This article advances two arguments. First, the domain of algebra teaching and research (especial... more This article advances two arguments. First, the domain of algebra teaching and research (especially at the secondary level) would be significantly enhanced if we developed a set of conceptual learning goals that could provide a language and focus for both researchers and practitioners. I characterize a conceptual learning goal as a statement of the concepts that serve as the target of instruction, where “concept” refers broadly to mathematical meanings, images, ideas, connections, ways of comprehending situations, and explanations regarding why particular procedures work, which can be leveraged productively in students’ mathematical development. Second, I argue that the empirical study of students’ thinking can play a role in the identification of conceptual learning goals. While there is a strong tradition of using psychological analysis to inform our models of what students know and how they learn, such analysis can also be used to help articulate what should be learned.
Although any mainstream thought is subject to theoretical challenges, the challenges to the
mains... more Although any mainstream thought is subject to theoretical challenges, the challenges to the mainstream cognitive perspective on transfer have had an unfortunate divisive effect. This article takes a pragmatic view that transfer perspectives are simply designed objects (Plomp & Nieveen, 2007), which provide different information for different purposes. Specifically, this paper compares one alternative approach—the actor-oriented transfer perspective—with the mainstream cognitive perspective on transfer, by examining the points of compatibility and tension across 5 dimensions. As a result, a space is opened up to explore 3 issues that are particularly well suited to an actor-oriented transfer approach: (a) how students interpret transfer situations, (b) the socially situated nature of transfer processes in classrooms, and (c) how contextual-sensitivity can play a productive role in the transfer of learning. Exploring the benefits and trade-offs of various approaches allows for greater understanding of the contributions of each perspective to educational research and practice.
This article is a response to Foundations for Success: The Final Report
of the National Mathemati... more This article is a response to Foundations for Success: The Final Report of the National Mathematics Advisory Panel (2008) and to one of the task group reports on which it was based, the report of the Task Group on Learning Processes. The author uses Maxwell’s two views of causality—regularity and process—to explore three major issues raised in the report: the nature of what is learned, how learning occurs, and the transfer of learning. She proposes alternative recommendations to those offered by the Panel, by drawing upon the mathematics education literature, which was largely excluded from the reviewed research. Furthermore, by neglecting research grounded in a process view of causality, the report excludes perspectives and findings that would have illuminated what it means to develop conceptual understanding in mathematics—one of three valued outcomes cited by the Panel.
JANUARY/FEBRUARY 2003 17
Limitations with current approaches to the investigation of the transfer... more JANUARY/FEBRUARY 2003 17 Limitations with current approaches to the investigation of the transfer of learning in design experiments constrain the type of information that is available to researchers as they make design decisions. This article addresses these limitations by presenting a reconceptualization of transfer, called actor-oriented transfer, which emerged from design experiment work. The merits of this alternative model are considered in terms of the information it provides to design experimenters.
Book Review of
Early Algebraization: A Global Dialogue From Multiple Perspectives. Jinfa Cai and... more Book Review of Early Algebraization: A Global Dialogue From Multiple Perspectives. Jinfa Cai and Eric Knuth (Eds., 2011). Heidelberg, Germany: Springer, 623 pp. ISBN 978-3642177347 (hb) $99.00, (e-book) $103.20.
We address the telling/not-telling dilemma in mathematics education. Telling is instructionally i... more We address the telling/not-telling dilemma in mathematics education. Telling is instructionally important, but has been downplayed due to both perceived inconsistencies with constructivism and attempts to develop pedagogical implications of constructivism. In response, we advance a theoretical reformulation of telling as the set of teaching actions that serve the function of stimulating students’ mathematical thoughts via the introduction of new ideas into a classroom conversation. We reformulate telling in three ways: (a) in terms of the function (which involves attention to the teacher’s intention, the nature of the teaching action, and the students’ interpretations of the action) rather than the form of teachers’ communicative acts; (b) in terms of the conceptual rather than procedural content of the new information; and (c) in terms of its relationship to other actions rather than as an isolated action. This reformulation resolves some of the concerns with teaching as telling and helps establish the legitimacy of providing new information within a constructivist perspective on learning.
Even in the resource-rich, more ideal conditions
of many design-based classroom interventions, un... more Even in the resource-rich, more ideal conditions of many design-based classroom interventions, unexpected events can lead to disappointing results in student learning. However, if later iterations in a design research study are more successful, the previous failures can provide opportunities for comparisons to reveal subtle differences in instruction that supported or constrained learning. This article presents design research in which a failure in an early iteration was addressed in a later iteration through an impromptu and unplanned response that better supported student learning. The responsive and interventionist nature of design research played a pivotal role in the emergence of the more successful outcome in the later iteration. The suitability of design research for investigating sociomathematical norms and for conducting conceptual analyses facilitated the deepening of our understanding of both the early failure and the later success. Specifically, our retrospective comparative analysis of these two non-consecutive iterations revealed that one sociomathematical norm better supported student learning of a particular mathematical concept then another norm, led to new insights into the conceptual complexity of division in rate situations, and revealed how the interrelatedness of sociomathematical norms and particular mathematical content can provide an important support for learning. We discuss how our cross-iteration analytic method extends existing approaches to leveraging failure in design research.
