Research

  • This design study tracks the development of student thinking about natural variation as late elementary grade students learned about distribution in the context of modeling plant growth at the population level. The data-modeling approach assisted children in coordinating their understanding of particular cases with an evolving notion of data as an aggregate of cases. Students learned to "read" shapes of distributions as signatures of prospective mechanisms of plant growth and conducted sampling investigations to represent repeated growth. These investigations, in turn, supported students' interpretations of the effects of added light and fertilizer. The authors argue for both the feasibility and importance of tools such as distribution and inference for supporting education that builds on children's own investigations of the world. (Fall 2004)

  • This dissertation investigated children's probabilistic reasoning during a twomonth<br>teaching experiment. As part of the research process, the researcher developed a<br>computer microworld environment, Probability Explorer, for children's explorations with<br>probability experiments. The design of the microworld is based on a constructivist theory<br>of learning, design of mathematical computer microworlds, and research on students'<br>understanding of probability and rational number concepts. Two major features in the<br>microworld include a dynamic link between numerical, graphical and iconic<br>representations of data that are updated simultaneously during a simulation, and the<br>ability to design experiments and assign probabilities to the possible outcomes.<br>The teaching experiment was conducted with three nine-year-old children. The<br>children participated in 10 hours of teaching sessions using the microworld. Each child<br>also participated in pre- and post- task-based interviews to assess their reasoning in<br>probabilistic situations. Each teaching session was videotaped, and computer interactions were recorded through internal mechanisms to create a video, including children's audio, of all actions in the microworld. These tapes provided the basis for analysis and interpretation of the children's development of probabilistic reasoning while using the microworld tools.<br>The individual case studies detail the children's probabilistic reasoning during the<br>pre-interview, teaching experiment, and post-interview. After extensive coding, several<br>themes were identified and discussed in each case study. Some of the major themes<br>included: understanding and interpretation of theoretical probability in equiprobable and<br>unequiproable situations; theories-in-action about the law of large numbers; and<br>development of part-whole reasoning as it relates to probability comparisons, a priori<br>predictions, and analysis of relative frequencies.<br>The children's development of probabilistic reasoning and their interactions with<br>the computer tools varied during the study. The children employed different strategies<br>and utilized different combinations of representations (e.g., numerical, graphical, iconic)<br>to make sense of the random data to enact their own theories-in-action. The results from<br>this study imply that open-ended microworld tools have the potential to act as agents for<br>children's development of intuitive-based probability conceptions. Dynamically linked<br>multiple representations and flexibility in designing experiments can facilitate an<br>exploratory approach to probability instruction and enhance children's meaning-making<br>activity and probabilistic reasoning.

  • Drawing from web-based materials previously developed to supplement on-campus offerings of an introductory statistics course for graduate students in education, the author made an initial offering of an on-line virtual class in the fall of 2001. Poor student performance and dissatisfaction with this initial course organization led to a deeper reading of the literature on online teaching in general and the online teaching of statistics in particular. A much greater degree of instructor supplied organization, direction and interaction were incorporated into other offerings of the course during the fall and spring of 2003. Final examination scores and course evaluations improved markedly over the first online offering but remained somewhat lower and more variable than the results from on-campus offerings of the same course. Suggestions for improvements are offered based on instructor observations and student feedback.

  • Statistical literacy is a complex developmental construct requiring both mathematical skills and contextual understanding. The development of statistical literacy is an important objective of classrooms where the curriculum is approached through considering problems that require the active engagement of learners with relevant social material. Such approaches are often advocated for the middle years of schooling. Little attention has been paid, however, to the effects of these approaches on male and female students. This paper reports on a study that considers Differential Item Functioning (DIF) with respect to gender of questions on a statistical literacy scale derived from archived data. Multi-faceted Rasch models were applied to polytomous data to determine the interactions between gender and item. Three criteria were applied to the results: statistical significance, replicability and substantive explanation of DIF. The results suggested that although there was no overall difference in the average performance of male and female students, items requiring numerical responses or calculations were less difficult for male students and, conversely, items demanding written explanations were less difficult for female students. The implications of these findings for both assessment and teaching are discussed.

