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.
A summary of the research review presented in the book.
This article highlights some research on several big ideas in statistics that seem particularly pertinent to school mathematics. A discussion of the notion of statistical thinking lays the foundation for discussing students' understanding of average, concept of variability and some important connections between proportional reasoning and statistical reasoning. In conclusion, some suggestions for teaching are offered from the research literature on the teaching and learning statistics.
This article describes two teaching experiments that focused on the development of statistical ideas and reasoning and illustrates different approaches (e.g. different rationale, activities, and technological tools). We focus on one aspect of each experiment: distribution and sample. Details of the experiments are provided to allow readers to appreciate some of the rich information that was gathered. Finally, suggestions for teachers are discussed.
A randomized trial of 265 consenting students was conducted within an introductory biostatistics course: 69 received eight small group cooperative learning sessions; 97 accessed internet learning sessions; 96 received no intervention. Effect on examination score (95% CI) was assessed by intent-to-treat analysis and by incorporating reported participation. No difference was found by intent-to-treat analysis. After incorporating reported participation, adjusted average improvement was 1.7 points (-1.8, 5.2) per cooperative session and 2.1 points (-1.4, 5.5) per internet session after one examination. After four examinations, adjusted average improvement for four study sessions was 5.3 points (0.4, 10.3) per examination for cooperative learning and 8.1 points (3.0, 13.2) for internet learning. Consistent participation in active learning may improve understanding beyond the traditional classroom.
What people mean by randomness should be taken into account when teaching statistical inference. This experiment explored subjective beliefs about randomness and probability through two successive tasks. Subjects were asked to categorize 16 familiar items: 8 real items from everyday life experiences, and 8 stochastic items involving a repeatable process. Three groups of subjects differing according to their background knowledge of probability theory were compared. An important finding is that the arguments used to judge if an event is random and those to judge if it is not random appear to be of different natures. While the concept of probability has been introduced to formalize randomness, a majority of individuals appeared to consider probability as a primary concept.
While other research has begun to contribute to our understanding of how pre-college students reason about variation, little has been published regarding pre-service teachers' statistical conceptions. This paper summarizes a framework useful in examining elementary pre-service teachers' conceptions of variation, and investigates the question of how a class of pre-service teachers' responses concerning variation in a probability context compare from before to after class interventions. The interventions comprised hands-on activities, computer simulations, and discussions that provided multiple opportunities to attend to variation. Results showed that there was overall class improvement regarding what subjects expected and why, in that more responses after the interventions included appropriate balancing of proportional thinking along with an appreciation of variation in expressing what was likely or probable.
The effect of educational technologies on learning is an area of active interest. We conducted an experiment to compare the impact of instructional software on student performance. We hypothesize that some of the impact on student performance may reflect the influence of the technology on student subject-related beliefs and that those beliefs may differ by gender. We desired to assess how course performance may be associated with student beliefs, and how the association may differ depending on instructional software environment and gender.
This paper is a personal exploration of where the ideas of "distribution" that we are trying to develop in students come from and are leading to, how they fit together, and where they are important and why. We need to have such considerations in the back of our minds when designing learning experiences. The notion of "distribution" as a lens through which statisticians look at the variation in data is developed. I explore the sources of variation in data, empirical versus theoretical distributions, the nature of statistical models, sampling distributions, the conditional nature of distributions used for modeling, and the underpinnings of inference.
Drawing conclusions from the comparison of datasets using informal statistical inference is a challenging task since the nature and type of reasoning expected is not fully understood. In this paper a secondary teacher's reasoning from the comparison of box plot distributions during the teaching of a Year 11 (15-year-old) class is analyzed. From the analysis a model incorporating ten distinguishable elements is established to describe her reasoning. The model highlights that reasoning in the sampling and referent elements is ill formed. The methods of instruction, and the difficulties and richness of verbalizing from the comparison of box plot distributions are discussed. Implications for research and educational practice are drawn.