As transfer researchers have begun to investigate a broader range of phenomena, they
have corresp... more As transfer researchers have begun to investigate a broader range of phenomena, they have correspondingly put forward new processes to provide explanatory accounts for the occurrence of transfer. This move coincides with a call to acknowledge the contribution of social interactions, language, cultural artifacts, and normed practices to the generalization of learning. In this article, we posit “noticing” as a plausible transfer process and investigate both individual noticing and the social organization of noticing via the focusing framework. Specifically, we relate the nature of students’ reasoning on transfer tasks with what students notice mathematically in classrooms when many sources of information compete for their attention, and then we account for noticing as socially situated in classroom discourse practices, features of mathematical tasks, and the nature of mathematical activity.
Despite recent research interest in student-created diagrams, little research
has systematically ... more Despite recent research interest in student-created diagrams, little research has systematically investigated students' diagram-construction processes, meaning the order and manner in which students create markings as they physically generate diagrams. In this study, we characterize the various processes students use to create diagrams that represent a quadratic motion situation involving increasing speed, and we explore how these diagram-construction processes are related to students' conceptions of speed as inferred from their explanations with their completed diagrams. Previous literature suggests contrasting predictions regarding whether or not students' diagramconstruction processes are closely related (from our perspective as researchers) to students' inferred conceptions. We see the study as having value for research and practice by raising new questions related to diagram-construction processes, pointing to the potential formative assessment value of attending to diagram-construction processes, and demonstrating the need for the development of theory to explain the relationships identified by this study.
features that students can attend to or notice. What students notice mathematically
has consequen... more features that students can attend to or notice. What students notice mathematically has consequences for their subsequent reasoning. By adapting work from both cognitive science and applied linguistics anthropology, we present a focusing framework, which treats noticing as a complex phenomenon that is distributed across individual cognition, social interactions, material resources, and normed practices. Specifically, this research demonstrates that different centers of focus emerged in two middle grades mathematics classes addressing the same content goals, which, in turn, were related conceptually to differences in student reasoning on subsequent interview tasks. Furthermore, differences in the discourse practices, features of the mathematical tasks, and the nature of the mathematical activity in the two classrooms were related to the different mathematical features that students appeared to notice.
Despite the proliferation of mathematics standards internationally and despite general agreement
... more Despite the proliferation of mathematics standards internationally and despite general agreement on the importance of teaching for conceptual understanding, conceptual learning goals for many K-12 mathematics topics have not been well-articulated. This article presents a coherent set of five conceptual learning goals for a complex mathematical domain, generated via a method of systematic empirical analysis of students’ reasoning. Specifically, we compared the reasoning of pairs of students who performed differentially on the same task and inferred the pivotal intermediate conceptions that afforded one student deeper engagement with the task than another student. In turn, each pivotal intermediate conception framed an associated conceptual learning goal. While the empirical analysis of student reasoning is typically used to understand how students learn, we argue that such analysis should also play an important role in determining what concepts students should learn.
This paper extends recent efforts to critique and reconceive transfer by using an empirical study... more This paper extends recent efforts to critique and reconceive transfer by using an empirical study to rethink the surface/structure distinction of the traditional transfer paradigm. The findings suggest that what researchers typically consider a surface feature can present conceptual complexities for students that are more structural in nature than previously understood. In particular,we investigated the quantitative reasoning (meaning reasoning with measurable properties of an object) that is involved in making sense of a typical transfer situation. Two related analyses—one focused on quantitative reasoning and one on transfer — were performed on a case study. The results document how a student reconstructed his understanding of the relationships among quantities in a complex transfer situation in such a way that he was able to see the situation as fundamentally proportional in nature and subsequently make connections with previous proportional-reasoning experiences from a teaching experiment. In our discussion of the findings, we identify four relationships between quantitative reasoning and transfer.
This article sets forth a way of connecting the classroom instructional environment
with individu... more This article sets forth a way of connecting the classroom instructional environment with individual students’ generalizations. To do so, we advance the notion of focusing phenomena, that is, regularities in the ways in which teachers, students, artifacts, and curricular materials act together to direct attention toward certain mathematical properties over others. The construct of focusing phenomena emerged from an empirical study conducted during a 5-week unit on slope and linear functions in a high school classroom using a reform curriculum. Qualitative evidence from interviews with 7 students revealed that students interpreted the m value in y=b+mx as a difference rather than a ratio as a result of counterproductive generalization afforded by focusing phenomena. Classroom analysis revealed 4 focusing phenomena, which regularly directed students’ attention to various sets of differences rather than to the coordination of quantities.
