# Peter LiljedahlSimon Fraser University · Faculty of Education

Peter Liljedahl

PhD

## About

134

Publications

91,912

Reads

**How we measure 'reads'**

A 'read' is counted each time someone views a publication summary (such as the title, abstract, and list of authors), clicks on a figure, or views or downloads the full-text. Learn more

2,065

Citations

Citations since 2017

Introduction

Additional affiliations

September 2004 - present

## Publications

Publications (134)

Mathematical creativity is an important topic both in research and in practice. Especially in research and practice on secondary school level, it is gaining increasing significance with a growing body of tasks, practices, and empirical studies being developed and conducted. At the same time, the growing field of research on mathematical creativity...

Many of the discourses on creativity, although not explicitly so, assumes that creativity is a solitary activity—a phenomenon that happens within an individual working in isolation away from other people and other resources. But this is not how most people work. In this chapter, I look at creativity as something that can, and does, occur within gro...

In this chapter, several critical foci are discussed in relation to the book. In particular, the framework for the book, development and mathematical creativity, is outlined so that readers understand what is being discussed. Conceptions and constructs are elucidated in this chapter in theory presentation. Existing theories in creativity are discus...

Research on creativity, invention, and innovation originates in different scientific disciplines and different decades. This diversity in theoretical backgrounds and definitions can still be seen in recent research, especially in mathematics education (see Joklitschke et al., 2021, for a review on notions of creativity in mathematics education rese...

In the COVID-19 era of adapting to pandemic lockdown protocol, teaching practices have become more negotiable and less tethered to the familiar and institutionally normative practices found in educational settings around the world. With a shift to online teaching, many practices are being adapted from face-to-face settings and being imported into o...

If we want to innovate mathematics education — if we want to achieve something beyond conformity and compliance in mathematics education — these institutional norms need to be challenged. In this paper, I look at the results-first research methodology in which the institutional norms are challenged by the simple goal of increasing student thinking...

In the spring of 2020, schools and universities around the world were closed because of the COVID-19 pandemic. The relative lockdown affected more than 1.5 billion learners as teachers and students sheltered at home for several weeks. As schooling moved online, teachers were forced to change how they taught. In the research presented here, we focus...

Problem solving and problem posing have long been of interest to the mathematics education community. In this survey paper we first look at some of the seminal moments in the history of research on the important topics. We then use this history to position the state-of-the-art research being done in both problem solving and problem posing, before i...

Many studies have investigated students’ problem-posing activities. However, there has not been a strong focus on affective aspects in mathematical problem-posing. Also, students’ understanding of the function concept whilst generating functions has not previously been investigated. This paper investigated students’ misconceptions of the function c...

The rise of social media has afforded new opportunities for professional activity around mathematics teaching. Thousands of users are posting publicly about their experiences with mathematics teaching on an ongoing basis at an unprecedented scale in an unprompted, unfunded, and unmandated setting. Given the challenges around engendering sustainable...

In this chapter we focus on emotions and their role in teaching and learning. Through the presentation and discussion of two episodes that took place in the context of teacher education and in-class problem-solving, respectively, we present the two mainstream approaches to emotions in affect-related research. We end the chapter with some methodolog...

Changes of beliefs do not happen arbitrarily; there are underlying mechanisms that enable the shift. This study outlines a problem-solving implementation in which two teachers experienced changes in their beliefs. We describe these belief changes and propose a new mechanism for the shift: first-person vicarious experiences. Our results suggest that...

Self-efficacy is commonly defined as the belief in one’s abilities to attain a goal or outcome. This has significance in classroom situations where students with low self-efficacy fall into a self-fulfilling feedback loop of low aspirations leading to low performance, leading to even lower aspirations. In this research outline, we present a context...

Changes of beliefs do not happen arbitrarily; there are underlying mechanisms that enable the shift from a preexisting belief to a new belief. This study outlines a problem-solving implementation in which we suggest participants experienced shifts in their beliefs. We describe these belief changes and propose a new mechanism for the shift: first-pe...

In this chapter we look at student engagement while doing a modelling task. We present the case of two grade 8 students (age 12–13) and look at their modelling behavior through the double lenses of flow and modelling. During their modelling process, these students experienced an imbalance between the challenges and skills presented to them. Results...

In this chapter, the engagement of 244 students is measured across three different types of mathematical problems (modelling problems, word problems, and mathematical exercises). We also investigate the potential of teaching modelling problems in changing students’ attitude towards mathematics. This research was conducted with a pre-test, followed...

