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Carolyn Maher

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Carolyn A. Maher is a Distinguished Professor of Mathematics Education at Rutgers, The State University of New Jersey. She is Editor of the Journal of Mathematics Behavior (JMB) and Director of the Robert B. Davis Institute for Learning (RBDIL) of the Graduate School of Education.

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University Of Education, Winneba

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National & Kapodistrian University of Athens

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Papers by Carolyn Maher

$Research paper thumbnail of Analysis of the argumentation of nine-year-olds engaged in discourse about comparing fraction models$

Analysis of the argumentation of nine-year-olds engaged in discourse about comparing fraction models

The Journal of Mathematical Behavior, Mar 1, 2019

In this paper we analyze the argumentation of nine-year-olds as they justify their reasoning for ... more In this paper we analyze the argumentation of nine-year-olds as they justify their reasoning for a fraction comparison problem. We illustrate how their proof-like arguments evolve in a class discussion and are backed by the rod models they produce, showing both valid and invalid arguments. It is well established that argumentation is an important aspect of K-12 education and that instructional programs should cultivate a classroom environment in which students are encouraged to evaluate arguments, ask questions to clarify arguments, and develop arguments that make sense (Krummheuer

Video Recordings as Pedagogical Tools in Mathematics Teacher Education

Proceedings of the Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Snowbird, Utah, October 18-21, 2001). Volume 1 [and] Volume 2

This book contains volume 1 of the proceedings of the 23rd annual meeting of the International Gr... more This book contains volume 1 of the proceedings of the 23rd annual meeting of the International Group for the Psychology of Mathematics Education (PME) held October, 2001 in Snowbird, Utah. Papers include: (1) "Opening the Dimensions of Mathematical Capability: The Development of Knowledge, Practice and Identity in Mathematics Classrooms" (Jo Boaler); (2) "Learning Mathematics as Developing a Discourse" (Anna Sfard); (3) "Gender and Mathematics: Themes Within Reflective Voices" (Diana B. Erchick and Linda Condron); .(4) "Representations and Mathematics Visualization" (Fernando Hitt); (5) "Models and Modeling" (Richard Lesh, Helen Doerr, Thomas Post, Judith S. Zawojewski, and Guadalupe Carmona); (6) "Geometry and Technology" (Douglas McDougall and Jean J. McGehee); (7) "The Complexity of Learning to Reason Probabilistically" (Carolyn A. Maher and Robert Speiser); and (8) "Problem Posing Research: Answered and Unanswered Questions" (Lyn D. English). Volume 2 of the proceedings of the 23rd annual meeting of the International Group for the Psychology of Mathematics Education (PME) held October, 2001 in Snowbird, Utah is also included. Papers include: (1) "Growth in Student Mathematical Understanding Through Precalculus Student and Teacher Interactions" (Daniel R. Ilaria and Carolyn A. Maher); (2) "Investigating the Teaching and Learning of Proof: First Year Results" (Tami S. Martin and Sharon Soucy McCrone); (3) "Beliefs About Proof in Collegiate Calculus" (Manya Raman); (4) "Metaphors of the Novice: Emergent Aspects of Calculus Students' Reasoning" (Mike Oehrtman); (5) "An Examination of the Interaction Patterns of a Single-Gender Mathematics Class" (Amy Burns Ellis); (6) "The Role of Intellectual Culture of Mathematics in Doctoral Student Attrition" (Abbe H. Herzig); (7) "Group Tests as a Context for / Learning How to Better Facilitate Small Group Discussion" (Judith Kysh); and (0 "Learning Paths to 5-and 10-Structured Understanding of Quantity: Addition and Subtraction Solution Strategies of Japanese Children" (Aki uLaLa aLen eproductions supplied by EDRS are the best that can be made from the original document

Rules Without Reason: Allowing Students to Rethink Previous Conceptions

The Mathematics Enthusiast, Jul 1, 2010

This paper reports on the strategies chosen by a group of sixth-grade students in an urban inform... more This paper reports on the strategies chosen by a group of sixth-grade students in an urban informal learning program as they worked to solve an open-ended, nonroutine task. In particular, the paper focuses on the ability of these students to rise above their previous, procedure-based misconceptions and arrive at a mathematically reasonable solution. Aspects of the problem task, the problem-solving environment, and, importantly, of the nature of the teacher's interventions are analyzed to determine the conditions that encouraged students to approach mathematics as a logical, meaningful, sense-making activity.

