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Association: Name: North American Chapter of the International Group for the Psychology of Mathematics Education URL: http://www.pmena.org
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Groth, Randall. "High School Students’ Levels of Thinking in Regard to Analyzing Univariate Data Sets" Paper presented at the annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Delta Chelsea Hotel, Toronto, Ontario, Canada, Oct 21, 2004 <Not Available>. 2009-05-26 <http://www.allacademic.com/meta/p117522_index.html>

Groth, R. E. , 2004-10-21 "High School Students’ Levels of Thinking in Regard to Analyzing Univariate Data Sets" Paper presented at the annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Delta Chelsea Hotel, Toronto, Ontario, Canada Online <.PDF>. 2009-05-26 from http://www.allacademic.com/meta/p117522_index.html

Publication Type: Conference Paper/Unpublished Manuscript Review Method: Peer Reviewed Abstract: Purpose of the Study The study of statistics is gaining momentum at the high school level. The College Entrance Examination Board has established an Advanced Placement Statistics course for high school that is growing rapidly in terms of numbers of students (College Entrance Examination Board, 2003). The National Science Foundation has funded the development of curricula that integrate data analysis with the teaching of topics in mathematics (Hirsch, Coxford, Fey, & Schoen, 1998; Alper, Fraser, Fendel, & Resek, 1998). As initiatives such as these become more widespread, it is incumbent upon the research community to continue to investigate and describe the statistical thinking of high school students, since knowledge of students’ thinking can help inform instruction (Fennema & Franke, 1992; Even & Tirosh, 2002). The present paper focuses upon describing high school students’ levels of thinking in regard to analyzing univariate data sets. The paper is a product of a larger study (Groth, 2003) which investigated high school students’ levels of thinking across several statistical thinking processes. The present paper addresses the following two research questions: (1)What levels of thinking can be identified in regard to analyzing the impact of increasing each value in a univariate data set by a constant? (2) What levels of thinking can be identified in regard to comparing sets of univariate data? Theoretical Perspective The present study made use of the SOLO Taxonomy of Biggs and Collis (1982, 1991) in order to differentiate among levels of statistical thinking. The Biggs and Collis theoretical perspective has been employed in describing levels of statistical thinking in several other studies (e.g., Watson, Collis, Callingham, & Moritz, 1995; Watson & Moritz, 1999; Jones et al, 2000; Mooney, 2002). Some of the studies contain descriptions of elementary and middle school students’ thinking in regard to analyzing sets of univariate data (Mooney, 2002; Jones et al, 2000; Watson & Moritz, 1999). The use of the SOLO Taxonomy helped to situate the results of the present study, which focused on high school age students, within this existing body of literature. Biggs (1999) described how the SOLO Taxonomy can be used to categorize responses to academic tasks. There are five levels in the taxonomy: prestructural, unistructural, multistructural, relational, and extended abstract. Prestructural responses show little evidence of learning relevant to the task at hand. Unistructural responses focus upon just one relevant aspect involved in completing a task. Multistructural responses incorporate more than one relevant aspect, but there is no unifying theme for the aspects. At the relational level, a unifying theme is apparent along with multiple relevant aspects. Responses at the extended abstract level are “breakthrough” responses that are not just coherent applications of academic learning, but go beyond the task at hand to apply the coherent whole to new areas. The middle three levels in SOLO (unistructural, multistructural, and relational) helped differentiate among levels of response in the present study. Methodology Participants Purposeful sampling (Patton, 1990) was used in the selection of study participants. In an attempt to observe a number of different patterns of thinking, participants were chosen on the basis of the mathematics courses they had taken while in high school. A total of 15 students were chosen for the study. Three had recently completed a semester-long high school statistics course, four had recently completed a year-long high school statistics course, one was currently enrolled in a year-long high school statistics course, and the remainder were high school students who had not yet taken a course focused solely upon statistics. The study participants were all volunteers. Procedure and instruments An interview protocol (Groth, 2003) designed to elicit statistical thinking was administered to each of the study participants. It took a cumulative total of 2-3 hours to administer the entire protocol to each of the students. This paper reports upon the patterns and levels of thinking that were evident in the responses to two parts of the interview protocol. One of the two parts asked students to determine what would happen to the center and spread of a given univariate data set if each of the values in the data set was increased by a given amount. The second of the two parts asked students to compare sets of univariate data. Each of the tasks was set in the context of test score results from students at a hypothetical school. Data Gathering and Analysis Data Gathering Data for the study were gathered during the clinical interviews. As students were interviewed, their responses were audiotaped, and the author took observational notes. The audiotapes were later transcribed for analysis. Any written work completed by the students during the interviews was also kept for analysis. Data Analysis Data analysis was advised by the constant comparative method described by Maykut and Morhouse (1994). One of the defining features of the constant comparative method is that data are analyzed as they are collected. Hence, as students’ interview responses were gathered, they were grouped into categories on the basis of the number of relevant attributes included in each response and whether or not connections were made among the relevant attributes incorporated. After categories of response had been formed