When technology is implemented in classrooms, students often
form ideas that are unexpected and u... more When technology is implemented in classrooms, students often form ideas that are unexpected and unwanted by the teachers and the designers of the technology. This article advances the notion of the focusing effect of technology as a way of systematically accounting for the role of technology in such situations. A focusing effect refers to the direction of students’ attention toward certain properties of the subject matter domain over others, brought about by the use of particular technological tools. Technology focuses students’ attention in ways that are often not anticipated in advance and can have unintended consequences for students’ conceptions. Vignettes are presented from two research studies, one involving graphing calculators and one involving mathematics software called SimCalc Mathworlds. The research findings are synthesized and reinterpreted in order to illustrate and develop the notion of the focusing effect of technology. The significance of this construct lies in the connection that it affords between individual students’ conceptions and the way technology is used in the instructional environment. Implications for teaching and for the preparation of teachers in the use of technology are discussed.
We use the notion offocusing phenomena to help explain how a teacher’s actions were connected to ... more We use the notion offocusing phenomena to help explain how a teacher’s actions were connected to her students’ interpretations of a linear equation. This study was conducted in a high-school classroom that regularly emphasised dependency relationships in real-world situations. Seven interviews revealed a majority view ofy = b + mx as astorage container—a place to insert b and m values—rather than as a relationship between x- and y-values. Classroom analysis revealed how the teacher directed attention away from functional relationships with increasing frequency as she moved from realistic situations to conventional representations.
This article advances two arguments. First, the domain of algebra teaching and research (especial... more This article advances two arguments. First, the domain of algebra teaching and research (especially at the secondary level) would be significantly enhanced if we developed a set of conceptual learning goals that could provide a language and focus for both researchers and practitioners. I characterize a conceptual learning goal as a statement of the concepts that serve as the target of instruction, where “concept” refers broadly to mathematical meanings, images, ideas, connections, ways of comprehending situations, and explanations regarding why particular procedures work, which can be leveraged productively in students’ mathematical development. Second, I argue that the empirical study of students’ thinking can play a role in the identification of conceptual learning goals. While there is a strong tradition of using psychological analysis to inform our models of what students know and how they learn, such analysis can also be used to help articulate what should be learned.
Although any mainstream thought is subject to theoretical challenges, the challenges to the
mains... more Although any mainstream thought is subject to theoretical challenges, the challenges to the mainstream cognitive perspective on transfer have had an unfortunate divisive effect. This article takes a pragmatic view that transfer perspectives are simply designed objects (Plomp & Nieveen, 2007), which provide different information for different purposes. Specifically, this paper compares one alternative approach—the actor-oriented transfer perspective—with the mainstream cognitive perspective on transfer, by examining the points of compatibility and tension across 5 dimensions. As a result, a space is opened up to explore 3 issues that are particularly well suited to an actor-oriented transfer approach: (a) how students interpret transfer situations, (b) the socially situated nature of transfer processes in classrooms, and (c) how contextual-sensitivity can play a productive role in the transfer of learning. Exploring the benefits and trade-offs of various approaches allows for greater understanding of the contributions of each perspective to educational research and practice.
This article is a response to Foundations for Success: The Final Report
of the National Mathemati... more This article is a response to Foundations for Success: The Final Report of the National Mathematics Advisory Panel (2008) and to one of the task group reports on which it was based, the report of the Task Group on Learning Processes. The author uses Maxwell’s two views of causality—regularity and process—to explore three major issues raised in the report: the nature of what is learned, how learning occurs, and the transfer of learning. She proposes alternative recommendations to those offered by the Panel, by drawing upon the mathematics education literature, which was largely excluded from the reviewed research. Furthermore, by neglecting research grounded in a process view of causality, the report excludes perspectives and findings that would have illuminated what it means to develop conceptual understanding in mathematics—one of three valued outcomes cited by the Panel.
JANUARY/FEBRUARY 2003 17
Limitations with current approaches to the investigation of the transfer... more JANUARY/FEBRUARY 2003 17 Limitations with current approaches to the investigation of the transfer of learning in design experiments constrain the type of information that is available to researchers as they make design decisions. This article addresses these limitations by presenting a reconceptualization of transfer, called actor-oriented transfer, which emerged from design experiment work. The merits of this alternative model are considered in terms of the information it provides to design experimenters.
Book Review of
Early Algebraization: A Global Dialogue From Multiple Perspectives. Jinfa Cai and... more Book Review of Early Algebraization: A Global Dialogue From Multiple Perspectives. Jinfa Cai and Eric Knuth (Eds., 2011). Heidelberg, Germany: Springer, 623 pp. ISBN 978-3642177347 (hb) $99.00, (e-book) $103.20.