Constructivist Views on the Teaching and Learning of Mathematics

National Council of Teachers of Mathematics is collaborating with JSTOR to digitize, preserve and... more National Council of Teachers of Mathematics is collaborating with JSTOR to digitize, preserve and extend access to Journal for Research in Mathematics Education. Monograph. "epistemic dependence," see Siegel, 1988.) To reject as knowledge a claim that the community accepts as true is a tricky business. How can constructivists reject such a knowledge claim? They might begin by saying that the student cannot give an adequate account of x. For example, how is x derived? But we do not expect students to be able to give this sort of account of most x 's. "How did you get it?" is the question we usually ask. The answers we accept to this question are based inevitably on the already established body of mathematical knowledge and the canons of thinking laid down by the mathematical community. The constructivist suggests that we can make a decision on the basis of whether or not x was constructed. But does this make sense? As a cognitive position, constructivism asserts that all mental activity is constructive. Even when students are in what look to be rote learning situations, they must perforce construct, because that is the way the mind operates. So it seems to me that constructivists should talk about weak and strong acts of construction rather than acts involving or not involving construction. In mathematical environments, strong acts of construction would no doubt be those recognized by mathematicians as mathematical; weak constructions would be those evaluated as limited in mathematical use. (I have already discussed those-like Benny's-that might be judged faulty because they do not belong at all to the applicable mathematical domain.) Some genuine and very tough questions about teaching follow. Might it not be the case, for example, that some students perform strong acts of construction no matter how the material is presented? And might it not also be the case that, while the teacher is encouraging exploration and genuine (strong) acts of construction, some students perform weak acts such as quietly waiting for group consensus and then noting the answer? Both aspects of the question about a knower's status need further analysis, elaboration, and clarification. It is by no means clear that the current use of epistemological language will be particularly helpful in this task. What is clear is that the emphasis on construction forces us to probe deeply into students' activity. How firm a grasp do they have on the material? What can they do with it? What misconceptions do they entertain? Even if they are producing wrong answers, are they constructing in a way that is mathematically recognizable? These are among the questions we need to ask in order to teach effectively, and they are not epistemological. Methodological Constructivism Acceptance of the premise that knowledge and (many constructivists would say) reality itself are constructed leads to methodological constructivism. In research this means that we have to investigate our subjects' perceptions, purposes, premises, and ways of working things out if Constructivist premises imply that there will be many roads to most solutions or instructional endpoints. We cannot, therefore, be sure that all students will find the anticipated skills necessary or even useful in their constructions. But there is a high probability that some particular skills will be needed in any given task. Any teacher who has conducted an overt thinking session sympathizes with students who must agonize over every step of a solution-often forgetting in the process why they are dividing, or solving a proportion, or factoring an unwieldy expression. My point here is that we need not discard all of the strategies recommended by theorists who espouse direct instruction (Brophy, 1986a; Good, Grouws, & Ebmeier, 1983) even if we disagree with them on fundamental cognitive premises (Noddings, 1986). Many mathematics educators recognize the power of "constructivist methods" in oneto-one situations, but they also see that schoolteachers cannot work continuously in such situations (Cobb, Wood, & Yackel, this volume). Classroom conditions force us to think about instructional economies. Constructivist teachers have to keep their basic premises in mind, but they should feel free to adapt a wide variety of methods for their own purposes. Given our premises, we need to get thinking out into the open, to encourage students to conceive their own mathematical purposes and execute their own plans, and to provide situations and objects that may trigger conflict (disequilibrium) and reflective abstraction. Consider, first, the common constructivist recommendation that teachers make heavy use of manipulatives. This recommendation was an early and plausible attempt to apply Piagetian theory directly to teaching. If reflective abstraction proceeds from the operations we perform on objects, then it makes sense to have our students work with objects. The difficulty, of course, is that students must have a purpose for engaging in the manipulation of objects. Otherwise, objects can be as mysterious as numerals; even Cuisenaire rods can become "symbols made of colored wood" (Holt, 1964). Understanding this possibility, we need, perhaps, to provide some direct instruction on the use of various manipulatives and then simply make them available. In actual problem solving situations, we probably should not guide students in their use. If we do, we are likely to detach students from their own purposes and set them blindly to work on our own. Caveats of this sort spring up everywhere in constructivist teaching. Students will construct, but we want their constructions to be guided by mathematical purposes, not by the need to figure out what teachers want or where they are headed. Next, because teachers have to work with many children, it makes sense to ask whether there is a way to approximate the one-to-one situation with a whole class. Can we elicit genuine student thinking in the whole class situation? Several promising models have appeared (see the essays in this volume and also Davis, 1984; Lampert, 1988; Schoenfeld, 1985; Steffe, Cobb, & von Glasersfeld, 1988). All of these methods share a common characteristic: They are all highly interactive. Teachers both model and elicit, but they model by asking questions, following leads, and conjecturing rather than presenting faultless products. Teaching this way requires considerable mathematical knowledge as well as pedagogical skill. How can teachers follow students' suggestions if they do not know enough mathematics to perceive where the suggestions may lead? This is a problem for teacher education. But a caveat again arises directly out of the constructivist framework. While a lesson of the sort advocated above is conducted, students will be "constructing." Some will be performing strong (mathematical) acts of construction. Hearing evidence of such thinking, teachers (and observers) may be delighted with the lesson. But other students (how many?) may be performing weak acts on the problem at hand, and some may be "chasing deer in the wildwood" as Virgil Mallory used to say. It would, therefore, seem unwise to rely on such lessons day in and day out. How else can we induce the engagement that is essential if students are to perform powerful constructions? One possibility is to increase the amount of time students spend working together (see Cobb, Wood, & Yackel; Maher & Alston, this volume). The use of small groups in cooperative learning is becoming a popular strategy, and there are sound cognitive reasons for allowing students to work together. Vygotsky (1978) posited group interaction as one source in the development of mental operations; that is, he suggested that children gradually internalize the talk that occurs in groups. They begin to challenge themselves, ask for reasons, and in general monitor their own mental work as others do their public speech. Cobb, Wood, and Yackel follow a line of thinking called "social constructivism" that puts great emphasis on the processes of communicating and negotiating in communities. A difficulty here is that, once again, we must somehow ensure that the community is a mathematical community. To assume that the deliberations of any community-conducted by some general plan of right action-will lead to acceptable mathematical results suggests a traditional epistemology. In a post-epistemological perspective, we recognize the canons and ways authorized by this particular community. (On the possibility of construing epistemology as itself the study of human understanding, see Toulmin, 1972.) But group work is not a pedagogical panacea in any case (see Noddings, 1989). Some students may participate eagerly while others sit out the session waiting for answers to develop. Assigning rigid roles so that everyone has to participate can distract students' attention from mathematics to the group process itself. Further, students can be rude and cruel to one another, and teachers have to watch group operations to be sure that students are learning to help and care for each other-not just to solve problems in expeditious ways. Here, too, there is a large part for teacher education to play. The literature on small groups and cooperative learning is growing rapidly. What kind of small group scheme should teachers use? How should groups be constituted? For what kinds of task? Should there be inter-group competition? Should teachers use individual or group evaluation? Will the endproduct be a group one or set of individual ones (Noddings, 1989)? Teachers need to be wellinformed in order to use these methods effectively. Further, constructivist teachers need practice in selecting and justifying the forms that are compatible with constructivist premises. The great strength of constructivism is that it leads us to think critically and imaginatively about the teaching-learning process. Believing the premises of constructivism, we no longer look for...

