can provide new and creative interpretations and extend our collective research power.


Numeracy Cognition Theory Teaching experiment Constructivism 



  1. Arzarello, F., Bosch, M., Lenfant, A., & Prediger, S. (2007). Different theoretical perspectives in research from teaching problems to research problems. In D. Pitta-Pantazi, G. Phillipou, et al. (Eds.), Proceedings of the 5th Congress of the European Society for Research in Mathematics Education (CERME 5) (pp. 1618–1627). Cyprus: ERME.Google Scholar
  2. Brownell, W. A. (1947). The place of meaning in the teaching of arithmetic. Elementary School Journal, 47(5), 256–265.Google Scholar
  3. Carpenter, T. P. (1985). Toward a theory of construction. Journal for Research in Mathematics Education, 16(1), 70–76.Google Scholar
  4. Chen, Z., & Siegler, R. S. (2000). Overlapping waves theory. Monographs of the Society for Research in Child Development, 65(2), 7–11.Google Scholar
  5. Clements, D. H. (1989). Review: consensus, more or less. Journal for Research in Mathematics Education, 20(1), 111–119.Google Scholar
  6. Clements, D. H. (1999). Subitizing: what is it? Why teach it? Teaching Children Mathematics, 5(7), 400–405.Google Scholar
  7. Clements, D. H., & Sarama, J. (2009). Learning and teaching early math: the learning trajectories approach. New York: Routledge.Google Scholar
  8. Cobb, P., Yackel, E., & Wood, T. (1992). A constructivist alternative to the representational view of mind. Journal for Research in Mathematics Education, 23(1), 2–33.Google Scholar
  9. Erickson, F. (2006). Definition and analysis of data from videotape: some research procedures and their rationales. In J. L. Green, G. Camilli, P. B. Elmore, A. Skukauskaite, & E. Grace (Eds.), Handbook of complementary methods in education research (pp. 177–192). Hillsdale: Erlbaum.Google Scholar
  10. Fazio, L. K., DeWolf, M., & Siegler, R. S. (2016). Strategy use and strategy choice in fraction magnitude comparison. Journal of Experimental Psychology, 42(1), 1–16.Google Scholar
  11. Fosnot, C., & Dolk, M. (2001). Young mathematicians at work: constructing number sense, addition and subtraction. Portsmouth: Heinemann.Google Scholar
  12. Fuson, K. C. (1982). An analysis of the counting—on solution procedure in addition. In T. P. Carpenter, J. M. Moser, & T. A. Romberg (Eds.), Addition and subtraction: a cognitive perspective (pp. 67–81). Hillsdale: Erlbaum.Google Scholar
  13. Fuson, K. C. (1988). Children’s counting and concepts of number. New York: Springer-Verlag.Google Scholar
  14. Fuson, K. C., Richards, J., & Briars, D. J. (1982). The acquisition and elaboration of the number word sequence. In C. J. Brainerd (Ed.), Children’s logical and mathematical cognition (pp. 33–92). New York: Springer.Google Scholar
  15. Fuson, K. C., Secada, W. G., & Hall, J. W. (1983). Matching, counting, and conservation of numerical equivalence. Child Development, 54(1), 91–97.Google Scholar
  16. Fuson, K. C., Pergament, G. G., & Lyons, B. G. (1985). Collection terms and preschoolers’ use of the cardinality rule. Cognitive Psychology, 17(6), 1429–1436.Google Scholar
  17. Gallistel, C. R., & Gelman, R. (1992). Preverbal and verbal counting and computation. Cognition, 44(1/2), 43–74.Google Scholar
  18. Gelman, R., & Gallistel, C. R. (1978). The child’s understanding of number. Cambridge: Harvard University Press.Google Scholar
  19. Gelman, R., & Meck, E. (1983). Preschoolers counting: principles before skill. Cognition, 13(3), 343–359.Google Scholar
  20. Gelman, R., & Tucker, M. F. (1975). Further investigations of the young child’s conception of number. Child Development, 46(1), 167–175.Google Scholar
  21. Glaser, B., & Strauss, A. (1967). The discovery of the grounded theory: strategies for qualitative research. New York: Aldine de Gruyter.Google Scholar
