A FRAMEWORK TO SUPPORT RESEARCH ON INFORMAL

A FRAMEWORK TO SUPPORT RESEARCH ON INFORMAL

INFERENTIAL REASONING

Andrew Zieffler

University of Minnesota

zief0002@umn.edu

Joan Garfield

University of Minnesota

jbg@umn.edu

Robert delMas

University of Minnesota

delma001@umn.edu

chris reading

University of New England

creading@une.edu.au

ABSTRACT

Informal inferential reasoning is a relatively recent concept in the research literature. Several research studies have defined this type of cognitive process in slightly different ways. In this paper, a working definition of informal inferential reasoning based on an analysis of the key aspects of statistical inference, and on research from educational psychology, science education, and mathematics education is presented. Based on the literature reviewed and the working definition, suggestions are made for the types of tasks that can be used to study the nature and development of informal inferential reasoning. Suggestions for future research are offered along with implications for teaching.

Keywords: Statistics education research; Inference; Informal reasoning; Introductory statistics course; Topic sequencing

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Statistics Education Research Journal, 7(2), 40-58, http://www.stat.auckland.ac.nz/serj

Ó International Association for Statistical Education (IASE/ISI), November, 2008

REFERENCES

Bakker, A. (2004). Reasoning about shape as a pattern in variability. Statistics Education Research Journal, 3(2), 64-83.

[Online: http://www.stat.auckland.ac.nz/~iase/serj/SERJ3(2)_Bakker.pdf]

Bakker, A., Derry, J., & Konold, C. (2006). Technology to support diagrammatic reasoning about center and variation. In A. Rossman & B. Chance (Eds.), Working Cooperatively in Statistics Education. Proceedings of the Seventh International Conference on Teaching Statistics, Salvador, Brazil. [CDROM]. Voorburg, The Netherlands: International Statistical Institute.

[Online: http://www.stat.auckland.ac.nz/~iase/publications/17/2D4_BAKK.pdf]

Bakker, A. & Gravemeijer, K. (2004). Learning to reason about distribution. In D. Ben-Zvi & J. Garfield (Eds.), The Challenge of Developing Statistical Literacy, Reasoning, and Thinking. Dordrecht, The Netherlands: Kluwer Academic Publishers.

Ben-Zvi, D. (2006). Scaffolding students’ informal inference and argumentation. In A. Rossman & B. Chance (Eds.), Working Cooperatively in Statistics Education. Proceedings of the Seventh International Conference on Teaching Statistics, Salvador, Brazil. [CDROM]. Voorburg, The Netherlands: International Statistical Institute.

[Online: http://www.stat.auckland.ac.nz/~iase/publications/17/2D1_BENZ.pdf]

Bransford J. D., Brown, A. L., & Cocking, R. R. (2000). How people learn: Brain, mind, experience, and school. Washington, D. C.: National Academy Press.

Carver, R. (2006, August). Ambiguity intolerance: An impediment to inferential reasoning? Paper presented at the Joint Statistics Meetings, Seattle, WA.

Cobb, P., & McClain, K. (2004). Principles of Instructional Design for Supporting the Development of Students’ Statistical Reasoning. In D. Ben-Zvi and J. Garfield (Eds.), The Challenge of Developing Statistical Literacy, Reasoning, and Thinking (pp. 375-396). Dordrecht, The Netherlands: Kluwer Academic Publishers.

Cobb, P., McClain, K., & Gravemeijer, K. P. E (2003). Learning about statistical covariation. Cognition and Instruction, 21, 1-78.

Cobb, P., Yackel, E., & Wood, T. (1992). A constructivist alternative to the representational view of mind in mathematics education. Journal of Research in Mathematics Education, 23(1), 23-33.

Cohen, J. (1994). The earth is round (p < .05). American Psychologist, 49(12), 997-1003.

Confrey, J. (1990). A review of the research on students’ conceptions in mathematics, science, and programming. Review of Research in Education, 16, 3-56.

delMas, R., Garfield, J., & Chance, B. (1999). A model of classroom research in action: Developing simulation activities to improve students’ statistical reasoning. Journal of Statistics Education, 7(3).

