Designing Effective Lessons on Probability: A Pilot Study Focused on the Illusion of Linearity Jean-Marc Miszaniec A Thesis in the Department of Mathematics and Statistics Presented in Partial Fulfillment of the Requirements For the degree of Master in the Teaching of Mathematics at Concordia University Montreal, Quebec, Canada APRIL 2016 © Jean-Marc Miszaniec i CONCORDIA UNIVERSITY School of Graduate Studies This is to certify that the thesis prepared By: Jean-Marc Miszaniec Entitled: Designing effective lessons on probability- A pilot study on the illusion of linearity and submitted in partial fulfillment of the requirements for the degree of Master in Teaching of Mathematics complies with the regulations of the University and meets the accepted standards with respect to originality and quality. Signed by the final Examining Committee: Dr. Nadia Hardy Chair Chair’s name Dr. Fred Szabo Examiner Examiner’s name Dr. Nadia Hardy Examiner Examiner’s name Dr. Anna Sierpinska Supervisor Supervisor’s name Approved by _______________________Dr. Nadia Hardy____________________________ Chair of Department or Graduate Program Director April 29th 2016 _____Dr. Anna Sierpiska_____ Dean of Faculty ii Abstract Designing Effective Lessons on Probability: A Pilot Study Focused on the Illusion of Linearity Jean-Marc Miszaniec Concordia University, 2016 This thesis is a summary of a pilot study adopting the design experiment methodology investigating alternative approaches to the teaching of probability – the classical approach (called “Laplace approach” in our study) and the axiomatic approach (called “Properties approach”) – and alternative pedagogical treatments of these approaches: with or without explicit refutations of common misconceptions by the teacher. Four short video-lessons were designed, focused on Bernoulli trials: “Laplace-exposition”; “Laplace-refutation”; “Properties-exposition” and “Properties-refutation “. The primary misconception addressed in the study was the erroneous application of proportionality models: the misconception called “illusion of linearity” in the context of probability. The design and layout of the videos’ visuals and audio were informed by the principles of cognitive load theory in multimedia to ensure optimal learning potential. Each lesson was tried with one volunteer participant, a post-college student. Each participant listened to the video-lesson assigned to him or her, solved six related probability questions, responded to a questionnaire and was interviewed one-on-one by the researcher. Data in the form of participants’ written responses, transcripts of the interviews and researcher’s notes were used for a thorough qualitative analysis and to inform future iterations of the study. Research suggests that the primary obstacles to probability education are misconceptions originating from cognitive biases and limitations. Often misconceptions in probability are attributed to intuitions. Intuitions are easily accessed and experience-based knowledge that can help or hinder individuals while problem solving. One question of this study was whether stating and refuting known misconceptions during a lecture-style lesson on Bernoulli trials (refutation treatment) promotes learning better than a lesson without the mention of misconceptions (exposition treatment). This study also intended to illuminate the nature of probability misconceptions, as research suggests that attributing them to “intuitions” may be misleading. “Intuition”, especially, “robust intuition”, usually refers to a connected set of beliefs, but our observations suggest that students’ predictions about probability of events are based on something much less homogeneous and connected. Participants in the study did not have consistent and robust intuitions about Bernoulli trials but rather a weaker form of basic knowledge called phenomenological primitives. Their intuitions would have to be iii developed, perhaps through a frequentist approach to probability. Participants had a high distrust in their initial guesses – a symptom of the weakness or lack of “robust intuition” – and this played an important role throughout the problem solving process. The Properties approach seemed to have more success than the Laplace approach because of its computational simplicity and because participants may have naturally implemented the Properties approach when thinking of chance events. The approaches did not seem to alter the use of misconceptions. The refutation treatment did not have the expected outcome on learning. Participants subjected to it performed better than those subjected to the exposition treatment but this could be because they “knew” (were told) – not necessarily “understood” – that changes to the number of trials and the number of desired events affect the probability of the desired event in a non-proportional manner. Future iterations of this study would explore further the above conjectures about the nature of the basis of students’ predictions about probability and the appropriateness of different approaches to probability to build on this basis. Also, other common probability misconceptions would be explored to determine their cognitive origins in order to inform the design of instructional content. iv Acknowledgements This thesis would not have been possible without the tremendous amount of effort and insight of Dr. Anna Sierpinska. I am thankful to have been her student, and I am grateful that she has kept me inspired and curious. v Table of Contents 1 Introduction .................................................................................................................................... 1 2 Sources of difficulties in the learning of probability .......................................................................... 5 2.1 Epistemological obstacles in the historical development of probability theory ......................... 6 2.1.1 THE OBSTACLE OF METAPHYSICAL DETERMINISM – GOD DOES NOT PLAY DICE ................ 6 2.1.2 THE OBSTACLE OF PREDICTABILITY – PROBABILITY THEORY DOES NOT REMOVE UNCERTAINTY ABOUT FUTURE EVENTS ............................................................................................ 7 2.1.3 THE OBSTACLE OF PROPORTIONALITY .............................................................................. 9 2.1.4 THE OBSTACLE OF PRESUMED EQUIPROBABILITY – ELEMENTARY EVENTS ARE PROBABLY EQUIPROBABLE .............................................................................................................................. 16 2.1.5 THE OBSTACLE OF INSUFFICIENT SPECIFICATION ............................................................ 21 2.2 Cognitive obstacles in the learning of probability theory ........................................................ 22 2.2.1 Obstacle of Prediction Schemas and Intuitions ............................................................... 23 2.2.2 Obstacle of Fixation ........................................................................................................ 27 2.2.1 Obstacle of Confirmation Bias in Learning ....................................................................... 28 2.2.2 Obstacle of the Unknown Nature of Probabilistic Misconceptions .................................. 29 2.2.3 Obstacle of Limited Cognitive Load ................................................................................. 30 2.3 Didactic obstacles in the learning of probability theory .......................................................... 32 2.3.1 Obstacle of Probability’s Assumed Didactic Structure ..................................................... 32 2.3.2 Obstacle of Ill-defined Basic Terminology ....................................................................... 36 3 Approaches to probability .............................................................................................................. 39 3.1 Classical (Laplace, Theoretical) Interpretation ........................................................................ 39 3.2 Frequentist (Empirical) Interpretation .................................................................................... 40 3.3 Subjective Interpretation ....................................................................................................... 40 3.4 Structural Interpretation (Properties) ..................................................................................... 41 vi 4 The Design of a teaching experiment ............................................................................................. 43 4.1 Objective of Study .................................................................................................................. 43 4.2 Material Content of Video Lessons ......................................................................................... 43 4.2.1 Layout of Lesson Material ............................................................................................... 44 4.2.2 Design of Script and Video .............................................................................................. 45 4.2.3 Measuring mental effort invested and confidence rating ................................................ 46 4.3 Misconceptions addressed in the “Refutation” treatments .................................................... 46 4.3.1 List of misconceptions and their refutations in the Laplace approach “Refutation” version 46 4.3.2 List of misconceptions and their refutations in the Properties approach “Refutation” version 48 4.4 Questions to test students’ understanding of