# Statistical reasoning at the secondary tertiary interface

Wilson, Therese Maree (2006) Statistical reasoning at the secondary tertiary interface. PhD thesis, Queensland University of Technology.

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## Abstract

Each year thousands of students enrol in introductory statistics courses at universities throughout Australia, bringing with them formal and informal statistical knowledge and reasoning, as well as a wide range of basic numeracy skills, mathematical inclinations and attitudes towards statistics, which have the potential to impact on their ability to develop statistically. This research develops and investigates measures of each of these components for students at the interface of secondary and tertiary education, and investigates the relationships that exist between them, and a range of background variables. The focus of the research is on measuring and analysing levels and abilities in statistical reasoning for a range of students at the tertiary interface, with particular interest also in investigating their basic numeracy skills and how these may or may not link with statistical reasoning allowing for other variables and factors. Information from three cohorts in an introductory data analysis course, whose focus is real data investigations, provides basis for the research. This course is compulsory for all students in degree programs associated with all sciences or mathematics.

The research discusses and reports on the development of questionnaires to measure numeracy and statistical reasoning and the students' attitudes and reflections on their prior school experiences with statistics. Students' attitudes are found to be generally positive, particularly with regard to their self-efficacy. They are also in no doubt as to the links that exist between mathematics and statistics.

The Numeracy Questionnaire, developed to measure pre-calculus skills relevant to an introductory data analysis course which emphasises real data investigations, demonstrates that many students who have completed a basic algebra and calculus senior school subject struggle with skills which are in the pre-senior curricula. Direct examination of the responses helps to understand where and why difficulties tend to occur. Rasch analysis is used to validate the questionnaire and assist in the description of levels of skill. General linear models demonstrate that a student's numeracy score depends on the result obtained in senior mathematics, whether or not the student is a mathematics student, gender, whether or not higher level mathematics has been studied, self-efficacy and year. The research indicates that either the pre-senior curricula need strengthening or that exposure to mathematics beyond the core senior course is required to establish confidence with basic skills particularly when applied to new contexts and multi- step situations.

The Statistical Reasoning Questionnaire (SRQ) is developed for use in the Australian context at the secondary/tertiary interface. As with the Numeracy Questionnaire, detailed examination of the responses provides much insight into the range and features of statistical reasoning at this level. Rasch analyses, both dichotomous and polychotomous, are used to establish the appropriateness of this instrument as a measuring tool at this level. The polychotomous, Rasch partial credit model is also used to define a new approach to scoring a statistical reasoning instrument and enables development and application of a hierarchical model and measures levels of statistical reasoning appropriate at the school/tertiary interface.

General linear models indicate that numeracy is a highly significant predictor of statistical reasoning allowing for all other variables including tertiary entrance score and students' backgrounds and self-efficacy. Further investigation demonstrates that this relationship is not limited to more difficult or overtly mathematical items on the SRQ.

Performance on the end of semester component of assessment in the course is shown to depend on statistical reasoning at the beginning of semester as measured by the partial credit model, allowing for all other variables. Because of the dominance of the relationship between statistical reasoning (as measured by the SRQ) and numeracy on entry, some further analysis of the end of semester assessment is carried out. This includes noting the higher attrition rates for students with less mathematical backgrounds and lower numeracy.

Impact and interest:

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