We address the telling/not-telling dilemma in mathematics education. Telling is instructionally i... more We address the telling/not-telling dilemma in mathematics education. Telling is instructionally important, but has been downplayed due to both perceived inconsistencies with constructivism and attempts to develop pedagogical implications of constructivism. In response, we advance a theoretical reformulation of telling as the set of teaching actions that serve the function of stimulating students’ mathematical thoughts via the introduction of new ideas into a classroom conversation. We reformulate telling in three ways: (a) in terms of the function (which involves attention to the teacher’s intention, the nature of the teaching action, and the students’ interpretations of the action) rather than the form of teachers’ communicative acts; (b) in terms of the conceptual rather than procedural content of the new information; and (c) in terms of its relationship to other actions rather than as an isolated action. This reformulation resolves some of the concerns with teaching as telling and helps establish the legitimacy of providing new information within a constructivist perspective on learning.
Even in the resource-rich, more ideal conditions
of many design-based classroom interventions, un... more Even in the resource-rich, more ideal conditions of many design-based classroom interventions, unexpected events can lead to disappointing results in student learning. However, if later iterations in a design research study are more successful, the previous failures can provide opportunities for comparisons to reveal subtle differences in instruction that supported or constrained learning. This article presents design research in which a failure in an early iteration was addressed in a later iteration through an impromptu and unplanned response that better supported student learning. The responsive and interventionist nature of design research played a pivotal role in the emergence of the more successful outcome in the later iteration. The suitability of design research for investigating sociomathematical norms and for conducting conceptual analyses facilitated the deepening of our understanding of both the early failure and the later success. Specifically, our retrospective comparative analysis of these two non-consecutive iterations revealed that one sociomathematical norm better supported student learning of a particular mathematical concept then another norm, led to new insights into the conceptual complexity of division in rate situations, and revealed how the interrelatedness of sociomathematical norms and particular mathematical content can provide an important support for learning. We discuss how our cross-iteration analytic method extends existing approaches to leveraging failure in design research.
As transfer researchers have begun to investigate a broader range of phenomena, they
have corresp... more As transfer researchers have begun to investigate a broader range of phenomena, they have correspondingly put forward new processes to provide explanatory accounts for the occurrence of transfer. This move coincides with a call to acknowledge the contribution of social interactions, language, cultural artifacts, and normed practices to the generalization of learning. In this article, we posit “noticing” as a plausible transfer process and investigate both individual noticing and the social organization of noticing via the focusing framework. Specifically, we relate the nature of students’ reasoning on transfer tasks with what students notice mathematically in classrooms when many sources of information compete for their attention, and then we account for noticing as socially situated in classroom discourse practices, features of mathematical tasks, and the nature of mathematical activity.
Despite recent research interest in student-created diagrams, little research
has systematically ... more Despite recent research interest in student-created diagrams, little research has systematically investigated students' diagram-construction processes, meaning the order and manner in which students create markings as they physically generate diagrams. In this study, we characterize the various processes students use to create diagrams that represent a quadratic motion situation involving increasing speed, and we explore how these diagram-construction processes are related to students' conceptions of speed as inferred from their explanations with their completed diagrams. Previous literature suggests contrasting predictions regarding whether or not students' diagramconstruction processes are closely related (from our perspective as researchers) to students' inferred conceptions. We see the study as having value for research and practice by raising new questions related to diagram-construction processes, pointing to the potential formative assessment value of attending to diagram-construction processes, and demonstrating the need for the development of theory to explain the relationships identified by this study.
features that students can attend to or notice. What students notice mathematically
has consequen... more features that students can attend to or notice. What students notice mathematically has consequences for their subsequent reasoning. By adapting work from both cognitive science and applied linguistics anthropology, we present a focusing framework, which treats noticing as a complex phenomenon that is distributed across individual cognition, social interactions, material resources, and normed practices. Specifically, this research demonstrates that different centers of focus emerged in two middle grades mathematics classes addressing the same content goals, which, in turn, were related conceptually to differences in student reasoning on subsequent interview tasks. Furthermore, differences in the discourse practices, features of the mathematical tasks, and the nature of the mathematical activity in the two classrooms were related to the different mathematical features that students appeared to notice.
Despite the proliferation of mathematics standards internationally and despite general agreement
... more Despite the proliferation of mathematics standards internationally and despite general agreement on the importance of teaching for conceptual understanding, conceptual learning goals for many K-12 mathematics topics have not been well-articulated. This article presents a coherent set of five conceptual learning goals for a complex mathematical domain, generated via a method of systematic empirical analysis of students’ reasoning. Specifically, we compared the reasoning of pairs of students who performed differentially on the same task and inferred the pivotal intermediate conceptions that afforded one student deeper engagement with the task than another student. In turn, each pivotal intermediate conception framed an associated conceptual learning goal. While the empirical analysis of student reasoning is typically used to understand how students learn, we argue that such analysis should also play an important role in determining what concepts students should learn.
