The catwalk task: Reflections and synthesis: Part 2

In this article we recount our experiences with a series of encounters with the catwalk task and reflect on the professional growth that these opportunities afforded. First, we individually reflect on our own mathematical work on the catwalk task. Second, we reflect on our experiences working with a group of community college students on the catwalk task and our interpretations of their mathematical thinking. In so doing we also detail a number of innovative and novel student-generated representations of the catwalk photos. Finally, we each individually reflect on the entire experience with the catwalk problem, as mathematics learners, as teachers, and as professionals.     

Multipurpose Professional Growth Sequence: The catwalk problem as a paradigmatic example

An important concern in mathematics teacher education is how to create learning opportunities for prospective and practicing teachers that make a difference in their professional growth as educators. The first purpose of this article is to describe one way of working with prospective and practicing teachers in a graduate mathematics education course that holds promise for positively influencing the way teachers think about mathematics, about student learning, and about mathematics teaching. Specifically, I use the “catwalk” task as an example of how a single problem can serve as the basis for a coherent sequence of professional learning experiences. A second purpose of this article is to provide background information that contextualizes the subsequent two articles, each of which details the positive influence of the catwalk task sequence on the authors’ professional growth.     

Perceptions of Professional Growth: A Mathematics Teacher Educator in Transition

To meet the need for reform in mathematics teacher preparation courses, two cycles of changes made in an elementary mathematics methods course are presented. Using action research, teaching approaches were developed, implemented, and evaluated as a meaningful way to continue my professional development. Results suggested that I improved my teaching practices and focused more on teaching tasks that engaged my students to “think like teachers.” Three critical components of teacher preparation courses are identified that are important for teacher educators to acknowledge when implementing change: (a) using reflective verbal and written communication, (b) establishing a collaborative mathematical community, and (c) focusing on a narrower selection of mathematical content.     

DESIGNING FISHERY MODELS: A PERSONAL ADVENTURE

ABSTRACT. This paper describes my experiences designing fishery models, starting from a mathematical background in the differential equations of theoretical physics. Three examples from my early research, cited by Quinn in the lead article for this issue, illustrate a historical approach to model design. Although such analytical results provide useful tools for thought, they sometimes gloss over important assumptions and limitations. I describe the series of questions that led me from simple models to a more complete statistical framework, involving state space models and Bayes statistics. Modern fishery models often grow into complex structures that depend on numerous arbitrary choices about underlying deterministic processes, process error, and measurement error. Given this inherent ambiguity and uncertainty, I discuss scientific limits to quantitative fishery models and future prospects for devising robust management algorithms.     

The emergence of women on the international stage of mathematics education

In this article, I consider the history of the International Commission on Mathematical Instruction (ICMI) from its inception until the International Congress on Mathematical Education (ICME) held in 1969. In this period, mathematics education developed as a scientific discipline. My aim is to study the presence and the contribution of women (if any) in this development. ICMI was founded in 1908, but my history starts before then, at the end of the nineteenth century, when the process of internationalization of mathematics began, thanks to the first International Congress of Mathematicians. Already in those years, the need for internationalizing the debate on mathematics teaching was spreading throughout the mathematical community. I use as my main sources of information the didactics sections in the proceedings of the International Congresses of Mathematicians and the proceedings of the first ICME. The data collected are complemented with information from the editorial board of two journals that for different reasons are linked to ICMI: L’Enseignement Mathématique and Educational Studies in Mathematics. In particular, as a result of my analyses, I have identified four women who may be considered as pioneer women in mathematics education. Some biographical notes on their professional life are included in the paper.     

Individual and Collective Mathematical Development: The Case of Statistical Data Analysis

In the first part of this article, I clarify how we analyze students' mathematical reasoning as acts of participation in the mathematical practices established by the classroom community. In doing so, I present episodes from a recently completed classroom teaching experiment that focused on statistics. Against the background of this analysis, I then broaden my focus in the final part of the article by developing the themes of change, diversity, and equity.     

