Shulman (1986, 1987) coined the term pedagogical content knowledge (PCK) to address what at that time had become increasingly evident – that content knowledge itself was not sufficient for teachers to be successful. Throughout the past two decades, researchers within the field of mathematics teacher education have been expanding the notion of PCK and developing more fine-grained conceptualizations of this knowledge for teaching mathematics. One such conceptualization that shows promise is mathematical knowledge for teaching – mathematical knowledge that is specifically useful in teaching mathematics. While mathematical knowledge for teaching has started to gain attention as an important concept in the mathematics teacher education research community, there is limited understanding of what it is, how one might recognize it, and how it might develop in the minds of teachers. In this article, we propose a framework for studying the development of mathematical knowledge for teaching that is grounded in research in both mathematics education and the learning sciences.

Lerman, in his challenge to radical constructivism, presented Vygotsky as an irreconcilable opponent to Piaget’s genetic epistemology and thus to von Glasersfeld’s radical constructivism. We argue that Lerman’s stance does not reflect von Glasersfeld’s opinion of Vygotsky’s work, nor does it reflect Vygotsky’s opinion of Piaget’s work. We question Lerman’s interpretation of radical constructivism and explain how the ideas of interaction, intersubjectivity, and social goals make sense in it. We then establish compatibility between the analytic units in Vygotsky’s and von Glasersfeld’s models and contrast them with Lerman’s analytic unit. Consequently, we question Lerman’s interpretation of Vygotsky. Finally, we question Lerman’s use of Vygotsky’s work in mathematics education, and we contrast that use with how we use von Glasersfeld’s radical constructivism.

A primary purpose for using teaching experiment methodology is for researchers to experience, firsthand, students’2 mathematical learning and reasoning. Without the experiences afforded by teaching, there would be no basis for coming to understand the powerful mathematical concepts and operations students construct or even for suspecting that these concepts and operations may be distinctly different from those of researchers. The constraints that researchers experience in teaching constitute a basis for understanding students’ mathematical constructions. As we, the authors, use it, “constraint” has a dual meaning. Researchers’ imputations to students of mathematical understandings and operations are constrained by the language and actions they are able to bring forth in students. They also are constrained by students’ mistakes, especially those mistakes that are essential; that is, mistakes that persist despite researchers’ best efforts to eliminate them. Sources of essential mistakes reside in students’ current mathematical knowledge. To experience constraints in these two senses is our primary reason for doing teaching experiments. The first type of constraint serves in building up a “mathematics of students” and the second type serves in circumscribing such a mathematics within conceptual boundaries.

Discusses criteria for evaluating clinical research in mathematics education. It is suggested that any set of criteria must be consistent with the underlying epistemology from which that research is carried out. The implications of environmentalist vs constructionist world views of mathematics education for establishing criteria are examined, and a set of criteria applicable to constuctivist mathematical-education research is presented.

Commentary on the paper sin the book “Epistemological foundations of mathematical experience” edited by Leslie Steffe.

Excerpt: Constructivism as a philosophical orientation has been widely accepted in mathematics and science education only since the early 1980s. As it became more broadly accepted, it also became clear that there were incongruous images of it. In 1984, Ernst von Glasersfeld introduced a distinction, echoed in Steier’s paper at this conference, between what he called “naive” constructivism and “radical” constructivism. At the risk of oversimplification, suffice it to say that naive constructivism is the acceptance that learners construct their own knowledge, while radical constructivism is the acceptance that naive constructivism applies to everyone – researchers and philosophers included. Von Glasersfeld’s distinction had a pejorative ring to it, and rightly so. Unreflective acceptance of naive constructivism easily became dogmatic ideology, which had and continues to have many unwanted consequences. On the other hand, I will attempt to make a case that to do research we must spend a good part of our time acting as naive constructivists, even when operating within a radical constructivist or ecological constructionist framework. To make clear that the orientation I have in mind is not unreflexive, I will call it “utilitarian” constructivism, and will use Steier’s and Spiro’s papers as a starting point in its explication.

Excerpt: I would like to bring the book full circle, returning to two points raised by Ernst von Glasersfeld in his opening chapter. These are the fact that radical constructivism is misinterpreted so persistently by its critics, and the need for radical constructivism to provide a clear model of social interaction. I return to these points not only to give the book a particular rhetorical structure, but because they penetrate many of the controversies both internal to mathematics and science education and at the boundaries of radical constructivism. At the same time, I will point out the importance of conceptual analysis in Glasersfeld’s method and urge more people to use it in mathematics and science education.

Excerpt: This chapter discusses ways in which conceptual analyses of mathematical ideas from a radical constructivist perspective complement Realistic Mathematics Education’s attention to emergent models, symbolization, and participation in classroom practices. The discussion draws on examples from research in quantitative reasoning, in which radical constructivism serves as a background theory. The function of a background theory is to constrain ways in which issues are conceived and types of explanations one gives, and to frame one’s descriptions of what needs explaining. The central claim of the chapter is that quantitative reasoning and realistic mathematics education provide complementary foci in both design of instruction and evaluation of it. A theory of quantitative reasoning enables one to describe mathematical understandings one hopes students will have, and ways students might express their understandings in action or communication. It is argued that conceptual analyses of mathematical ideas cannot be carried out abstractly. In contrast, it is found to be highly useful to imagine students thinking about something in discussions of it. In relation to this, the focus is on what one imagines to be the “something” teachers and students discuss, and on the nature of the discussions surrounding it. This type of conceptual analyses overlaps considerably with the Realistic Mathematics Education notion of emergent models in instructional design. There is, however, a difference in one respect; Realistic Mathematics Education attends to tools which will influence students’ activity, while from a quantitative-reasoning perspective the focus is more on things students might re-perceive and things about which a teacher might hold fruitful discussions.

Mathematics during the late 18th century through the early 20th century experienced a period of turmoil and renewal that was rooted in a variety of attempts to put mathematics on solid conceptual footing. Taken-for-granted meanings of concept after concept, from number to function to system, came under increasing scrutiny because they could not carry the weight of new ways of thinking. In a very real sense, that period of time can be characterized as mathematicians’ search for broad, encompassing coherence among foundational mathematical meanings. Part of the resolution of this quest was the realization that meanings can be designed. We can decide what an idea will mean according to how well it coheres with other meanings to which we have also committed, and we can adjust meanings systematically to produce the desired coherence. Mathematics education is in the early stages of a similar period. Competing curricula and standards can be seen as expressions of competing systems of meanings--but the meanings themselves remain tacit and therefore competing systems of meanings cannot be compared objectively. I propose a method by which mathematics educators can make tacit meanings explicit and thereby address problems of instruction and curricula in a new light.