  22. Groen, G. J., & Parkman, J. M. (1972). A chronometric analysis of simple addition. Psychological Review, 79(4), 329–343.Google Scholar
  23. Ilg, F., & Ames, L. B. (1951). Developmental trends in arithmetic. Journal of Genetic Pschology, 79(1), 3–28.Google Scholar
  24. Karp, A., & Schubring, G. (Eds.). (2014). Handbook on the history of mathematics education. New York: Springer.Google Scholar
  25. Kosslyn, S. M. (1980). Image and mind. Cambridge: Harvard University Press.Google Scholar
  26. Kosslyn, S. M. (2005). Mental images and the brain. Cognitive Neuropsychology, 22(3/4), 333–347.Google Scholar
  27. Moschkovich, J. N., & Brenner, M. E. (2000). Integrating a naturalistic paradigm into research on mathematics and science cognition and learning. In R. Lesh & A. E. Kelly (Eds.), Research design in mathematics and science education (pp. 457–486). Hillsdale: Erlbaum.Google Scholar
  28. Olive, J. (2001). Children's number sequences: an explanation of Steffe's constructs and an extrapolation to rational numbers of arithmetic. The Mathematics Educator, 11(1), 4–9.Google Scholar
  29. Piaget, J. (1952). The child 's concept of number. London: Routledge.Google Scholar
  30. Prediger, S., Bikner-Ahsbahs, A., & Arzarello, F. (2008). Networking strategies and methods for connecting theoretical approaches: first steps towards a conceptual framework. ZDM, 40(2), 165–178.Google Scholar
  31. Putnam, H. (1988). Representation and reality. Cambridge: Bradford.Google Scholar
  32. Pylyshyn, Z. W. (1974). What the mind’s eye tells the mind’s brain: a critique of mental imagery. In J. M. Nichols (Ed.), Images, perceptions, and knowledge (pp. 1–36). New York: Springer.Google Scholar
  33. Pylyshyn, Z. W. (2002). Mental imagery: In search of a theory. Behavioral and Brain Sciences, 25(2), 157–238.Google Scholar
  34. Rathgeb-Schnierer, E., & Green, M. (2013). Flexibility in mental calculation in elementary students from different math classes. In B. Ubuz, C. Haser, & M. A. Mariotti (Eds.), Proceedings of the eighth congress of the European Society for Research in Mathematics Education (pp. 353–362). Ankara: Middle East Technical University.Google Scholar
  35. Rorty, R. (1979). Philosophy and the mirror of our nature. Princeton: Princeton University Press.Google Scholar
  36. Seigler, R. S. (2000). The rebirth of children's learning. Child Development, 71(1), 26–35.Google Scholar
  37. Siegler, R. S. (1987). Strategy choices in subtraction. In J. A. Sloboda & D. Rogers (Eds.), Cognitive processes in mathematics (pp. 81–106). Oxford: Clarendon Press.Google Scholar
  38. Siegler, R. S. (1994). Cognitive variability: a key to understanding cognitive development. Current Directions in Psychological Science, 3(1), 1–5.Google Scholar
  39. Siegler, R.S. & Crowley, K. (1994). Constraints on nonprivileged domains. Cognitive Psychology, 27(2), 194-226.Google Scholar
  40. Siegler, R. S. (2006). Microgenetic analyses of learning. In D. Kuhn, R. S. Siegler, W. Damon, & R. M. Lerner (Eds.), Handbook of child psychology: Vol. 2. Cognition, perception, and language (6th ed., pp. 464–510). Hoboken: Wiley.Google Scholar
  41. Siegler, R. S., & Robinson, M. (1982). The development of numerical understandings. In H. W. Reese & L. P. Lipsett (Eds.), Advances in child development and behavior (Vol. 16, pp. 242–312). New York: Academic Press.Google Scholar
  42. Siegler, R. S., & Shrager, J. (1984). Strategy choices in addition and subtraction: How do children know what to do? In C. Sophian (Ed.), The origins of cognitive skills (pp. 229–293). Hillsdale: Erlbaum.Google Scholar
  43. Smith, T. M., Cobb, P., Farran, D. C., Cordray, D. S., & Munter, C. (2013). Evaluating Math Recovery: assessing the causal impact of a diagnostic tutoring program on student achievement. American Education Research Journal, 50(2), 397–428.Google Scholar
  44. Sophian, C. (2007). The origins of mathematical knowledge in childhood. New York: Lawrence Erlbaum.Google Scholar
