Probability is generally considered one of the most challenging areas to teach in mathematics education due to its intricate nature. However, the simulation-based teaching method can increase students' accessibility significantly to the probability problems because it enables students to resolve the problems with minimal mathematical skills. By…

Random sampling and random allocation are essential processes in the practice of inferential statistics. These processes ensure that all members of a population are equally likely to be selected, and that all possible allocations in an experiment are equally likely. It is these characteristics that allow the validity of the subsequent calculations…

In this paper, the probability content in senior secondary mathematics Victorian curriculum between 1978 and 2016 was classified and compared, by content, context, procedural complexity, the SOLO thinking frameworks and use of technology. While probability continues to form an important and increasing component of the curriculum, it has moved from…

Automatic answer assessment systems typically apply semantic similarity methods where student responses are compared with some reference answers in order to access their correctness. But student responses in dialogue based tutoring systems are often grammatically and semantically incomplete and additional information (e.g., dialogue history) is…

This paper attempts to quantify the accuracy limit of "nextitem-correct" prediction by using numerical optimization to estimate the student's probability of getting each question correct given a complete sequence of item responses. This optimization is performed without an explicit parameterized model of student behavior, but with the…

Online adaptive tutoring systems are increasingly being used in classrooms as a way to provide guided learning for students. Such tutors have the potential to provide tailored feedback based on specific student needs and misunderstandings. Bayesian knowledge tracing (BKT) is used to model student knowledge when knowledge is assumed to be changing…

In this paper, we introduce a mathematical model for collaborative learning and the answering process for multiple-choice questions. The collaborative learning model is inspired by the Ising spin model and the model for answering multiple-choice questions is based on their difficulty level. An intensive simulation study predicts the possibility of…

This paper reports the results of a research about the strategies and difficulties developed by university students in the process of modeling and simulating of random phenomena in an environment of a spreadsheet. The results indicate that students had difficulties to identify key components of the problems, which are crucial to formulate a…

Basic issues in understanding Item Response Theory (IRT), or Latent Trait Theory, measurement models are discussed. These theories have gained popularity because of their promise to provide greater precision and control in measurement involving both achievement and attitude instruments. IRT models implement probabilistic techniques that yield…

Previous work on the effects of dimensionality on parameter estimation was extended from dichotomous models to the polytomous graded response (GR) model. A multidimensional GR model was developed to generate data in one-, two-, and three-dimensions, with two- and three-dimensional conditions varying in their interdimensional associations. Test…

Two experiments reproduced in a simulated computerized test-taking situation the effect of two of the main determinants in answering an item in a test: the difficulty of the item and the time available to answer it. A model is proposed for the time to respond or abandon an item and for the probability of abandoning it or answering it correctly. In…

To generalize test results beyond the particular test administration, an examinee's ability estimate must be independent of the particular items attempted, and the item difficulty calibrations must be independent of the particular sample of people attempting the items. This stability is a key concept of the Rasch model, a latent trait model of…

The analysis of fit of data to a measurement model for graded responses is described. The model is an extension of Rasch's dichotomous model to formats which provide more than two levels of response to items. The model contains one parameter for each person and one parameter for each "step" in an item. A dichotomously-scored item…

A tailored testing model employing the beta distribution, whose mean equals the difficulty of an item and whose variance is approximately equal to the sampling variance of the item difficulty, and employing conditional item difficulties, is proposed. The model provides a procedure by which a minimum number of items of a test, consisting of a set…

The development of a probabilistic model for validating Gange's learning hierarchies is described. Learning hierarchies are defined as paired networks of intellectual tasks arranged so that a substantial amount of positive transfer occurs from tasks in a lower position to connected ones in a higher position. This probabilistic validation technique…