[Online: http://www.amstat.org/publications/jse/secure/v7n3/delmas.cfm]

Driver, R., Newton, P., & Osborne, J. (2000), Establishing the Norms of Scientific Argumentation in Classrooms, Science Education, 84, 287-312.

Evans, J. St. B. T., Newstead, S. E., & Byrne, R. M. J. (1993). Human reasoning: The psychology of deduction. Hove (UK): Lawrence Erlbaum Associates Ltd.

Falk, R., & Greenbaum, C. W. (1995). Significance tests die hard: The amazing persistence of a probabilistic misconception. Theory and Psychology, 5, 75-98.

Garfield, J., & Ben-Zvi, D. (2008). Developing students’ statistical reasoning: Connecting research and teaching. New York: Springer.

Garfield, J., delMas, R., & Chance, B. (2007). Using students’ informal notions of variability to develop an understanding of formal measures of variability. In M. Lovett and P. Shah (Eds.), Thinking with Data (Proceedings of the 33^rd Carnegie Symposium on Cognition) (pp. 117-147). New York: Erlbaum.

Gravemeijer, K., & Doorman, K. (1999). Context problems in realistic mathematics education: A calculus course example. Educational Studies in Mathematics, 39(1/3), 111-129.

Jones, G. A., Langrall, C. E., Mooney, E. S., & Thornton, C. A. (2005). Models of development in statistical reasoning. In D. Ben-Zvi and J. Garfield (Eds.), The Challenge of Developing Statistical Literacy, Reasoning, and Thinking (pp. 97-117). Dordrecht, The Netherlands: Kluwer Academic Publishers.

Kahneman, D., Slovic, S., & Tversky, A. (1982). Judgment under uncertainty: Heuristics and biases. Cambridge: Cambridge University Press.

Klahr, D., Fay, A., & Dunbar, K. (1993). Heuristics for scientific experimentation: A developmental study. Cognitive Psychology, 25, 111-146.

Kuhn, D. (1991). The skills of argument. Cambridge, UK: Cambridge University Press.

Kuhn, D., Garcia-Mila, M., Zohar, A., & Andersen, C. (1995). Strategies of knowledge acquisition. Monographs of the Society for Research in Child Development, 60(4, Serial No. 245).

Lipson, K. (2003). The role of the sampling distribution in understanding statistical inference. Mathematics Education Research Journal, 15(3), 270-287.

Mack, N. K. (1995). Confounding whole-number and fraction concepts when building on informal knowledge. Journal of Research in Mathematics Education, 26(5), 422-441.

Means, M. L., & Voss, J. F. (1996). Who reasons well? Two studies of informal reasoning among children of different grade, ability, and knowledge levels. Cognition and Instruction, 14(2), 139-178.

Miller-Jones, D. (1991). Informal reasoning in inner-city children. In J. F. Voss, D. N. Perkins, and J. Segal (Eds.), Informal reasoning and education (pp. 107-130). Hillsdale, NJ: Lawrence Erlbaum Associates.

Moore, D.S. (1990). Uncertainty. In L.A. Steen (Ed.), On the shoulders of giants (pp. 95-173). Washington, DC: National Academy Press.

Moore, D.S. (2004). The basic practice of statistics (3^rd ed.). New York: W.H. Freeman.

Nickerson, R. S. (2000). Null hypothesis significance testing: A review of an old and continuing controversy. Psychological Methods, 5(2), 241-301.

Nickerson, R., Perkins, D. N., & Smith, E. (1985). The teaching of thinking. Hillsdale, NJ: Lawrence Erlbaum Associates.

Papert, S., & Harel, I. (Eds.) (1991). Constructionism. Norwood, NJ: Ablex Publishing.

Pegg, J. (2003). Assessment in mathematics: A developmental approach. In J. Royer (Ed.), Mathematical Cognition (pp. 227-259). Greenwich, CT: Information Age Publishing.

Perkins, D. N. (1985a). Postprimary education has little impact on informal reasoning. Journal of Educational Psychology, 77(5), 562-571.

Perkins, D. N. (1985b). Reasoning as imagination. Interchange, 16(1), 14-26.

