A framework for teachers’ knowledge of mathematical reasoning

Exploring and developing primary teachers’ understanding of mathematical reasoning was the focus of the Mathematical Reasoning Professional Learning Research Program. Twenty-four primary teachers were interviewed after engagement in the first stage of the program incorporating demonstration lessons focused on reasoning conducted in their schools. Phenomenographic analysis of interview transcripts exploring variations in primary teachers’ perceptions of mathematical reasoning revealed seven categories of description based on four dimensions of variation, establishing a framework to evaluate development in understanding of reasoning

Revealing conceptions of rate of change

Rate of change is an important mathematical concept. Research referring to students&rsquo; difficulties with this concept spans more than twenty years. Research suggests that problems experienced by some calculus students are likely a result of pre-existing limited or incorrect conceptions of rate of change. This study investigated 23 Victorian Year 10 students&rsquo; understanding of rate as revealed by phenomenographic analysis of interviews. Eight conceptions of rate of change emerged. Four important aspects of the concept were identified and gaps in students&rsquo; thinking defined. In addition, the employment of phenomenography, to reveal conceptions of rate, is described in detail.<br /

Impact of context and representation on year 10 students\u27 expression of conceptions of rate

Rate is an important, but difficult mathematical concept. More than twenty years of research, especially with calculus students, report difficulties with this concept. This paper reports on an alternative analysis, from the perspective of multiple representations and context, of interviews probing twenty Victorian Year 10 students&rsquo; conceptions of rate. This analysis shows that multiple representations of functions provide different rate-relatedinformation for different students. Understandings of rate in one representation or context are not necessarily transferred to another representation or context

Gesture types for functions

This paper reports on the different gesture types employed by twenty-three Year 10 students as they endeavoured to explain their understanding of rate of change associated with the functions resulting from two different computer simulations. These gestures also have application to revealing students&rsquo; understanding of functions. However, interpretation of gesture is problematic but classification of gestures assisted in the analysis of the videorecorded interviews probing participants&rsquo; conceptions of rate of change. This paper builds on the classifications reported in previous research. Five additional gesture types are presented, which provide insights into students&rsquo; thinking about rate of change, and hence functions

Where is the rate in the rule?

A well-developed understanding of rate is foundational to conceptual understanding of introductory calculus. Many students achieve procedural competence with the application of rules for differentiation without developing an awareness of the connection between derivative and rate. In addition, rate-related reasoning is needed to make informed decisions in many everyday applications of rate. This paper reports on additional data collected during interviews for a project investigating the different ways rate may be experienced by pre-calculus students. Many researchers (for example Kaput, 1999) have suggested that the conceptual understanding of function may be enhanced through the presentation and exploration of multiple representations of a variety of functions. In this paper, one section of each interview is considered in detail to evaluate the participants&rsquo; understanding in a specific rate context. Participants were asked to discuss a dynamic geometry simulation of a blind on two different windows one rectangular and the other not. Detailed analysis of the video-record of each participant&rsquo;s interview provides insights into their perceptions of rate in several different representations. In the sections below, the conceptual framework is described; details of the interviews and the computer-based simulation are provided; and the analysis of the data is discussed.<br /

Overcoming Challenges in Assessing Mathematical Reasoning

Despite mathematical reasoning being necessary for in-depth understanding of mathematical concepts, many teacher experience difficulty in assessing it. Data were collected from 34 primary teachers at 4 Victorian government schools at two post- lesson reflective sessions following lessons with a focus on reasoning. These sessions facilitated teachers’ collaborative efforts to assess their students’ reasoning from students’ work samples. The data included transcripts of all the reflective sessions; written work samples; and associated completed rubrics. Analysis of these data enabled identification of seven challenges teachers experienced in assessing reasoning: Limited guidance provided by curriculum documents; Teachers’ knowledge of reasoning; Teacher noticing and interpretation of student reasoning; Students’ difficulties in articulating their reasoning; Assessing progress in reasoning; Inadequacy of work samples; and Challenges in tracking and reporting student progress in reasoning. The discussion presents strategies to overcome these challenges

Opportunities to promote mathematical content knowledge for primary teaching

Understanding the development of pre-service teachers&rsquo; mathematical content knowledge (MCK) is important for improving primary mathematics&rsquo; teacher education. This paper reports on a case study, Rose , and her opportunities to develop MCK during the four years of her program. Program opportunities to promote MCK when planning and practicing primary teaching included: coursework experiences and responding to assessment requirements. Discussion includes the Knowledge Quartet: foundation knowledge, transformation, connection and contingency. By fourth-year, Rose demonstrated development of different categories of MCK when practicing her teaching because of her program experiences

Potential of technology and a familiar context to enhance students\u27 concept of rate of change

Students\u27 concept image of rate of change may be incomplete or erroneous. This paper reports a pilot study, with secondary school students, which explores the potential of technology (JavaMathWorlds), depicting a familiar context of motion, to develop students\u27 existing schema of informal understandings of rate of change to more formal mathematical representations. Students developed numerous \u27models of\u27 rate of change in a motion context which then transferred to serve as a \u27model for\u27 rate of change in other contexts.<br /

Novices no longer : computer education for rural adults

This paper reports on &quot;Introduction to Computer&quot; classes conducted in Ballarat, Victoria as part of Adult Learners\u27 Week, 2002. It outlines the background to the classes, topics covered, participants\u27 reflections and further actions taken. The paper reveals the social and learning outcomes experienced by adults who participated in the computer classes. In addition, it explains the role of Graduate Diploma, Secondary, Information Technology Education students in planning and evaluating their teaching practice.<br /