This paper extends recent efforts to critique and reconceive transfer by using an empirical study... more This paper extends recent efforts to critique and reconceive transfer by using an empirical study to rethink the surface/structure distinction of the traditional transfer paradigm. The findings suggest that what researchers typically consider a surface feature can present conceptual complexities for students that are more structural in nature than previously understood. In particular,we investigated the quantitative reasoning (meaning reasoning with measurable properties of an object) that is involved in making sense of a typical transfer situation. Two related analyses—one focused on quantitative reasoning and one on transfer — were performed on a case study. The results document how a student reconstructed his understanding of the relationships among quantities in a complex transfer situation in such a way that he was able to see the situation as fundamentally proportional in nature and subsequently make connections with previous proportional-reasoning experiences from a teaching experiment. In our discussion of the findings, we identify four relationships between quantitative reasoning and transfer.
This article sets forth a way of connecting the classroom instructional environment
with individu... more This article sets forth a way of connecting the classroom instructional environment with individual students’ generalizations. To do so, we advance the notion of focusing phenomena, that is, regularities in the ways in which teachers, students, artifacts, and curricular materials act together to direct attention toward certain mathematical properties over others. The construct of focusing phenomena emerged from an empirical study conducted during a 5-week unit on slope and linear functions in a high school classroom using a reform curriculum. Qualitative evidence from interviews with 7 students revealed that students interpreted the m value in y=b+mx as a difference rather than a ratio as a result of counterproductive generalization afforded by focusing phenomena. Classroom analysis revealed 4 focusing phenomena, which regularly directed students’ attention to various sets of differences rather than to the coordination of quantities.
When technology is implemented in classrooms, students often
form ideas that are unexpected and u... more When technology is implemented in classrooms, students often form ideas that are unexpected and unwanted by the teachers and the designers of the technology. This article advances the notion of the focusing effect of technology as a way of systematically accounting for the role of technology in such situations. A focusing effect refers to the direction of students’ attention toward certain properties of the subject matter domain over others, brought about by the use of particular technological tools. Technology focuses students’ attention in ways that are often not anticipated in advance and can have unintended consequences for students’ conceptions. Vignettes are presented from two research studies, one involving graphing calculators and one involving mathematics software called SimCalc Mathworlds. The research findings are synthesized and reinterpreted in order to illustrate and develop the notion of the focusing effect of technology. The significance of this construct lies in the connection that it affords between individual students’ conceptions and the way technology is used in the instructional environment. Implications for teaching and for the preparation of teachers in the use of technology are discussed.
We use the notion offocusing phenomena to help explain how a teacher’s actions were connected to ... more We use the notion offocusing phenomena to help explain how a teacher’s actions were connected to her students’ interpretations of a linear equation. This study was conducted in a high-school classroom that regularly emphasised dependency relationships in real-world situations. Seven interviews revealed a majority view ofy = b + mx as astorage container—a place to insert b and m values—rather than as a relationship between x- and y-values. Classroom analysis revealed how the teacher directed attention away from functional relationships with increasing frequency as she moved from realistic situations to conventional representations.
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Papers by Joanne Lobato
mainstream cognitive perspective on transfer have had an unfortunate divisive effect. This
article takes a pragmatic view that transfer perspectives are simply designed objects (Plomp
& Nieveen, 2007), which provide different information for different purposes. Specifically,
this paper compares one alternative approach—the actor-oriented transfer perspective—with
the mainstream cognitive perspective on transfer, by examining the points of compatibility
and tension across 5 dimensions. As a result, a space is opened up to explore 3 issues that
are particularly well suited to an actor-oriented transfer approach: (a) how students interpret
transfer situations, (b) the socially situated nature of transfer processes in classrooms, and (c)
how contextual-sensitivity can play a productive role in the transfer of learning. Exploring
the benefits and trade-offs of various approaches allows for greater understanding of the
contributions of each perspective to educational research and practice.
of the National Mathematics Advisory Panel (2008) and to one of the
task group reports on which it was based, the report of the Task
Group on Learning Processes. The author uses Maxwell’s two views
of causality—regularity and process—to explore three major issues
raised in the report: the nature of what is learned, how learning
occurs, and the transfer of learning. She proposes alternative
recommendations to those offered by the Panel, by drawing upon the
mathematics education literature, which was largely excluded from
the reviewed research. Furthermore, by neglecting research
grounded in a process view of causality, the report excludes perspectives
and findings that would have illuminated what it means to
develop conceptual understanding in mathematics—one of three
valued outcomes cited by the Panel.
