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# Chapter

• ### How do teachers foster students' understanding of probability?

This chapter reviews the research on what students know (and do not know) about probability and chance and the role of technology in fostering students' understanding of probability, specifically in the connection between randomness, the law of large numbers and the notion of distribution. Implications for pedagogy is considered.

• ### Dealing with compound events.

This chapter will analyze elementary and middle school students' ability to generate sets of outcomes associated with compound events, and will examine some research on the impact of instruction on the learning of both theoretical and experimental probability. Also, the learning experiences that might be used to support the development of students' thinking in dealing with compound events will be explored. Specifically, the focus will be on understanding students' probabilistic thinking when dealing with compound and simple events in both interview and instructional designs.

• ### How can teachers build notions of conditional probability and independence?

The focus of this chapter is on research in conditional probability and independence that uses both with- and without-replacement tasks. Probabilistic thinking about conditional occurrences as well as independence are explored. A framework postulates that middle school students' thinking in conditional probability and independence could be described and predicted across four levels that represent a continuum from subjective thinking to numerical reasoning. Then implications for teaching and learning are considered, emphasizing the fostering of understanding.

• ### What is the nature of high school students' conceptions and misconceptions about probability?

This is a review of the research that focuses specifically on the probabilistic thinking of secondary school students (14 - 18 years) and the relation to curriculum expectations. In particular, we will look at research associated with some key elements for the probability curriculum: combinatorial reasoning and problem solving, randomness, probability misconceptions, conditional variables and probability distributions, sampling and inference, and simulation. We will also consider the implication of this research for teaching probability in the secondary school.

• ### Probability and statistical inference: How can teachers enable learners to make the connection?

This chapter considers a possible pathway to formal inference by first drawing on, as an illustration, a case study that involved students in drawing informal inferences form the comparison of boxplots. Second, ways that students could be helped towards formal inference are suggested, and finally two possible pathways to formal inference, theoretical or simulation, are discussed.

• ### Teaching and learning the mathematization of uncertainty: Historical, cultural, social and political contexts.

We deal with the conceptual development of probability as part of mathematics that grew historically in intimate relationship with its applications. As well, we consider the role of probability in contemporary society. We use these analyses to present the arguments for the importance of education for understanding probabilistic thinking as a tool for understanding the physical and social worlds. Lastly, we consider the challenges facing this endeavor, and offer suggestions for meeting these challenges.

• ### Assessing probabilistic thinking and reasoning.

This chapter considers the assessment of probabilistic thinking and reasoning via informal monitoring as well as through formal tasks. Such assessment are considered in the context of the purposes of, and frameworks for, assessment. Specific tasks are examined as to whether they assess thinking and reasoning. Suggestions on improvement to the quality of the assessment instruments are made in light of research studies on the understanding of probability concepts. Alternative assessment strategies are suggested.

• ### Probability in teacher education and development.

The purpose of this chapter is to investigate issues concerning the nature and development of teachers' probability understanding. An outline of the central issues that affect teachers' efforts to facilitate students' probabilistic understanding is given. Then, I examine teachers' knowledge and beliefs about probability, their ability to teach probabilistic ideas, and lessons learned from programs in teacher education that have aimed at developing teachers' knowledge about probability.

• ### Reflections.

A summary of the research review presented in the book.

• ### Graphing with four-year-olds: Exploring the possibilities through staff development.

This article deals with professional development strategies for teaches to use in initiating and supporting mathematical thinking through data-collecting experiences in contexts that are meaning to the children, as well as nurturing the children's efforts in recording and reasoning with data.