Promoting Student Reasoning Through Careful Task Design: A Comparison of Three Studies

Jornal Internacional de Estudos em Educação Matemática, Jun 18, 2015

Researchers have found that students as young as elementary school can engage in mathematical rea... more Researchers have found that students as young as elementary school can engage in mathematical reasoning. Specifically, particular tasks tend to encourage this reasoning. This paper provides insight into some general characteristics of tasks that may lead to arguments that represent varied forms of reasoning. In this paper we report on arguments built by diverse student groups, of different ages, that were used to justify their solutions to problems from the fraction and counting strands of longitudinal and cross-sectional studies. We compare the characteristics of the two tasks and suggest how the implementation of tasks such as these can help elicit varied student reasoning.

Challenging Beliefs About Children's Mathematics Learning Through Video Study

This study examines the extent to which pre-service teacher beliefs about learning mathematics ar... more This study examines the extent to which pre-service teacher beliefs about learning mathematics are impacted by studying videos of children’s learning. Elementary pre-service teachers from three universities participated and studied videos of children’s learning stored at the Video Mosaic Repository of Rutgers University as an intervention unit in a required mathematics education course. A pre and post beliefs assessment was administered to participants as well as to comparison groups. A statistical analysis of beliefs data showed that across all sites there was a significantly greater change in experimental pre-service teacher beliefs about student learning as compared to the comparison groups.

Digitalized Interactive Components in Computer-Based-Assessment in Mathematics for K12 Students: A Research Synthesis

Computers in the schools, Oct 3, 2022

Meaningful Assessments of Students Who Struggle to Learn Mathematics

Springer eBooks, 2022

Proceedings of the Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (11th, New Brunswick, New Jersey, September 20-23, 1989), Volume 1: Research Reports

Reproductions supplied by EDRS are the best that can be made from the original document.

Eliciting Student Reasoning Through Problem Solving

Researchers have found that students as young as elementary school can engage in mathematical rea... more Researchers have found that students as young as elementary school can engage in mathematical reasoning. However, particular tasks tend to encourage this reasoning. This paper provides insight into the characteristics of tasks that lead to arguments that represent certain forms of reasoning. In this paper we report on arguments built by diverse student groups, of different ages, that were used to justify their solutions to problems from the fraction and counting strands of longitudinal and cross-sectional studies.

Chapter 5: What Do We Do When We "Do Mathematics"?

Journal for research in mathematics education, 1990

"Every theory," Albert Einstein once said, "should be as simple as possible, but n... more "Every theory," Albert Einstein once said, "should be as simple as possible, but no simpler." We argue here that those of us who study the learning and teaching of mathematics are handicapped by encounters with theories that are, indeed, simpler than the allowable minimum. The antidote, we argue, consists of two necessary steps: careful observation of the details of people who are dealing with mathematical situations of various sorts, and the postulation of somewhat more elaborated theories.

