Making Sense of Probability Through Paradox and Programming: A Case Study in a Connected Mathematics Framework*

The disciplines of probability and statistics have fundamentally changed the way we do science and the way we think about our world. Many scholars have argued (e.g., Cohen, 1990; Gigerenzer, 1990; Hacking, 1990) that a probabilistic revolution has occurred in our century and that notions of randomness and uncertainty have opened up whole new areas of mathematics and science. This has released a ground swell of interest in subjects such as complexity, chaos, and artificial life. Statistical methods are ubiquitous in the scientific literature. Courses in probability and statistics are required for virtually all students in the natural and social sciences. Our daily newspapers are full of statistics about such matters as lung cancer risks, divorce rates, birth control failure rates, variation in temperature, the purity of soap, etc.

[1]  Frank Tsai,et al.  A method for evaluating the effectiveness of educational software , 1994 .

[2]  A. A..  Probability, Statistics and Truth , 1940, Nature.

[3]  W. Thurston.  On Proof and Progress in Mathematics , 1994, math/9404236.

[4]  D. Ball.  With an Eye on the Mathematical Horizon: Dilemmas of Teaching Elementary School Mathematics , 1993, The Elementary School Journal.

[5]  Stephen Jay Gould,et al.  The Streak of Streaks , 1989 .

[6]  Luis Rico Romero,et al.  Intuition in Science and Mathematics: an educational approach , 1988 .

[7]  Uri Leron,et al.  Structuring Mathematical Proofs. , 1983 .

[8]  Uri Leron,et al.  Of geometry, turtles, and groups , 1992 .

[9]  Mitchel Resnick,et al.  Beyond the Centralized Mindset , 1996 .

[10]  L. Marinoff,et al.  A Resolution of Bertrand's Paradox , 1994, Philosophy of Science.

[11]  Mitchel Resnick,et al.  Animal simulations with Logo: Massive parallelism for the masses , 1991 .

[12]  A. Tversky,et al.  On the psychology of prediction , 1973 .

[13]  C. Gilligan.  In a Different Voice: Psychological Theory and Women’s Development , 2009 .

[14]  S. Turkle,et al.  Epistemological Pluralism and the Revaluation of the Concrete. , 1992 .

[15]  V. Rich.  Personal communication , 1989, Nature.

[16]  A. Schoenfeld.  On mathematics as sense-making: An informal attack on the unfortunate divorce of formal and informal mathematics. , 1991 .

[17]  L. Ross,et al.  Human Inference: Strategies and Shortcomings of Social Judgment. , 1981 .

[18]  Preliminaries to any Theory of Representation , 2008 .

[19]  Seymour Papert,et al.  Software Design as a Learning Environment , 1990, Interact. Learn. Environ..

[20]  Harald Bergstriim.  Mathematical Theory of Probability and Statistics , 1966 .

[21]  Magdalene Lampert,et al.  When the Problem Is Not the Question and the Solution Is Not the Answer: Mathematical Knowing and Teaching , 1990 .

[22]  Uriel Joseph Wilensky,et al.  Connected mathematics : builiding concrete relationships with mathematical knowledge , 1993 .

[23]  A. Tversky,et al.  On the study of statistical intuitions , 1982, Cognition.

[24]  J. Jordan,et al.  Women's Growth In Connection: Writings from the Stone Center , 1991 .

[25]  Elliot Soloway,et al.  Should we teach students to program? , 1993, CACM.

[26]  A. Tversky,et al.  Extensional versus intuitive reasoning: the conjunction fallacy in probability judgment , 1983 .

[27]  E. Keller,et al.  A feeling for the organism : the life and work of Barbara McClintock , 1985 .

[28]  Israel Scheffler,et al.  In Praise of the Cognitive Emotions , 1977, Teachers College Record: The Voice of Scholarship in Education.

[29]  J. S. Evans,et al.  Bias and rationality. , 1993 .

[30]  Uri Wilensky.  Learning Probability through Building Computational Models , 1995 .

[31]  James G. Greeno,et al.  Number sense as situated knowing in a conceptual domain , 1990 .

[32]  The probabilistic revolution in psychology: An overview , 1987 .

[33]  U. Wilensky.  Abstract Meditations on the Concrete and Concrete Implications for Mathematics Education , 1991 .

[34]  Daniel Kahneman,et al.  Availability: A heuristic for judging frequency and probability , 1973 .

[35]  Gregory J. Chaitin,et al.  Information, Randomness and Incompleteness - Papers on Algorithmic Information Theory; 2nd Edition , 1987, World Scientific Series in Computer Science.

[36]  Kenneth Tobin,et al.  The Practice of constructivism in science education , 1993 .

[37]  D. Krantz,et al.  The use of statistical heuristics in everyday inductive reasoning , 1983 .

[38]  A. N. Kolmogorov,et al.  Foundations of the theory of probability , 1960 .

[39]  J. Surrey.  Working Paper Series Working Paper Series Work In Progress Relationship and Empowerment , 2003 .

[40]  L. Cohen,et al.  On the psychology of prediction: Whose is the fallacy? , 1979, Cognition.

[41]  C. Hirsch.  Curriculum and Evaluation Standards for School Mathematics , 1988 .

[42]  I. Lakatos.  PROOFS AND REFUTATIONS (I)*† , 1963, The British Journal for the Philosophy of Science.

[43]  Richard J. Cleary,et al.  How to Think about Statistics , 2000 .

[44]  Mitchel Resnick,et al.  Beyond the centralized mindset-explorations in massively-parallel microworlds , 1992 .

[45]  A. Tversky,et al.  Causal Schemata in Judgments under Uncertainty , 1982 .

[46]  Uri Wilensky,et al.  What is Normal Anyway? Therapy for Epistemological Anxiety , 1997 .

[47]  Daniel Kahneman,et al.  Judgment and Decision Making: A Personal View , 1991 .

[48]  Laurie D. Edwards,et al.  Microworlds as Representations , 1995 .

[49]  J. Uspensky,et al.  Introduction to Mathematical Probability , 1938, Nature.

[50]  Leonard Gillman,et al.  The car and the goats , 1992 .

[51]  Seymour Papert,et al.  Teaching Children to be Mathematicians vs. Teaching About Mathematics. Artificial Intelligence Memo Number 249. , 1971 .

[52]  A. Tversky,et al.  Judgment under Uncertainty: Heuristics and Biases , 1974, Science.

[53]  Clifford Konold,et al.  Understanding Students’ Beliefs About Probability , 1991 .

[54]  Harold Abelson,et al.  Turtle geometry : the computer as a medium for exploring mathematics , 1983 .

[55]  Patrick Love,et al.  Women's Ways of Knowing. , 1999 .

[56]  C. Geertz,et al.  The Interpretation of Cultures , 1973 .

[57]  Seymour Papert,et al.  Mindstorms: Children, Computers, and Powerful Ideas , 1981 .

[58]  P. L. Adams.  THE ORIGINS OF INTELLIGENCE IN CHILDREN , 1976 .

[59]  Judah L. Schwartz,et al.  The Geometric Supposer: An Intellectual Prosthesis for Making Conjectures. , 1987 .