Results 1  10
of
94
Learning to think Mathematically: Problem solving, metacognition, and sense making in mathematics
 in Grouws (ed), ‘Handbook of research on mathematics teaching and learning’, NCTM
, 1992
"... Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, ..."
Abstract

Cited by 349 (6 self)
 Add to MetaCart
Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving,
With an eye to the mathematical horizon: Dilemmas of teaching. Paper presented at the annual meeting of the American Educational Research Association
 In J. Brophy (Ed.), Advances in research on teaching
, 1990
"... to this work. 3We begin with the hypothesis that any subject can be taught effectively in some intellectually honest form to any child at any stage of development. It is a bold hypothesis and an essential one in thinking about the nature of a curriculum. No evidence exists to contradict it; consider ..."
Abstract

Cited by 167 (6 self)
 Add to MetaCart
(Show Context)
to this work. 3We begin with the hypothesis that any subject can be taught effectively in some intellectually honest form to any child at any stage of development. It is a bold hypothesis and an essential one in thinking about the nature of a curriculum. No evidence exists to contradict it; considerable evidence is being amassed that supports it. (Bruner, 1960, p. 33) The tendrils of this famous passage still wind around current discourse about the improvement of instruction. This paper revisits Bruner's oftquoted assertion that "any subject can be taught effectively in some intellectually honest form. " While my aim is not to suggest that he was wrong, I seek to persuade the reader that figuring out what it might mean to create a practice of teaching that is "intellectually honest " is a project laden with thorny dilemmas and that teachers need to be prepared to face off with the uncertainties inherent in the goal. The new mathematics, science, and history curricula that swept the United States during the 1960s in the wake of Bruner's hypothesis gave us ample evidence that acting on his claim is not easy. This paper takes up the challenge in the particular context of elementary school mathematics: How can and should mathematics as a school subject be connected with mathematics as a discipline? Much current educational discourse centers on the importance of teachers ' subject matter
Building student capacity for mathematical thinking and reasoning: An analysis of mathematical tasks used in reform classrooms
 American Educational Research Journal
, 1996
"... This article focuses on mathematical tasks as important vehicles for building student capacity for mathematical thinking and reasoning. A stratified random sample of 144 mathematical tasks used during reformoriented instruction was analyzed in terms of (a) task features (number of solution strategi ..."
Abstract

Cited by 97 (3 self)
 Add to MetaCart
This article focuses on mathematical tasks as important vehicles for building student capacity for mathematical thinking and reasoning. A stratified random sample of 144 mathematical tasks used during reformoriented instruction was analyzed in terms of (a) task features (number of solution strategies, number and kind of representations, and communication requirements) and (b) cognitive demands (e.g., memorization, the use of procedures with [and without] connections to concepts, the "doing of mathematics "). The findings suggest that teachers were selecting and setting up the kinds of tasks that reformers argue should lead to the development of students' thinking capacities. During task implementation, the task features tended to remain consistent with how they were set up, but the cognitive demands of highlevel tasks had a tendency to decline. The ways in which highlevel tasks declined as well as factors associated with task changes from the setup to implementation phase were explored. MARY KAY STEIN is a Research Associate at the Learning Research and Develop
Clarifying the meaning of mathematical objects as a priority area of research in Mathematics Education
 In A. Sierpinska & J. Kilpatrick (Eds.), Mathematics
, 1998
"... ABSTRACT. The main thesis of this chapter is that the epistemological and psychological analyses concerning the nature of mathematical objects play a fundamental role in addressing certain research questions in mathematics education. In particular the question of assessment of students ’ knowledge, ..."
Abstract

Cited by 31 (19 self)
 Add to MetaCart
(Show Context)
ABSTRACT. The main thesis of this chapter is that the epistemological and psychological analyses concerning the nature of mathematical objects play a fundamental role in addressing certain research questions in mathematics education. In particular the question of assessment of students ’ knowledge, and that of the selection of didactical situations. The thesis is justified within the framework of a pragmatic and relativist theory of meaning of mathematical objects, outlined in the chapter. The framework emphasizes the complex and systemic nature of the meaning of mathematical objects and stresses the institutional and cultural contexts of the teaching and learning processes in mathematics.
Reasoning With Tools and Inscriptions
 Journal of the Learning Sciences
, 2002
"... The unit of analysis that I use when discussing the 2 sample episodes is that of a classroom mathematical practice together with the students ’ diverse ways of contributing to its continual regeneration. Analyses cast in terms of this unit account for the mathematical learning of the classroom com ..."
Abstract

