Advertisement

Toward a theoretical structure to characterize early probabilistic thinking

  • Randall E. GrothEmail author
  • Jathan W. Austin
  • Madeline Naumann
  • Megan Rickards
Original Article

Abstract

The role of probability in curricula for children has fluctuated greatly over the past several decades. Recently, some countries have removed probability from their preschool and primary curricula, and others have retained it. One reason for such lack of agreement is that theory about early probability learning is still relatively new and under development. The purpose of this report is to sketch a tentative theoretical structure with the potential to anchor curricular decisions and inform further research on early probability learning. Toward that end, we begin by reviewing existing literature. We attend, in particular, to the probabilistic thinking tendencies exhibited by children whom researchers consider to be in the earliest stages of learning the subject. We then use these tendencies to posit two different cycles for early probabilistic thinking. One of these cycles is compatible with disciplinary norms and supports normative thinking; thinking tendencies in this normative compatible cycle include attending to the position of objects in a container, forming images of random generators, attending to the operation of random generators, and thinking about past experiences playing games of chance. Thinking tendencies in the other posited cycle lead to belief systems that conflict with normative disciplinary practice; these belief systems include elements such as myths, superstitions, animism, and determinism. We illustrate the two cycles using empirical data from design-based research. We then reflect on how the two cycles comprising our structure of early probabilistic thinking (SEPT) framework can provide a basis for further curricular and theoretical work.

Keywords

Probability SOLO taxonomy Cognition Beliefs 

Notes

Funding information

This material is based upon work supported by the National Science Foundation under Grant Number DRL-1356001. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

