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«Birgit Christina Aquilonius - How Do College Students Reason About Hypothesis Testing in Introductory Statistics Courses? UNIVERSITY OF CALIFORNIA ...»

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Birgit Christina Aquilonius - How Do College Students Reason About Hypothesis Testing in Introductory Statistics Courses?

UNIVERSITY OF CALIFORNIA

Santa Barbara

How Do College Students Reason About Hypothesis Testing in

Introductory Statistics Courses?

A Dissertation submitted in partial satisfaction of the requirement for

the degree of Doctor of Philosophy

in Education

by

Birgit Christina Aquilonius

Committee in charge:

Professor Mary E. Brenner, Chair Professor Richard Durán Professor Yukari Okamoto March 2005 The dissertation of Birgit Christina Aquilonius is approved Richard Durán Yukari Okamoto ___________________________________________________________

Mary E. Brenner, Committee Chair January 2005 How Do College Students Reason About Hypothesis Testing in Introductory Statistics Courses?

Copyright © 2005 by Birgit Christina Aquilonius iii

ACKNOWLEDGEMENTS

First I want to thank my advisor, Professor Betsy Brenner, for all her support and expert advice. She gave me great direction without restraining me from what I wanted to do. She also helped me with language at some critical junctions. I also want to thank Professor Richard Durán for encouraging me to apply to the UCSB School of Education and for letting me have a head start on my dissertation in his cognition class. I am grateful to Professor Yukari Okamoto for her detailed suggestions at the dissertation proposal meeting, particularly regarding probability education literature.

The professional discussions about statistics education with colleagues at West Valley College, particularly Jim Wilczak, were invaluable for this dissertation.

Thank you, Alyson Clark, Cathy van Hook, Jim Wilczak and Rebecca Wong for letting me recruit participants for my study from your classes. Thank you Wade Ellis for your long-time support and for introducing me to graphing calculators when they first appeared in the educational landscape.

My year with the Functions Group at UC Berkeley's School of Education opened my eyes to the possibilities in analyzing student conversations. Thank you Professor Rogers Hall, Dr. Cathy Kessel, and Professor Alan Schoenfeld for your full inclusion of me in the EMST program at UC Berkeley during 1993-1994.

Thank you Lee Kanner, for all your support, good advice and also for reading and commenting on my work. Your belief in my work was very important to me.

Thank you Lee Knutsen for your thorough proofreading of my manuscripts and for your friendship.

My most heartfelt thanks go to my husband Lasse Bergman. His constant support and our frequent discussions about my work made all the difference. Thanks also to our sons Thomas and Ted for their interest, support and inspiration.

–  –  –

EDUCATION Physical Therapy Diploma, Karolinska Institutet, Stockholm, Sweden, June 1970 Bachelor of Arts in Mathematics, University of California, Santa Cruz, June 1979 Master of Arts in Mathematics, University of California, Santa Cruz, June 1981 Master of Arts in Mathematics, University of California, Santa Barbara, June 2001 Master of Arts in Education, University of California, Santa Barbara, March 2002 Doctor of Philosophy in Education, University of California, Santa Barbara, March 2005 (expected)

PROFESSIONAL EMPLOYMENT

1970-72 Physical Therapist, Jönköping County Hospital, Sweden 1980-82 Teaching Assistant, Mathematics Department, UC Santa Cruz 1982 Lecturer in Statistics, Summer Session, UC Santa Cruz 1982-83 Mathematics Instructor, Cabrillo College, Aptos and Monterey Peninsula College (part-time) 1984-present Mathematics Instructor (full-time) West Valley College, Saratoga 1995-97 Teaching Assistant/ Associate in Mathematics, UC Santa Barbara

PUBLICATION

"Students' Peer Discussions of Statistics. How Do Students Learn From Them?" Unpublished Master's project submitted in partial fulfillment of the requirements for the Master of Arts degree in Education, University of California, Santa Barbara, 2002

PRESENTATION TO PROFESSIONAL ORGANIZATION

"Students Talk About Hypothesis Testing." Presentation at the California Mathematics Council, Community Colleges Annual Meeting, Monterey, December 2004.

AWARDS Alpha Gamma Sigma Teacher Award, West Valley College, Saratoga, 1990, 1991 and 1992 Office of the President Community College Research Assistantship, UCSB, 2000-01

–  –  –

Many college students are required to take statistics courses for their majors.

Hypothesis testing is often taught as the last part of such a course and in a sense becomes the goal of the course. Statistics instructors receive mixed messages about their students' understanding of hypothesis testing. The students in their classes sometimes say or do things that make instructors believe that students have good understanding of the topic. At other times, the same students make mistakes on tests and homework that make the instructor doubt their understanding. In this study, present technology allowed me to go one layer below what can be observed in the classroom. By videotaping students' statistical conversations and viewing them on DVDs, time after time, I was able to analyze students' reasoning at more depth and observe more closely what students understand and do not understand. Two statistics instructors and eight pairs of community college students were asked to solve hypothesis test problems and answer questions about their work.





Regarding sample and population students were able to reason competently in general terms. They knew why one takes samples and about the importance of unbiased samples. They did not realize the mathematical character of random

–  –  –

reasoning they did not exhibit understanding of the qualitative difference between sample mean and population mean which is inherent in the theory of hypothesis testing.

Students' approach to p-values in hypothesis testing was procedural. They considered p-values as something that one compares to alpha-values in order to arrive at an answer. Students did not attach much meaning to p-values as an independent concept. Therefore it is not surprising that though their p-values gave them valid statistical conclusions, they often were puzzled over how to translate the statistical answer to an answer of the question asked in the problem. Their textbooks and instructors gave students scripts to help them formulate their answers. Those scripts were helpful to some students but did not always lead them to the right answer.

