Take a family of independent events. If some of these events, or all of them, are replaced by their complements, then independence still holds. This fact, which is agreed upon by the members of the statistical/probability communities, is tremendously well known, is fairly intuitive and has always been frequently used for easing probability…

To design and improve instruction in mathematical proof, mathematics educators require an adequate definition of proof that is faithful to mathematical practice and relevant to pedagogical situations. In both mathematics education and the philosophy of mathematics, mathematical proof is typically defined as a type of justification that satisfies a…

The authors present a new classroom-tested lesson that was designed to engage students in the joy of mathematical inquiry through a card game, Frustration, while building number sense, understanding of uncertainty, statistical reasoning, and discourse skills. The purpose in developing and sharing these lessons is to highlight how games of chance…

To facilitate people using mathematical reasoning to compare risks associated with the COVID-19 pandemic, the National Science Foundation (NSF) funded the COViD-TASER (Creation of Visualizations of Data: The Application of STEM Education Research) research team to create the COVID-19 Relative Risk Tool (RRT). The RRT uses an interactive bar chart…

The purpose of this research is to construct a preliminary genetic decomposition delineating the mental constructions underlying probabilistic independence. This delineation is considered within the framework of APOS theory. While the use of the term independence in probability is often conflated with causation, the definition relies instead upon…

The aim of this paper is to review some studies conducted with different learning areas in which the schemes of different participants emerge. Also it is about to show how mathematical proofs are handled in these studies by considering Harel and Sowder's classification of proof schemes with specific examples. As a result, it was seen that the…

To be literate in a society where the information shared online is often exploited, learners should be exposed to multiple aspects of contemporary predictive modeling. This article explores an activity in which grade 10 students learned how a famous AI algorithm (the Apriori algorithm) uses conditional probability to automate the process of…

Finding intersections, unions, and complements of sets is an essential issue in elementary mathematics. It builds the foundation for set theory, probability, logic, and other topics. It is commonly recognized that drawing a Venn diagram, which was first introduced by the British philosopher and mathematician John Venn in 1881, is a classic and…

Randomness and distribution are important concepts underpinning the ability to think and reason probabilistically. Traditional approaches to teaching the Poisson distribution focus on mathematical definitions and formulae which obscure the randomness intrinsic in this process. Advances in technology have made it possible for students learning…

Teachers often use number talks at the beginning of class to hear the ways that students understand mathematical topics so that they can base future instruction on student thinking. Students who engage in number talks become accustomed to thinking about mathematics in multiple ways, sharing their thoughts and critiquing the thoughts of others. The…

In this paper, we give possible suggestions for a classroom lesson about an application of probability using basic mathematical notions. We will approach to some combinatoric results without using "induction", "polynomial identities" nor "generating functions", and will give a proof of the "Vandermonde…

We have extended two perspectives of proportional reasoning to solve problems based in probability. Four future middle grade teachers were enrolled in a mathematics content course that emphasized reasoning about multiplication with quantities. The course expected future teachers to generate and explain methods for solving proportions. Probability…

Informal inference is a critical practice for students to engage in if they are to understand formal statistical methods. However, during informal inference students often utilize complex ideas that many in-service teachers are not prepared for as they have not had the opportunity to think deeply about statistics and develop statistical knowledge…

White and Gorard make important and relevant criticisms of some of the methods commonly used in social science research, but go further by criticising the logical basis for inferential statistical tests. This paper comments briefly on matters we broadly agree on with them and more fully on matters where we disagree. We agree that too little…

Calculating Laplace transforms from the definition often requires tedious integrations. This paper provides an integration-free technique for calculating Laplace transforms of many familiar functions. It also shows how the technique can be applied to probability theory.