We examine probabilistic intuitions and concepts as they relate to children aged 5 to 11 from research over the past 50 years. In the first section, we review the research pertaining to specific concepts and skills associated with probabilistic reasoning. The other section presents a discussion of theoretical perspectives on instruction in probability.
Some elementary ideas of combinatorics and its role in supporting children's development of beginning probability ideas and problem-solving skills are explored. A review of studies that addressed children's combinatorial reasoning is then presented. To conclude, ways in which children's access to powerful ideas are considered.
This chapter covers a wide range of ideas associated with probabilistic thinking and provides some examples of the potential development in understanding that occurs in students in Grades 6 through 9. The first sections include intuitive ideas on luck and fairness, as well as chance language. Secondly, various types of probabilistic events, ranging from simple events to conjunction events are explored. Finally, appreciation of random behavior, sampling and variation in a probabilistic setting, the equiprobability bias, and the importance of dealing with probability in context, are considered in the last four sections.
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.
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.
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.
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.
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.
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.
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.