In this commentary, I reflect on the chapters by De Simone; Khalil, Lake, and Johnson; and Montoro and Gil (this volume) and how they contribute to our understanding of engagement within mathematics education. I begin by exploring how engagement has been pursued in mathematics education in particular, and in education in general, and argue that eac...

Three hundred manuscripts on mathematics education are submitted for review to the International Journal of Science and Mathematics Education (IJSME) every year. The vast majority of these are rejected. In many cases, manuscripts that are being rejected are based on good research on interesting topics, but are being rejected because the author has...

Tensions are an integral part of mathematics teachers’ practice and have been recognised as playing a pivotal role in guiding teachers’ choices, and in driving teacher change. Professional development particularly elicits dilemmas. Tensions appear to be a crucial aspect of mathematics teacher education and therefore a deeper understanding of tensio...

This book contributes to the field of mathematical problem solving by exploring current themes, trends and research perspectives. It does so by addressing five broad and related dimensions: problem solving heuristics, problem solving and technology, inquiry and problem posing in mathematics education, assessment of and through problem solving, and...

For many teachers, the incorporation of problem solving into their practice is often met with difficulty as students who are not accustomed to this form of mathematics teaching struggle with, and resist, these efforts. In this chapter, I present results from two research projects in which I studied the effects of various teaching methods on student...

Recent research in problem solving has shifted its focus to actual classroom implementation and what is really going on during problem solving when it is used regularly in classroom. This book seeks to stay on top of that trend by approaching diverse aspects of current problem solving research, covering three broad themes. Firstly, it explores the...

Engagement in mathematical problem-solving is an aspect of problem-solving that is often overlooked in our efforts to improve students’ problem-solving abilities. In this chapter, I look at these constructs through the lens of Csíkszentmihályi’s theory of flow. Studying the problem-solving habits of students within a problem-solving environment des...

In previous research [19, 20], seven mathematics professors shared their views on positive and negative aspects of oral and written assessments in mathematics and types of knowledge and understanding in mathematics that can be assessed on written and oral exams. These participants are coming from Bosnia, Poland, Romania, Ukraine, Canada, the United...

The preparation of graduate students to become independent researchers is a central goal of all doctoral programs. In this paper, I present the results of a study looking at how this move to independence is viewed by graduate students enrolled in a mathematics education master’s or PhD program. Results indicate that graduate students have very diff...

Students do not always behave the way we intend for them to. Sometimes they substitute the learning opportunities we offer them with behaviours that, on the surface, seem to be conducive to learning but in reality stand in place of learning. In this paper I explore teachers’ beliefs of such behaviour, called proxies for learning, and compare them t...

In this chapter I first introduce the notion of a thinking classroom and then present the results of over 10 years of research done on the development and maintenance of thinking classrooms. Using a narrative style I tell the story of how this research began and led first to the notion of a thinking classroom and then to a research project designed...

Research in the affective domain has often been restricted to focused attention on a single affective variable. This is ironic given that we know that affective variables tend to cluster. Perhaps the reason for this is that we lack theories for thinking about affective clusters. In this paper I use Green’s theory of a belief cluster (1971) as the f...

Pre-service teachers come to mathematics method courses with well-established conceptions of what it means to teach and learn mathematics. Images of teaching reinforced by their own lived experiences shape their pedagogy. This can be problematic for a teacher educator for whom it may be necessary to offer a way of reframing traditional notions of t...

Stimulating sustainable mathematics teacher collaboration can be challenging in many commonly found professional development contexts. Despite this, an unprompted, unfunded, unmandated, and largely unstudied mathematics teacher community has emerged where mathematics teachers use social media to communicate about the teaching and learning of mathem...

Over the last 15 years, numeracy has become more and more prominent in curriculum initiatives around the world. Yet, the notion of numeracy is still not well defined, and as such, often not well understood by the teachers who are charged with the responsibility of helping our students to develop these skills. In this article I explore the work of a...

On the last day at MAVI 21 Lovisa Sumpter gave a presentation that began with a story about how her talk came out of a talk that I had given at MAVI 20. This sort of things happens at MAVI often. MAVI is informal enough and small enough that we can start a presentation by talking about each other’s past research. MAVI also has a consistent enough p...