Cited by 24 (0 self)
 Add to MetaCart
(Show Context)
The unit of analysis that I use when discussing the 2 sample episodes is that of a classroom mathematical practice together with the students ’ diverse ways of contributing to its continual regeneration. Analyses cast in terms of this unit account for the mathematical learning of the classroom community. As I clarify, a classroom mathematical practice is itself composed of 3 interrelated types of norms: a normative purpose, normative standards of argumentation, and normative ways of reasoning with tools and inscriptions. In keeping with the theme of this special issue, I step back from the sample analysis by focusing on the last of these 3 aspects. In doing so, I introduce the notion of a chain of signification to illustrate a way of accounting for mathematical learning in semiotic terms. The dominant view of mathematical symbols as external representations has been challenged in recent years by an alternative perspective that emphasizes the activity of symbolizing. In this newer perspective, the focus of investigations shifts away from the analysis of symbols as external supports for reasoning and moves toward students ’ participation in practices that involve symbolizing. Rather than describing the properties of tools such as physical devices, computer icons, and notations independently of their use, this perspective treats symbolizing as an integral aspect
Evolution of the function concept: A brief survey
 The College Mathematics Journal
, 1989
"... received his Ph.D. in ring theory at McGill University, and has been at York University for over twenty years. He has been involved in teacher education at the undergraduate and graduate levels and has given numerous talks to high school students and teachers. One of his major interests is the histo ..."
Abstract

Cited by 16 (0 self)
 Add to MetaCart
(Show Context)
received his Ph.D. in ring theory at McGill University, and has been at York University for over twenty years. He has been involved in teacher education at the undergraduate and graduate levels and has given numerous talks to high school students and teachers. One of his major interests is the history of mathematics and its use in the teaching of mathematics. Introduction. The evolution of the concept of function goes back 4000 years; 3700 of these consist of anticipations. The idea evolved for close to 300 years in intimate connection with problems in calculus and analysis. (A onesentence definition of analysis as the study of properties of various classes of functions would not be far off the mark.) In fact, the concept of function is one of the distinguishing features of
Mathematical explanation and the theory of whyquestions
 British Journal for the Philosophy of Science
, 1998
"... Van Fraassen and others have urged that judgements of explanations are relative to whyquestions; explanations should be considered good in so far as they effectively answer whyquestions. In this paper, I evaluate van Fraassen's theory with respect to mathematical explanation. I show that his ..."
Abstract

Cited by 10 (0 self)
 Add to MetaCart
Van Fraassen and others have urged that judgements of explanations are relative to whyquestions; explanations should be considered good in so far as they effectively answer whyquestions. In this paper, I evaluate van Fraassen's theory with respect to mathematical explanation. I show that his theory cannot recognize any proofs as explanatory. I also present an example that contradicts the main thesis of the whyquestion approach—an explanation that appears explanatory despite its inability to answer the whyquestion that motivated it. This example shows how explanatory judgements can be contextdependent without being whyquestionrelative. 1
Computers, Reasoning and Mathematical Practice
"... ion in itself is not the goal: for Whitehead [117]"it is the large generalisation, limited by a happy particularity, which is the fruitful conception." As an example consider the theorem in ring theory, which states that if R is a ring, f(x) is a polynomial over R and f(r) = 0 for every e ..."
Abstract

Cited by 7 (3 self)
 Add to MetaCart
ion in itself is not the goal: for Whitehead [117]"it is the large generalisation, limited by a happy particularity, which is the fruitful conception." As an example consider the theorem in ring theory, which states that if R is a ring, f(x) is a polynomial over R and f(r) = 0 for every element of r of R then R is commutative. Special cases of this, for example f(x) is x 2 \Gamma x or x 3 \Gamma x, can be given a first order proof in a few lines of symbol manipulation. The usual proof of the general result [20] (which takes a semester's postgraduate course to develop from scratch) is a corollary of other results: we prove that rings satisfying the condition are semisimple artinian, apply a theorem which shows that all such rings are matrix rings over division rings, and eventually obtain the result by showing that all finite division rings are fields, and hence commutative. This displays von Neumann's architectural qualities: it is "deep" in a way in which the symbol manipulati...
Who's Afraid Of Undermining? Why the Principal Principle need not contradict Humean Supervenience
"... The Principal Principle (PP) says that, for any proposition A , given any admissible evidence and the proposition that the chance of A is x%, one's conditional credence in A should be x%. Humean Supervenience (HS) claims that, among possible worlds like ours, no two differ without differing in ..."
Abstract

Cited by 6 (0 self)
 Add to MetaCart
(Show Context)
The Principal Principle (PP) says that, for any proposition A , given any admissible evidence and the proposition that the chance of A is x%, one's conditional credence in A should be x%. Humean Supervenience (HS) claims that, among possible worlds like ours, no two differ without differing in the spacetimepointbyspacetimepoint arrangement of local properties. David Lewis (1986b, 1994) has argued that PP contradicts HS, and his argument has been accepted by Bigelow, Collins, and Pargetter (1993), Thau (1994), Hall (1994), Strevens (1995), Ismael (1996), and Hoefer (1997). Against this consensus, I argue that PP need not contradict HS. 1.
On The Purpose of Mathematics Education Research: Making productive contributions to policy and practice
 In L. English (Ed.), Handbook of International Research on Mathematics Education
, 2002
"... Lester and Dylan Wiliam and published in 2002. The complete reference citation for the ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
(Show Context)
Lester and Dylan Wiliam and published in 2002. The complete reference citation for the