# Research

• ### On learning the relationship between quantitative properties and symbolic representations

Many students do not understand what representational problems a particular notation solves, thus limiting their ability to use the notation, as well as their understanding of the problem situation it applies to. Forty-six undergraduates completed a lesson designed to help them understand variance and its notation. Students in the invention group were asked to create a procedure for calculating the variance of contrasting distributions of numbers; students in the procedural group were presented with a procedure for calculating variance and asked to practice it on the numbers. Results indicate that invention students learned to reflect on the quantitative properties of distributions, and to evaluate statistical procedures in terms of their ability to differentiate those properties. Students in the procedural condition tended to evaluate a procedure simply in terms of whether or not it was like the "correct" procedure. We plan to extend this instructional method to facilitate classroom conversations and as a platform for a complementary intelligent instructional system.

• ### Using technology to promote students' construction of knowledge

In this talk I will address how the participants in my dissertation research employed technology as a way for students to interact with statistics and focus on how this interaction with statistics through technology allows students to construct their own knowledge and understanding of statistics.

• ### The use of implicative and correspondence analysis for assessing pupil's combinatorial reasoning

In this paper we present the results of applying implicative and correpondence analysis to pupils' responses to a questionnaire aimed at assessing combinatorial reasoning in secondary school pupils. We also show the effect of some task variables on pupils' errors, as well as their evolution following instruction.

• ### Intuitive strategies and preconceptions about association in contingency tables

The aim of this research was to identify students' preconceptions concerning statistical association in contingency tables. An experimental study was carried out with 213 preuniversity students, and it was based on students' responses to a written questionnaire including 2 x 2, 2 x 3, and 3 x 3 contingency tables. In this article, the students' judgments of association and solution strategies are compared with the findings of previous psychological research on 2 x 2 contingency tables. We also present an original classification of students' strategies, from a mathematical point of view. Correspondence analysis is used to show the effect of item task variables on students' strategies. Finally, we include a qualitative analysis of the strategies of 51 students, which has served to characterize three misconceptions concerning statistical association.

• ### A comparative study of two instruments for evaluating primary probabilistic reasoning

Here we approach the complex nature of probabilistic reasoning, even at its most basic level, and its evaluation through written texts. We also present the conclusions of a theoretical and experimental study of two tests (Green, 1983 and Fischbein, 1984) designed to evaluate primary probabilistic intuitions, reaching the conclusion of the need for them to be mutually complementary in order to improve the validity of their content, both for including items from the different components of probabilistic reasoning and from the universe of task and contextual variables.

• ### Identification of misconceptions in the central limit theorem and related concepts and evaluation of computer media as a remedial tool

In this paper, we will break down the central limit theorem into several components, point out the misconceptions in each part, and evaluate the appropriateness of computer simulation in the context of instructional strategies. The position of this paper is that even with the aid of computer simulations, instructors should explicitly explain the correct and incorrect concepts of each component of the CLT.

• ### Learning by problem solving versus examples: The benefits of generating and receiving information

This experiment contrasts learning by solving problems with learning by studying examples, while attempting to control for the elaborations that accompany each solution step. Subjects were given different instruction materials for a set of probability problems. They were either provided with or asked to generate solutions, and they were either provided with or asked to create their own explanations for the solutions. Subjects were then tested on a set of related problems. Subjects in all four conditions exhibited good performance on the near transfer test problems. On the far transfer problems, however, subjects in two cells exhibited stronger performance: those solving and elaborating on their own and those recieving both solutions and elaborations from the experimenter. There also was an indication of a generation effect in the far transfer case, benefiting subjects who generated their own solutions. In addition, subjects' self-explanations on a particular concept were predictive of good performance on the corresponding subtask of the test problems.

• ### Using students' probabilistic thinking to inform instruction

This study addresses the development and evaluation of an instuctional program guided by research-based knowledge of students' probabilistic thinking. In particular, it seeks to (a) use a framework that describes and predicts students' thinking in probability to construct a third-grade instructional program, and (b) evaluate the effect of two different sequences of the instructional program on students' thinking in probability.

• ### Using small groups to promote active learning in the introductory statistics course: A report from the field

Over several semesters, we changed form the traditional lecture approach to cooperative learning. After some initial difficulty, we found procedures that work in classes of 40 to 100 students. Data consist of final grade distributions, the number of students retained in the class, and responses on a questionnaire that asked students' attitudes towards the group activities. Working in cooperative groups resulted in higher final scores in two experimental sections than in a comparison course section. A higher percentage of students successfully completed the course in the experimental sections, and student attitudes toward the cooperative group experience were positive.

• ### Understanding repeated choices under uncertainty

Assessed understanding of decision making about repeated uncertain events, using hypothetical and in-vivo prediction tasks modeled after those used in probabilitiy-learning research. Previous studies have been unclear on whether suboptimal choices reflected reasoning errors and lack of strategic thinking, or confounding factors. Quantitative and qualitative analyses of subjects' choices and strategy explanation showed that up to 50% of college students did not fully understand the relative value of different strategies. Only 5 % of subjects preferred a "true" probability matching strategy, included on an in-vivo task. High-school students showed greater misunderstandings. Large gender differences in prediction strategies and in related computational skills were observed. Understanding was discussed in terms of subjects' inferences from knowledge of independence of events, (lack of) computational skills, and correlates of the quantitative nature of prediction tasks. Implications for research on decision making, including need to address individual differences, are discussed.