–  –  –

The introductory statistics course, usually called Elementary Statistics, has become a pivotal course in many community college students lives. Most students enter the community college with the goal of transferring to a four-year college or university, and many departments at those educational institutions require that transfer students take a statistics course prior to upper division entry. Even when a student s transfer department does not require statistics, the transfer student often uses statistics to satisfy the four-year institution s quantitative reasoning requirement.

Because of the increasing student demand for statistics courses, those courses are now one of the most ubiquitous mathematics offerings at the community college level.

Hypothesis testing is usually taught as the last third of the Elementary Statistics course. Hypothesis testing therefore in some sense becomes the goal for the course. The central place that hypothesis testing has in the statistics courses makes sense if you consider how students might use their knowledge later in their academic career or as informed citizens.

(1) They might read research reports in their upper division classes, for example in sociology or psychology, and need to interpret the results or read the results with a critical stance.

(2) They will read reports in newspapers or magazines, or hear reports on radio and TV, and need to interpret the results or to read those results with a critical stance.

(3) They will take a more advanced statistics course for which the Elementary Statistics course will serve as a foundation.

(4) They will carry out some small study as part of their upper division or graduate work.

For each of those purposes the students need to have a rudimentary understanding of inferential statistics. Competence in carrying out computational procedures is not going to be sufficient. Recognizing the student need to understand inferential statistics, most introductory statistics books end with a treatment of hypothesis tests. Hypothesis tests naturally become the goal of the Elementary Statistics course, and research on student understanding of hypothesis tests becomes an important research subject in statistics education.

1.1 Overview of Problem Two statements, when juxtaposed, coming from the two major review articles on statistics learning research, put the spotlight on the situation of the Elementary Statistics student: Garfield and Ahlgren (1988) write, Over the past 20 years, most of the literature on teaching stochastics has been at the college level. This literature has been filled with comments by instructors about students not attaining an adequate understanding of basic statistical concepts and not being able to solve applied statistical problems (Duchastel, 1974; Joliff, 1976; Kalton, 1973; Urcuhart, 1971). The experience of most members in education and the social sciences is that a large proportion of university students in introductory statistics courses do not understand many of the concepts they are studying (p. 46).

and Shaughnessy (1992) writes, Most of the courses in probability and statistics that are offered at the university level continue to be either rule bound recipe-type courses for calculating statistics, or overly mathematized introductions to statistical probability that were the norm a decade ago. Thus, college level students with all their prior beliefs and conceptual misunderstanding about stochastics, rarely get the opportunity to improve their statistical intuition or to see the applicability of the subject as undergraduates. University courses may, therefore only make a bad situation worse, by masking conceptual and psychological complexities of the subject (p. 466).

Thus, on the one hand, statistics instructors have for a long time complained that students do not understand the deeper meaning of statistics. On the other hand, available curriculum materials have encouraged teaching introductory statistics courses in a way that prevents such deeper understanding to develop.

Some recent developments might open the way for better teaching and learning of introductory statistics courses such as Elementary Statistics. The use of handheld calculators such as the TI-83, with its built-in statistical functions, is dramatically reducing the time students need to spend on routine computations. The TI-83 calculator will, for example, compute the p-value for all the hypothesis tests normally taught in an introductory statistics course.

The use of hands-on simulations and more frequent use of real life data to build statistical and probabilistic intuition are other important new directions in introductory statistics curriculum development. Workshop Statistics (Rossman, Chance & von Oehsen, 2002) by Key Curriculum Press provides an example of using hands-on simulations and real data to build students statistical understanding. Some students participating in my Master s project were taught from the Minitab version of Workshop Statistics (Rossman & Chance, 2000). I later taught from the TI-83 version of Workshop Statistics (Rossman et al., 2002) during two subsequent semesters in my Elementary Statistics courses at a community college. Then I found that students in Workshop Statistics classes were more at ease when talking about statistical concepts than were students, whom I had taught from traditional textbooks. The Workshop Statistics curriculum encourages writing directly in the textbook, which might help student verbalize their statistical work.

Observing students writing and talking about their statistical work made it more transparent to me that the majority of introductory statistics students are in the process of gaining statistical understanding. In each class, there might be a few students who do not at all understand, for example, hypothesis testing. There also might be one or more students, who have a solid understanding of the concept.

However, most students can be found on a continuum between those two extremes.

Students who have listened to the same lectures, read the same textbook and been assigned the same homework, have widely varying understanding of the material covered in those lectures and in that textbook. Moreover, the same student can, even within the confines of one test, show a varying degree of understanding. However, there is very little research that details what introductory statistics students understand and do not understand. This study will contribute to building a knowledge base of introductory statistics students' understanding. If statistics teaching is to improve, more knowledge is needed about how students think and reason about concepts such as hypothesis testing at different stages of their learning process.

1.2 Research Questions For my Master s project (Aquilonius, 2002), which also served as a pilot study for my dissertation, I videotaped pairs of community college students solving statistical problems. The focus of that study was on peer interactions in statistics problem solving. The mechanisms through which stronger and weaker students supported each other were analyzed. Though the purpose of the Master s project was to study peer interactions, its corpus of data also raised issues about student reasoning about hypothesis testing, a topic that I already was considering as a dissertation topic.

When searching data bases such as ERIC and Psychinfo, I discovered that very little research had been done concerning students' reasoning about hypothesis testing. The only article, which directly treated the subject, was Falk's (1986) article about misconceptions of statistical significance. So, there was a lack of knowledge regarding students' reasoning about hypothesis testing. Thus there was a need for a study like mine that more comprehensively looked at students' reasoning about hypothesis testing.



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