Limitations with current approaches to the investigation of the transfer
of learning in design experiments constrain the type of information
that is available to researchers as they make design decisions. This article
addresses these limitations by presenting a reconceptualization
of transfer, called actor-oriented transfer, which emerged from design
experiment work. The merits of this alternative model are considered
in terms of the information it provides to design experimenters.
Early Algebraization: A Global Dialogue From Multiple Perspectives. Jinfa Cai and
Eric Knuth (Eds., 2011). Heidelberg, Germany: Springer, 623 pp. ISBN
978-3642177347 (hb) $99.00, (e-book) $103.20.
of many design-based classroom interventions, unexpected
events can lead to disappointing results in student
learning. However, if later iterations in a design research
study are more successful, the previous failures can provide
opportunities for comparisons to reveal subtle differences
in instruction that supported or constrained learning. This
article presents design research in which a failure in an
early iteration was addressed in a later iteration through an
impromptu and unplanned response that better supported
student learning. The responsive and interventionist nature
of design research played a pivotal role in the emergence
of the more successful outcome in the later iteration. The
suitability of design research for investigating sociomathematical
norms and for conducting conceptual analyses
facilitated the deepening of our understanding of both the
early failure and the later success. Specifically, our retrospective
comparative analysis of these two non-consecutive
iterations revealed that one sociomathematical norm better
supported student learning of a particular mathematical
concept then another norm, led to new insights into
the conceptual complexity of division in rate situations,
and revealed how the interrelatedness of sociomathematical
norms and particular mathematical content can provide
an important support for learning. We discuss how our cross-iteration analytic method extends existing approaches
to leveraging failure in design research.
have correspondingly put forward new processes to provide explanatory accounts
for the occurrence of transfer. This move coincides with a call to acknowledge the
contribution of social interactions, language, cultural artifacts, and normed practices
to the generalization of learning. In this article, we posit “noticing” as a
plausible transfer process and investigate both individual noticing and the social
organization of noticing via the focusing framework. Specifically, we relate the
nature of students’ reasoning on transfer tasks with what students notice mathematically
in classrooms when many sources of information compete for their
attention, and then we account for noticing as socially situated in classroom discourse
practices, features of mathematical tasks, and the nature of mathematical
activity.
has systematically investigated students' diagram-construction processes, meaning the
order and manner in which students create markings as they physically generate
diagrams. In this study, we characterize the various processes students use to create
diagrams that represent a quadratic motion situation involving increasing speed, and we
explore how these diagram-construction processes are related to students' conceptions
of speed as inferred from their explanations with their completed diagrams. Previous
literature suggests contrasting predictions regarding whether or not students' diagramconstruction
processes are closely related (from our perspective as researchers) to
students' inferred conceptions. We see the study as having value for research and
practice by raising new questions related to diagram-construction processes, pointing
to the potential formative assessment value of attending to diagram-construction
processes, and demonstrating the need for the development of theory to explain the
relationships identified by this study.
has consequences for their subsequent reasoning. By adapting work from both cognitive
science and applied linguistics anthropology, we present a focusing framework,
which treats noticing as a complex phenomenon that is distributed across individual
cognition, social interactions, material resources, and normed practices. Specifically,
this research demonstrates that different centers of focus emerged in two middle
grades mathematics classes addressing the same content goals, which, in turn, were
related conceptually to differences in student reasoning on subsequent interview tasks.
Furthermore, differences in the discourse practices, features of the mathematical tasks,
and the nature of the mathematical activity in the two classrooms were related to the
different mathematical features that students appeared to notice.
on the importance of teaching for conceptual understanding, conceptual learning goals for many
K-12 mathematics topics have not been well-articulated. This article presents a coherent set of five
conceptual learning goals for a complex mathematical domain, generated via a method of systematic
empirical analysis of students’ reasoning. Specifically, we compared the reasoning of pairs of
students who performed differentially on the same task and inferred the pivotal intermediate conceptions
that afforded one student deeper engagement with the task than another student. In turn, each
pivotal intermediate conception framed an associated conceptual learning goal. While the empirical
analysis of student reasoning is typically used to understand how students learn, we argue that such
analysis should also play an important role in determining what concepts students should learn.
surface/structure distinction of the traditional transfer paradigm. The findings suggest that what researchers typically
consider a surface feature can present conceptual complexities for students that are more structural in nature than
previously understood. In particular,we investigated the quantitative reasoning (meaning reasoning with measurable
properties of an object) that is involved in making sense of a typical transfer situation. Two related analyses—one
focused on quantitative reasoning and one on transfer — were performed on a case study. The results document
how a student reconstructed his understanding of the relationships among quantities in a complex transfer situation
in such a way that he was able to see the situation as fundamentally proportional in nature and subsequently make
connections with previous proportional-reasoning experiences from a teaching experiment. In our discussion of the
findings, we identify four relationships between quantitative reasoning and transfer.
