Michael O. J. Thomas

Verified

Michael verified their affiliation via an institutional email.

Learn more

Verified

Michael verified their affiliation via an institutional email.

Learn more

• PhD
• Professor Emeritus at University of Auckland

The literature on gestures describes how they often comprise iconic, deictic and metaphoric dimensions, but the interplay between these dimensions can be very subtle and nuanced. Due to the abstract nature of the subject, the use of gestures in the learning of mathematics means that the metaphoric dimension is often prominent. However, iconic and d...

Many beginning university students struggle with the new approaches to mathematics that they find in their courses due to a shift in presentation of mathematical ideas, from a procedural approach to concept definitions and deductive derivations, and ideas building upon each other in quick succession. This paper highlights this struggle by consideri...

Several theories have been proposed to describe the transition from process to object in mathematical thinking. Yet, what is the nature of this “object” produced by the “encapsulation” of a process? Here, we outline the development of some of the theories (including Piaget, Dienes, Davis, Greeno, Dubinsky, Sfard, Gray, and Tall) and consider the na...

The role that visualization plays in mathematical learning is still unclear in spite of much research. In this research study, a group of students with mathematics difficulties was divided in two on the basis of their visuo-spatial abilities. There was no difference between the mathematical performance of the two groups when assessed on standard ma...

Objective: To examine whether or not university mathematics students semantically process gestures depicting mathematical functions (mathematical gestures) similarly to the way they process action gestures and sentences. Semantic processing was indexed by the N400 effect. Results: The N400 effect elicited by words primed with mathematical gestur...

In this chapter, I will consider my personal experience of many years of collaborative work in research and teaching with mathematicians. The phrase “mind the gap”, originally arising in 1968 from the gap between the platform and the train on the London underground, has become synonymous with the idea of avoiding the danger inherent in such a gap....

The use of digital technology has the potential to support students’ understanding in the mathematics classroom with the teacher playing a vital role. However, teaching with digital technology is not trivial, especially for teachers who are new to this. In this paper, we present an analysis of the enactment of a function lesson of a Sri Lankan math...

In this paper we follow a hypothetical mathematician who is working on a problem that is eventually solved. We treat this problem as if it were difficult for the mathematician. In following the mathematician’s work, we note both what she does and what she doesn’t do in the process. By the latter, we consider the times when progress is not being mad...

This study set out to evaluate an intervention that introduced a period of non-routine problem-solving into tertiary STEM lectures at four tertiary institutions in New Zealand for 683 students. The aim was twofold: to attempt to increase student engagement and to introduce them to the kind of domain-free abstract reasoning that involves critical, c...

This study describes an intervention that introduced a period of solving non-routine problems into tertiary STEM lectures. The aim was twofold: to attempt to increase student engagement and to introduce them to the kind of domain-free abstract reasoning that involves critical, creative and innovative thinking. The study involved over 600 STEM parti...

Article is Open access at: https://link.springer.com/article/10.1007%2Fs11858-020-01196-0 In this survey paper, we describe the state of the field of research on teaching mathematics with technology with an emphasis on the secondary school phase. We synthesize themes, questions, results and perspectives emphasized in the articles that appear in t...

This study involved an intervention designed to examine whether employability prospects for STEM students studying mathematics could be improved. It incorporated use of non-routine problem solving in second year mathematics courses at two New Zealand universities. From a theoretical standpoint, we conceptualised a novel construct called lateral thi...

The primary purpose of this study is to investigate students' perceptions about the characteristics of creativity and engagement in solving non-routine problems. It involved 64 science, technology, engineering, and mathematics (STEM) university students, who participated in a two-year research project in New Zealand during which participants were g...

A crucial step in improving the use of digital technology (DT) for learning mathematical concepts in the classroom appears to be increasing teacher involvement in task development. Hence, this research considered the effect of a professional development (PD) programme designed to assist teachers with DT task production for implementation in their c...

Proof has a prominent place in the linear algebra curriculum, teaching and learning but in first-year courses it continues to be challenging for both instructors and students. While an introduction to new concepts through definitions and theorems adds to the complexity of the course, proof remains the number one hurdle for many students. How do stu...