Paradigm shifts in mathematical learning and teaching: The legacy of Robert B. Davis, Founding Editor: The Journal of Mathematical Behavior

The Journal of Mathematical Behavior, 2021

Imaginative Learning

Encyclopedia of the Sciences of Learning, 2012

Information-Processing Approach

Encyclopedia of the Sciences of Learning, 2012

Various methods are employed to communicate pharmaceutical information to the general public. Inf... more Various methods are employed to communicate pharmaceutical information to the general public. Information may be conveyed by a variety of sources including labeling on the product container itself, on enclosures such as patient product inserts (PPis), on exterior packaging, through advertising, or via direct communication with medical professionals. In many instances, the printed material supplied with a pharmaceutical product may be the only medium used to educate consumers on information associated with the product. Using these printed materials to communicate pharmaceutical information to older adults is a practice that is receiving greater attention. Interest is due in large part to the fact that older adults tend to consume more medications than other population groups. This fact, combined with the onset of visual and cognitive difficulties that accompany the aging process, make the creation of effective methods of communicating medication information to adults in general, and older adults in particular, an important challenge. Pharmaceutical labels are used to disseminate information about medication uses, indications, benefits, and potential hazards. Effective medication labels serve

Schools, mathematics, and the world of reality

Proceedings of the Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (11th, New Brunswick, New Jersey, September 20-23, 1989), Volume 2: Plenary Lectures and Symposia

IliwiMLOiu.Pn , logico-mathematical abstraction refers to the construction of logicomathematical ... more IliwiMLOiu.Pn , logico-mathematical abstraction refers to the construction of logicomathematical invariants together with the relevant logico-physical invariants, the reversibility and composition of logico-mathematical transformations and operations, and their generalization; formalization refers to its usual interpretations, that of axiomatization and formal proof which at the elementary level could be viewed as the discovery of axioms and the elaboration of logical mathematical justifications. Two additional meanings are assigned to formalization: that of enclosing a mathematical notion into a formal definition, and that of using mathematical symbolization for notions for which prior procedural understanding or abstraction already exist to some degree. This model suggests a distinction between on one hand logico-physical understanding which results from thinking about procedures applied to physical objects and about spatio-physical transformations of these objects, and on the other hand logico-mathematical understanding which results from thinking applied to procedures and transformations dealing with mathematical objects. In this framework, one can contend with reflective abstraction of actions operating in the physical realm without necessarily describing it as somehow having to be mathematical. This is not too different from the views Piaget developed in his later years. Piaget has acknowledged this when he suggested two forms of reflective abstraction: We will speak in this case of "pseudo-empirical abstractions" since the information is based on the objects; however, the information regarding their properties results from the subject's actions on these objects. And this initial form of reflective abstraction plays a fundamental psycho-genetic role in all logico-mathematical learning, as long as the subject requires concrete manipulations in order to understand certain structures that might be considered too 'abstract'. Piaget, J. (1974).Adaptation vitale et psychologie de l'intelligence.Paris: Hermann, 83-84. (our translation) The existence of two tiers in our model takes into account the subject's action on his or her physical environment. The two forms of reflective abstraction are comparable to the two aspects of understanding in our model: Piaget's pseudoempirical abstraction is equivalent to our logico-physical abstraction, his logicomathematical abstraction is the same as the one we mention in our second tier. The following diagram gives a global view of the model: 9 xr ( UNDERSTANDING OF PRELIMINARY PHYSICAL CONCEPT Intuitive .procedural Logico-physical understanding Logico-physical abstraction abstraction V Logico-math, procedural .....Logico-math. understanding abstraction IF Formalization UNDERSTANDING OF EMERGING MATHEMATICAL CONCEPT It should be noted that the three levels of understanding included in the first tier are essentially linear. Without prior intuitive understanding, the acquisition of concrete procedures could hardly qualify as understanding. Similarly, one cannot expect the child to achieve any logico-physical abstraction without being able to reflect on the procedures used . Nevertheless, the model as a whole is not linear. The aspects of understanding identified in the second tier need not await the completion of the physical tier. Well before they achieve logico-physical abstraction, children can start acquiring the various relevant arithmetic procedures by the quantification of problems introduced in the first tier. Formalization need not await the completion of logico-mathematical abstraction; for instance, the formalization of the arithmetic procedures will occur much earlier than formalization of the axioms. Let us remember that the objective here is to develop a model that will adequately describe the construction of a conceptual scheme. We now will look at three such applications.

Teachers attending to student reasoning: Do beliefs matter?

The Journal of Mathematical Behavior, Mar 1, 2023

Implementing the “Professional Standards for Teaching Mathematics”: Teachers Paying Attention to Students' Thinking

The Arithmetic Teacher, May 1, 1992

The Professional Standards for Teaching Mathematics in its call for reform underscores the import... more The Professional Standards for Teaching Mathematics in its call for reform underscores the importance of teachers&amp;#39; knowledge of how students build their mathematical ideas. In our own work we have come to stress the importance of teachers&amp;#39; awareness of students&amp;#39; thought processes (Davis 1984; Davis and Maher 1990; Maher, Davis, and Alston 1991; Maher and Davis 1990). Teachers&amp;#39; knowledge of students&amp;#39; thinking is an important guide in planning effective lessons.

