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# Journal Article

• ### Developing a Questionnaire to Assess the Probability Content Knowledge of Prospective Primary School Teachers

In this paper we describe the development of a questionnaire designed to assess the
probability content knowledge of prospective primary school teachers. Three components of mathematical knowledge for teaching and three different meanings of probability (classical, frequentist and subjective) are considered. The questionnaire
content is based on curricular guidelines and primary school textbooks in Spain. The
items were selected and adapted, after expert judgment, from previous research. The
responses of 157 prospective primary school teachers were used to analyze the
psychometric properties of the questionnaire and to provide information about various
aspects of participants’ probability content knowledge.

• ### What’s Missing in Teaching Probability and Statistics: Building Cognitive Schema for Understanding Random Phenomena

Teaching probability and statistics is more than teaching the mathematics itself. Historically, the mathematics of probability and statistics was first developed through analyzing games of chance such as the rolling of dice. This article makes the case that the understanding of probability and statistics is dependent upon building a “mature” understanding of common random phenomena such as the rolling of dice or the blind drawing of balls from an urn. An analysis of the verbalizations of 24 college students, who interact with random phenomena involving the mixture of colored marbles, is presented, using cognitive schema to represent the subjects’ expressed understanding. A cognitive schema representing a mature understanding is contrasted to a diversity of observed immature understandings. Teaching to explicitly build the mature cognitive schema is proposed.

• ### A Mediation Model to Explain the Role of Mathematics Skills and Probabilistic Reasoning on Statistics Achievement

Among the wide range of factors related to the acquisition of statistical knowledge, competence in basic mathematics, including basic probability, has received much attention. In this study, a mediation model was estimated to derive the total, direct, and indirect effects of mathematical competence on statistics achievement taking into account probabilistic reasoning ability. The participants were psychology students enrolled in an undergraduate introductory statistics course. At the beginning of the course, all students completed a questionnaire to measure their proficiency in the basic probabilistic reasoning and mathematics skills required in introductory statistics courses. At the end of the course, the students’ final grades were collected. The hypothesized mediation model was tested using a bootstrapping procedure (with 5000 bootstrap samples). Results showed a significant positive, indirect effect of mathematical competence on students’ final grades, with probabilistic reasoning ability acting as a mediator. This study suggests that interventions designed to promote the mathematical prerequisites necessary to probabilistic reasoning can have a positive effect on achievement in statistics.

• ### Flipping Between Languages? An Exploratory Analysis of the Usage by Spanish-Speaking English Language Learner Tertiary Students of a Bilingual Probability Applet

English language learners (ELLs) are a rapidly growing part of the student population in many countries. Studies on resources for language learners—especially Spanish speaking ELLs—have focused on areas such as reading, writing, and mathematics, but not introductory probability and statistics. Semi-structured qualitative interviews investigated how a purposeful sample of six (Spanish-speaking) ELLs experienced a bilingual coin-flipping simulation applet (NLVM, 2015) and how students might use such resource to confront content misconceptions and language misunderstandings related to probability concepts covered in college introductory statistics courses. We discuss findings, limitations, directions for future research, and implications for teaching, such as handling the phrases “in the long run” and “longest run”.

• ### “Probability from a Socio-Cultural Perspective”

There exists considerable and rich literature on students’ misconceptions about probability; less attention has been paid to the development of students’ probabilistic thinking in the classroom. Grounded in an analysis of the literature, this article offers a lesson sequence for developing students’ probabilistic understanding. In particular, a context familiar to teachers—exploring compound events that occur in a game of chance—is presented, and it is demonstrated how the context can be used to explore the relationship between experimental and theoretical probabilities in a classroom setting. The approach integrates both the content and the language of probability and is grounded in sociocultural theory.

• ### Secondary School Students’ Reasoning about Conditional Probability, Samples, and Sampling Procedures

In the Australian mathematics curriculum, Year 12 students (aged 16-17) are asked to solve conditional probability problems that involve the representation of the problem situation with two-way tables or three-dimensional diagrams and consider sampling procedures that result in different correct answers. In a small exploratory study, we investigate three Year 12 students’ conceptions and reasoning about conditional probability, samples, and sampling procedures. Through interviews with the students, supported by analysis of their work investigating probabilities using tabular representations, we investigate the ways in which these students perceive, express, and answer conditional probability questions from statistics, and also how they reason about the importance of taking into account what is being sampled and how it is being sampled. We report on insights gained about these students’ reasoning with different conditional probability problems, including how they interpret, analyse, solve, and communicate problems of conditional probability.

• ### Conceptual Issues in Quantifying Unusualness and Conceiving Stochastic Experiments: Insights from Students’ Experiences in Designing Sampling Simulations

This article reports on a classroom teaching experiment that engaged a group of high school students in designing sampling simulations within a computer microworld. The simulation-design activities aimed to foster students’ abilities to conceive of contextual situations as stochastic experiments, and to engage them with the logic of hypothesis testing. This scheme of ideas involves imagining a population and a sample drawn from it, and an image of repeated sampling as a basis for quantifying a sampling outcome’s unusualness in terms of long-run relative frequency under an assumption about the population’s composition. The study highlights challenges that students experienced, and sheds light on aspects of conceiving stochastic experiments and conceiving a sampling outcome’s unusualness as a probabilistic quantity.

• ### Students’ Expressions of Uncertainty in Making Informal Inference when Engaged in a Statistical Investigation using Tinkerplots

This paper reports on the results of a study investigating the potential to embed Informal Statistical Inference in statistical investigations, using TinkerPlots, for assisting 8th grade students’ informal inferential reasoning to emerge, particularly their articulations of uncertainty. Data collection included students’ written work on a statistical investigation as well as audio and screen records. Results show students’ ability to draw conclusions based on data, recognizing that these are constrained by uncertainty, and to use them to make inferences. However, few students used probabilistic language for describing their generalizations. These results highlight the need for working on probabilistic ideas within statistics, helping students to evolve from a deterministic perspective of inference to include uncertainty in their statements.

• ### Students’ Informal Inference about the Binomial Distribution of “Bunny Hops”: A Dialogic Perspective

The study explores the development of 11-year-old students’ informal inference about random bunny hops through student talk and use of computer simulation tools. Our aim in this paper is to draw on dialogic theory to explain how students make shifts in perspective, from intuition-based reasoning to more powerful, formal ways of using probabilistic ideas. Findings from the study suggest that dialogic talk facilitated students’ reasoning as it was supported by the use of simulation tools available in the software. It appears that the interaction of using simulation tools, talk between students, and teacher prompts helps students develop their understanding of probabilistic ideas in the context of making inferences about the distribution of random bunny hops.

• ### Fostering Self-concept and Interest for Statistics through Specific Learning Environments

Supporting motivational variables such as self-concept or interest is an important goal of schooling as they relate to learning and achievement. In this study, we investigated whether specific interest and self-concept related to the domains of statistics and mathematics can be fostered through a four-lesson intervention focusing on statistics. Data about these motivational variables and achievement related to statistics were gathered from 503 eighth graders. Our results indicate that students perceived mathematics and statistics differently with respect to their self-concept and interest. Moreover, statistics-related self-concept and interest could be fostered through the domain-specific intervention, whereby a greater increase was found among students with higher prior achievement in the domain of statistics.