A way to improve conceptual use of digital technology (DT) in secondary mathematics teaching may be to increase teacher involvement in task development. This research considered the documentational genesis of four groups of teachers designing and implementing DT tasks. We analysed the tasks they produced and considered factors involved in their doc...

The ways in which digital technology is often used in university teaching of mathematics can be quite different from how it is employed in schools. This has the potential to form a discontinuity between school and university, making the transition less than smooth for students. In this chapter we consider several examples of how digital technology...

Algebra is often thought of as a ‘gatekeeper’ in school mathematics, being crucial to further study in mathematics as well as to future educational and employment opportunities. However, a large number of studies have highlighted the difficulties and cognitive obstacles that students face when they learn algebra. In response to growing concerns abo...

Many have thought deeply about the construction of the school algebra curriculum, but the question remains as to why we teach the topics we do in the manner we do, stressing manipulations of symbols, and why some other avenues are ignored. In this chapter we consider the basic constructs in the school algebra curriculum and the procedural approach...

This paper discusses the results of an intervention designed to enable richer digital technology (DT) task design by secondary school teachers. Working in groups of three the teachers designed their own tasks. They were then introduced to some theoretical design principles, following which they further developed their tasks. The results show that t...

Enacting the work of diagnostic teaching is challenging and demands that teachers pay attention to mathematical details when designing tasks, orchestrating discussions and reflecting on their lessons. This chapter presents the FOCUS Framework on teacher noticing, which can be used to characterise teachers’ efforts to notice productively during all...

A crucial step towards improving the conceptual use of digital technology (DT) in the mathematics classroom is to increase teacher involvement in the development of tasks. Hence, this research considers some teacher factors that might influence DT algebra task development and implementation in secondary schools. We observed and assisted four groups...

This paper reports on a professional development workshop in University level mathematics. The workshop was grounded in Schoenfeld’s (How we think. A theory of goal-oriented decision-making and its educational applications, 2010) theoretical approach, emphasizing the roles of Resources (primarily, knowledge), Orientations (primarily, beliefs), and...

Linear algebra is one of the first abstract mathematics courses that students encounter at university. Research shows that many students find the dense presentation of definitions, theorems and proofs difficult to comprehend. Using a case study approach, we report on a teaching intervention based on Tall’s three worlds (embodied, symbolic and forma...

The papers in this issue describe recent collaborative research into the role of inhibition of intuitive thinking in mathematics education. This commentary reflects on this research from a mathematics education perspective and draws attention to some of the challenges that arise in collaboration between research fields with different cultures, incl...

In a 2-year project, we developed and trialled a mode of lecturing professional development amongst staff in our department of mathematics. Theoretically grounded in Schoenfeld’s resources, orientations, and goals (ROG) model of teacher action, a group met regularly to discuss both the video excerpts of themselves lecturing along with written pre-...

Recent studies of the transition from school to university mathematics have identified a number of epistemological gaps, including the need to change from an emphasis on equality to that of inequality. Another crucial epistemological change during this transition involves the movement from the pointwise and global perspectives of functions usually...

This report from the ICME12 Survey Team 4 examines issues in the transition from secondary school to university mathematics with a particular focus on mathematical concepts and aspects of mathematical thinking. It comprises a survey of the recent research related to: calculus and analysis; the algebra of generalised arithmetic and abstract algebra;...

A key variable in the use of digital technology in the mathematics classroom is the teacher. In this chapter we examine research that identifies some of the obstacles to, and constraints on, secondary teachers’ implementation of digital technology. While a lack of physical resources is still a major extrinsic concern we introduce a framework for, a...

In this paper we describe the design and implementation of a technology-active introductory first year university mathematics course. The design principles underpinning the course are presented. The results of the implementation show some areas where the technology-active approach has proven of value, as well as improvements that can be made for th...

This Research Forum highlights the most recent research on the development of the role of the teacher of mathematics within mathematics classrooms that involve the use of technological tools, with an emphasis on teachers’ experiences within both formal and informal professional development programmes. We foreground the theoretical ideas and methodo...