Inquiry into the interlocution of students engaged with mathematics: appreciating links between research and practice

$Research paper thumbnail of Analysis of the argumentation of nine-year-olds engaged in discourse about comparing fraction models$

Analysis of the argumentation of nine-year-olds engaged in discourse about comparing fraction models

The Journal of Mathematical Behavior, Mar 1, 2019

In this paper we analyze the argumentation of nine-year-olds as they justify their reasoning for ... more In this paper we analyze the argumentation of nine-year-olds as they justify their reasoning for a fraction comparison problem. We illustrate how their proof-like arguments evolve in a class discussion and are backed by the rod models they produce, showing both valid and invalid arguments. It is well established that argumentation is an important aspect of K-12 education and that instructional programs should cultivate a classroom environment in which students are encouraged to evaluate arguments, ask questions to clarify arguments, and develop arguments that make sense (Krummheuer

Video Recordings as Pedagogical Tools in Mathematics Teacher Education

Proceedings of the Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Snowbird, Utah, October 18-21, 2001). Volume 1 [and] Volume 2

This book contains volume 1 of the proceedings of the 23rd annual meeting of the International Gr... more This book contains volume 1 of the proceedings of the 23rd annual meeting of the International Group for the Psychology of Mathematics Education (PME) held October, 2001 in Snowbird, Utah. Papers include: (1) "Opening the Dimensions of Mathematical Capability: The Development of Knowledge, Practice and Identity in Mathematics Classrooms" (Jo Boaler); (2) "Learning Mathematics as Developing a Discourse" (Anna Sfard); (3) "Gender and Mathematics: Themes Within Reflective Voices" (Diana B. Erchick and Linda Condron); .(4) "Representations and Mathematics Visualization" (Fernando Hitt); (5) "Models and Modeling" (Richard Lesh, Helen Doerr, Thomas Post, Judith S. Zawojewski, and Guadalupe Carmona); (6) "Geometry and Technology" (Douglas McDougall and Jean J. McGehee); (7) "The Complexity of Learning to Reason Probabilistically" (Carolyn A. Maher and Robert Speiser); and (8) "Problem Posing Research: Answered and Unanswered Questions" (Lyn D. English). Volume 2 of the proceedings of the 23rd annual meeting of the International Group for the Psychology of Mathematics Education (PME) held October, 2001 in Snowbird, Utah is also included. Papers include: (1) "Growth in Student Mathematical Understanding Through Precalculus Student and Teacher Interactions" (Daniel R. Ilaria and Carolyn A. Maher); (2) "Investigating the Teaching and Learning of Proof: First Year Results" (Tami S. Martin and Sharon Soucy McCrone); (3) "Beliefs About Proof in Collegiate Calculus" (Manya Raman); (4) "Metaphors of the Novice: Emergent Aspects of Calculus Students' Reasoning" (Mike Oehrtman); (5) "An Examination of the Interaction Patterns of a Single-Gender Mathematics Class" (Amy Burns Ellis); (6) "The Role of Intellectual Culture of Mathematics in Doctoral Student Attrition" (Abbe H. Herzig); (7) "Group Tests as a Context for / Learning How to Better Facilitate Small Group Discussion" (Judith Kysh); and (0 "Learning Paths to 5-and 10-Structured Understanding of Quantity: Addition and Subtraction Solution Strategies of Japanese Children" (Aki uLaLa aLen eproductions supplied by EDRS are the best that can be made from the original document

Rules Without Reason: Allowing Students to Rethink Previous Conceptions

The Mathematics Enthusiast, Jul 1, 2010

This paper reports on the strategies chosen by a group of sixth-grade students in an urban inform... more This paper reports on the strategies chosen by a group of sixth-grade students in an urban informal learning program as they worked to solve an open-ended, nonroutine task. In particular, the paper focuses on the ability of these students to rise above their previous, procedure-based misconceptions and arrive at a mathematically reasonable solution. Aspects of the problem task, the problem-solving environment, and, importantly, of the nature of the teacher's interventions are analyzed to determine the conditions that encouraged students to approach mathematics as a logical, meaningful, sense-making activity.

