NSF DUE Collaborative Research: Adaptation & Implementation of Activity & Web-Based Materials Into Post-Calculus Introductory Probability and Statistics Courses Tracy Goodson-Espy M. Leigh Lunsford Ginger Holmes Rowell

NSF DUE A Collaborative Approach Athens State Univ. M. Leigh Lunsford Middle Tenn. St. Univ. Ginger Holmes Rowell Univ. of Alabama, Huntsville Tracy Goodson-Espy A&I Materials into Post Calculus Prob/Stat Courses Provide Objective Independent Assessment of A&I

NSF DUE Project* Objectives To improve post-calculus students learning of probability & statistics. To provide students with better preparation for their future careers in mathematics & statistics, mathematics education, and computer science. *This project is partially supported by the National Science Foundation. The project started in June, 2002 and continues through August, 2004.

NSF DUE The Materials for A&I “A Data-Oriented, Active Learning, Post- Calculus Introduction to Statistical Concepts, Methods, and Theory (SCMT)”“A Data-Oriented, Active Learning, Post- Calculus Introduction to Statistical Concepts, Methods, and Theory (SCMT)” A. Rossman, B. Chance, K. Ballman NSF DUE “Virtual Laboratories in Probability and Statistics (VLPS)”“Virtual Laboratories in Probability and Statistics (VLPS)” K. Siegrist NSF DUE

NSF DUE Statistical Concepts, Methods, and Theory (SCMT): A Small Sample of Materials ActivityContextConcepts Description Random- ization Test Friendly Observers Randomization, simulation, p-value Uses cards (23 per student) and Minitab to simulate a randomization test to estimate p- value from a 2x2 table for a psychology study. Equal Likeliness Random Babies Sample space, long-run relative frequency, random variable, expected value, simulation Uses index cards (4 per student) and Minitab to simulate the matching problem and develops probability calculations with equally likely outcomes. Fishers Exact Test Friendly Observers Counting rules, hypergeometric probabilities Exact probabilities for simulation in Randomization Test General vs. Specific The Birthday Problem Applications of counting techniques, complement rule Does calculations for the birthday problem (using a spreadsheet) contrasting any birthday vs. a specific birthday Prob. Rules 100 top films, 2000 Michigan primary Variety of basic probability rules Discovery approach through two-way tables and some Venn diagrams. HW is very interesting

NSF DUE Virtual Laboratories in Probability & Statistics: An Example Games of Chance Contents 1.PokerPoker 2.Poker Dice and Chuck-a-LuckPoker Dice and Chuck-a-Luck 3.CrapsCraps 4.IntroductionIntroduction 5.RouletteRoulette 6.The Monty Hall ProblemThe Monty Hall Problem 7.LotteriesLotteries 8.NotesNotes Applets Poker Experiment Poker Dice Experiment Chuck-a-Luck Experiment Craps Experiment Roulette Experiment Monty Hall Game Monty Hall Experiment Poker Experiment Applet

NSF DUE The A&I Cycle SCMT VLPS Materials Use Materials in Courses Evaluate A&I Refine Adaptation Provide Feedback to Material Developers Use Independent Assessment Results to Improve A&I Develop Best Practices for Use of Materials Co-jointly Receive Training on New Materials

NSF DUE Courses for A&I ASU: Applied Statistics & Probability I (3 hrs) Clientele: CS, MA, Math. Ed. Majors Prereq: Calculus II MTSU: Probability & Statistics (3 hrs) Data Analysis (1 hr) Clientele: CS, Math. Ed. Majors Prereq: Calculus I

NSF DUE Topics in ASU’s Course Topics Included: Random Experiments, Sample Spaces, Random Samples Basic Descriptive Statistics (Mean, Var., Std. Dev., Sample Mean and Var.) Probability Theory: Laws of Probability, Conditional Probability, Independence, Law of Total Probability, Bayes’ Theorem Discrete and Continuous Random Variables Uniform (discrete & continuous), Binomial, Hypergeometric, Geometric, Normal, pdf vs. cdf. Expected Value Central Limit Theorem Confidence Intervals for Means & Proportions Basic Concepts in Hypothesis Testing Topics Not Included: Multivariate Distributions Moment Generating Functions NOTE: Topics not covered linearly!

NSF DUE An Example Friendly Observers Experiment*: Researchers investigated a conjecture that having an observer with a vested interest would decrease subjects’ performance on a skill-based task. Subjects given time to practice task. Subjects randomly assigned to one of two groups: Group (A) was told that the participant and observer would each win $3 if the participant beat a certain threshold time. Group (B) was told only that the participant would win $3 if the threshold were beaten. Threshold chosen to be a time that participant beat in 30% of their practice turns. A: observer shares prizeB: no sharing of prizeTotal Beat threshold38 11 Do not beat threshold94 13 Total *Journal of Personality and Social Psychology (Butler and Baumeister, 1998)

NSF DUE An Example Week 1: Activity with the Friendly Observers Experiment Concepts: Randomness, random variable, empirical probability distribution, p-value. (Hypothesis testing) Uses a tactile simulation (in-class) and Minitab (assignment) to determine the empirical probability distribution for the random variable X (the number of winners assigned to group A by chance). Students start writing a report detailing their simulation results and their empirical estimate of the p-value. Week 3: Follow-up activity includes Friendly Observers Exper. Concepts: Hypergeometic probabilities and the hypergeometric distribution. Students apply theory to Friendly Observers Experiment above to compute the probability distribution of X and the actual p-value of the experiment. Uses the Ball and Urn Applet from the Virtual Labs. Students finish report.

NSF DUE Project Goals and Outcomes The development of post-calculus probability and statistics courses that produce well-educated students. The integration of technology and group-based activity work into the courses for the purpose of enhancing student learning. The enhancement of student communication skills through oral and written reports and presentations. The improvement and implementation of non-traditional assessment techniques for evaluating students. A contribution to the mathematics community discussion/research concerning what topics/materials/methods should be included in reform- oriented probability and statistics courses to improve overall student understanding of the subject.

NSF DUE Assessment of A&I of Materials Action Research ModelWill Use an Action Research Model* –What is the problem? I.e., what is not working in the classroom? –What technique can be used to address the learning problem? –What type of evidence can be gathered to show whether the implementation is effective? –What should be done next, based on what was learned? * R. delMas, J. Garfield, B. Chance

NSF DUE Teaching Experiment Cycle Teaching Hypotheses; Curricular & Instructional Choices Class Implementation & Feedback Instructors’ Reflections and Curricular Modifications

NSF DUE Preliminary Survey Students in the project classes were given mid-term and final Class Activities Surveys. These surveys consisted of three parts: Section One asked a series of questions concerning student’s beliefs concerning his/her understanding of specific mathematics concepts covered in the course such as sample space, conditional probability, independence of events, probability laws, etc. The student was asked to rate their understanding on a 1-5 scale(L-H). Section Two included a series of questions that asked students to rate the functioning of the class in terms of class dynamics, group dynamics, instructional strategies used, amount of technology used, and the effectiveness of the technology for conveying ideas. Section Three included open-ended questions that solicited student opinions. This survey was also given to pre-project classes in Spring 2002.

NSF DUE Further Data Acquisition During the spring term 2003, each project class will be observed repeatedly by the project evaluator. Individual videotaped teaching interviews will be conducted with selected students from each class and case studies will be developed from these interviews and the written artifacts of student work including, tests, homework, and reports.

NSF DUE Dissemination Presentations at Professional Conferences In-Service Training for High School Statistics Teachers (Spring 2004) Summer Workshop for College Faculty (Summer 2004) Papers in Mathematics Education Journals Project Website: