Development of Informal Inferential Statistical Reasoning among Young Children in Research-based Learning Environment Einat Gil

ABSTRACT

Statistical Inference is central to the statistical thinking and practice. Informal Inferential Statistics is a concept which gets a lot of attention in the last years in the academic community of statistics education. The purpose is to develop informal reasoning, regarding statistical inference among elementary school students, without the use of complex mathematical computations and formalization, in order to make the "big ideas" of statistics accessible for young students. The present study follows the first steps in the development of Informal Inferential Statistical Reasoning (IIR), among sixth grade students, in a research-based learning environment in the discipline of Exploratory Data Analysis (EDA; Statistics). The research was conducted within Kishurim project (Ben-Zvi, Gil & Apel, 2007), in which an experimental curriculum was designed for the development of IIR. During a five weeks intervention, the students experienced a rich and diverse learning trajectory. It included collaborative learning in groups, class discussions, the use of an innovative software TinkerPlots (Konold & Miller, 2005), and authentic exploratory data inquiry activities, which were based on the expanded statistical inquiry cycle (Gil, 2007) that focuses on IIR from a sample to a population. The researchers' approach to the design of the learning environment was socio-constructivist, an approach which emphasized the emergence of ideas (in contrast to defining and memorizing concepts), fostering community of learners, authentic learning, inquiry-based learning and project-based learning. TinkerPlots, the software used by the students in some of the activities, is a technological tool that helps to develop reasoning in EDA among students (Ben-Zvi, 2006; Paparistodemou, & Meletiou-Mavrotheris, 2007). Organizing data in a dynamic way, allows to pose questions and to research them, examine trends and relations between variables, and to compare groups, in easy to use, children friendly educational software.

Gil, E. (2008). Developing informal inferential statistical reasoning among young children in research-based learning environment. Thesis for M.A. degree, supervised by Dr. Dani Ben-Zvi. Faculty of Education, University of Haifa, Haifa.

The purpose of the study is to describe and to characterize the first steps of sixth grade students, in the development of IIR. The study focuses on the cognitive aspects of the emergence of ideas and concepts related to sample and sampling and to IIR, as well as on the design aspect of the learning environment. Three research questions are at the heart of this study: 1. What characterizes the development of reasoning regarding sample and sampling in relation to IIR among sixth grade students in this learning environment? 2. What characterizes the development of Informal Inferential Statistical Reasoning among sixth grade students in this learning environment? 3. What are the characteristics of the learning environment that support the development of IIR? The research, as a design experiment, (Cobb et al., 2003) combines qualitative, quantitative and design-based approaches to the data analysis. The research tools are mostly qualitative (e.g. observations and interviews that were videotaped, a researcher journal, and learning outputs), as well as some quantitative tools (pre-post tests). In that it belongs to mixed methods research (Tashakkori & Teddlie, 2003). The purpose of the data analysis, which was done mainly according to the "interpretive micro-analysis" approach (Meira, 1995), was to trace the development of students' reasoning related to sample and sampling as well as to IIR. In addition, the design aspect of the curriculum development was examined through the design documents, the researcher journal and the interviews and observations. The data analysis and the presentation of the results were divided to three sections according to the research questions. In the first part, the starting point of students' reasoning regarding sample and sampling was examined as well as the development of this reasoning in the course of learning. This issue was examined through in-class discussions and through intuitive sampling activity planned and performed by the students, in order to infer from the sample to the population regarding a research question related to their lives in school. The main results in this section point to a significant development which occurred in the students' thinking regarding sample and sampling, during the project. This development was expressed in several components of sample and sampling, which are defined in the theoretical section of the present paper (BenZvi et al., 2007).

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It was found that students tended to represent the diversity and variety in the population in the samples they have sampled, by using non-random strata samples. They have divided the population to segments according to grade or gender. While working with these samples, the students did not make use of the concept "random sample" and showed a partial understanding or a misconception of the statistical concept "random", which was explained by them as "by chance" ("I ask the first student I see"). In addition, there was no reference by the students to the potential of biased samples as well as biased sampling methods. These results affected the program design and brought the inclusion of a new structured activity, which dealt with a random sample and a bias in sampling, in a hands-on way. The activity was accompanied by in-class discussions. It was found that this activity, as well as other open inquiry activities, which emphasized IIR, brought a development in the understanding of the concepts of sample and sampling. The tracing in the second part of the study has focused on one group of students, "the triplet group", and examined the development of IIR in its statistical inquiries. The students explored questions which they have phrased regarding the transition to junior high school and sportsmanship by sampling of random samples (n=20; n=30) out of the 206 sixth and seventh grade students' population in their school. The students performed two inquiries. In the first inquiry they examined the relations between the grade level (sixth versus seventh) and the homework load, in two different samples. They inferred regarding the population, from each of the samples, substantiated their inferences and compared the two samples. In the second inquiry the students examined the long jump differences between sixth and seventh grade students, as well as the relation between the preferred sports and the long jump results. The students reached conclusions regarding the population, basing them on the data analysis, while performing inquiry cycles regarding two different random samples. The following IIR components were examined in the students' work: inferences, substantiating of inferences, explanations of inferences, relating the statistical inference to a random sample and sampling, and the level of confidence of the inferences. In the framework of this analysis, we distinguished between inference that relates to inferring from a sample to a wider population, and interpretation, which relates to the sample alone. A development in the referential processes of the "triplet group" was found, which was expressed by the complexity of inferences and their substantiation. The substantiating of inferences has confronted the students with "surprise" in the results, which created a cognitive conflict and allowed an accumulated substantiation of inferences. This included re-sampling a

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random sample, the use of diversified representations for examining a hypothesis, and performing secondary inquiry cycles. A development was also found in their ability to relate the statistical inference to the random sample and sampling during the inference process, through an internal conflict and discussions related to issues such as: reliability of the random sample, level of confidence in the inferences and a potential sample bias. The discussions of the students, regarding these issues, which are described in this study, point to the significant progress, made by the students, starting with the lack of understanding of the concept of a random sample and of the concept of a biased sample, at the beginning of the project, and ending with the understanding, though a partial one, of central concepts in statistical inference. A development in the understanding of the level of confidence in the inferences was also found, which show, in our opinion, the students' growing confidence regarding the random sampling process and statistical inference. Their ability to accept the discernible trends presented in the random sample data, although they contradict their own logic, was evident. The quantitative analysis results confirm the IIR significant development in two out of the three classes, which participated in the project, regarding sample, sampling and statistical inference (p<0.001). In the third part of the study, the design aspect of the learning environment was examined. Unique characteristics of the learning environment which was created in the course of a design experiment study were found, including: a learning trajectory consisting of eight activities, which took place in class and included discussions, collaborative learning, inquiry activities with and without the use of computers, and presentation of results and inferences; The extended inquiry statistical cycle as a basis for activities, the use of the TinkerPlots software and advanced tools within it, for encouraging statistical inference and argumentation, scaffolding learning with activities sheets, inquiry of different data bases, which are related to the children's world, etc'. The main conclusions of the study are: 1. In regard to the first research question, our conclusion is that a significant development has occurred in the understanding of random sample and sampling related to IIR. A development has occurred in the understanding and the use of the concepts: a random sample, random sampling and a biased sample, which was confirmed by the pre-post tests (in two out of the three classes).

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2. In regard to the second research question, at the level of the case study of one group of students, a development has occurred in the understanding and implementation of the following issues: choosing a topic for study, drawing inferences, substantiating inferences, understanding the relations between statistical inference and random sample and sampling, and level of confidence in the inferences. In spite of the fact that no development was found regarding the quality of explanations to the inferences given by the students, it was found that the explanations were related to creativity in IIR (Ben-Zvi, Gil & Apel, forthcoming). 3. In regard to the third research question, we have pointed to several characteristics of the Kishurim learning environment for the sixth grade, which support the emergence of ideas and the development of the IIR, including the suggested learning trajectory, an extended inquiry cycle and working with TinkerPlots. The present study has significant practical and pedagogical implications. It is suggested to extend the research regarding the development of IIR and regarding the development of a conceptual understanding of main concepts within it. In addition it is suggested to study socio-cultural aspects of the IIR in-class development in the course of the project. From the curricular point of view, the study supports the inclusion of IIR in the curriculum of elementary schools (sixth grade), as a means for familiarity with the "great statistical ideas" and readiness for the encounter with the formal statistical inference subject at an older age. Using the program in other schools and in different circumstances can contribute to the updating of the design of the learning environment and to examining its suitability to different populations. It is also recommended to extend the study and the use of TinkerPlots to data inquiry teaching programs to young students.

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