Constructivist Views on the Teaching and Learning of Mathematics

National Council of Teachers of Mathematics is collaborating with JSTOR to digitize, preserve and... more National Council of Teachers of Mathematics is collaborating with JSTOR to digitize, preserve and extend access to Journal for Research in Mathematics Education. Monograph. "epistemic dependence," see Siegel, 1988.) To reject as knowledge a claim that the community accepts as true is a tricky business. How can constructivists reject such a knowledge claim? They might begin by saying that the student cannot give an adequate account of x. For example, how is x derived? But we do not expect students to be able to give this sort of account of most x 's. "How did you get it?" is the question we usually ask. The answers we accept to this question are based inevitably on the already established body of mathematical knowledge and the canons of thinking laid down by the mathematical community. The constructivist suggests that we can make a decision on the basis of whether or not x was constructed. But does this make sense? As a cognitive position, constructivism asserts that all mental activity is constructive. Even when students are in what look to be rote learning situations, they must perforce construct, because that is the way the mind operates. So it seems to me that constructivists should talk about weak and strong acts of construction rather than acts involving or not involving construction. In mathematical environments, strong acts of construction would no doubt be those recognized by mathematicians as mathematical; weak constructions would be those evaluated as limited in mathematical use. (I have already discussed those-like Benny's-that might be judged faulty because they do not belong at all to the applicable mathematical domain.) Some genuine and very tough questions about teaching follow. Might it not be the case, for example, that some students perform strong acts of construction no matter how the material is presented? And might it not also be the case that, while the teacher is encouraging exploration and genuine (strong) acts of construction, some students perform weak acts such as quietly waiting for group consensus and then noting the answer? Both aspects of the question about a knower's status need further analysis, elaboration, and clarification. It is by no means clear that the current use of epistemological language will be particularly helpful in this task. What is clear is that the emphasis on construction forces us to probe deeply into students' activity. How firm a grasp do they have on the material? What can they do with it? What misconceptions do they entertain? Even if they are producing wrong answers, are they constructing in a way that is mathematically recognizable? These are among the questions we need to ask in order to teach effectively, and they are not epistemological. Methodological Constructivism Acceptance of the premise that knowledge and (many constructivists would say) reality itself are constructed leads to methodological constructivism. In research this means that we have to investigate our subjects' perceptions, purposes, premises, and ways of working things out if Constructivist premises imply that there will be many roads to most solutions or instructional endpoints. We cannot, therefore, be sure that all students will find the anticipated skills necessary or even useful in their constructions. But there is a high probability that some particular skills will be needed in any given task. Any teacher who has conducted an overt thinking session sympathizes with students who must agonize over every step of a solution-often forgetting in the process why they are dividing, or solving a proportion, or factoring an unwieldy expression. My point here is that we need not discard all of the strategies recommended by theorists who espouse direct instruction (Brophy, 1986a; Good, Grouws, & Ebmeier, 1983) even if we disagree with them on fundamental cognitive premises (Noddings, 1986). Many mathematics educators recognize the power of "constructivist methods" in oneto-one situations, but they also see that schoolteachers cannot work continuously in such situations (Cobb, Wood, & Yackel, this volume). Classroom conditions force us to think about instructional economies. Constructivist teachers have to keep their basic premises in mind, but they should feel free to adapt a wide variety of methods for their own purposes. Given our premises, we need to get thinking out into the open, to encourage students to conceive their own mathematical purposes and execute their own plans, and to provide situations and objects that may trigger conflict (disequilibrium) and reflective abstraction. Consider, first, the common constructivist recommendation that teachers make heavy use of manipulatives. This recommendation was an early and plausible attempt to apply Piagetian theory directly to teaching. If reflective abstraction proceeds from the operations we perform on objects, then it makes sense to have our students work with objects. The difficulty, of course, is that students must have a purpose for engaging in the manipulation of objects. Otherwise, objects can be as mysterious as numerals; even Cuisenaire rods can become "symbols made of colored wood" (Holt, 1964). Understanding this possibility, we need, perhaps, to provide some direct instruction on the use of various manipulatives and then simply make them available. In actual problem solving situations, we probably should not guide students in their use. If we do, we are likely to detach students from their own purposes and set them blindly to work on our own. Caveats of this sort spring up everywhere in constructivist teaching. Students will construct, but we want their constructions to be guided by mathematical purposes, not by the need to figure out what teachers want or where they are headed. Next, because teachers have to work with many children, it makes sense to ask whether there is a way to approximate the one-to-one situation with a whole class. Can we elicit genuine student thinking in the whole class situation? Several promising models have appeared (see the essays in this volume and also Davis, 1984; Lampert, 1988; Schoenfeld, 1985; Steffe, Cobb, & von Glasersfeld, 1988). All of these methods share a common characteristic: They are all highly interactive. Teachers both model and elicit, but they model by asking questions, following leads, and conjecturing rather than presenting faultless products. Teaching this way requires considerable mathematical knowledge as well as pedagogical skill. How can teachers follow students' suggestions if they do not know enough mathematics to perceive where the suggestions may lead? This is a problem for teacher education. But a caveat again arises directly out of the constructivist framework. While a lesson of the sort advocated above is conducted, students will be "constructing." Some will be performing strong (mathematical) acts of construction. Hearing evidence of such thinking, teachers (and observers) may be delighted with the lesson. But other students (how many?) may be performing weak acts on the problem at hand, and some may be "chasing deer in the wildwood" as Virgil Mallory used to say. It would, therefore, seem unwise to rely on such lessons day in and day out. How else can we induce the engagement that is essential if students are to perform powerful constructions? One possibility is to increase the amount of time students spend working together (see Cobb, Wood, & Yackel; Maher & Alston, this volume). The use of small groups in cooperative learning is becoming a popular strategy, and there are sound cognitive reasons for allowing students to work together. Vygotsky (1978) posited group interaction as one source in the development of mental operations; that is, he suggested that children gradually internalize the talk that occurs in groups. They begin to challenge themselves, ask for reasons, and in general monitor their own mental work as others do their public speech. Cobb, Wood, and Yackel follow a line of thinking called "social constructivism" that puts great emphasis on the processes of communicating and negotiating in communities. A difficulty here is that, once again, we must somehow ensure that the community is a mathematical community. To assume that the deliberations of any community-conducted by some general plan of right action-will lead to acceptable mathematical results suggests a traditional epistemology. In a post-epistemological perspective, we recognize the canons and ways authorized by this particular community. (On the possibility of construing epistemology as itself the study of human understanding, see Toulmin, 1972.) But group work is not a pedagogical panacea in any case (see Noddings, 1989). Some students may participate eagerly while others sit out the session waiting for answers to develop. Assigning rigid roles so that everyone has to participate can distract students' attention from mathematics to the group process itself. Further, students can be rude and cruel to one another, and teachers have to watch group operations to be sure that students are learning to help and care for each other-not just to solve problems in expeditious ways. Here, too, there is a large part for teacher education to play. The literature on small groups and cooperative learning is growing rapidly. What kind of small group scheme should teachers use? How should groups be constituted? For what kinds of task? Should there be inter-group competition? Should teachers use individual or group evaluation? Will the endproduct be a group one or set of individual ones (Noddings, 1989)? Teachers need to be wellinformed in order to use these methods effectively. Further, constructivist teachers need practice in selecting and justifying the forms that are compatible with constructivist premises. The great strength of constructivism is that it leads us to think critically and imaginatively about the teaching-learning process. Believing the premises of constructivism, we no longer look for...

