THE COMPLEXITY OF LEARNING TO REASON PROBABILISTICALLY

PME-NA XXII Working Group Organizer

Robert Spieser

Robert B. Davis Mathematics Education Institute

Rspeiser@e-mail.rci.rutgers.edu

Over the last several years, we have given serious attention to how students learn to reason probabilistically; that is, how learners build mathematical models, and how these models interrelate with each other and with data.Ê A special focus has been on how students build and work with information.Ê Some of work along these lines has been reported and discussed at Singapore (ICOTS-5, June 21-26, 1998), at PME-NA 20 (North Carolina State University, Raleigh, North Carolina, October 31öNovember 3, 1998), at the third Robert B. Davis (RBD) Working Conference (Snowbird, Utah, May 22-26, 1999) and at PME-NA 21 (Cuernavaca, Mexico, October 23-26, 1999).Ê Continuing discussion, investigation and collaboration draw on work at sites around the world.

Issues

At PMEöNA 20 (Raleigh, 1998), the Working Group began to formulate a joint agenda for research, discussion and investigation.Ê At Cuernavaca, the Working Group at PME-NA 21 (Cuernavaca) developed this agenda further, side by side with research presentations.Ê Central issues that the group discussed include:

(1) Study of how learners work with data, through analyses of learnersâ images, data representations, models, arguments and generalizations.Ê Attention to learnersâ success as well as learnersâ difficulties, in the unifying context of research on the development of learnersâ understanding.

(2) Attention to how models, reasoning and thinking function in communities of learners, teachers and researchers.Ê Examination of the roles of given tasks, of classroom environments, of student-teacher interactions, and of how learners, in a range of settings, share ideas, reasoning, and information.

(3) Emphasis upon development of mathematical ideas through time, with learners of different cultures, ages, social backgrounds, and with different prior mathematical and scientific experience.Ê Analysis of learnersâ and researchersâ changing views of underlying mathematical and scientific issues.

To help focus and develop this agenda, the Cuernavaca discussion took as starting points the interplay of combinatorial and probabilistic reasoning for constructing images and models in the course of task investigations.

Theoretical Framework

Recent research emphasizes the complexity and subtlety of probabilistic reasoning, even in very basic situations.Ê Models can extend distortions, even as they help support the growth of understanding.Ê Indeed, the variety of representations which learners find useful, and the complex relationships among the models learners build and data which they seek to explicate provide rich opportunities for research investigations focused centrally on sense and meaning.Ê Given its complexity, the development of probabilistic thinking entails building over time, in which earlier inquiries are revisited, reconsidered, extended and reformulated.Ê The tools available, the ways the tools are used, the ways in which ideas and information move among the learners, the teacherâs questions, ideas and interventions, all contribute (or fail to contribute) in important ways.Ê In our view, both research and teaching need to take the long-term building and the complexity into account.

Background

Related cross-cultural research on particular dice games, by researchers in several countries, using different methods of analysis across a range of settings and learner populations, was reported in joint sessions at the International Conference on the Teaching of Statistics (ICOTS-5, Singapore, June 21-26, 1998).Ê The Singapore reports (Amit, 1998; Fainguelernt & Frant, 1998; Maher, 1998; Speiser & Walter, 1998; Vidakovic, Berenson & Brandsma, 1998) helped motivate the work at Raleigh.Ê Further discussions at the third RBD Working Conference (Snowbird, Utah, June 1999) addressed important aspects of the Working Groupâs agenda in the context of the growth of understanding.

The present Working Group, first at Raleigh, then at Cuernavaca, built upon this shared research, enlisted new collaborators, and helped initiate further discussion.Ê An incomplete but perhaps somewhat representative list of active members of the Working Group would include Sylvia Alatorre and Araceli Limon Segovia, both from Mexico; and Alice Alston, Sally Berenson, George Bright, Susan Friel, Regina Kiczek, Clifford Konold, Carolyn A. Maher, Robert Speiser, Draga Vidakovic and Charles Walter from the United States.Ê Further colleagues, in several countries, are engaged in work related to the Groupâs agenda and concerns.

Plan for Involvement of Participants

At Raleigh, the Working Group considered data drawn from sixth-gradersâ work on two dice games (Maher, Speiser, Friel & Konold, 1998) which led to an extremely rich discussion.Ê Based on this experience, a list evolved now including four tasks that we invite participants at different sites to explore with diverse learner populations.Ê Here are current versions of these tasks.