References

  1. Amir, G. S., & Williams, J. S. (1999). Cultural influences on children’s probabilistic thinking. Journal of Mathematical Behavior, 18, 85–107.  https://doi.org/10.1016/S0732-3123(99)00018-8.Google Scholar
  2. Australian Education Council. (1991). A national statement on mathematics for Australian schools. Carlton: Curriculum Corporation.Google Scholar
  3. Batanero, C., & Serrano, L. (1999). The meaning of randomness for secondary school students. Journal for Research in Mathematics Education, 30, 558–567.Google Scholar
  4. Ben-Zvi, D. (2018). Foreword. In A. Leavy, M. Meletiou-Mavrotheris, & E. Paparistodemou (Eds.), Statistics in early childhood and primary education: supporting early statistical and probabilistic thinking (pp. vii–viii). Singapore: Springer.Google Scholar
  5. Biggs, J. B., & Collis, K. F. (1991). Multimodal learning and the quality of intelligent behavior. In H. A. H. Rowe (Ed.), Intelligence: reconceptualization and measurement (pp. 57–76). Hillsdale: Erlbaum.Google Scholar
  6. Callingham, R. A. (1997). Teachers’ multimodal functioning in relation to the concept of average. Mathematics Education Research Journal, 9, 205–224.Google Scholar
  7. Chinn, C. A., & Brewer, W. F. (1993). The role of anomalous data in knowledge acquisition: a theoretical framework and implications for science instruction. Review of Educational Research, 63(1), 1–49.Google Scholar
  8. Cobb, P., Jackson, K., & Sharpe, C. (2017). Conducting design studies to investigate and support mathematics students’ and teachers’ learning. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 208–233). Reston: National Council of Teachers of Mathematics.Google Scholar
  9. Collis, K. F., & Romberg, T. A. (1990). “The standards”: theme and assessment. In K. Milton & H. McCann (Eds.), Mathematical turning points—strategies for the 1990s (pp. 173–189). Hobart: Australian Association of Mathematics Teachers.Google Scholar
  10. Common Core State Standards Initiative. (2010). Common core state standards for mathematics. Retrieved from http://www.corestandards.org. Accessed 20 Aug 2019.
  11. DfEE. (1999). The national curriculum: mathematics. London: DfEE Publication.Google Scholar
  12. English, L. D., & Watson, J. M. (2016). Development of probabilistic understanding in fourth grade. Journal for Research in Mathematics Education, 47, 28–62.Google Scholar
  13. Fischbein, E. (1975). The intuitive sources of probabilistic thinking in children. Dordrecht: Reidel.Google Scholar
  14. Garfield, J. B. (2002). The challenge of developing statistical reasoning. Journal of Statistics Education, 10(3), Retrieved from ww2.amstat.org/publications/jse/v10n3/garfield.html. Accessed 20 Aug 2019.
  15. Greer, B. (2014). Commentary on perspective II: psychology. In E. J. Chernoff & B. Sriraman (Eds.), Probabilistic thinking: presenting plural perspectives (pp. 299–309). Dordrecht: Springer.Google Scholar
  16. Groth, R. E. (2017). Developing statistical knowledge for teaching during design-based research. Statistics Education Research Journal, 16(2), 376–396. Retrieved from https://iase-web.org/documents/SERJ/SERJ16(2)_Groth.pdf. Accessed 20 Aug 2019.
  17. Groth, R. E., Bergner, J. A., Burgess, C. R., Austin, J. W., & Holdai, V. (2016). Re-imagining education of mathematics teachers through undergraduate research. Council on Undergraduate Research (CUR) Quarterly, 36(3), 41–46.Google Scholar
  18. Groth, R. E., Jones, M., & Knaub, M. (2018). A framework for characterizing students' cognitive processes related to informal best fit lines. Mathematical Thinking and Learning, 20(4), 251–276.Google Scholar
  19. Groth, R. E., Austin, J. W., Naumann, M., Rickards, M. (2019). Probability puppets. Teaching Statistics, 41(2), 54–57Google Scholar
  20. Inhelder, B., & Piaget, J. (1958). The growth of logical thinking from childhood to adolescence. (A Parsons & S. Milgram, Trans.). New York: Basic Books.Google Scholar
  21. Jones, G. A., & Thornton, C. A. (2005). An overview of research into the learning and teaching of probability. In G. A. Jones (Ed.), Exploring probability in school: challenges for teaching and learning (pp. 65–92). New York: Springer.Google Scholar
  22. Jones, G. A., Langrall, C. W., Thornton, C. A., & Mogill, A. T. (1997). A framework for assessing and nurturing young children’s thinking in probability. Educational Studies in Mathematics, 32, 101–125.  https://doi.org/10.1023/A:1002981520728.Google Scholar
  23. Jones, G. A., Langrall, C. W., Thornton, C. A., & Mogill, A. T. (1999a). Students’ probabilistic thinking in instruction. Journal for Research in Mathematics Education, 30, 487–519.Google Scholar
  24. Jones, G. A., Thornton, C. A., Langrall, C. W., & Tarr, J. E. (1999b). Understanding students’ probabilistic reasoning. In L. V. Stiff & F. R. Curcio (Eds.), Developing mathematical reasoning in grades K-12 (1999 yearbook) (pp. 146–155). Reston: National Council of Teachers of Mathematics.Google Scholar
  25. Jones, G. A., Thornton, C. A., Langrall, C. W., Mooney, E. S., Perry, B., & Putt, I. J. (2000). A framework for characterizing children’s statistical thinking. Mathematical Thinking and Learning, 2, 269–307.  https://doi.org/10.1207/S15327833MTL0204_3.Google Scholar
  26. Langrall, C. W. (2018). The status of probability in the elementary and lower secondary school mathematics curriculum: the rise and fall of probability in school mathematics in the United States. In C. Batanero & E. Chernoff (Eds.), Teaching and learning stochastics. ICME-13 monographs (pp. 39–50). Springer: Cham.  https://doi.org/10.1007/978-3-319-72871-1_3.Google Scholar
  27. Langrall, C. W., & Mooney, E. S. (2005). Characteristics of elementary school students’ probabilistic thinking. In G. A. Jones (Ed.), Exploring probability in school: challenges for teaching and learning (pp. 95–120). New York: Springer.Google Scholar