with individual students’ generalizations. To do so, we advance the notion of focusing
phenomena, that is, regularities in the ways in which teachers, students, artifacts,
and curricular materials act together to direct attention toward certain mathematical
properties over others. The construct of focusing phenomena emerged from an empirical
study conducted during a 5-week unit on slope and linear functions in a high
school classroom using a reform curriculum. Qualitative evidence from interviews
with 7 students revealed that students interpreted the m value in y=b+mx as a difference
rather than a ratio as a result of counterproductive generalization afforded by focusing
phenomena. Classroom analysis revealed 4 focusing phenomena, which regularly
directed students’ attention to various sets of differences rather than to the
coordination of quantities.
form ideas that are unexpected and unwanted by the
teachers and the designers of the technology. This article advances
the notion of the focusing effect of technology as a
way of systematically accounting for the role of technology
in such situations. A focusing effect refers to the direction of
students’ attention toward certain properties of the subject
matter domain over others, brought about by the use of particular
technological tools. Technology focuses students’ attention
in ways that are often not anticipated in advance and
can have unintended consequences for students’ conceptions.
Vignettes are presented from two research studies, one
involving graphing calculators and one involving mathematics
software called SimCalc Mathworlds. The research findings
are synthesized and reinterpreted in order to illustrate
and develop the notion of the focusing effect of technology.
The significance of this construct lies in the connection that
it affords between individual students’ conceptions and the
way technology is used in the instructional environment. Implications
for teaching and for the preparation of teachers in
the use of technology are discussed.
mainstream cognitive perspective on transfer have had an unfortunate divisive effect. This
article takes a pragmatic view that transfer perspectives are simply designed objects (Plomp
& Nieveen, 2007), which provide different information for different purposes. Specifically,
this paper compares one alternative approach—the actor-oriented transfer perspective—with
the mainstream cognitive perspective on transfer, by examining the points of compatibility
and tension across 5 dimensions. As a result, a space is opened up to explore 3 issues that
are particularly well suited to an actor-oriented transfer approach: (a) how students interpret
transfer situations, (b) the socially situated nature of transfer processes in classrooms, and (c)
how contextual-sensitivity can play a productive role in the transfer of learning. Exploring
the benefits and trade-offs of various approaches allows for greater understanding of the
contributions of each perspective to educational research and practice.
of the National Mathematics Advisory Panel (2008) and to one of the
task group reports on which it was based, the report of the Task
Group on Learning Processes. The author uses Maxwell’s two views
of causality—regularity and process—to explore three major issues
raised in the report: the nature of what is learned, how learning
occurs, and the transfer of learning. She proposes alternative
recommendations to those offered by the Panel, by drawing upon the
mathematics education literature, which was largely excluded from
the reviewed research. Furthermore, by neglecting research
grounded in a process view of causality, the report excludes perspectives
and findings that would have illuminated what it means to
develop conceptual understanding in mathematics—one of three
valued outcomes cited by the Panel.
Limitations with current approaches to the investigation of the transfer
of learning in design experiments constrain the type of information
that is available to researchers as they make design decisions. This article
addresses these limitations by presenting a reconceptualization
of transfer, called actor-oriented transfer, which emerged from design
experiment work. The merits of this alternative model are considered
in terms of the information it provides to design experimenters.
Early Algebraization: A Global Dialogue From Multiple Perspectives. Jinfa Cai and
Eric Knuth (Eds., 2011). Heidelberg, Germany: Springer, 623 pp. ISBN
978-3642177347 (hb) $99.00, (e-book) $103.20.
of many design-based classroom interventions, unexpected
events can lead to disappointing results in student
learning. However, if later iterations in a design research
study are more successful, the previous failures can provide
opportunities for comparisons to reveal subtle differences
in instruction that supported or constrained learning. This
article presents design research in which a failure in an
early iteration was addressed in a later iteration through an
impromptu and unplanned response that better supported
student learning. The responsive and interventionist nature
of design research played a pivotal role in the emergence
of the more successful outcome in the later iteration. The
suitability of design research for investigating sociomathematical
norms and for conducting conceptual analyses
facilitated the deepening of our understanding of both the
early failure and the later success. Specifically, our retrospective
comparative analysis of these two non-consecutive
iterations revealed that one sociomathematical norm better
supported student learning of a particular mathematical
concept then another norm, led to new insights into
the conceptual complexity of division in rate situations,
and revealed how the interrelatedness of sociomathematical
norms and particular mathematical content can provide
an important support for learning. We discuss how our cross-iteration analytic method extends existing approaches
to leveraging failure in design research.