Tensions are endemic to the teaching profession. Viewed as dichotomous forces, tensions shape the experiences of mathematics teachers, affecting both their practice and professional growth. In this article, we identify and examine some of the tensions experienced by Naomi in her practice of teaching mathematics. While previous research presents the...

Although contemporary thinking on the teaching of mathematics calls for an abundance of teacher orchestrated collaborative discovery learning, the reality is that most mathematics classrooms are dominated by healthy doses of direct instruction, teacher led exemplification, followed by the assignment of homework. This results in students coming to s...

The book presents a selection of the most relevant talks given at the 21st MAVI conference, held at the Politecnico di Milano. The first section is dedicated to classroom practices and beliefs regarding those practices, taking a look at prospective or practicing teachers’ views of different practices such as decision-making, the roles of explanatio...

This survey book reviews four interrelated areas: (i) the relevance of heuristics in problem-solving approaches – why they are important and what research tells us about their use; (ii) the need to characterize and foster creative problem-solving approaches – what type of heuristics helps learners devise and practice creative solutions; (iii) the i...

Problem solving in mathematics education has been a prominent research field that aims at understanding and relating the processes involved in solving problems to students’ development of mathematical knowledge and problem solving competencies. The accumulated knowledge and field developments include conceptual frameworks to characterize learners’...

In this chapter I first introduce the notion of a thinking classroom and then present the results of over ten years of research done on the development and maintenance of thinking classrooms. Using a narrative style I tell the story of how a series of failed experiences in promoting problem solving in the classroom led first to the notion of a thin...

In the first Handbook of Research on the Psychology of Mathematics Education (Gutiérrez & Boero, 2006) Gilah Leder and Helen Forgasz began the chapter on Affect and Mathematics Education with an overview of research on affect outside and before PME.

Mathematics teachers do not come to their professional growth opportunities as
blank slates. They come to them carrying a complex array of tensions that af ect their
intentions and actions as a teacher, and are often the very reason that they are
seeking out professional growth opportunities. In this article we explore some of
these tensions in the...

Over the last 15 years, numeracy has become more and more prominent in curriculum initiatives around the world. Yet, the notion of numeracy is still not well defined, and as such, often not well understood by the teachers who are charged with the responsibility of helping our students to develop their numeracy skills. In this article I explore the...

This theoretical contribution comes from a broader study that investigates undergraduate students' conceptions of inequalities. History of inequalities is looked into in a search for an answer to the question: Why are inequalities hard to meaningfully manipulate and understand? Memorable dates in the development of inequalities and the symbols for...

Group interaction is gaining more and more prominence in curricula around the world, for many reasons ranging from teaching to acknowledging the value of dialogue as scaffolded metacognition (Sfard, 2001). But group interactions are complex socially and affectively charged environments wherein affect cannot be separated out from learning. Roth & Ra...

Group work has become a staple in many progressive mathematics classrooms. These groups are often set objectives by the teacher in order to meet specific pedagogical or social goals. These goals, however, are rarely the same as the goals of the students visa -vis group work. As such, the strategic setting of groups, either by teachers or by student...

Lesson play is a novel construct in research and teachers' professional development in mathematics education. Lesson play refers to a lesson or part of a lesson presented in dialogue form-inspired in part by Lakatos's evocative Proofs and Refutations-featuring imagined interactions between a teacher and her/his students. We have been using and refi...

Teachers do not come to professional learning opportunities as blank slates. Instead, they come to these settings with a complex collection of wants and needs. The research presented here takes a closer look at these wants across five different professional learning settings distilling form the data a taxonomy of five categories of wants that teach...

The lesson play idea that we have been exploring in this book is an elaboration on a lesson or part of a lesson in a form of a script that presents interaction between a teacher and students, and among students. As a construct in teacher education, it provides various opportunities for the prospective (or practicing) teachers who write the plays, a...

In this chapter, we discuss lesson plays composed by prospective elementary school teachers that related to linear measurement. The particular focus is on using a ruler in determining a length of an object.

In this chapter, we discuss lesson plays composed by prospective elementary school teachers that are related to number sense and elementary number theory. The particular focus on the property of divisibility and divisibility tests.

This chapter concerns a variation of the lesson play assignment that focuses on the critiquing rather than the writing of a classroom interaction. In particular, we discuss reactions of a group of experienced elementary and middle school teachers to part of a play written by a prospective elementary school teacher. They were asked to consider what...