by the author, they were compared against the categories formed by another researcher who had analyzed a random sample of the data. The categorizations reported upon in this paper were influenced by the discussions with the researcher who conducted the separate analysis. Results and Conclusions Different levels of thinking were discerned in the responses to the interview questions. There were three levels of thinking apparent for the impact of data transformation upon the center and spread of a univariate data set. Likewise, there were three levels of thinking apparent in the comparison of univariate data sets. Levels of thinking in regard to the impact of data transformation upon center and spread The first level of thinking in regard to the impact of data transformation upon center and spread of a univariate data set was that at which students were able to describe the impact on only one of the two measures: either center or spread. Some students responding at the first level did not understand that the spread of a univariate data set remains unchanged after each value of the set is increased by the same amount. Others at the first level recognized that the spread was unchanged, but did not realize that the center would be changed. This first level of thinking was considered unistructural, since the impact upon only one of the measures, either center or spread, was recognized. The second level of thinking about impact upon center and spread was that at which students recognized that the spread remained unchanged and the center increased. Responses fitting this level, however, did not quantify the change upon the center of the data set. They simply included a statement that the measure of center would increase. Since the impact on more than one measure was understood, the responses were considered multistructural. The responses were not considered relational because of the fact that the impact on center was left unquantified. The third and final level of thinking about impact upon center and spread was that at which it was recognized that the spread would remain unchanged after data transformation, and the amount by which the center would increased was specified. Since the impact on both relevant measures of spread and center was recognized, and the impact upon center was quantified, the third level of response was considered relational. Levels of thinking in regard to comparing data sets Like the first level of thinking in regard to impact on center and spread, the first level of thinking for comparing data sets was considered unistructural in nature. Responses at this level incorporated only one strategy, that of point-by-point comparisons, for comparing the given sets of data. Responses categorized at the second level of thinking were multistructural in nature. At this level, point-by-point strategies were not the only ones used. They were supplemented by the use of at least one relevant attribute of the aggregate data sets, such as shape, center, or spread. At this level, comparisons based on multiple attributes of the aggregate data sets were not yet incorporated. Responses categorized at the highest level of thinking for comparing univariate data sets were those which made comparisons based on multiple relevant attributes of the data sets. Students demonstrated the ability to compare data sets based on at least two of the following: shape, center, and spread. Since the view of data sets as aggregates rather than just collections of individual points seemed to be fully developed in responses at the third level, they were considered relational in nature. Conclusion The present study holds implications for both teaching practice and further research. High school teachers need to be aware of the fact that some students do not demonstrate the ability to view data sets as aggregates, and that instruction needs to be adjusted accordingly. Directions for further research suggested by the study include follow-up studies where the impact of context upon levels of response is investigated, and also studies in which the impact of different instructional techniques upon levels of response is described. References Alper, L., Fraser, S., Fendel, D., & Resek, D. (1998). Interactive mathematics program. Emeryville, CA: Key Curriculum Press. Biggs, J.B. (1999). Teaching for quality learning at university. Philadelphia: Open University Press. Biggs, J.B., & Collis, K.F. (1982). Evaluating the quality of learning: The SOLO taxonomy. New York: Academic. Biggs, J.B. & Collis, K.F. (1991). Multimodal learning and quality of intelligent behavior. In H.A.H. Rowe (Ed.), Intelligence: Reconceptualization and measurement (pp. 57-66). Hillsdale, NJ: Erlbaum. College Entrance Examination Board (2003). Course description: AP Statistics. New York: College Board. Even, R. & Tirosh, D. (2002). Teacher knowledge and understanding of students’ mathematical learning. In L.D. English (Ed.), Handbook of international research in mathematics education (pp. 219-240). Mahwah, NJ: Lawrence Erlbaum Associates. Fennema, E., & Franke, M.L. (1992). Teachers’ knowledge and its impact. In D.A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 147-164). New York, NY: Macmillan. Groth, R.E. (2003). Development of a high school statistical thinking framework. Unpublished doctoral dissertation: Illinois State University. Hirsch, C.R., Coxford, A.F., Fey, J.T., Schoen, H.L. (1998). Contemporary Mathematics in Context: A unified approach. New York: Glencoe/McGraw-Hill. Jones, G.A., Thornton, C.A., Langrall, C.W., Mooney, E.S., Perry, B., & Putt, I.J. (2000). A framework for characterizing children’s statistical thinking. Mathematical Thinking and Learning, 2, 269-307. Maykut, P., & Morehouse, R. (1994). Beginning qualitative research: A philosophic and practical guide. London: The Falmer Press. Mooney, E.S. (2002). A framework for characterizing middle school students’ statistical thinking. Mathematical Thinking and Learning, 4, 23-64. Patton, M. (1990). Qualitative evaluation and research methods (2nd ed.). Newbury Park, CA: Sage. Watson, J.M., Collis, K.F., Callingham, R.A., & Moritz, J.B. (1995). A model for assessing higher order thinking in statistics. Educational Research and Evaluation, 1, 247-275. Watson, J.M., & Moritz, J.B. (1999). The beginning of statistical inference: Comparing two data sets. Educational Studies in Mathematics, 37, 145-168.

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