Describing the plastic deformation of aluminium softball bats

Hollow aluminium bats were introduced over 30 years ago to provide improved durability over wooden bats. Since their introduction, however, interest in hollow bats has focused almost exclusively around their hitting performance. The aim of this study was to take advantage of the progress that has been made in predicting bat performance using finite elements and apply it to describe bat durability. Accordingly, the plastic deformation from a ball impact of a single-wall aluminum bat was numerically modelled. The bat deformation from the finite-element analysis was then compared with experiment using a high-speed bat test machine. The ball was modelled as an isotropic, homogeneous, viscoelastic sphere. The viscoelastic parameters of the ball model were found from instrumented, high-speed, rigid-wall ball impacts. The rigid-wall ball impacts were modelled numerically and showed good agreement with the experimentally obtained response. The strain response of the combined bat-ball model was verified with a strain-gauged bat at intermediate ball impact speeds in the elastic range. The strain response of the bat-ball model exhibited positive correlation with the experimental measurements. High-speed bat-ball impacts were performed experimentally and simulated numerically at increasing impact speeds which induced correspondingly increased dent sizes in the bat. The plastic deformation from the numerical model found good agreement with experiment provided the aluminium work hardening and strain rate effects were appropriately described. The inclusion of strain rate effects was shown to have a significant effect on the bat deformations produced in the finite-element simulations. They also helped explain the existence of high bat stresses found in many performance models.     

Understanding a High School Physics Teacher’s Pedagogical Content Knowledge of Argumentation

Scientific argumentation is an important learning objective in science education. It is also an effective instructional approach to constructivist science learning. The implementation of argumentation in school settings requires science teachers, who are pivotal agents of transforming classroom practices, to develop sophisticated knowledge of argumentation. However, there is a lack of understanding about science teachers’ knowledge of argumentation, especially the dialogic meaning of argumentation. In this case study, we closely examine a high school physics teacher’s argumentation-related pedagogic content knowledge (PCK) in the context of dialogic argumentation. We synthesize the teacher’s performed PCK from his argumentation practices and narrated PCK from his reflection on the argumentation practices, from which we summarize his PCK of argumentation from the perspectives of orientation, instructional strategies, students, curriculum, and assessment. Finally, we describe the teacher’s perception and adaption of argumentation in his class. We also identity the barriers to argumentation implementation in this particular case and suggest solutions to overcome these barriers.     

Socio-economic status,cultural diversity and the aspirations of secondary students in the Western Suburbs of Melbourne,Australia

Using data from a recent survey of Australian secondary students, we find that those from higher socio-economic backgrounds are more likely to aspire to attend university. The same can be said for students who do not speak English at home. We find that students with an ethnic minority background are more likely to perceive higher levels of support from parents. However, we find that all students believe they receive encouragement from their parents to do well at school (rather than discouragement or disinterest), and that there is little difference in the level of importance placed on the views of parents between students from English and non-English speaking background. While interest in university education is strong across all socio-economic groups, particularly for students who do not speak English at home, there is a considerable gap between aspirations and enrolment levels. We suggest that this ‘aspirations gap’ is larger for students from low socio-economic backgrounds. This analysis also supports growing evidence that the postcode methodology for allocating socio-economic status to individuals is unreliable.     

Types of reasoning in 3D geometry thinking and their relation with spatial ability

The aim of this study is to describe and analyse the structure of 3D geometry thinking by identifying different types of reasoning and to examine their relation with spatial ability. To achieve this goal, two tests were administered to students in grades 5 to 9. The results of the study showed that 3D geometry thinking could be described by four distinct types of reasoning which refer to the representation of 3D objects, spatial structuring, conceptualisation of mathematical properties and measurement. The analysis of the study also showed that 3D geometry types of reasoning and spatial abilities should be modelled as different constructs. Finally, it was concluded that students’ spatial abilities, which consist of spatial visualisation, spatial orientation and spatial relations factors, are a strong predictive factor of the four types of reasoning in 3D geometry thinking.     

