A 1960 "Mathematical Games" column describes the problem, now known as the Secretary Problem, which asks how someone interviewing candidates for a position should maximize the chance of hiring the best applicant. This note looks at how an applicant should respond, if they know the interviewer uses this optimal strategy. We show that all but the…

Mathematics and proof are two closely related concepts. Mathematics not only shows what is right or wrong, but it also teaches that it is not enough to know the latest formulas and results should be explained with causality. In this context, students learn the underlying meaning behind what mathematicians do by way of proofs. Accordingly, this…

Multi-voxel pattern recognition techniques combined with Hidden Markov models can be used to discover the mental states that people go through in performing a task. The combined method identifies both the mental states and how their durations vary with experimental conditions. We apply this method to a task where participants solve novel…

This paper will offer an analysis from a theoretical point of view of mathematical modelling, applications and inverse problems of both causation and specification types. Inverse modelling problems give the opportunity to establish connections between theory and practice and to show this fact, a simple linear algebra example in two different…

In an earlier paper, we presented methods for estimating achievement gaps when test scores are coarsened into a small number of ordered categories, preventing fine-grained distinctions between individual scores. We demonstrated that gaps can nonetheless be estimated with minimal bias across a broad range of simulated and real coarsened data…

Ho and Reardon (2012) present methods for estimating achievement gaps when test scores are coarsened into a small number of ordered categories, preventing fine-grained distinctions between individual scores. They demonstrate that gaps can nonetheless be estimated with minimal bias across a broad range of simulated and real coarsened data…

We investigated human performance on the Euclidean Traveling Salesperson Problem (TSP) and Euclidean Minimum Spanning Tree Problem (MST-P) in regards to a factor that has previously received little attention within the literature: the spatial distributions of TSP and MST-P stimuli. First, we describe a method for quantifying the relative degree of…

Rhetoric about the importance of students being equipped to apply mathematics to relevant problems arising in their lives, individually, as citizens, and in the workplace has never been matched by serious policy or curricular support. This paper identifies and elaborates authenticity implications for addressing this issue, and describes aspects of…

The authors illustrate three basic types of reasoning used in mathematics by showing how they operate in practical and mathematical situations. The importance and function of the different types of reasoning in each situation is outlined. As a consequence the authors note that while introducing new techniques by example is good from a pedagogical…

A concept of major importance in human electrophysiology studies is the process by which activation of an excitable cell results in a rapid rise and fall of the electrical membrane potential, the so-called action potential. Hodgkin and Huxley proposed a model to explain the ionic mechanisms underlying the formation of action potentials. However,…

This paper demonstrates the use of programing software that provides the student programmer visual cues to construct the code to a student programming assignment. This method does not disregard or minimize the syntax or required logical constructs. The student can concentrate more on the logic and less on the language itself.

Grading practices can send a powerful message to students about course expectations. A study by Henderson et al. ("American Journal of Physics" 72:164-169, 2004) in physics education has identified a misalignment between what college instructors say they value and their actual scoring of quantitative student solutions. This work identified three…

Thirty-six secondary school students aged 14-16 were interviewed while they chose between a table, a graph or a formula to solve three linear function problems. The justifications for their choices were classified as (1) task-related if they explicitly mentioned the to-be-solved problem, (2) subject-related if students mentioned their own…

This study was designed according to the mixed research method in which quantitative and qualitative research methods were used in order to identify the challenges confronted by classroom teacher candidates in solving mathematical problems and the factors affecting how they choose these representations. The population of this study consisted of…

Most mathematics assignments consist of a group of problems requiring the same strategy. For example, a lesson on the quadratic formula is typically followed by a block of problems requiring students to use the quadratic formula, which means that students know the appropriate strategy before they read each problem. In an alternative approach,…