In this chapter, we develop in detail the construct of '‘lesson play’' as an imagined interaction between a teacher and his/her students. We provide an example of a lesson play and describe the affordances of the method.

In this chapter we describe the evolution of the Lesson Play Task for prospective teachers, as a regular assignment in our ‘methods’ courses.

In this chapter we discuss lesson plays composed by prospective elementary school teachers that are related to understanding fractions. The particular focus is on different methods of comparing fractions and on testing conjectures.

The “lesson plan” has been a staple of pre-service teacher education for many decades. In fact, almost everyone who has undergone a formal teacher education program has had to devise a lesson plan according to some prescribed format. Indeed, it is hard to imagine what teachers did before they used lesson plans! In this chapter, we describe the emer...

In this chapter we return to lesson plays composed by prospective elementary school teachers that are related to measurement. The particular focus is on the concept of area and the relationship between area and perimeter.

In this chapter, we continue the discussion of lesson plays composed by prospective elementary school teachers that are related to number sense and elementary number theory. The particular focus is on prime numbers and on ways of determining whether a number is prime.

Lesson plays provide a clear image of how prospective teachers imagine their teaching, focusing on their interaction with students as well as on interaction among students. Looking back at Chaps. 4–11, we first present several themes that are associated with ‘reform’ teaching, and then turn to images that are in accord with more ‘traditional’ appro...

Lack of appropriate and adequate mathematical knowledge in elementary teachers is a major concern in mathematics education. At their first meeting in Kalamazoo the working group on Developing Elementary Teachers’ Mathematical Knowledge for Teaching identified five significant issues to explore from multiple, diverse perspectives: (1) selecting/crea...

The study explores how students, who had completed the AP calculus course, mathematized the optimal navigation real-life problem simulated in the Second Life Virtual Environment. The particular research interest was to investigate whether/how students’ empirical activity in VE influences the way of their mathematizing.

Lesson play is a novel construct in research and teachers� professional development in mathematics education. Lesson play refers to a lesson or part of a lesson presented in dialogue form�inspired in part by Lakatos�s evocative Proofs and Refutations�featuring imagined interactions between a teacher and her/his students. We have been using and refi...

In this chapter we discuss lesson plays composed by prospective elementary school teachers that are related to patterns and regularities. The particular focus is on repeating patterns and problem solving strategies.

This chapter addresses language choices that can be observed in the lesson plays, choices that seem independent of the actual mathematical problem being discussed, but central to the relationships arising among the teacher, the students, and mathematics.

The field of mathematics education has assumed for too long that stability is an inherent and definable characteristic of beliefs. In this article we explore the validity of this claim through the critical analysis of 92 journal articles, conference papers, and book chapters. Using a stringent definition of what it means for a belief to be stable w...

The study explores how students, who had completed the AP calculus course, applied their knowledge, mathematizing a situation simulated in the Second Life Virtual Environment.

What is the nature of illumination in mathematics? That is, what is it that sets illumination apart from other mathematical experiences? In this article the answer to this question is pursued through a qualitative study that seeks to compare and contrast the AHA! experiences of preservice teachers with those of prominent research mathematicians. Us...

We present the perspectives of teachers and others involved in the collaborative design of teaching and learning artifacts across three cases: (a) an independent group participating in lesson study; (b) teachers participating in professional development programs; and (c) a district initiative for producing numeracy tasks. Among the results we found...

Change in mathematics teachers’ practice is often characterized as something that takes time and sustained intervention. In
this article, I present the results of research that highlights a different kind of change—a profound change that takes place
very quickly. Based on the analysis of 42 cases of such rapid and profound change, I also present a...

In my work with teachers I have observed a phenomenon of profound and rapid changes in practice. A deeper look at this phenomenon has revealed that there are nuances to it that go beyond the nature of the change itself. In a significant number of these cases the phenomenon can be attributed more to the teachers’ classroom experiences than to their...

Initial mathematics teacher education is primarily concerned with knowledge–the acquisition of knowledge required for the
teaching of mathematics. Opinions as to what exactly comprises this knowledge and how it is best delivered and best learned
varies widely across different contexts. In what follows, we will look more closely at this concept of t...

In 1943 Jacques Hadamard gave a series of lectures on mathematical invention at the École Libre des Hautes Etudes in New York City. These talks were subsequently published as The Psychology of Mathematical Invention in the Mathematical Field (Hadamard,1945). In this chapter I present a study that mirrors the work of Hadamard. Results both confirm a...