An empirical study of using history as a ‘goal’

This article discusses an empirical study on the use of history as a goal. A historical module is designed and implemented in a Danish upper secondary class in order to study the students’ capabilities at engaging in meta-issue discussions and reflections on mathematics and its history. Based on videos of the implementation, students’ hand-in essay assignments, questionnaires, and follow-up interviews, the conditions, sense, and extent to which the students are able to perform such discussions and reflections are analyzed using a described theoretical framework.     

Thinking aloud together: A teacher’s semiotic mediation of a whole-class conversation about percents

How does classroom interaction support students’ apprenticeship into the ways of speaking, writing, and diagramming that constitute the practice of mathematics? We address this problem through an interpretative analysis of a whole-group conversation about alternative ways of solving a problem involving percent discounts that occurred in a sixth grade classroom. This research study draws upon Dewey’s theory of inquiry, Vygotsky’s cultural–historical psychology, Freudenthal’s realistic mathematics education, and Halliday’s systemic functional linguistics (SFL). From Freudenthal, we borrow the notions of mathematizing and guided reinvention—the former notion offers a view of mathematics as an activity of structuring subject matter and the latter one provides insights into the processes whereby mathematizing is learned and taught in the classroom. We glean from Dewey his view of reflective thinking as inquiry and the role that conversations may serve therein. We rely upon Vygotsky’s notions of a verbal thinking plane and a social phase of learning in order to reconsider the function of whole-class interaction in apprenticing students into mathematizing. Finally, SFL provides us with tools for explaining the choices of grammar and vocabulary students and teachers make as they realize meanings in whole-group conversations. Treating the selected whole-class conversation as a text, we focus our analysis on how this text came to mean what it did. Our central questions are as follows: What meanings were realized in the whole-class conversation by teacher and students and how were these meanings realized? How did the teacher’s lexico-grammatical choices guide the students’ choices? In addressing these questions, we advance an interpretation of the conversation as paradigmatic of students and teacher thinking aloud together about percents.     

To see or not to see: analyzing difficulties in geometry from the perspective of visual perception

In this paper, we consider theories about processes of visual perception and perception-based knowledge representation (VPR) in order to explain difficulties encountered in figural processing in junior high school geometry tasks. In order to analyze such difficulties, we take advantage of the following perspectives of VPR: (1) Perceptual organization: Gestalt principles, (2) recognition: bottom-up and top-down processing; and (3) representation of perception-based knowledge: verbal vs. pictorial representation, mental images and hierarchical structure of images. Examples given in the paper were mostly taken from Gal's study (2005) which aimed at identifying and analyzing Problematic Learning Situations (after Gal & Linchevski, 2000) in junior high school geometry classes. Gal's study (2005) suggests that while this theoretical perspective became part of teachers' pedagogic content knowledge, the teachers were aware of their students' thinking processes and their ability to analyze and cope with their students’ difficulties in geometry was improved.     

The transitional stages in the PhD degree in mathematics in terms of students’ motivation

This paper presents results of a longitudinal study in the transition to independent graduate studies in mathematics. The analysis of the data collected from 24 students doing a PhD in mathematics revealed the existence of three transitional stages within the PhD degree, namely Adjustment, Expertise and Articulation. The focus is on the first two transitional stages, since the data collection focused mainly on these. Based on the first two transitional stages and the students’ ways of dealing with them, which were called ‘survival strategies’, three types of students were identified. The importance of motivation for each transitional stage and the successful transition overall are considered as well.     

Investigating further preservice teachers’ conceptions of multidigit whole numbers: refining a framework

This study was designed to investigate preservice elementary school teachers’ (PSTs’) responses to written standard place-value-operation tasks (addition and subtraction). Previous research established that PSTs can often perform but not explain algorithms and provided a four-category framework for PSTs’ conceptions, two correct and two incorrect. Previous findings are replicated for PSTs toward the end of their college careers, and two conceptions are further analyzed to yield three categories of incorrect views of regrouped digits: (a) consistently as 1 value (all as 1 or all as 10), (b) consistently within but not across contexts (i.e., all as 10 in addition but all as 1 in subtraction), and (c) inconsistently (depending on the task).