  28. Leavy, A., Meletiou-Mavrotheris, M., & Paparistodemou, E. (2018). Preface. In A. Leavy, M. Meletiou-Mavrotheris, & E. Paparistodemou (Eds.), Statistics in early childhood and primary education: supporting early statistical and probabilistic thinking (pp. ix–xxii). Singapore: Springer.Google Scholar
  29. Makar, K., & Rubin, A. (2009). A framework for thinking about informal statistical inference. Statistics Education Research Journal, 8(1), 82–105 Retrieved from https://iase-web.org/documents/SERJ/SERJ8(1)_Makar_Rubin.pdf. Accessed 20 Aug 2019.Google Scholar
  30. Martignon, L. (2014). Fostering children’s probabilistic reasoning and first elements of risk evaluation. In E. J. Chernoff & B. Sriraman (Eds.), Probabilistic thinking: presenting plural perspectives (pp. 149–160). Dordrecht: Springer.Google Scholar
  31. McClain, K., & Cobb, P. (2001). Supporting students’ ability to reason about data. Educational Studies in Mathematics, 45(1), 103–129.  https://doi.org/10.1023/A:1013874514650.Google Scholar
  32. Metz, K. E. (1998). Emergent understanding and attribution of randomness: comparative analysis of reasoning of primary grade children and undergraduates. Cognition and Instruction, 16, 285–365.  https://doi.org/10.1207/s1532690xci1603_3.Google Scholar
  33. Mooney, E. S., Langrall, C. W., & Hertel, J. T. (2014). A practitional perspective on probabilistic thinking models and frameworks. In E. J. Chernoff & B. Sriraman (Eds.), Probabilistic thinking: presenting plural perspectives (pp. 495–507). Dordrecht: Springer.Google Scholar
  34. Moore, D. S. (1990). Uncertainty. In L. A. Steen (Ed.), On the shoulders of giants: new approaches to numeracy (pp. 95–137). Washington, D.C.: National Academy Press.Google Scholar
  35. National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston: Author.Google Scholar
  36. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston: Author.Google Scholar
  37. Nikiforidou, Z. (2018). Probabilistic thinking and young children: theory and pedagogy. In A. Leavy, M. Meletiou-Mavrotheris, & E. Paparistodemou (Eds.), Statistics in early childhood and primary education: supporting early statistical and probabilistic thinking (pp. 21–34). Singapore: Springer.Google Scholar
  38. Pegg, J., & Davey, G. (1998). Interpreting student understanding of geometry: a synthesis of two models. In R. Lehrer & D. Chazan (Eds.), Designing learning environments for developing understanding of geometry and space (pp. 109–135). Mahwah: Lawrence Erlbaum Associates.Google Scholar
  39. Shaughnessy, J. M. (2007). Research on statistics learning and reasoning. In F. K. Lester Jr. (Ed.), Second handbook of research on mathematics teaching and learning (pp. 957–1009). Charlotte: Information Age & NCTM.Google Scholar
  40. Smit, J., & van Eerde, H. A. A. (2011). A teacher’s learning process in dual design research: learning to scaffold language in a multilingual mathematics classroom. ZDM – The International Journal on Mathematics Education, 43(6), 889–900.  https://doi.org/10.1007/s11858-011-0350-5.Google Scholar
  41. Tarr, J. E., & Jones, G. A. (1997). A framework for assessing middle school students’ thinking in conditional probability and independence. Mathematics Education Research Journal, 9(1), 39–59.  https://doi.org/10.1007/BF03217301.Google Scholar
  42. Truran, K. (1995). Animism: a view of probability behavior. In B. Atweh & S. Flavel (Eds.), Proceedings of the 18 th annual conference of the Mathematics Education Research Group of Australasia (pp. 537–541). Darwin: MERGA.Google Scholar
  43. U.S. Department of Education, Institute of Education Sciences, & National Center for Education Statistics. (2017). NAEP Questions Tool. Retrieved from http://nces.ed.gov/nationsreportcard/itmrlsx. Accessed 20 Aug 2019.
  44. Watson, J. M. (2018). Variation and expectation for six-year-olds. In A. Leavy, M. Meletiou-Mavrotheris, & E. Paparistodemou (Eds.), Statistics in early childhood and primary education: supporting early statistical and probabilistic thinking (pp. 55–73). Singapore: Springer.Google Scholar
  45. Watson, J. M., & Collis, K. (1994). Multimodal functioning in understanding chance and data concepts. In J. P. da Ponte & J. F. Matos (Eds.), Proceedings of the 18th Conference of the International Group for the Psychology of Mathematics Education: Volume 4 (pp. 396–376). Portugal: Lisbon.Google Scholar
  46. Watson, J. M., & Moritz, J. B. (2000). Developing concepts of sampling. Journal for Research in Mathematics Education, 31, 44–70.Google Scholar
  47. Watson, J. M., & Moritz, J. B. (2003). Fairness of dice: a longitudinal study of students’ beliefs and strategies for making judgments. Journal for Research in Mathematics Education, 34, 270–304.Google Scholar
  48. Watson, J. M., Collis, K. F., Callingham, R. A., & Moritz, J. B. (1995). A model for assessing higher-order thinking in statistics. Educational Research and Evaluation, 1, 247–275.Google Scholar
  49. Watson, J. M., Collis, K. F., & Moritz, J. B. (1997). The development of chance measurement. Mathematics Education Research Journal, 9, 60–82.  https://doi.org/10.1007/BF03217302.Google Scholar
  50. Williams, A., & Nisbet, S. (2014). Primary school students’ attitudes to and beliefs about probability. In E. J. Chernoff & B. Sriraman (Eds.), Probabilistic thinking: presenting plural perspectives (pp. 683–708). Dordrecht: Springer.Google Scholar

Copyright information

© Mathematics Education Research Group of Australasia, Inc. 2019

Authors and Affiliations

  1. 1.Seidel School of EducationSalisbury UniversitySalisburyUSA
  2. 2.Department of Mathematics and Computer ScienceSalisbury UniversitySalisburyUSA

Personalised recommendations