Perkins, D. N. (1989). Reasoning as it is and could be. In D. Topping, D. Crowell, & V. Kobayashi (Eds.), Thinking: The third international conference (pp. 175-194). Hillsdale, NJ: Lawrence Erlbaum Associates.

Perkins, D. N., Bushey, B., & Farady, M. (1986). Learning to reason (Final report for grant no. NIE-G-83_0028). Cambridge, MA: Harvard Graduate School of Education.

Perkins, D. N., Farady, M., & Bushey, B. (1991). Everyday reasoning and the roots of intelligence. In J. F. Voss, D. N. Perkins, and J. Segal (Eds.), Informal reasoning and education (pp. 83-105). Hillsdale, NJ: Lawrence Erlbaum Associates.

Pfannkuch, M. (2005). Probability and statistical inference: How can teachers enable learners to make the connection? In G. A. Jones (Ed.), Exploring probability in school: Challenges for teaching and learning (pp. 267-294). New York: Springer.

Pfannkuch, M. (2006). Informal inferential reasoning. In A. Rossman & B. Chance (Eds.), Working Cooperatively in Statistics Education. Proceedings of the Seventh International Conference on Teaching Statistics, Salvador, Brazil. [CDROM]. Voorburg, The Netherlands: International Statistical Institute.

[Online: http://www.stat.auckland.ac.nz/~iase/publications/17/6A2_PFAN.pdf]

Quilici, J. L., & Mayer, R. E. (2002). Teaching students to recognize structural similarities between statistics word problems. Applied Cognitive Psychology, 16, 325-342.

Reading, C. (2007, August). Cognitive development of reasoning about inference. Discussant reaction presented at the Fifth International Research Forum on Statistical Reasoning, Thinking and Literacy (SRTL-5), University of Warwick, UK.

Rossman, A. (2007, August). A statistician’s view on the concept of inferential reasoning. Paper presented at the Fifth International Research Forum on Statistical Reasoning, Thinking and Literacy (SRTL-5), University of Warwick, UK.

Rubin, A., Hammerman, J., & Konold, C. (2006). Exploring informal inference with interactive visualization software. In A. Rossman & B. Chance (Eds.), Working Cooperatively in Statistics Education. Proceedings of the Seventh International Conference on Teaching Statistics, Salvador, Brazil. [CDROM]. Voorburg, The Netherlands: International Statistical Institute.

[Online: http://www.stat.auckland.ac.nz/~iase/publications/17/2D3_RUBI.pdf]

Sadler, T. D. (2004). Informal reasoning regarding socioscientific issues: A critical review of research. Journal of Research in Science Teaching, 41(5), 513-536.

Sadler, T. D., & Zeidler, D. L. (2004). The significance of content knowledge for informal reasoning regarding socioscientific issues: Applying genetics knowledge to genetic engineering issues. Science Education, 89, 71-93.

Saldanha, L. A. & Thompson, P. W. (2002). Conceptions of sample and their relationship to statistical inference. Educational Studies in Mathematics, 51(3), 257-270.

Saldanha, L. A. & Thompson, P. W. (2006). Investigating statistical unusualness in the context of resampling: Students exploring connections between sampling distributions and statistical inference. In A. Rossman & B. Chance (Eds.), Working Cooperatively in Statistics Education. Proceedings of the Seventh International Conference on Teaching Statistics, Salvador, Brazil. [CDROM]. Voorburg, The Netherlands: International Statistical Institute.

[Online: http://www.stat.auckland.ac.nz/~iase/publications/17/6A3_SALD.pdf]

Schauble, L. (1990). Belief revision in children: The role of prior knowledge and strategies for generating evidence. Journal of Experimental Child Psychology, 49, 31-57.

Schauble, L. (1996). The development of scientific reasoning in knowledge-rich contexts. Developmental Psychology, 32, 102-119.

Schauble, L., & Glaser, R. (1990). Scientific thinking in children and adults. In D. Kuhn (Ed.), Developmental perspectives on teaching and learning thinking skills (pp. 9-27). Basel, Switzerland: Karger.

