2013 Conference Proceedings - University of Nevada, Las Vegas

At the highest level of reasoning about the sampling methods, students understood theimportance of having a sample that is representative of the larger population. However,misconceptions of what makes a sample representative persisted. For example, some studentsperceived diversity in the sample as a sign that it is representative. Other students held the beliefthat as long as the entire population was available to be chosen for the sample, then the samplewas representative. Very few students recognized the need for all members of the population tohave an equal chance of being a part of the sample in order for the sample to be truly random (asimple random sample) with a high probability of being representative.A majority of students in this study showed progress toward deepening their statisticalreasoning from pre to post interview. This observed progress followed paths along the threetieredstructure described above, suggesting a hypothetical learning path (Clements & Sarama,2004) along which students may tend to progress as they develop the necessary understanding ofwhat it means to have a good sampling plan.ConclusionStudents can acknowledge the importance of random sampling in a statistical study yet havea limited understanding of what this means. Indeed, understanding can be confounded by the factthat it is often difficult or impossible to conduct true random samples of populations in specificcontexts (e.g. the common context of predicting elections). In the midst of this messiness, howcan teachers help students conceptualize appropriate sampling methods for a research study?This project sought to answer this question by developing KCS with regard to how studentsunderstand sampling methods ideas. The results of this study suggest a hypothetical learningtrajectory along which students may travel as they think through critical issues associated withchoosing representative random samples from populations.ReferencesBall, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makesit special? Journal of Teacher Education, 59(5), 389–407.Brown, J. S., & Burton, R. R. (1978). Diagnostic models for procedural bugs in basicmathematical skills. Cognitive Science, 2(2), 155–192. doi:10.1207/s15516709cog0202_4Burton, R. (1982). Diagnosing bugs in a simple procedural skill. In J. Brown (Ed.), IntelligentTutoring Systems. London: Academic Press, Inc.Alf, C., & Lohr, S. (2007). Sampling assumptions in introductory statistics classes. AmericanStatistician, 61(1), 71–77.Clements, D. H., & Sarama, J. (2004). Learning trajectories in mathematics education.Mathematical Thinking & Learning, 6(2), 81–89.Proceedings of the 40 th Annual Meeting of the Research Council on Mathematics Learning 2013 82