Promoting Student Reasoning Through Careful Task Design: A Comparison of Three Studies

Jornal Internacional de Estudos em Educação Matemática, Jun 18, 2015

Researchers have found that students as young as elementary school can engage in mathematical rea... more Researchers have found that students as young as elementary school can engage in mathematical reasoning. Specifically, particular tasks tend to encourage this reasoning. This paper provides insight into some general characteristics of tasks that may lead to arguments that represent varied forms of reasoning. In this paper we report on arguments built by diverse student groups, of different ages, that were used to justify their solutions to problems from the fraction and counting strands of longitudinal and cross-sectional studies. We compare the characteristics of the two tasks and suggest how the implementation of tasks such as these can help elicit varied student reasoning.

Challenging Beliefs About Children's Mathematics Learning Through Video Study

This study examines the extent to which pre-service teacher beliefs about learning mathematics ar... more This study examines the extent to which pre-service teacher beliefs about learning mathematics are impacted by studying videos of children’s learning. Elementary pre-service teachers from three universities participated and studied videos of children’s learning stored at the Video Mosaic Repository of Rutgers University as an intervention unit in a required mathematics education course. A pre and post beliefs assessment was administered to participants as well as to comparison groups. A statistical analysis of beliefs data showed that across all sites there was a significantly greater change in experimental pre-service teacher beliefs about student learning as compared to the comparison groups.

Digitalized Interactive Components in Computer-Based-Assessment in Mathematics for K12 Students: A Research Synthesis

Computers in the schools, Oct 3, 2022

Meaningful Assessments of Students Who Struggle to Learn Mathematics

Springer eBooks, 2022

Proceedings of the Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (11th, New Brunswick, New Jersey, September 20-23, 1989), Volume 1: Research Reports

Reproductions supplied by EDRS are the best that can be made from the original document.

Eliciting Student Reasoning Through Problem Solving

Researchers have found that students as young as elementary school can engage in mathematical rea... more Researchers have found that students as young as elementary school can engage in mathematical reasoning. However, particular tasks tend to encourage this reasoning. This paper provides insight into the characteristics of tasks that lead to arguments that represent certain forms of reasoning. In this paper we report on arguments built by diverse student groups, of different ages, that were used to justify their solutions to problems from the fraction and counting strands of longitudinal and cross-sectional studies.

Chapter 5: What Do We Do When We "Do Mathematics"?

Journal for research in mathematics education, 1990

"Every theory," Albert Einstein once said, "should be as simple as possible, but n... more "Every theory," Albert Einstein once said, "should be as simple as possible, but no simpler." We argue here that those of us who study the learning and teaching of mathematics are handicapped by encounters with theories that are, indeed, simpler than the allowable minimum. The antidote, we argue, consists of two necessary steps: careful observation of the details of people who are dealing with mathematical situations of various sorts, and the postulation of somewhat more elaborated theories.