Schoenfeld, A. H. (1982). Measures of problem-solving performance and of problem-solving instruction. Journal for Research in Mathematics Education, 13(1), 31-49.

Schoenfeld, A. H. (1991). On mathematics as sense-making: An informal attack on the unfortunate divorce of formal and informal mathematics. In J. F. Voss, D. N. Perkins, and J. Segal (Eds.), Informal reasoning and education (pp. vii-xvii). Hillsdale, New Jersey: Lawrence Erlbaum Associates.

Schoenfeld, A. H., & Herrmann, D. J. (1982). Problem perception and knowledge structure in expert and novice mathematical problem solvers. Journal of Experimental Psychology: Learning, Memory, and Cognition, 8, 484-494.

Schwarz, D. L., Sears, D., & Chang, J. (2007). Reconsidering prior knowledge. In M. Lovett and P. Shah (Eds.), Proceedings of the 33^rd Carnegie Symposium on Cognition: Thinking with Data. Mahweh, NJ: Erlbaum.

Smith, J. P., diSessa, A. A., & Roshelle, J. (1993/1994). Misconceptions reconceived: A constructivist analysis of knowledge in transition. The Journal of the Learning Sciences, 3(2), 115-163.

Sotos, A. E. C., Vanhoof, S., Van den Noortgate, W., & Onghena, P. (2007). Students’ misconceptions of statistical inference: A review of the empirical evidence from research on statistics education. Educational Research Review, 2, 98-113.

Stohl, H. & Tarr, J. E. (2002). Developing notions of inference using probability simulation tools. Journal of Mathematical Behavior, 21, 319-337.

Tarr, J. E., Stohl Lee, H., & Rider, R. (2006). When data and chance collide: Drawing inferences from simulation data. In G. Burrill (Ed.), National Council of Teachers' of Mathematics' 2006 Yearbook: Reasoning with Data and Chance (pp. 139-150). Reston, VA: National Council of Teachers of Mathematics.

Thompson, P., Saldanha, L., & Liu, Y. (2004, April). Why statistical inference is hard to understand. Paper presented at the Annual Meeting of the American Educational Research Association, San Diego.

Toulmin, S. (1958). The uses of argument. Cambridge, UK: Cambridge University Press.

van Eemeren, F. H., Grootendorst, R., Henkemans, F. S., Blair, J. A., Johnson, R. H., Krabbe, E. C. W., et al. (1996). Fundamentals of argumentation theory: A handbook of historical backgrounds and contemporary developments. Mahweh, NJ: Erlbaum.

Voss, J. F., Blais, J., Means, M. L., ‌Greene, T. R., & Ahwesh, E. (1986).‌ Informal reasoning and subject matter knowledge in the solving of economics problems by naive and novice individuals. Cognition and Instruction, 3(3), 269-302.

Voss, J. F., Perkins, D. N., & Segal, J. W. (1991). Preface. In J. F. Voss, D. N. Perkins, and J. Segal (Eds.), Informal reasoning and education (pp. vii-xvii). Hillsdale, NJ: Lawrence Erlbaum Associates.

Walton, D. N. (1989). Informal logic: A handbook for critical argumentation. Cambridge, UK: Cambridge University Press.

Watson, J.M. (2004). Developing reasoning about samples. In D. Ben-Zvi & J. Garfield (Eds.), The Challenge of Developing Statistical Literacy, Reasoning, and Thinking (pp. 277–294). Dordrecht, The Netherlands: Kluwer Academic Publishers.

Watson, J. M., & Moritz, J. B. (1999). The beginnings of statistical inference: Comparing two data sets. Educational Studies in Mathematics, 37, 145-168.

Wild, C., & Pfannkuch, M. (1999). Statistical thinking in empirical enquiry. International Statistical Review, 67(3), 223-265.

Zieffler, A., delMas, R., Garfield, J., & Gould, R. (2007, August). Studying the development of college students’ reasoning about statistical inference. Paper presented at the Fifth International Research Forum on Statistical Reasoning, Thinking and Literacy (SRTL-5), University of Warwick, UK.

Andrew Zieffler

Educational Psychology

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