A game for two players.Ê Roll one die.Ê If the die lands on 1, 2, 3 or 4, Player A gets one point (and Player B gets 0).Ê If the die lands on 5 or 6, Player B gets one point (and Player A gets 0).Ê Continue rolling the die.Ê The first player to get 10 points is the winner.Ê Is this game fair? Why or why not?

Another game for two players.Ê Roll two dice.Ê If the sum of the two is 2, 3, 4, 11 or 12, Player A gets one point (and Player B gets 0).Ê If the sum is 5, 6, 7, 8 or 9, Player B gets one point (and Player A gets 0).Ê Continue rolling the dice.Ê The first player to get 10 points is the winner.Ê Is this game fair? Why or why not?

The World Series Problem.Ê In a "world series" two teams play each other in at least four and at most seven games.Ê The first team to win four games is the winner of the "world series."Ê Assuming that both teams are equally matched, what is the probability that a "world series" will be won: (a) in four games? (b) in five games? (c) in six games? (d) in seven games?

The problem of points.Ê Pascal and Fermat, in correspondence, discuss a simple game.Ê They toss a coin.Ê If the coin comes up heads, Fermat receives a point.Ê If tails, Pascal receives a point.Ê The first player to receive four points wins the game.Ê Each player stakes fifty francs, so that the winner stands to gain one hundred francs, and then they play.Ê Suppose, however, that the players need to terminate the game before a winner is determined.Ê Further, suppose this happens at a moment when Fermat is ahead, two points to one.Ê In correspondence, Pascal and Fermat discuss the question:Ê How should the 100 francs be divided?

These tasks were developed by Carolyn A. Maher and her collaborators in the Rutgers-Kenilworth longitudinal study.Ê The first two tasks were developed for sixth-graders.Ê The last two tasks were developed later, initially for eleventh-graders.Ê Related research includes (Kiczek & Maher, 1998; Maher & Martino, 1997; Maher & Martino, 1996; Maher, Davis, & Alston, 1991; Maher & Speiser, 1997; Martino, 1992; Martino & Maher, 1999; Muter, 1999; Muter & Maher, 1998).

Parallel research on several of these tasks has taken place at several sites around the world.Ê Work in Brazil (Fainguelernt & Frant, 1998), in Israel (Amit, 1998), and in at least four places in the United States (Berenson, 1999), (Kiczek & Maher, 1998), (Maher, 1998), (Speiser & Walter, 1998), (Vidakovic, Berenson & Brandsma, 1998) has already been reported.Ê Closely related findings, including (Alatorre, 1999) and (Berenson, 1999), were discussed in detail by the Working Group at Cuernavaca.Ê

At the Tucson sessions of the Working Group, additional research will be presented and discussed, including resent work by G. Bright, S. Friel and F. Curcio, and suggestions for continued investigation, discusiion and collaboration will be invited.

Anticipated Follow-Up Activities

Collaborative work, based on case studies drawing on a focused set of tasks, and upon related research, from a variety of points of view, in different sites in several countries, has already helped to focus and extend discussion and collaboration.Ê Based on discussions to take place atÊ Tucson, further work with learners, in a range of settings, as well as further sharing and collaboration, will be initiated.Ê We cordially invite further participants to join a growing and productive enterprise.

Connections to the Goals of PME

This Working Group has emphasized research into the nature and development of probabilistic and statistical understanding, based on collaboration between researchers in several countries, focused by a shared, continually developing research agenda.Ê Recent work by members of the group, which draws on a rich background of psychological, pedagogical and mathematical ideas, has opened opportunities for further study and collaboration.Ê In all these ways, the Working Group supports the aims of PME.

References

Alatorre, S. (1999).Ê Adultsâ Intuitive Answers to Probability Problems: Methodology.Ê In F. Hitt & M. Santos (Eds.), Proceedings of the Twenty-First Annual Meeting of the North American Group for the Psychology of Mathematics Education, v. 2, (pp. 451-458). Columbus, Ohio: ERIC.

Berenson, S. (1999).Ê Studentsâ Representations and Trajectories of Probabilistic Thinking.Ê In F. Hitt & M. Santos (Eds.), Proceedings of the Twenty-First Annual Meeting of the North American Group for the Psychology of Mathematics Education, v. 2, (pp. 459-465).Ê Columbus, OH: ERIC.