Embodied experience of velocity and acceleration: a narrative

Constructing a link between what a student is learning and personal experience is an important, and sometimes difficult task. I present here a narrative of my own experience as a mathematics and physics teacher trying to create an embodied sense of motion in my students by actually putting them in motion. I use the story to present the difficulty of teaching motion in the absence of the embodiment of motion as well as the tension that is created between an embodied sense of motion and the static representations used to describe it.     

A combinatorial mathematician in Norway: some personal reflections

In the paper, I first try to give some impression of Norwegian contributions to combinatorics in the 20th century. This is followed by some remarks on my own combinatorial experiences.     

A quote a day educates

Conclusion  I often ponder on my duties as a teacher of the subject I love. I feel I am responsible for more than simply transmitting knowledge. I wish I could help my students see mathematics from various vantage points. One of these should be from a point high enough to afford a full, sweeping view of the mathematical valley below—maybe missing the details we strive to convey in class-but seeing thelandscape of mathematics. Claude Bragdon said, “Mathematics is the handwriting on the human consciousness of the very Spirit of Life itself.” I want my students to consider that such a bold statement might actually be true.     

Faces of mathematical modeling

In this paper I will discuss and exemplify my perspectives on how to teach mathematical modeling, as well as discuss quite different faces of mathematical modeling. The field of mathematical modeling is so enormous and vastly outspread and just not possible to comprehend in one single paper; or in one single book, or even in one single book shelf. Nevertheless, I have found that the more I can illuminate some of the various interpretations and perceptions of mathematical modeling which exists in the world around us when introducing and starting a course in mathematical modeling, the more benefit I will have during the course when discussing the need and purpose of mathematical modeling with the students. The fact that only some models fit within the practical teaching and assessing of a course in mathematical modeling, does not exclude the importance to illustrate that the world of today cannot go on without mathematical modeling. Students are nevertheless much more charmed with some models of reality than others.     

You want them to remember? Then make it memorable! Means for enhancing operations research education

In this article I summarize the main points I made in the keynote presentation of the same title I gave at the EURO XXIV conference in Lisbon, Portugal in July of 2010. Each of these points deals in some way with making communications between an operations research professional (academic or practitioner) and a student, client, subordinate, supervisor, or colleague more effective. Furthermore, each point is directly related to some realization (or epiphany) that I have had with regard to communication since I began teaching ORMS in 1984. It is noteworthy that these communications share a common objective; we are trying to facilitate learning. Since I have spent most of my career in academia, my primary emphasis is on communication with students (particularly those enrolled in introductory ORMS courses). However, I have also spent a great deal of time working on operations research problems outside of academia, either as an employee in private industry or as an operations research consultant to corporations and not-for-profit organizations, and I hope as a consequence my discussion is also relevant to those working in the practice of ORMS.     

Mathematics in economics: Achievements,difficulties, perspectives

I am deeply moved by this high honor which has fallen my lot to receive and I am happy to have the opportunity to appear here as a participant in this honorable series of lectures.In our time mathematics has penetrated into economics so solidly and widely and my theme is connected with such a variety of facts and problems that it brings me to cite the words of Kozma Prutkov, which are very popular in my country: One cannot embrace the unembraceable. The appropriateness of this wise sentence is not diminished by the fact that that great thinker is only a pen name.So, I want to restrict my theme to topics which are nearer to me, mainly to optimization models and their use in the control of the economy in order to best use resources for obtaining best results. I shall touch mainly on the problems and experiences of a planned economy, especially of the Soviet economy. Even within these limits I will succeed to consider only a few problems.Copyright © the Nobel Foundation 1975.     

Reasoning about relationships between quantities to reorganize inverse function meanings: The case of Arya

Researchers have argued high school students, college students, pre-service teachers, and in-service teachers do not construct productive inverse function meanings. In this report, I first summarize the literature examining students’ and teachers’ inverse function meanings. I then provide my theoretical perspective, including my use of the terms understanding and meaning and my operationalization of productive inverse function meanings. I describe a conceptual analysis of ways students may reorganize their limited inverse function meanings into productive meanings via reasoning about relationships between covarying quantities. I then present one pre-service teacher’s activity in a semester long teaching experiment to characterize how her quantitative, covariational, and bidirectional reasoning supported her in reorganizing her limited inverse function meanings into more productive meanings. I describe how this reorganization required her to reconstruct her meanings for various related mathematical ideas. I conclude with research and pedagogical implications stemming from this work and directions for future research.     

