Open ended tasks and barriers to learning : teachers\u27 perspectives

Examines barriers to learning mathematics when using an open-ended task teaching strategy. Features of an open-ended task; Benefits of an open-ended task; Problems concerning multiplicity of responses and contexts in an open-ended task.<br /

Planning and teaching mathematics lessons as a dynamic, interactive process

We are researching actions that teachers can take to improve mathematics learning for all students, with particular attention to specific groups of students who might experience difficulty. After identifying possible barriers to learning, we offered teachers mathematics lessons structured in a particular way. Teachers&rsquo; use of the model outlined in this paper seemed productive and their resulting planning and teaching proved to be dynamic and interactive. This paper uses excerpts from a conversation between two teachers to illustrate specific aspects of the model.<br /

Making the pedagogic relay inclusive for indigenous Australian students in mathematics classrooms

Many students are unsuccessful in the study of school mathematics, not because of some innate ability, but because of pedagogical practices. Bernstein (1996) has argued that pedagogy serves as a mechanism for cultural reproduction, so that for those students whose cultures are different from that represented in and through pedagogy, the task of constructing school mathematics is made more difficult. The paper explores the ways in which a teacher changes the pedagogic relay in order to be more inclusive of her students. Her practice is informed by understanding the ways in which pedagogy is a subtle tool for marginalization in mathematics. <br /

Alternative learning trajectories

Lesson planning usually involves the generation of a hypothetical learning trajectory. This paper illustrates a teaching strategy that is one focus of a major research project. Alternative learning trajectories with different entry level prompts were used to enable students to access the concepts and procedures necessary for their joining the main learning trajectory. The strategy is being trialled in primary classrooms that have a large proportion of lower SES students, with the aim of maximising success in mathematics for all students.<br /

Keeping all students on the learning path

A powerful notion to guide thinking about whole-class mathematics teaching is Vygotsky&rsquo;s zone of proximal development (ZPD). Our research with primary and secondary teachers over the last six years has identified roles of teachers in relation to the ZPD, and ways of overcoming some typical barriers to students&rsquo; movement through their zones. Methods have included focus groups of experts, video analysis of classroom interactions, classroom observation, and analysis of lesson plans and teachers&rsquo; reflections teaching processes their outcomes. The research has involved the gradual development, trailing, evaluation, and adjustment of a six-component model for planning and teaching mathematics. The focus of this paper is on the use of one of its components, &ldquo;differentiated learning trajectories&rdquo;.<br /

Developing guidelines for teachers helping students experiencing difficulty in learning mathematics

As part of an ongoing project, we have developed a model of planning and teaching that is designed to assist teachers to help students overcome barriers they might experience in learning mathematics. The following is a discussion of one aspect of the model that we term &ldquo;enabling prompts&rdquo;. These refer to the directions, invitations, or questions that a teacher offers when interacting one-on-one with students experiencing difficulties. We argue that teachers should plan to pose subsidiary questions in the first instance, rather than, for example, offering further explanations. We outline our overall planning and teaching model, we present some examples of enabling prompts used by our project teachers, and we propose some considerations for teachers when structuring their own enabling prompts.<br /

Alternative Learning Trajectories

Lesson planning usually involves the generation of a hypothetical learning trajectory. This paper illustrates a teaching strategy that is one focus of a major research project. Alternative learning trajectories with different entry level prompts were used to enable students to access the concepts and procedures necessary for their joining the main learning trajectory. The strategy is being trialled in primary classrooms that have a large proportion of lower SES students, with the aim of maximising success in mathematics for all students. Defining Learning Trajectories A variety of images for teachers&apos; roles in scaffolding learning have been presented. In describing quality teaching, Wood (1991, p. 109) used the term &quot;leading by following&quot;, noting that the most effective scaffolding draws on the interests and understandings of the child. Cobb and McClain (1999) described an instructional sequence that follows a conjectured learning trajectory that &quot;culminates with the mathematical ideas that constitute our overall instructional intent&quot; (p. 24). Simon (1995) demonstrated how the continually changing knowledge of the teacher creates change in expectations of how students might learn a specific idea. A hypothetical learning trajectory provides the teacher with a rationale for choosing a particular instructional design; thus, I (as a teacher) make my design decisions based on my best guess of how learning might proceed. This can be seen in the thinking and planning that preceded my instructional interventions … as well as the spontaneous decisions that I make in response to students &apos; thinking. (pp. 135-136) Simon used the word &quot;hypothetical&quot; to suggest that all three parts of the trajectory are likely to be somewhat flexible, with teachers changing the learning goals and adapting 37

The contexts of mathematics tasks and the context of the classroom: are we including all students?

Mathematics teachers are encouraged to use realistic contexts in order to make mathematics more meaningful and accessible for all students. However, the focus group research reported in this article shows that decisions on the suitability of contexts are complex and multidimensional. Similarly, the way the task contexts are presented, and the way the tasks are incorporated into classroom routines have potential to alienate some groups of students. We suggest that teachers and researchers should be sensitive to difficulties that students might experience as a result of both the task and classroom contexts, and take specific steps to avoid or overcome the difficulties.<br /

Recommendations for measuring tennis racket parameters

Tennis rackets have advanced significantly since the invention of the game in 1874, including innovations in both shape and materials. Advances in these design parameters have implications for racket performance, especially swing speed. This study tested one hundred rackets, spanning brands and eras, using simple, portable instruments in order to pilot protocols and make recommendations for streamlining testing procedures for tennis rackets. A wide range of properties were measured and documented for each racket. We suggest that since Transverse and Lateral Moment of Inertia are well correlated, measuring both is not necessary when processing a large number of rackets. In addition, it is also possible to predict the Transverse Moment of Inertia well from models that use simple dimension and mass measurements, which may be preferable in larger studies. Exploring the use of more complex modelling will allow us to better understand the impact of tennis racket design on performance in the future

Recommendations for estimating the moments of inertia of a tennis racket

Tennis racket properties are of interest to sports engineers and designers as it allows them to evaluate performance, review trends and compare designs. This study explored mathematical models that correlated to the mass moments of inertia of a tennis racket, both about an axis through the butt and about the longitudinal axis, using its dimensions, mass and centre of mass location. The models were tested on 416 rackets, dating from 1874 to 2017. Results showed that moments of inertia about the butt and longitudinal axis can be estimated to within − 4 to 5% and − 11 to 12% of measured values, respectively, using the proposed models on original rackets. When rackets were customised, with 30 g of additional mass, moment of inertia about the butt could be estimated within 6%, but the model for moment of inertia about the longitudinal axis was less accurate (largest error at 25%). A Stepwise Linear Regression model indicated that racket mass and then centre of mass location had the largest effect on moment of inertia about the handle, with head width having the largest effect on moment of inertia about the longitudinal axis