Mohan Chinnappan

Teachers function schemas and their role in modelling

An important element in teaching is the quality of content and pedagogical knowledge that teachers use in the design and delivery of their lessons. In this paper we present a framework for investigating how this knowledge is structured and how it relates to the mathematical modelling process. The framework is then used to compare an experienced teacher’s knowledge and teaching of functions with that of four trainee teachers. The data show that the experienced teacher has built up knowledge that is dominated by conceptual rather than procedural aspects of functions, whereas the prospective teachers have structures often lacking a strong conceptual base.

Preservice teachers’ understanding and representation of fractions in a JavaBars environment

In recent years, considerable research effort has been invested in identifying the nature of the knowledge that drives mathematics teachers’ actions in the classroom. While this investigation has generated a useful body of information, there has been little information about changes in the character of this knowledge when teaching involves the use of technology. In this paper, I address this issue by examining a group of preservice primary mathematics teachers’ understanding of fractions. The participants were required to order fractions within software called JavaBars. The results suggest that, while the preservice teachers had built up robust knowledge about fractions, they experienced difficulty in translating this knowledge in the JavaBars environment.

Using the Productive Pedagogies Framework to Build a Community of Learners Online in Mathematics Education

Productive pedagogies (PP) is an influential framework for teaching and learning that has featured well in current reforms of teacher education. The present study was designed to examine principles of PP adopted by a cohort of beginning mathematics teachers. A networked online learning environment, WebCT™, facilitated the teachers’ deliberations about PP. The uptake of PP by these new teachers was examined in the context of a collaborative activity in which the teachers were required to reflect on, critique, and share views with peers about a lesson on the teaching of multiplication. These online interactions are argued to provide opportunities for shared learning and activity. Qualitative analyses of the results show that participants espoused and embraced dimension of PP in varying degrees. The implications for using WebCT™ and similar online course management systems to support teacher reflections on best practice are discussed.

Cognitive load and modelling of an algebra problem

In the present study, I examine a modelling strategy as employed by a teacher in the context of an algebra lesson. The actions of this teacher suggest that a modelling approach will have a greater impact on enriching student learning if we do not lose sight of the need to manage associated cognitive loads that could either aid or hinder the integration of core concepts with processes that are at play. Results here also show that modelling a problem that is set within an authentic context helps learners develop a better appreciation of variables and relations that constitute the model. The teacher’s scaffolding actions revealed the use of strategies that foster the development of connected, meaningful and more useable algebraic knowledge.

The accessing of geometry schemas by high school students

In this study I examine the question, what is the nature of prior mathematical knowledge that facilitates the construction of useful problem representations in the domain of geometry? The quality of prior knowledge is analysed in terms ofschemas that provide a measure of the degree of organisation of prior knowledge. Problem-solving performance and schema activation of a group of high- and low-achieving students were compared. As expected, the high achievers produced more correct answers than the low achievers. More significantly, schema comparison indicated that the high achievers accessed more problem-relevant schemas than the low achievers. In a related task which focused on the problem diagram, both groups accessed almost equal numbers of geometry schemas. The results are interpreted as suggesting that high achievers build schemas that are qualitatively more sophisticated than low achievers which in turn helps them construct representations that are conducive to understanding the structure of geometry problems.

The use of microcomputers in the analysis of mathematical knowledge schemas

An important question about the organizational aspects of mathematical knowledge is how does the structure of the knowledge influence the accessing and subsequent use of that knowledge in mathematical tasks. In this paper we report the design and trialing of a HyperCard testing procedure known as the Recognition and Testing System (RATS) that will enable mathematics teachers and researchers to generate data relating to domain knowledge representation and its access. RATS is designed to provide three types of information. First it provides an estimate of what a student knows about an area of subject matter content. In our work we have focused on students’ knowledge of plane geometry and trigonometry, though many other areas could be examined. The program identifies and records the subject matter knowledge that can be recognized and labelled by a student. A second type of information recorded by the RATS program is the level of assistance required by the student to access these knowledge components. RATS is...

Relationship between mathematics teacher subject matter knowledge, pedagogical content knowledge and professional development needs

Two key variables emerged from the literature review is that Specific Matter Knowledge [SMK] and Pedagogical Content Knowledge [PCK] can influence the mathematics teachers’ Professional Development [PD] needs. However, the key variables of SMK and PCK that were being investigated were not defined clearly. Empirical evidence that support relationship between SMK and PD and PCK and PD were not verified. In addition, how does PCK mediate SMK and PD is not clear and somewhat lacking. Therefore, the purpose of this paper was to examine the relationship between primary mathematics teacher’s SMK, PCK and PD needs. Results of path analysis with SmartPLS indicated that the direct effect of SMK on PD was mediated via PCK. This data provide support for the claim that PD programs for future teachers of primary mathematics should be driven by a more nuanced understanding of the link between SMK and PCK.

Prospective primary teachers’ perceptions about the use of mathematics homework

ABSTRACT The benefit of homework (HW) has been the subject of ongoing debate among various stakeholders. Within this debate, and in relation to teacher development, prospective teachers’ views of HW have received limited attention. In this study, we survey primary pre-service teachers’ (‘PSTs’) views of HW use (n = 45 teaching grades 2–5; n = 39 teaching grades 6 and 7) in a single discipline (mathematics) at one large Australian university. The literature suggests students in upper primary and secondary grades will benefit from the use of mathematics HW. Results of the study, in contrast, suggest our PSTs teaching early primary grades planned to give proportionally (but not significantly) more mathematics HW than those teaching later grades. Reasons for this disparity are discussed. We argue there is a need in teacher development to equip PSTs with a better understanding of how classroom and home-based learning work together in the development of mathematical understanding.

A Review of Fully Online Undergraduate Mathematics Instruction through the Lens of Large Scale Research (2000-2015)

Abstract The use of fully online instruction has been proliferating in higher education. In undergraduate mathematics, arguments about the effectiveness of fully online instruction have been met with contrasting claims about its potential. Some assert that fully online instruction can promote a deeper and richer conceptual understanding of mathematical ideas while others claim it is difficult to teach mathematics effectively in a fully online context. The aim of this study is to examine these claims by documenting relevant empirical evidence, the findings of which we hope will inform current efforts, by both faculty and administrators, to improve fully online mathematics instruction.

Symbiosis Between Subject Matter and Pedagogical Knowledge in Geometry

Teacher knowledge that supports effective mathematics teaching has come under scrutiny alongside associated theoretical developments in the education field. Amongst these developments, the Mathematics Knowledge for Teaching (MKT) framework by Ball et al. (J Teacher Educ 59(5):389–407, 2008) has been one of the most influential. While MKT has been useful in helping us identify the knowledge strands teachers need for effective practice, the interplay among MKT’s knowledge strands during the course of teaching has received less attention. In this study, we address this issue by exploring interaction between Subject Matter Knowledge (SMK) and Pedagogical Content Knowledge (PCK) in the domain of secondary geometry. We provide results of a preliminary study of SMK and PCK in the context of a teacher teaching students how to construct and bisect an acute angle with the aid of compass and ruler only. Our analysis suggests future research needs to consider (a) the particular characteristics of the discipline of geometry and (b) the developmental knowledge trajectories of teachers of geometry in order to better understand how teachers’ SMK influences and influenced by PCK.