Research

  • The purpose of this study was to investigate the effectiveness of computer manipulatives compared to concrete manipulatives in teaching selected elementary probability topics. With the growing availability of computers in the classroom and the advancements in technological capabilities, computer manipulatives have the potential to have the same benefits of concrete manipulatives. It has been well documented that when used properly, concrete manipulatives benefit student's mathematical learning. Despite this fact, few teachers use concrete manipulatives because of classroom management issues. Several studies have shown that computer manipulatives are more manageable compared to their concrete counterparts and that computer manipulatives can facilitate students' mental operations better with the movements on screen. Thirteen fourth-grade students and two teachers were participants in the study. The students were separated into two groups of comparable ability. All students were to complete two activities which addressed nine probability-related target objectives. The first activity involved number cubes while the second involved spinners. When performing the number cube activity, half the students completed the activity using concrete cubes, the other half using computer cubes. To complete the second activity involving spinners, students who had used the computer number cubes for the first activity now used concrete spinners, and students who had used concrete number cubes for the first activity now used computer spinners. Students and teachers filled out a questionnaire and were interviewed at the completion of the study. Several comparisons showed that students using concrete manipulatives did just as well as those using computer manipulatives. Two out of four comparisons showed that students using concrete manipulatives scored better than those using computer manipulatives. Students and teachers reported that they enjoyed using computer manipulatives, and found them easy to use. Eight out of thirteen students saw no difference between the manipulatives with respect to their contribution to their learning, while about four out of thirteen students believed that concrete manipulatives were better for learning. Teachers did not change their belief that computer manipulatives are one of many tools that could be used to teach concepts however they reported that computer manipulatives will not replace concrete manipulatives.

  • The purpose of this study was to investigate the conceptions secondary school students have when dealing with stochastic questions and the heuristics these students use to solve stochastic questions. The second purpose of this study was to determine if there were any effects of gender, grade level, mathematical placement, reading ability and prior stochastic experience on the students' stochastic achievement. The students' stochastic achievement was based on the percentage correct on a multiple-choice stochastic test and the students' conceptions and the heuristics they used were based on the answers students gave on a stochastic reasoning test. The analysis sample for the study consisted of 392 secondary school mathematics students in the Toms River, New Jersey, school district who took the multiple-choice stochastic test. Eighteen of the 392 students volunteered to take the reasoning test, where six students were from each group of students who scored in the top third, middle third and bottom third of the multiple-choice test. Statistical methods were used to test if there were any effects of the variables mentioned earlier on students' stochastic achievement, and whether there was a difference in the proportion of correctly answered questions on the multiple-choice test between probability and statistics questions. The results indicated that, at the 0.05 significance level, reading ability, grade level (Grade 9), the interaction between gender and mathematical placement (track 3), and the interaction between reading ability and stochastic experience had a significant effect on students' stochastic achievement. In addition, there was a significant difference in the proportion of correct answers between probability and statistics questions. Another question that was investigated in this study was if secondary school students use heuristics to solve stochastic questions. This question was qualitatively researched. From the results of the reasoning test, it was concluded that secondary school students use the following heuristics to solve stochastic problems: Belief Strategy, Equiprobable Bias, Bigger is Better, Prior Experience and Normative Reasoning. Belief strategy was used more often than the other heuristics. Also, it was determined that students do not always use the same heuristics to solve similar types of problems.

  • The major assumption underlying this research is that all knowledge and understanding about statistics is constructed. Given that students construct their own knowledge, teaching must be designed to support knowledge construction. In this context, the global purpose addressed in this research is: "How do accomplished statistics educators support knowledge construction in their introductory statistics courses?" This global purpose is studied by attending to two more manageable questions: 1) What instructional strategies are being used in and around the statistics classroom?, and 2) What are the results of an analysis of these instructional strategies when the analysis is grounded in a constructivist perspective? "The Quest for the Constructivist Statistics Classroom" is a qualitative research study that investigated the teaching of four accomplished statistics educators (Paul Velleman at Cornell University, David Moore at Purdue University, Gudmund Iversen at Swarthmore College, and Beth Chance at California Polytechnic State University). Data collection methods included e-mail questionnaires, on-site interviews, and classroom observations of the participants. Instructional strategies employed by the participants were grouped into categories: strategies for how students come to know statistics; strategies involving technology; and, strategies for assessing student learning. For the purpose of data analysis, the following definition of constructivism was used: Constructivism is a theory of learning that allows students to develop and construct their own understanding of the material based upon their own knowledge and beliefs and experiences in concert with new knowledge presented in the classroom. During the analysis, it was decided that the instructional strategies being used in the participants' classrooms did not dichotomously support or not support constructivism, but rather supported constructivism to varying degrees. Some findings of the study included: 1) all four participants supported student construction of knowledge to some degree; 2) each of the participants employed multiple instructional strategies to involve the students in the learning process; and, 3) class size impacted the ability of the instructors to employ instructional strategies that were more supportive of knowledge construction. In addition, a series of questions intended to inspire further thought and research emerged from the study.

  • This study investigated the following three questions: What are the Chinese students main misconceptions of probability? What is the developmental structure of students' understanding of probability? Can an activity-based short-term teaching programme improve ordinary grade 8 students' understanding of probability? The first two questions were answered in the main study. The sample was 567 Chinese students from three grades (6, 8 and 12) and two school streams (ordinary and advanced). After one year, six activity-based lessons which focused on empirical probability were given to two grade 8 classes in an ordinary school. The approaches were parallel except that one class had the opportunity to see computer simulations of a long series of experiments, while the other class was given the data in written form. All the students were tested and interviewed both prior to and after the teaching intervention. Fourteen groups of misconceptions were observed in this study. The outcome approach, chance cannot be measured mathematically, compound approach and equiprobability were the main misconceptions for each grade and each stream of students. The context and data used in an item were found to play a role in eliciting some misconceptions. Using the SOLO taxonomy, it was found that, generally, there was no improvement in developmental level at grades 6 and 8, the two grades without any formal probability training. Grade 12 students have a better understanding than the younger students. The results of the teaching show that a short intervention can help students overcome some of their misconceptions. Students in the two classes improved substantially in their answers and reasoning but no statistically significant difference was found between the classes.

  • This study documented the learning of a research team as it engaged in the process of instructional design. An 11-week classroom teaching experiment conducted in a seventh-grade classroom and the planning year prior to the teaching experiment were the sites for the research team's instructional design investigation. The goal of the teaching experiment was to support students' development of statistical understandings related to data analysis through the design of an instructional sequence. Two computer-based data analysis tools were integral aspects of the instructional sequence and served as primary means of supporting the students' learning. This dissertation clarified the instructional design decisions made by the research team and described how those decisions created learning opportunities. These decisions emerged as the research team continually tested and revised its conjectures about how to support students' mathematical development as it designed the instructional sequence. To this end, this dissertation focused on critical issues that guided the research team in its initial attempts at instructional design. These critical issues were tracked from the planning year throughout the classroom teaching experiment in order to understand what the research team learned about (a) the mathematics involved in teaching and learning statistics and (b) how to support students' development of ways to reason statistically while engaging in data analysis.

  • The purpose of the study was to identify the important aspects of statistical knowledge needed for teaching at the middle school level and to assess prospective teachers' conceptions and misconceptions of statistics related to teaching data analysis. An analytic study of the current literature, including state and national standards, was conducted to identify the important aspects of statistical knowledge for teaching. A written assessment instrument was developed and administered to a sample of 42 prospective middle school teachers. The purpose of the instrument was to gather data in order to describe teachers' conceptions for teaching data analysis and statistics. A subset of the sample (n = 7) was interviewed to provide deeper insight into their conceptions and to assure reliability of the instrument.<br><br>Results show that state and national standards differ greatly on their expectations of what students and teachers should know about data analysis and statistics. The variation is also large for the emphasis or importance given to the content. The average emphasis of all the documents reviewed is given to the selection and proper use of graphical representations of data, and measures of center and spread. Important aspects of knowledge applied to teaching are proper selection and use of teaching strategies and inferring students' understanding from their work and discourse.<br><br>Prospective teachers that participated in this study performed better at the level of pure statistical knowledge than at the level of application of this knowledge to teaching. In particular, they showed abilities on reading, interpreting, and constructing graphical representations, and computing measures of center and spread. Difficulties were shown in judging students' comments and identifying students' mistakes.

  • The focus of this chapter is on the nature of mathematical and statistical reasoning. The chapter begins with a description of the general nature of human reasoning. This is followed by a description of mathematical reasoning as described by mathematicians along with recommendations by mathematics educators regarding educational experiences to improve mathematical reasoning. The literature on statistical reasoning is reviewed and findings from the general literature on reasoning are used to identify areas of statistical reasoning that students find most challenging. Statistical reasoning and mathematical reasoning are compared and contrasted, and implications for instruction and research are suggested.

  • This chapter examines cognitive models of development in statistical reasoning and the role they can play in statistical education. The meaning of statistical reasoning is explored and cognitive models of development are examined. Cognitive models of development include both maturational and interactionist effects. This chapter uses a developmental model to analyze different aspects of statistical reasoning, such as, reasoning about center, spread, and association. Implications for curriculum design, instruction, and assessment are also discussed.

  • The purpose of this chapter is to describe and analyze the ways in which middle school students begin to reason about data and come to understand exploratory data analysis (EDA). The process of developing reasoning about data while learning skills, procedures, and concepts is described. In addition, the students are observed as they begin to adopt and exercise some of the habits and points of view that are associated with statistical thinking. The first case study focuses on the development of a global view of data and data representations. The second case study concentrates on design of a meaningful EDA learning environment that promotes statistical reasoning about data analysis. In light of the analysis, a description of what it may mean to learn to reason about data analysis is proposed and educational and curricular implications are drawn.

  • The purpose of this chapter is to explore how informal reasoning about distribution can be developed in a technological learning environment. The development of reasoning about distribution in seventh-grade classes is described in three stages as students' reason about different representations. It is shown how specially designed software tools, students' created graphs, and prediction tasks supported the learning of different aspects of distribution. In this process, several students came to reason about the shape of a distribution using the term bump along with statistical notions such as outliers and sample size.<br>This type of research, referred to as "design research," was inspired by that of Cobb, Gravemeijer, McClain, and colleagues (see Chapter 16). After exploratory interviews and a small field test, we conducted teaching experiments of 12 to 15 lessons in 4 seventh-grade classes in the Netherlands. The design research cycles consisted of three main phases: design of instructional materials, classroom-based teaching experiments, and retrospective analyses. For the retrospective analysis of the data, we used a constant comparative method similar to the methods of Glaser and Strauss (Strauss &amp; Corbin, 1998) and Cobb and Whitenack (1996) to continually generate and test conjectures about students' learning processes.

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