  • The Statistics Anxiety Rating Scale has 51 items, each scored on a 5-point rating scale to measure statistics anxiety with six subscales, Worth of Statistics, Interpretation Anxiety, Test and Class Anxiety, Computational Self-concept, Fear of Asking for Help, and Fear of Statistics Teachers. Psychometric properties included analyses of construct and concurrent validities an internal consistency and test-retest reliability. 221 college students (74% women; M age=28 yr.) in elementary statistics courses at several southwestern state universities participated. The findings are consistent with previous reports and indicate adequate concurrent validity, internal consistency, and split-half reliability, but for construct validity confirmatory factor analysis yielded marginal support.

  • The authors (teachers-researcher) carried out an exploratory study in which they designed , on the basis of a didactic analysis similar to the one proposed by Vallecillos (1996), three problems of hypothesis tests. Our purpose was to analyze the effects of graphic calculators use in the application and understanding of the concepts of p-value and significance level in the solving of these problems. The study was done with students of a statistics course for Social Sciences at the university level.<br>We studied, on the basis of the problem solving activities done by the students, their errors and difficulties concerning the concepts of significance lebel and p-value. We thought that the graphic calculator use was going to promote the use of the graphic and numeric representations. However, we found that the graphic calculator was used only in order to represent the p-value numeric representation. Even though each student "seems to understand" the concepts of p-value or significance level because he or she can resolve a problem, we find that little changes in data, (i.e., change in test laterality), generate new errors.<br>This result and other similar ones suggest the need to reflect about the phenomena that put into play the concepts in question, the kind of diactical activities that are designed and used in order to work with these concepts, and the conceptions and obstacles which are behind the errors made by the students.

  • The problem which inspired the present research referred to the relationships between schemata and intuitions. Schemata are defined following the Piagetian line of thought, as programs of processing information and controlling adaptive reactions. Intuitions are defined as self-evident, global, immediate cognitions.<br>Our main hypothesis was that intuitions are generally based on certain structural schemata. In the present research this hypothesis was checked with regard to intuitive solutions of combinatorial problems.

  • Recently there has been a trend towards admitting expert statistical evidence in UK court cases. There have been a number of cases, however, in which outcomes have been distorted by statistical or probabilistic misconceptions and by faulty inference. Typically, lawyers receive no training in these areas apart from their compulsory school mathematical education. In this study, data was taken from five groups of trainee lawyers. This demonstrated that they made errors in assessing likelihoods, irrespective of the level and type of mthematical education that they have received. The typical approaches and content of mathematical education at school or college need to be re-considered. Data from two other groups of subjects (one of statistical educators) with different types of mathematical backgrounds were available for comparison purposes.

  • This paper discusses some characteristic ways of reasoning within the discipline of statistics from the perspective of someone who is both a practising statstician and teaching statistician. It is conjectured that recognition of variation and critically evaluating and distinguishing the types of variation are essential components in the statistical reasoning process. Statistical thinking appears to be the interaction between the real situation and the stiatistical model. The role of variation in staistical thinking and the implications for teaching are also discussed.

  • Our comparative studies investigate the influence of different representations (i.e. formulas or graphical models and numeric formats) on the understanding of "big ideas" in stochastics (e.g. characteristics of probability, conditional probability, distribution, significance). We know from previous work (e.g. Sedlmeier &amp; Gigerenzer, 2001) that special tree-representations combined with frequency-formats increase the performance in understanding dramatically. Another aspect of the experiments is the utility of different presenting-modes (e.g. static vs. dynamic, imitation vs. learning by doing). The pupils of age 15-19 receive a computer-based training with different representations resp. modes on basic probability tasks. The effects of the training are measured by subsequent tests. Thus we obtain insight, if they succeed easily in using the learned representations and if they benefit from it. The first results support the assumption that groups of pupils trained with frequency-representations have a better understanding of key-problems in stochastics.

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