have correspondingly put forward new processes to provide explanatory accounts
for the occurrence of transfer. This move coincides with a call to acknowledge the
contribution of social interactions, language, cultural artifacts, and normed practices
to the generalization of learning. In this article, we posit “noticing” as a
plausible transfer process and investigate both individual noticing and the social
organization of noticing via the focusing framework. Specifically, we relate the
nature of students’ reasoning on transfer tasks with what students notice mathematically
in classrooms when many sources of information compete for their
attention, and then we account for noticing as socially situated in classroom discourse
practices, features of mathematical tasks, and the nature of mathematical
activity.
has systematically investigated students' diagram-construction processes, meaning the
order and manner in which students create markings as they physically generate
diagrams. In this study, we characterize the various processes students use to create
diagrams that represent a quadratic motion situation involving increasing speed, and we
explore how these diagram-construction processes are related to students' conceptions
of speed as inferred from their explanations with their completed diagrams. Previous
literature suggests contrasting predictions regarding whether or not students' diagramconstruction
processes are closely related (from our perspective as researchers) to
students' inferred conceptions. We see the study as having value for research and
practice by raising new questions related to diagram-construction processes, pointing
to the potential formative assessment value of attending to diagram-construction
processes, and demonstrating the need for the development of theory to explain the
relationships identified by this study.
has consequences for their subsequent reasoning. By adapting work from both cognitive
science and applied linguistics anthropology, we present a focusing framework,
which treats noticing as a complex phenomenon that is distributed across individual
cognition, social interactions, material resources, and normed practices. Specifically,
this research demonstrates that different centers of focus emerged in two middle
grades mathematics classes addressing the same content goals, which, in turn, were
related conceptually to differences in student reasoning on subsequent interview tasks.
Furthermore, differences in the discourse practices, features of the mathematical tasks,
and the nature of the mathematical activity in the two classrooms were related to the
different mathematical features that students appeared to notice.
on the importance of teaching for conceptual understanding, conceptual learning goals for many
K-12 mathematics topics have not been well-articulated. This article presents a coherent set of five
conceptual learning goals for a complex mathematical domain, generated via a method of systematic
empirical analysis of students’ reasoning. Specifically, we compared the reasoning of pairs of
students who performed differentially on the same task and inferred the pivotal intermediate conceptions
that afforded one student deeper engagement with the task than another student. In turn, each
pivotal intermediate conception framed an associated conceptual learning goal. While the empirical
analysis of student reasoning is typically used to understand how students learn, we argue that such
analysis should also play an important role in determining what concepts students should learn.
surface/structure distinction of the traditional transfer paradigm. The findings suggest that what researchers typically
consider a surface feature can present conceptual complexities for students that are more structural in nature than
previously understood. In particular,we investigated the quantitative reasoning (meaning reasoning with measurable
properties of an object) that is involved in making sense of a typical transfer situation. Two related analyses—one
focused on quantitative reasoning and one on transfer — were performed on a case study. The results document
how a student reconstructed his understanding of the relationships among quantities in a complex transfer situation
in such a way that he was able to see the situation as fundamentally proportional in nature and subsequently make
connections with previous proportional-reasoning experiences from a teaching experiment. In our discussion of the
findings, we identify four relationships between quantitative reasoning and transfer.
with individual students’ generalizations. To do so, we advance the notion of focusing
phenomena, that is, regularities in the ways in which teachers, students, artifacts,
and curricular materials act together to direct attention toward certain mathematical
properties over others. The construct of focusing phenomena emerged from an empirical
study conducted during a 5-week unit on slope and linear functions in a high
school classroom using a reform curriculum. Qualitative evidence from interviews
with 7 students revealed that students interpreted the m value in y=b+mx as a difference
rather than a ratio as a result of counterproductive generalization afforded by focusing
phenomena. Classroom analysis revealed 4 focusing phenomena, which regularly
directed students’ attention to various sets of differences rather than to the
coordination of quantities.
form ideas that are unexpected and unwanted by the
teachers and the designers of the technology. This article advances
the notion of the focusing effect of technology as a
way of systematically accounting for the role of technology
in such situations. A focusing effect refers to the direction of
students’ attention toward certain properties of the subject
matter domain over others, brought about by the use of particular
technological tools. Technology focuses students’ attention
in ways that are often not anticipated in advance and
can have unintended consequences for students’ conceptions.
Vignettes are presented from two research studies, one
involving graphing calculators and one involving mathematics
software called SimCalc Mathworlds. The research findings
are synthesized and reinterpreted in order to illustrate
and develop the notion of the focusing effect of technology.
The significance of this construct lies in the connection that
it affords between individual students’ conceptions and the
way technology is used in the instructional environment. Implications
for teaching and for the preparation of teachers in
the use of technology are discussed.