Using the Monte Carlo Method to Teach Probabilistic Problem Solving to Ninth Grade General Mathematics Students

564 p.Thesis (Educat.D.)--University of Illinois at Urbana-Champaign, 1980.For over fifty years, mathematical curriculum reform studies throughout the United States (and other countries) have advocated the inclusion of topics from probability and statistics within the secondary school mathematics curriculum. The advanced nature of the mathematics required for conventional treatment of these topics, however, has generally forced enrollment in such courses to be restricted to a few mathematically talented students at the upper levels of secondary education.In the present study, a unit on probability was developed and tested which was suitable for use with secondary school students of limited mathematical skills and aptitudes. In lieu of traditional analytic methods, a simulation-based approach using Monte Carlo techniques was taught for solving probabilistic problems. Random data for problem solving were generated by students using common hands-on manipulative devices (such as coins, dice, and playing cards), with individualized computer-generated sets of data available when needed. Data for many trials, conducted by all students in a class, were combined to produce more reliable estimates of problem solutions.The experimental unit was taught to four ninth grade general mathematics classes in a suburban Chicago public high school, with two additional classes serving as a control group. Two of the experimental classes were taught by the experimenter, and the other two were taught by their regular classroom teacher (who was a novice in the use of simulation techniques). Measures of problem-solving skills and attitudes toward mathematics were obtained at three time levels--prior to instruction, at the end of the experimental unit, and at the end of the school year (approximately twelve weeks after the conclusion of the experimental instruction).It was found that students studying the experimental unit did learn significantly more about probability than did those in the control sections (who studied a traditional general mathematics unit on consumer education), confirming that students of below-average ability can succeed in the study of probability. It was further noted that the majority of the students in the experimental classes attained a 'passing' score (70% or greater) on the final exam administered at the conclusion of the unit. It was thus determined that the study of probability using simulation techniques is pedagogically appropriate for inclusion in the mathematics curriculum taught to students of below-average achievement. It was also found that specialized teacher training or advanced study in probability was not necessary in order to teach the concepts of this unit successfully.The experimental unit was found not to differ from the control unit in developing computational skills in the use of common and decimal fractions, in increasing the students' willingness to use common fractions, or in developing more favorable attitudes toward mathematics. It was thus shown that arguments to include the study of probability in the curriculum must rest upon the pedagogical value of the subject matter, and not on claims of possible effects upon attitudes or concomitant skills.Based upon the findings of the study and an extensive review of the literature dealing with the teaching of probability and statistics in American and European schools, as well as with earlier uses of the Monte Carlo method in the teaching of statistics, chemistry, and physics, recommendations have been made concerning the development and use of Monte Carlo-based probability and statistics units throughout the elementary and secondary curricula.U of I OnlyRestricted to the U of I community idenfinitely during batch ingest of legacy ETD