The Stochastic Goal Programming Model: Theory and Applications

Supported by a network of researchers and practitioners, the goal programming (GP) model is alive today more than ever and is continually fed with theoretical developments and new applications with resounding success. The standard formulation of the GP model was introduced in the earliest of 1960s, and since then, important extensions and numerous applications have been proposed. One of these variants is the stochastic GP model that deals with the uncertainty of some decision‐making situations by using stochastic calculus. In such a situation, the decision maker is not able to assess with certainty the different parameters. However, he or she can provide some information regarding the likelihood of occurrence of the decision‐making parameter values. The aim of this paper is to highlight the main methodological developments of the stochastic GP model and to present an overview of its applications in several domains. Copyright © 2012 John Wiley & Sons, Ltd.     

‘MusiMath’ and ‘Academic Music’ – Two music‐based intervention programs for fractions learning in fourth grade students

Music and mathematics require abstract thinking and using symbolic notations. Controversy exists regarding transfer from musical training to math achievements. The current study examined the effect of two integrated intervention programs representing holistic versus acoustic approaches, on fraction knowledge. Three classes of fourth graders attended 12 lessons on fractions: One class attended the ‘MusiMath’ holistic program (n = 30) focusing on rhythm within the melody. Another class attended the ‘Academic Music’ acoustic program (Courey et al., Educ Stud Math 81:251, 2012) (n = 25) which uses rhythm only. The third class received regular fraction lessons (comparison group, n = 22). Students in both music programs learned to write musical notes and perform rhythmic patterns through clapping and drumming as part of their fraction lessons. They worked toward adding musical notes to produce a number (fraction), and created addition/subtraction problems with musical notes. The music programs used a 4/4 time signature with whole, half, quarter and eighth notes. In the math lessons, the students learned the analogy between musical durations and fractions, but also practiced fractions other than . Music and math were assessed before, immediately following, and 3‐ and 6‐months post‐intervention. Pre‐ to post‐intervention analyses indicated that only the ‘MusiMath’ group showed greater transfer to intervention‐trained and untrained fractions than the comparison group. The ‘Academic Music’ group showed a trend on trained fractions. Although both music groups outperformed the comparison group 3‐ and 6‐months post‐intervention on trained fractions, only the ‘MusiMath’ group demonstrated greater gains in untrained fractions. Gains were more evident in trained than in untrained fractions. A video abstract of this article can be viewed at https://youtu.be/uJ_KWWDO624