This paper describes part of an international project considering graphical construction of antiderivative functions in the secondary mathematics classroom. We use Schoenfeld’s resources, orientations, and goals (ROGs) framework to analyse the decisions made by a teacher, Adam, during a lesson on graphical antiderivatives. We present details of Ada...

A team led by Bill Barton and Judy Paterson at The University of Auckland is investigating different approaches to the teaching and learning of undergraduate mathematics, in a project funded by Ako Aoteoroa called Capturing Learning in Undergraduate Mathematics. A large part of the study centres on the development of three innovative approaches, on...

Linear algebra with its rich theoretical nature is a first step towards advanced mathematical thinking for many undergraduate students. In this paper, we consider the teaching approach of an experienced mathematician as he attempts to engage his students with the key ideas embedded in a second-year course in linear algebra. We describe his approach...

Graphic calculators (GC) have been widely used in teaching for around 20 years, and yet many teachers still appear to be either unconvinced, or unaware, of how to release the potential of such technology to assist students to learn mathematics. This research describes a study addressing aspects of the teacher's role in integrating technology into t...

Computer Algebra Systems (CAS) are software systems with the capability of symbolic manipulation linked with graphical, numerical, and tabular utilities, and increasingly include interactive symbolic links to spreadsheets and dynamical geometry programs. School classrooms that incorporate CAS allow for new explorations of mathematical invariants, a...

Issues arising in the transition from secondary school to tertiary mathematics study are increasingly coming under scrutiny. In this paper, we analyse two practical aspects of the school–tertiary interface: teaching style; and assessment. We present some of the findings arising from a 2-year national project in New Zealand titled “Analysing the Tra...

The research study described in this paper investigated whether the use of computer algebra systems (CAS), such as Derive on the TI-92 supercalculator, could provide a significant advantage to students sitting standard university entrance calculus examinations. It also investigated the feasibility of calculatorneutral questions. The results showed...

The importance of technology in facilitating the teaching and learning process has been increasingly recognised by the mathematics education community, both in the Australasian region and elsewhere. This growth has been encouraged by two noticeable movements relating to technology in mathematics education in the last ten years. Firstly, there has b...

This paper examines how a person’s gesture space can become endowed with mathematical meaning associated with mathematical spaces and how the resulting mathematical gesture space can be used to communicate and interpret mathematical features of gestures. We use the theory of grounded blends to analyse a case study of two teachers who used gestures...

A group of mathematicians and mathematics educators are collaborating in the fine-grained examination of selected 'slices' of video recordings of lectures, drawing on Schoenfeld's Resources, Orientations and Goals framework of teaching-in-context. In the larger project, we are exploring ways in which this model can be extended to examine university...

What role do gestures play in advanced mathematical thinking? We argue that the role of gestures goes beyond merely communicating thought and supporting understanding – in some cases, gestures can help generate new mathematical insights. Gestures feature prominently in a case study of two participants working on a sequence of calculus activities. O...

This article reports on some findings from the project ‘Analysing the Transition from Secondary to Tertiary Education in Mathematics’. One of the key variables in the school to university transition is the teacher/lecturer, and here, we deal with the data analysing secondary teachers’ and tertiary lecturers’ responses to four mathematics questions....

This article continues a fairly recent trend of research examining the teaching practice of university mathematics lecturers. A lecturer's pedagogical practices in a course in linear algebra were discussed via a supportive community of inquiry. We use Schoenfeld's framework describing the relationship of resources, orientations and goals to decisio...

In this paper we consider the professional development of university mathematics lecturers. We describe two exemplars from a two-year research process to engage mathematics educators and research mathematicians in a constructive dialogue about teaching. In this study lectures were video recorded and then discussed in a supportive community of pract...

This paper describes part of an international project considering graphical construction of antiderivative functions in the secondary mathematics classroom. We use Schoenfeld's Resources, Orientations and Goals (ROG) framework to analyse the decisions made in part of a lesson of a teacher, Adam. In this he discusses with his students constructs ari...

A group of mathematicians and mathematics educators are collaborating in the fine-grained examination of selected ‘slices’ of video recordings of lectures drawing on Schoenfeld’s ROG framework of teaching-in-context. We seek to examine ways in which this model can be extended to examine university lecturing. In the process we have identified a numb...

