A software framework based on a conceptual unified model for evolutionary multiobjective optimization: ParadisEO-MOEO

This paper presents a general-purpose software framework dedicated to the design and the implementation of evolutionary multiobjective optimization techniques: ParadisEO-MOEO. A concise overview of evolutionary algorithms for multiobjective optimization is given. A substantial number of methods has been proposed so far, and an attempt of conceptually unifying existing approaches is presented here. Based on a fine-grained decomposition and following the main issues of fitness assignment, diversity preservation and elitism, a conceptual model is proposed and is validated by regarding a number of state-of-the-art algorithms as simple variants of the same structure. This model is then incorporated into the ParadisEO-MOEO software framework. This framework has proven its validity and high flexibility by enabling the resolution of many academic, real-world and hard multiobjective optimization problems.     

Obstacles and challenges in preservice teachers’ explorations with fractions: A view from a small-scale intervention study

In this study, we implemented one-on-one fractions instruction to eight preservice teachers. The intervention, which was based on the principle of Progressive Formalization (Freudenthal, 1983), was centered on problem solving and on progressively formalizing the participants’ intuitive knowledge of fractions. The objectives of the study were to examine the potential effects of the intervention and to uncover specific difficulties experienced by the preservice teachers during instruction. Results revealed improvement on one measure of conceptual knowledge, but not on a transfer task, which required the teachers to generate word problems for number sentences involving fractions. In addition, the qualitative analysis of the videotaped instructional sessions revealed a number of cognitive obstacles encountered by the participants as they attempted to construct meaningful solutions and represent those solutions symbolically. Based on the findings, specific suggestions for modifying the intervention are provided for mathematics teacher educators.     

Envisioning the infinite by projecting finite properties

We analyze interviews with 24 post-secondary students as they reason about infinite processes in the context of the tricky Tennis Ball Problem. By metaphorically projecting various properties from the finite states such as counting and indexing, participants envisioned widely varying final states for the infinite process. Depending on which properties they attended to, some participants recognized contradictions arising from these infinite projections and sought to resolve the contradictions in non-dismissive ways. The findings indicate that increasing their meta-level awareness of the features of their mental constructions may help students overcome common difficulties with the infinite.     

Fifth graders’ additive and multiplicative reasoning: Establishing connections across conceptual fields using a graph

This paper looks at 21 fifth grade students as they discuss a linear graph in the Cartesian plane. The problem presented to students depicted a graph showing distance as a function of elapsed time for a person walking at a constant rate of 5 miles/h. The question asked students to consider how many more hours, after having already walked 4 h, would be required to reach 35 miles. To answer this question, the students needed to extend the graph that was presented, either mentally or on paper, as the axes did not go up to 7 h or 35 miles. They also needed to be able to consider not only the total number of hours to reach 35 miles, but also the interval of time after 4 h. The purpose of this paper is to consider the student responses from the viewpoint of multiplicative and additive reasoning, and specifically within Vergnaud's framework of multiplicative and additive conceptual fields and scalar and functional approaches to linear relationships (Vergnaud, 1994). The analysis shows that: some student answers cannot be classified as either scalar or functional; some students combined several kinds of approaches in their explanations; and that the representation of the problem using a graph may have facilitated responses that are different from those typically found when the representation presented is a function table.     

What does it mean for a student to understand the first-year calculus? Perspectives of 24 experts

This article presents the views of 24 nationally recognized authorities in the field of mathematics, and in particular the calculus, on student understanding of the first-year calculus. A framework emerged that includes four overarching end goals for understanding of the first-year calculus: (a) mastery of the fundamental concepts and-or skills of the first-year calculus, (b) construction of connections and relationships between and among concepts and skills, (c) the ability to use the ideas of the first-year calculus, and (d) a deep sense of the context and purpose of the calculus. Each end goal for student understanding is explored in detail and the potential for using the framework as an organizational tool is discussed.     

Observation‐based missing data methods for exploratory data analysis to unveil the connection between observations and variables in latent subspace models

This paper introduces a class of methods to infer the relationship between observations and variables in latent subspace models. The approach is a modification of the recently proposed missing data methods for exploratory data analysis (MEDA). MEDA is useful to identify the structure in the data and also to interpret the contribution of each latent variable. In this paper, MEDA is augmented with dummy variables to find the data variables related to a given deviation detected among observations, for instance, the difference between one cluster of observations and the bulk of the data. The MEDA extension, referred to as observation‐based MEDA or oMEDA, can be performed in several ways, one of which is theoretically shown to be equivalent to a comparison of means between groups. The use of the proposed approach is demonstrated with a number of examples with simulated data and a real data set of archeological artifacts. Copyright © 2011 John Wiley & Sons, Ltd.     

Fractions as Subtraction: An Activity‐Oriented Perspective from Elementary Children

A sample of third‐, fourth‐, and fifth‐grade student responses to the question “What is a fraction?” were examined to gain an understanding of how children in upper elementary grades make sense of fractions. Rather than measure children's understanding of fractions relative to mathematically conventional part–whole constructions of fractions, we attempted to understand children's actions and processes. A small but nontrivial group of children used subtraction (takeaway and removal) as a framework for understanding how fractions were created and written. An analysis of the content of their responses as well as a comparison of the performance of these children with that of children who used other ways of describing fractions suggests that the use of subtraction may be a reasonable (or at least not harmful) way for children to begin to access concepts related to fractions. Also, this study suggests that attention to children's understanding through the lens of children's activity might reveal ways of thinking and insights that are masked when we compare children's thinking in more structured research settings.     

Investigating statistical approximations with computer graphics

This paper describes the use of interactive computer graphics software to explore the notion of the approximation of one statistical distribution by another. In particular, the authors illustrate how such programs can be used to investigate the normal approximation to the binomial distribution, the Poisson approximation to the binomial distribution, the convergence of t‐distributions to the normal distribution, and the limiting behaviour of the chi‐squared distribution. The discussions include possible methods using such programs, either in a classroom setting on a demonstration basis or on an individual student basis for independent study in a computer laboratory setting.     

Experienced engineers becoming mathematics teachers: preliminary perceptions of mathematics teaching

Engineers who choose to change careers and become mathematics teachers are a specific group as far as their mathematics learning in the context of engineering and their previous work experience are concerned. Regarding mathematics, they mainly engaged in applied mathematics associated with engineering, which is a highly practical field. This research explores experienced engineers’ perceptions of mathematics teaching-related topics, before starting their studies in a pre-service mathematics teacher preparation programme. This research explores their perceptions of mathematics as a discipline, mathematics teaching and mathematical understanding. The qualitative research involves three mechanical engineers, two industrial management engineers, and an electrical engineer. Semi-structured interviews were conducted before the beginning of the programme, and analysed qualitatively. The participants view engineering as an applied and changing discipline while perceiving mathematics as closed, rigorous, accurate, systematic, theoretical and as a tool for engineering. They mostly address general features of mathematics teaching while expressing a more multifaceted view of mathematical understanding. Due to the specific characteristics of the participants, this study may contribute to planning mathematics teacher preparation programmes for engineers.     

The numerical evaluation of the error term in Gaussian quadrature rules

A method for the numerical evaluation of the error term in Gaussian quadrature rules is derived by means of Chebyshev polynomials of the first kind.