Amit, M. (1998).Ê Learning Probability Concepts through Games.Ê In Pereira-Mendoza, Kea, Kee & Wong (Eds.) Proceedings of the International Conference on the Teaching of Statistics (ICOTS-5), v. 1, (pp. 45-48). Singapore.

Fainguelernt, E. & Frant, J. (1998).Ê The Emergence of Statistical Reasoning in Brazilian School Children.Ê In Pereira-Mendoza, Kea, Kee & Wong (Eds.) Proceedings of the International Conference on the Teaching of Statistics (ICOTS-5) v. 1, (pp. 49-52).Ê Singapore.

Kiczek, R. and Maher, C. A. (1998).Ê Tracing the origins and extensions of mathematical ideas.Ê In S. Berenson, K. Dawkins, M. Blanton, W. Columbe, J. Kolb, & K. Norwood (Eds.), Proceedings of the Twentieth Annual Meeting of the North American Group for the Psychology of Mathematics Education (pp. 377-382).Ê Columbus, OH: ERIC.

Maher, C.A. (1998).Ê Is This Game Fair? TheÊ Emergence of Statistical Reasoning in Young Children.Ê In Pereira-Mendoza, Kea, Kee & Wong (Eds.), Proceedings of the International Conference on the Teaching of Statistics (ICOTS-5) Singapore, v. 1, (pp. 53-60).

Maher, C. A., Davis, R. B., & Alston, A. (1991).Ê Implementing a thinking curriculum in mathematics.Ê Journal of Mathematical Behavior, 10, 219-224.

Maher, C. A. and Martino, A. M. (1997).Ê Conditions for conceptual change: From pattern recognition to theory posing.Ê In H. Mansfield & N. H. Pateman (Eds.), Young children and mathematics: Concepts and their representations.Ê Sydney, Australia: Australian Association of Mathematics Teachers.

Maher, C. A. and Martino, A. M. (1996).Ê The development of the idea of proof: A five year case study.Ê Journal for Research in Mathematics Education, 27(2), 194-219.

Maher, C. A. and Speiser, R. (1997).Ê How far can you go with block towers?Ê Journal of Mathematical Behavior, 16 (2), 125-132.

Maher, C.A., Speiser, R., Friel, S. and Konold, C. (1998).Ê Learning to Reason Probabilistically.Ê In S. Berenson, K. Dawkins, M. Blanton, W. Columbe, J. Kolb, & K. Norwood (Eds.), Proceedings of the Twentieth Annual Meeting of the North American Group for the Psychology of Mathematics Education, v. 1, (pp. 82-87).Ê Columbus, OH: ERIC.

Martino, A. M. (1992).Ê Elementary studentsâ construction of mathematical knowledge: Analysis by profile (Doctoral dissertation, Rutgers University, 1992)Ê Dissertation Abstracts International, 53, 1833a.

Martino, A. M. & Maher, C. A. (1999).Ê Teacher questioning to stimulate justification and generalization in mathematics: What research practice has taught us.Ê Journal of Mathematical Behavior, 18(1).

Muter, E. M. (1999).Ê The development of student ideas in combinatorics and proof: A six year study. Unpublished doctoral dissertation, Rutgers University.

Muter, E. M. and Maher, C. A. (1998).Ê Recognizing Isomorphism and Building Proof:Ê Revisiting Earlier Ideas.Ê In S. Berenson, K. Dawkins, M. Blanton, W. Columbe, J. Kolb, & K. Norwood (Eds.), Proceedings of the Twentieth Annual Meeting of the North American Group for the Psychology of Mathematics Education, vol. 1, (pp. 461-467).Ê Columbus, OH: ERIC.

Speiser, R. & Walter, C. (1998).Ê Two Dice, Two Sample Spaces.Ê In Pereira-Mendoza, Kea, Kee & Wong (Eds.) Proceedings of the International Conference On the Teaching of Statistics (ICOTS-5) v. 1, (pp.61-66). Singapore

Vidakovic, D., Berenson, S. & Brandsma, J. (1998).Ê Childrenâs Intuitions of Probabilistic Concepts Emerging from Fair Play.Ê In Pereira-Mendoza, Kea, Kee & Wong (Eds.), Proceedings of the International Conference On the Teaching of Statistics (ICOTS-5) v. 1, (pp. 67-73).Ê Singapore.