  45. Steffe, L. (1992). Learning stages in the construction of the number sequence. In J. Bideaud, C. Meljac, & J. Fischer (Eds.), Pathways to number: children’s developing numerical abilities (pp. 83–88). Hillsdale: Lawrence Erlbaum.Google Scholar
  46. Steffe, L. P. (2013). Establishing mathematics education as an academic field: a constructive odyssey. Journal for Research in Mathematics Education, 44(2), 354–370.Google Scholar
  47. Steffe, L. P., & Thompson, P. W. (2000). Teaching experiment methodology: underlying principles and essential elements. In A. E. Kelly & R. A. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 267–307). Mahwah: Erlbaum.Google Scholar
  48. Steffe, L. P., von Glasersfeld, E., Richards, J., & Cobb, P. (1983). Children’s counting types: philosophy, theory, and application. New York: Praeger Scientific.Google Scholar
  49. Steffe, L. P., Cobb, P., & von Glasersfeld, E. (1988). Construction of arithmetical meanings and strategies. New York: Springer-Verlag.Google Scholar
  50. Svenson, O., & Sjöberg, K. (1983). Evolution of cognitive processes for solving simple additions during the first three school years. Scandinavian Journal of Psychology, 24(1), 117–124.Google Scholar
  51. Thomas, J. & Tabor, P.D. (2012). Developing Quantitative Mental Imagery. Teaching Children Mathematics, 19(3), 174-183. Google Scholar
  52. Thomas, J. & Harkness, S. S. (2013). Implications for intervention: Categorizing the quantitative mental imagery of children. Mathematics Education Research Journal, 25(2), 231-256. Google Scholar
  53. Thomas, J. & Harkness, S.S. (2016). Patterns of Non-verbal Social Interaction within Intensive Mathematics Intervention Contexts. Mathematics Education Research Journal, 28(2), 277-302.Google Scholar
  54. Thomas, J., Tabor, P. D., & Wright, R. J. (2010). Three aspects of first-graders' number knowledge: Observations and instructional implications. Teaching Children Mathematics, 17 (5), 299-308. Google Scholar
  55. Thompson, P. W. (1979). The Soviet-style teaching experiment in mathematics education. Paper presented at the Annual Research Meeting of the National Council of Teachers of Mathematics, Boston, MA.Google Scholar
  56. Thompson, P. W. (1982). Were lions to speak, we wouldn't understand. Journal of Mathematical Behavior, 3(2), 147–165.Google Scholar
  57. Threlfall, J. (2002). Flexible mental calculation. Educational Studies in Mathematics, 50(1), 29–47.Google Scholar
  58. Van de Walle, J. A., Karp, K. S., & Bay-Williams, J. M. (2016). Elementary and middle school mathematics: teaching developmentally (9th ed.). New York: Pearson.Google Scholar
  59. Wirszup, I., & Kilpatrick, J. (Eds.). (1975). Soviet studies in the psychology of mathematics education (Vol. 1-14). Palo Alto and Reston: School Mathematics Study Group and National Council of Teachers of Mathematics.Google Scholar
  60. Wright, R. J. (1994). A study of the numerical development of 5-year-olds and 6-year-olds. Educational Studies in Mathematics, 26(1), 25–44.Google Scholar
  61. Wright, R.J., Ellemor-Collins, D. (2016). The Learning Framework in Number: Pedagogical Tools for Assessment and Instruction. London: Paul Chapman Publications/Sage. Google Scholar
  62. Wright, R. J., Martland, J., Stafford, A., & Stanger, G. (2002). Teaching number: advancing children’s skills and strategies. London: Paul Chapman Publications/Sage.Google Scholar
  63. Wright, R. J., Martland, J., & Stafford, A. (2006). Early numeracy: assessment for teaching and intervention (2nd ed.). London: Paul Chapman publications/Sage.Google Scholar

Copyright information

© This is a U.S. Government work and not under copyright protection in the US; foreign copyright protection may apply 2019

Authors and Affiliations

  1. 1.University of KentuckyLexingtonUSA
  2. 2.University of CincinnatiCincinnatiUSA

Personalised recommendations