This article presents the partial results obtained in the first stage of the research, which sought to answer the following questions: (a) What is the role of intuition in university students' solutions to optimization problems? (b) What is the role of rigor in university students' solutions to optimization problems? (c) How is the combination of…

Previous research suggests that vocational skills training for individuals with autism may increase the likelihood that they may be effectively employed. In the present study, a multiple-baseline-across-participants design was used to assess the effects of graduated guidance, scripts, and script fading to teach four adolescents with autism in a…

Traditional mathematics education focuses on teaching rote procedures to solve problems, though these procedures are not usually motivated by goals. As a result, students have trouble flexibly using procedures and generalizing their knowledge to solve novel problems that differ from the problems they practice during instruction. In the following…

What is the role of patterns in developing algebraic reasoning? This important question deserves thoughtful attention. In response, this article examines some differing views of algebraic reasoning, discusses a controversy regarding patterns, and describes how three types of patterns--in contextual problems, in growing geometric figures, and in…

The traveler's dilemma is a generalization of the prisoner's dilemma which shows clearly a paradox of game theory. In the traveler's dilemma, the strategy chosen by analysis and theory seems obviously wrong intuitively. Here we develop a measure of evolutionary stability and show that the evolutionarily stable equilibrium is in some sense not very…

A. Lobb discovered an interesting generalization of Catalan's parenthesization problem, namely: Find the number L(n, m) of arrangements of n + m positive ones and n - m negative ones such that every partial sum is nonnegative, where 0 = m = n. This article uses Lobb's formula, L(n, m) = (2m + 1)/(n + m + 1) C(2n, n + m), where C is the usual…

This note presents demonstrations of mathematics that emerges when problems are posed with respect to a combined 12 x 12 multiplication table showing multiplier and multiplicand. Through processes such as recognizing and extending patterns, specializing and generalizing particular functional relationships between the diagonal and row sequences are…

A generalization of a well-known result for the arctangent function poses a number of interesting questions concerning the existence of integer solutions of related problems.

Problem solving allows students to use what they know to achieve a goal when no solution is apparent. Traditional educational models evolved from an earlier system, based on rote memorization and designed to produce employees for industry. The workforce of tomorrow must move beyond rote learning by both applying current knowledge and using…

Problem-solving instruction facilitates children in becoming successful real-world problem solvers. Research that incorporates problem-solving instruction has been limited for students with mild and moderate intellectual disabilities. However, this population of students needs increased opportunities to learn the skills of problem solving. Using a…

Script concordance test (SCT) is a case based assessment format of clinical reasoning in which questions are nested into several cases. Recent results using Q4 format suggest that nested questions contribute more to reliability of measure than cases. The present study aims at documenting variance components associated with SCT cases and nested…

Positive sums count. Alternating sums match. Alternating sums of binomial coefficients, Fibonacci numbers, and other combinatorial quantities are analyzed using sign-reversing involutions. In particular, we describe the quantity being considered, match positive and negative terms through an Involution, and count the Exceptions to the matching rule…

We are currently at an exciting juncture in developing effective means for assessing so-called 21st-century skills in an innovative yet reliable fashion. One of these avenues leads through the world of "epistemic games" (Shaffer, 2006a), which are games designed to give learners the rich experience of professional practica within a discipline.…

Differentiation and integration played large roles within classic developmental theories but have been relegated to obscurity within contemporary theories. However, they may have a useful role to play in modern theories as well, if conceptualized as guiding principles for analyzing change rather than as real-time mechanisms. In the present study,…

The note introduces sequences having M-bonacci property. Two summation formulas for sequences with M-bonacci property are derived. The formulas are generalizations of corresponding summation formulas for both M-bonacci numbers and Fibonacci numbers that have appeared previously in the literature. Applications to the Arithmetic series, "m"th "g -…