Paradigm shifts in mathematical learning and teaching: The legacy of Robert B. Davis, Founding Editor: The Journal of Mathematical Behavior

The Journal of Mathematical Behavior, 2021

Imaginative Learning

Encyclopedia of the Sciences of Learning, 2012

Information-Processing Approach

Encyclopedia of the Sciences of Learning, 2012

Various methods are employed to communicate pharmaceutical information to the general public. Inf... more Various methods are employed to communicate pharmaceutical information to the general public. Information may be conveyed by a variety of sources including labeling on the product container itself, on enclosures such as patient product inserts (PPis), on exterior packaging, through advertising, or via direct communication with medical professionals. In many instances, the printed material supplied with a pharmaceutical product may be the only medium used to educate consumers on information associated with the product. Using these printed materials to communicate pharmaceutical information to older adults is a practice that is receiving greater attention. Interest is due in large part to the fact that older adults tend to consume more medications than other population groups. This fact, combined with the onset of visual and cognitive difficulties that accompany the aging process, make the creation of effective methods of communicating medication information to adults in general, and older adults in particular, an important challenge. Pharmaceutical labels are used to disseminate information about medication uses, indications, benefits, and potential hazards. Effective medication labels serve

Schools, mathematics, and the world of reality

Proceedings of the Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (11th, New Brunswick, New Jersey, September 20-23, 1989), Volume 2: Plenary Lectures and Symposia

IliwiMLOiu.Pn , logico-mathematical abstraction refers to the construction of logicomathematical ... more IliwiMLOiu.Pn , logico-mathematical abstraction refers to the construction of logicomathematical invariants together with the relevant logico-physical invariants, the reversibility and composition of logico-mathematical transformations and operations, and their generalization; formalization refers to its usual interpretations, that of axiomatization and formal proof which at the elementary level could be viewed as the discovery of axioms and the elaboration of logical mathematical justifications. Two additional meanings are assigned to formalization: that of enclosing a mathematical notion into a formal definition, and that of using mathematical symbolization for notions for which prior procedural understanding or abstraction already exist to some degree. This model suggests a distinction between on one hand logico-physical understanding which results from thinking about procedures applied to physical objects and about spatio-physical transformations of these objects, and on the other hand logico-mathematical understanding which results from thinking applied to procedures and transformations dealing with mathematical objects. In this framework, one can contend with reflective abstraction of actions operating in the physical realm without necessarily describing it as somehow having to be mathematical. This is not too different from the views Piaget developed in his later years. Piaget has acknowledged this when he suggested two forms of reflective abstraction: We will speak in this case of "pseudo-empirical abstractions" since the information is based on the objects; however, the information regarding their properties results from the subject's actions on these objects. And this initial form of reflective abstraction plays a fundamental psycho-genetic role in all logico-mathematical learning, as long as the subject requires concrete manipulations in order to understand certain structures that might be considered too 'abstract'. Piaget, J. (1974).Adaptation vitale et psychologie de l'intelligence.Paris: Hermann, 83-84. (our translation) The existence of two tiers in our model takes into account the subject's action on his or her physical environment. The two forms of reflective abstraction are comparable to the two aspects of understanding in our model: Piaget's pseudoempirical abstraction is equivalent to our logico-physical abstraction, his logicomathematical abstraction is the same as the one we mention in our second tier. The following diagram gives a global view of the model: 9 xr ( UNDERSTANDING OF PRELIMINARY PHYSICAL CONCEPT Intuitive .procedural Logico-physical understanding Logico-physical abstraction abstraction V Logico-math, procedural .....Logico-math. understanding abstraction IF Formalization UNDERSTANDING OF EMERGING MATHEMATICAL CONCEPT It should be noted that the three levels of understanding included in the first tier are essentially linear. Without prior intuitive understanding, the acquisition of concrete procedures could hardly qualify as understanding. Similarly, one cannot expect the child to achieve any logico-physical abstraction without being able to reflect on the procedures used . Nevertheless, the model as a whole is not linear. The aspects of understanding identified in the second tier need not await the completion of the physical tier. Well before they achieve logico-physical abstraction, children can start acquiring the various relevant arithmetic procedures by the quantification of problems introduced in the first tier. Formalization need not await the completion of logico-mathematical abstraction; for instance, the formalization of the arithmetic procedures will occur much earlier than formalization of the axioms. Let us remember that the objective here is to develop a model that will adequately describe the construction of a conceptual scheme. We now will look at three such applications.

Teachers attending to student reasoning: Do beliefs matter?

The Journal of Mathematical Behavior, Mar 1, 2023

Implementing the “Professional Standards for Teaching Mathematics”: Teachers Paying Attention to Students' Thinking

The Arithmetic Teacher, May 1, 1992

The Professional Standards for Teaching Mathematics in its call for reform underscores the import... more The Professional Standards for Teaching Mathematics in its call for reform underscores the importance of teachers&amp;#39; knowledge of how students build their mathematical ideas. In our own work we have come to stress the importance of teachers&amp;#39; awareness of students&amp;#39; thought processes (Davis 1984; Davis and Maher 1990; Maher, Davis, and Alston 1991; Maher and Davis 1990). Teachers&amp;#39; knowledge of students&amp;#39; thinking is an important guide in planning effective lessons.

Inquiry into the interlocution of students engaged with mathematics: appreciating links between research and practice