Norma Presmeg

Mathematics Learners in Transition

Formal, non-formal and informal mathematics education practices continue to evolve through globalisation and through the use of technology and the WWW. They do so in response to the need for more mathematics to be learnt by increasing numbers of students, both school students and adults. As these practices develop, and as adult education and life-long education grow in importance, along with their mathematical versions, there is an increasing need for mathematics education to move away from ideas and practices based on traditional child development theories and normative ideas. This is particularly important if research in mathematics education is to continue to have relevance and influence in these new and diverse fields of activity. In the last two decades educational and psychological research studies on social, cultural and political aspects of mathematics learning, have raised awareness of the complexities of the process of learning and using mathematics in specific sociocultural practices (see for instance, Bishop, 1988a, 1988b, 1994; Secada, 1992; Van Oers & Forman, 1998, Cobb & Bauersfeld, 1995; Lerman, 1994). On the other hand such studies have also indicated the potential of this field for informing and developing teaching practices at all levels of mathematics education.

A Dance of Instruction with Construction in Mathematics Education

As the field of mathematics education research matures, the examples proliferate of the dual nature of learning as an individual construction and as a social endeavor guided by instruction. I call this interaction of individual volition and social engagement the dance of instruction with construction, and in this introductory Chap. I discuss some of the implications and give a few examples of this dance and its significance.

Metaphor and Metonymy in Processes of Semiosis in Mathematics Education

Building on Michael Otte’s insights regarding the roles of icon, index, and symbol in mathematical signification, definitions of these categories of representation are explored in terms of metaphors and metonymies. A nested model of signs, based on Peirce’s triadic formulation, is described, along with his trichotomic distinction among interpretants that are intentional, effectual, and communicational (leading to the commens). The theoretical argument and its utility is illustrated in terms of an episode of creating a proof in a college geometry class. The significance of the theoretical notions for creativity in mathematics is seen to reside in metaphorical and metonymical processes.

Creativity, mathematizing, and didactizing: leen streefland's work continues

This reaction to the papers in this PME Special Issue of Educational Studies in Mathematics draws a wider perspective on the issues addressed and some of the constructs used in research in Realistic Mathematics Education (RME). In particular, it tries to show that while the problems addressed existed within the world-wide arena of mathematics education and were not unique to the Dutch educational system, the methods used at the Freudenthal Institute to address them were uniquely adapted to that system yet foreshadowed developments in the wider field of mathematics education. The predictive aspects of mathematizing, didactizing, and guided reinvention, in which models-of become models-for on various levels, resonate with trends in mathematics education in recent years, including those promoted by the National Council of Teachers of Mathematics in the USA. Research methodologies, too, have broadened to include more humanistic qualitative methods. Developmental research as epitomized in the RME tradition makes the distinction between quantitative and qualitative research obsolete, because there is no restriction on research methods that may be useful in investigating how to improve the teaching and learning of mathematics, and in the designing of mathematics curricula. Thus some aspects of this research resonate with what have come to be known as multitiered teaching experiments. However, in RME there is also a special content-oriented didactical approach that harmonizes with an emphasis on didactics (rather than pedagogy)in several other European countries. Some implications are drawn for future research directions.

A Summary of Results

Within the constraints of this Topical Survey, we have of necessity concentrated on the many theoretical constructs that are relevant to semiotics in mathematics education.

Mathematics at the Center of Distinct Fields: A Response to Michael and Ted

This response to Ted and Michael points out that we are unified in considering mathematics to be central in the work of mathematicians, mathematics educators, and mathematics education researchers. However, there are distinctions between the fields of pure mathematics research, the teaching and learning of mathematics, and research in mathematics education, and unless these differences are honored it is possible for researchers to talk past one another. The case of Swedish mathematics education research is examined to exemplify the distinctions. Another distinction is that between “training” and “education”. To further characterize mathematics education research, submission of manuscripts to Educational Studies in Mathematics is explored. Values and aesthetics in various relevant fields are touched upon. Finally, an example is given of the mutual enhancement that exists when mathematicians and mathematics education researchers work together in university mathematics departments.

Preface to Part IV

Stephen Lerman has done the community of mathematics education researchers a service by opening up some of the issues that arise as a result of the proliferation of theories that concern the learning and teaching of mathematics. We teach research students in mathematics education that it is necessary to be guided by one or more theories in designing and carrying out a research study in this field. For the coherence of the research, the particular conceptual framework developed from the theoretical considerations should inform the overall design as well as every detail of the decisions made regarding methodology, participants, specific methods of data collection, plans for analysis once the data are collected, and finally the reporting of the results. Every research decision should have a rationale that is grounded in the conceptual framework. Thus theory is eminently important. But why is it important? And as Lerman asks, is it a problem that there is a growing plethora of theories that have potential uses in our field? He comes to the ultimate conclusion that it is not a problem, and I agree with him.

Introduction: What Is Semiotics and Why Is It Important for Mathematics Education?

Over the last three decades, semiotics has gained the attention of researchers interested in furthering the understanding of processes involved in the learning and teaching of mathematics

An Investigation of High School Students’ Mathematical Problem Posing in the United States and China

In the literature, problem posing is claimed to be important in learning mathematics. This study investigated US and Chinese high school students’ attitudes and abilities in posing mathematical problems. All of the participants were taking advanced mathematics in high school. A mathematics content test and a mathematical problem-posing test were administered to the students. The mathematical content test was adapted from the National Assessment of Educational Progress for 12th graders. The problem-posing test included three situations, namely a free problem-posing situation, a semi-structured problem-posing situation, and a structured problem-posing situation. Students who scored 39 or above out of 50 points were interviewed. During the interviews, the majority of the students reported that they had not had any prior experience in posing mathematical problems. Many students did not have a specific strategy for posing problems, and many had difficulty explaining their problem-posing processes. Most of the US students for various reasons said that problem posing was important in mathematics. Most Chinese students said that problem posing was not important in high school learning because of college entrance examinations.

Overcoming Pedagogical Barriers Associated with Exploratory Tasks in a College Geometry Course

With the advent of dynamic geometry software, and a current trend towards the exploration that such software facilitates, textbook authors have in some cases introduced geometric concepts intended for college students by means of exploratory tasks. The strengths of this approach are offset by the disadvantage that many students are thereby convinced of the veracity of a geometric statement without feeling the need for a proof, and without having the knowledge of inner structure that will illuminate why the statement is true. This chapter explores examples of exploratory tasks from a college geometry textbook with a view to connecting these tasks with formal geometric proofs that will not only prove but explain principles that underlie the explorations.