Using traditional educational research methods, it is difficult to assess students’ understanding of mathematical concepts, even though qualitative methods such as task observation and interviews provide some useful information. It has now become possible to use functional magnetic resonance imaging (fMRI) to observe brain activity whilst students...

Two complementary processes involved in mathematical modelling are mathematising a realistic situation and applying a mathematical technique to a given realistic situation. We present and analyse work from two undergraduate students and two secondary school teachers who engaged in both processes during a mathematical modelling task that required th...

One of the earlier, more challenging concepts in linear algebra at university is that of basis. Students are often taught procedurally how to find a basis for a subspace using matrix manipulation, but may struggle with understanding the construct of basis, making further progress harder. We believe one reason for this is because students have major...

The transition from school to tertiary study of mathematics comes under increasing scrutiny in research. This article reports on some findings from a project analysing the transition from secondary to tertiary education in mathematics. One key variable in this transition is the teacher or lecturer. This article deals with a small part of the data f...

Linear algebra is one of the unavoidable advanced courses that many mathematics students encounter at university level. The research reported here was part of the first author's recent PhD study, where she created and applied a theoretical framework combining the strengths of two major mathematics education theories in order to investigate the lear...

Statistics was for a long time a domain where calculation dominated to the detriment of statistical thinking. In recent years, the latter concept has come much more to the fore, and is now being both researched and promoted in school and tertiary courses. In this study, we consider the application of the concept of flexible or versatile thinking to...

The use of the historical development of mathematical concepts to inform current teaching and learning has been a debatable process. Historically, in Indian mathematics (and Mayan), a study of large numbers seems to have provided the impetus for the development of a place value number system. Present day students do not have to create a number syst...

This paper reports on some findings from the project 'Analysing the Transition from Secondary to Tertiary Education in Mathematics'. A key variable in the school to university transition is the teacher/lecturer, and here we deal with data analysing secondary teachers' responses to four mathematics questions. Elsewhere we consider a comparison of te...

We introduce the notion of a virtual space, which is a mathematical and cognitive space that is created through gestures. The notion of a virtual space is illustrated in a case study of two teachers who create a virtual space while constructing the graphical antiderivative of a function. We discuss some features of a virtual space, including the wa...

Linear algebra is one of the unavoidable advanced courses that many mathematics students encounter at university level. The research reported here is as part of the first named author's recent PhD thesis where she created and applied a theoretical framework combining the strengths of two major mathematics education theories in order to investigate...

The transition from school to tertiary study of mat hematics is rightly coming under increasing scrutiny in research. This paper employs Tall's model of the three worlds of mathematical thinking to examine key variables in teaching and l earning as they relate to this transition. One key variable in the transition is clearly the teacher/l ecturer a...

The transition from school to tertiary study of mathematics is coming under increasing scrutiny in research. This paper reports on some findings from a project analysing the transition from secondary to tertiary education in mathematics. One key variable in this transition is the teacher/lecturer. This paper deals with a small part of the data from...

In 11 chapters, this book presents research across a wide range of topics relevant to those teaching secondary (high) school matheamics and statistics. Topics include promoting thinking, culturally responsive mathematics teaching, classroom mathematics talk, modelling, statistical thinking, learning about probability, using hand-held technology, ma...

It has been recognised that many student perspectives on equations and their use of the equals sign have not mirrored those that mathematicians would like to see in tertiary students. This paper tracks transition of understanding of the equals sign by comparing secondary school students’ thinking with that of first year university students. We anal...

One of the earlier, challenging concepts in linear algebra at university is that of basis. Students are taught procedurally how to find a basis for a subspace using matrix manipulation, but may struggle with understanding the construct of basis, making further progress harder. In this research we sought to apply an APOS theory framework, in the con...

This paper documents some of the historical steps in the development of the concept of procept. It then describes how these ideas have led to the construction of an emerging theoretical perspective on what constitutes versatile mathematical thinking and learning. Three parts of this theory are presented: process/object versatility; visuo/analytic v...