Developing a theory of mathematical growth

In this paper I formulate a basic theoretical framework for the ways in which mathematical thinking grows as the child develops and matures into an adult. There is an essential need to focus on important phenomena, to name them and reflect on them to build rich concepts that are both powerful in use and yet simple to connect to other concepts. The child begins with human perception and action, linking them together in a coherent way. Symbols are introduced to denote mathematical processes (such as addition) that can be compressed as mathematical concepts (such as sum) to give symbols that operate flexibly as process and concept (procept). Knowledge becomes more sophisticated through building on experiences met before, focussing on relationships between properties, leading eventually to the advanced mathematics of concept definition and deduction. This gives a theoretical framework in which three modes of operation develop and grow in sophistication from conceptual-embodiment using thought experiments, to proceptual-symbolism using computation and symbol manipulation, then on to axiomatic-formalism based on concept definitions and formal proof.     

Investigating the transformation of a secondary teacher’s knowledge of trigonometric functions

Shulman (1987) defined pedagogical content knowledge as the knowledge required to transform subject-matter knowledge into curricular material and pedagogical representations. This paper presents the results of an exploratory case study that examined a secondary teacher’s knowledge of sine and cosine values in both clinical and professional settings to discern the characteristics of mathematical schemes that facilitate their transformation into learning artifacts and experiences for students. My analysis revealed that the teacher’s knowledge of sine and cosine values consisted of uncoordinated quantitative and arithmetic schemes and that he was cognizant only of the behavioral proficiencies these schemes enable, not the mental actions and conceptual operations they entail. Based on these findings, I hypothesize that the extent to which a teacher is consciously aware of the mental activity that comprises their mathematical conceptions influences their capacity to transform their mathematical knowledge into curricular material and pedagogical representations to effectively support students’ conceptual learning.     

All for one and one for all: Theoretical models,sociological theory,and mathematical sociology

I make the argument that mathematical sociology uniquely contributes to sociological theory through the theoretical models it develops to bridge the gap between the ideas of sociological theorists and data relevant to their empirical evaluation. My work on intergroup association and social integration, the fruit of a long-time collaboration with Fararo and more recently with Karpiński, is used throughout to illustrate my points.     

Reflections on learning and cognition

The occasion of my 9th ICME—the first being in Berkeley in 1980, the most recent being in Seoul in 2012—provides an opportunity for reflecting on changes in the field over more than 30 years. “Learning and cognition” have a very different meaning now than they did in 1980. I argue that in various ways, the papers in this volume (derived from the ICME 12 Topic Study Group on Learning and Cognition) represent a significant evolution of the field—with mathematical sense making being a central conception, and with the evolution of the very notions of learning and cognition to include embodied, sociocultural, and historical perspectives. In this volume one sees a focus on classroom activities as they engender aspects of sense making, framed in ways that were not even part of the discourse on learning in 1980; one also sees a widely varied set of research methods for addressing such issues. I reflect on the state of the art, and then discuss some possibly productive directions with regard to the characterization and support of mathematically productive classrooms.     

Statistics in the computer age: personal reflections

It is a trivial observation that the computers have changed the way statistics is practiced. But has it also changed the theory of statistics and the way we teach it? I think yes—even if the changes appear to be surprisingly small in some contexts. This is an attempt to give a more detailed answer, based on experiences from my own corner of the world from 1964 till now.     

Part-whole number knowledge in preschool children

Ability to reflect on a number as an object of thought, and to isolate its constituent parts, is basic to a deep knowledge of arithmetic, as well as much practical and applied mathematical problem solving. Part-whole reasoning and counting are closely related in children’s numerical development. The mathematical behavior of young children in part-whole problem settings was examined by using a dynamic problem situation, in which a small set of items was partitioned such that one of the subsets remained perceptually inaccessible. Issues addressed include the problem solving strategies successful children used, adaptations children make in response to successive administrations of the task over time, and characterizations of children’s mathematical thinking based on their responses to the task.