Difference between pages "Cobb, Stephan, McClain, & Gravemeijer (2001)" and "Brown (2009)"

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The article ''Participating in Classroom Mathematical Practices'' was written by [[Paul Cobb]], [[Michelle Stephan]], [[Kay McClain]], and [[Koeno Gravemeijer]] and published in ''[[The Journal of the Learning Sciences]]'' in 2001. The article is available from Taylor & Francis Online at [http://www.tandfonline.com/doi/abs/10.1207/S15327809JLS10-1-2_6 http://www.tandfonline.com/doi/abs/10.1207/S15327809JLS10-1-2_6] and JSTOR at [http://www.jstor.org/stable/1466831 http://www.jstor.org/stable/1466831].
{{Title|The Teacher-Tool Relationship: Theorizing the Design and Use of Curriculum Materials}}


== Abstract ==
* Author: [[Matthew Brown]]
* Book: ''[[Mathematics Teachers at Work: Connecting Curriculum Materials and Classroom Instruction]]''
* Year: 2009
* Source: http://www.routledge.com/books/details/9780415990103/


<blockquote>
In this article, we describe a methodology for analyzing the collective learning of the classroom community in terms of the evolution of classroom mathematical practices. To develop the rationale for this approach, we first ground the discussion in our work as mathematics educators who conduct classroom-based design research. We then present a sample analysis taken from a 1st-grade classroom teaching experiment that focused on linear measurement to illustrate how we coordinate a social perspective on communal practices with a psychological perspective on individual students' diverse ways of reasoning as they participate in those practices. In the concluding sections of the article, we frame the sample analysis as a paradigm case in which to clarify aspects of the methodology and consider its usefulness for design research.
</blockquote>


== Detailed Summary of ''Participating in Classroom Mathematical Practices'' ==
== Outline of Headings ==


In this article, Cobb et al. describe the type of design research they conducted with 1st grade students to study the evolution of classroom mathematical practices. Using a design research approach and teaching experiments, the authors developed both sequences of instructional activities as well as a theory and process for interpreting classroom events from both a social and a psychological perspective.
* Teaching as Design
* Theoretical Background
** Artifacts can Extend Human Capacities
* Curriculum Materials as Artifacts
* How Curriculum Artifacts Influence Instruction
* How Teachers Interpret and Use Curriculum Artifacts
* How Teaching is Design
* Analyzing Teacher Use of Curriculum Artifacts
* Types of Curriculum Use: Offloading, Adapting, and Improvising
* Facets of the Teacher-Tool Relationship: The Design Capacity for Enactment Framework
** Applying the Design Capacity Enactment Framework
* Pedagogical Design Capacity
* Design Implications
* The Design of Materials
** Multiple Points of Access
** Resource-Centric Material Design
** Creating Reusable Resources and Supporting Customization
* The Design of Professional Development


=== Design Research ===
== Summary ==


Rooting themselves in more than a decade of research involving students' mathematical learning in the social context of classrooms, Cobb et al. use a design research approach (which both Cobb and Gravemeijer had previously called developmental research) to develop instructional activities and study student learning. Design research is based on iterative cycles of design and analysis, with Cobb et al. basing their instructional design on the theories of [[Realistic Mathematics Education]] (RME). The theories of RME inform the design of activities along with the researcher's predictions about how the activity will further student learning. These predictions are tested upon enactment of the activity, which generates data to inform future development and refinement of instructional activities. Cobb et al. list three criteria for their analytical approach:
Brown begins this chapter comparing the relationship between teachers and curriculum materials to those of musicians and their music; just as the same song played by different musicians takes on its own character, teachers interpret and adapt curriculum materials in ways that make their practice unique, even if there are similarities across classrooms. Curriculum materials are often used to promote educational reforms and the results of such efforts have been mixed ({{Cite|Ball & Cohen|1996}}; [[Cohen (1988) Tech|Cohen, 1988]]; Cuban [[Cuban (1992)|1992]], [[Cuban (1993)|1993]]; {{Cite|Snyder, Bolin, & Zumwalt|1992}}), the reasons of which have been attributed to practitioners ({{Cite|Cohen|1990}}; {{Cite|Spillane|1999}}), policies ({{Cite|Spillane|1998}}), and professional development ({{Cite|Putnam & Borko|2000}}; {{Cite|Wilson & Berne|1999}}). Other studies have focused on how teachers interpret curriculum materials ({{Cite|Ben-Peretz|1990}}; {{Cite|Brown|2002}}; {{Cite|Brown & Edelson|2003}}; {{Cite|Lloyd|1999}}; Remillard [[Remillard (2000)|2000]], [[Remillard (2005)|2005]]; {{Cite|Wiley|2001}}) and how curriculum materials might be better designed to meet the need of teachers ({{Cite|Brown|2002}}; {{Cite|Brown & Edelson|2003}}; {{Cite|Davis & Krajcik|2005}}; {{Cite|Davis & Varma|2008}}; {{Cite|Schneider & Krajcik|2002}}). Brown's purpose in writing this chapter is to describe a theoretical framework for the relationship between curriculum materials and teacher practice. In doing so, Brown conceives of teaching as a design activity in which teachers evaluate their resources and make decisions in an effort to achieve instructional goals. Tensions between teachers (the agent) and their curriculum materials (the tool) build on a well-established body of learning theory ({{Cite|Gibson|1977}}; {{Cite|Hutchins|1996}}; Norman [[Norman, (1988)|1988]], [[Norman (1991)|1991]]; {{Cite|Pea|1993}}; Wertsch, [[Wertsch (1991)|1991]], [[Wertsch (1998)|1998]]), which Brown uses to highlight three key points:


<blockquote>
# Curriculum materials play an important role in affording and constraining teachers' actions.
# It should enable us to document the collective mathematical development of the classroom community over the extended periods of time covered by instructional sequences.
# Teachers notice and use such artifacts differently given their experience, intentions, and abilities.
# It should enable us to document the developing mathematical reasoning of individual students as they participate in the practices of the classroom community.
# "Teaching by design" is not so much a conscious choice as an inevitable reality (p. 19).
# It should result in analyses that feed back to inform the improvement of our instructional designs. (p. 116)
</blockquote>


Cobb et al. carefully worded the first of the criteria to focus on the classroom community, as predictions and instructional design cannot possibly account for each and every individual student's learning at a given point in time. The second criteria, however, takes advantage of individual student thinking not in the planning of activities, but in the enactment, where observed differences in student reasoning help to highlight key ideas in the mathematics. The third criteria is perhaps the most obvious in design research, where data collected in each cycle is used to inform the design of the next.
[[Wartofsky (1973)]] theorized artifacts as human-created tools that play a prominent role in our survival. Artifacts can be transmitted or preserved across place and time and our progress is inseparable from the artifacts we use ({{Cite|Wertsch|1998}}). Artifacts mediate human activity ({{Cite|Vygotsky|1978}}; Wertsch, [[Wertsch (1991)|1991]], [[Wertsch (1998)|1998]]) in ways shaped by the artifact's affordances and constraints. Affordances might be attributed to the functional properties of the artifact ({{Cite|Gibson|1977}}) or perceptual cues that suggest how they might be used ({{Cite|Norman|1988}}). Constraints of artifacts restrict the ways in which we act and what we see as possible ({{Cite|Burke|1966}}; {{Cite|Wertsch|1998}}).


=== Interpretative Framework ===
[[Norman (1988)]] suggested that humans often design artifacts to afford and constrain human activities in particular ways. In the case of curriculum, materials are designed to promote certain instructional, assessment, and student practices while discouraging the use of others. Often, curriculum materials allow teachers to engage in practices they could not have done alone, thus representing a shared capacity across teachers and their materials. Artifacts are also not limited to physical materials ({{Cite|Wartofsky|1973}}); our cultural and historical ideas and practices also act to shape our activity. Brown enumerates six characteristics of curriculum materials that influence instruction:


Cobb et al. take two theoretical viewpoints in this paper: a psychological perspective for thinking about and describing individual student learning, and a social perspective for thinking about and describing the classroom microculture.
# They are static representations of abstract concepts and dynamic activities &mdash; a means for transmitting and producing activity, not the activity itself.
# They are intended to convey rich ideas and dynamic practices, yet they do so through succinct shorthand that relies heavily on interpretation.
# They observe a number of culturally shared notational rules, norms, and conventions in their representations &mdash; although fewer consistently and conventions in their representations &mdash; although fewer consistently used conventions exist for curriculum materials than for sheet music.
# They may reflect common or existing practices and at the same time aim to shape innovative or new practices.
# They represent an interface between the knowledge, goals, and values of the author and the user.
# They require craft in their use; they are inert objects that come alive only through interpretation and use by a practitioner (pp. 21-22).


==== Social and Psychological Perspectives ====
In addition to the influence of curriculum materials on teachers, Brown notes the importance of understanding the "dynamic and constructive ways" (p. 22) teachers interpret and use curriculum materials ({{Cite|Barab & Luehmann|2003}}; {{Cite|Brown|2002}}; {{Cite|Davis & Krajcik|2005}}; {{Cite|Matese|2005}}; {{Cite|Remillard|2005}}). Even when curriculum materials are provided, teachers still ''select'' materials according to their beliefs, skills, knowledge, and goals ({{Cite|Freeman & Porter|1989}}; {{Cite|Tarr, Reys, Reys, Chavez, Shih, & Osterlind|2008}}) or resist scripted or otherwise inflexible materials ({{Cite|Cohen|1990}}; {{Cite|Remillard|1992}}; {{Cite|Wilson|1990}}) based on their goals and beliefs (Cohen, [[Cohen (1988) Practice|1988]], [[Cohen (1990)|1990]]; {{Cite|Lloyd|1999}}; {{Cite|Lloyd & Wilson|1998}}; {{Cite|Wilson & Goldenberg|1998}}). Next, teachers ''interpret'' materials in planning and instruction ({{Cite|Ben-Peretz|1990}}; {{Cite|Stein, Remillard, & Smith|2007}}). Then teachers ''reconcile'' those interpretations with their instructional goals ({{Cite|Ben-Peretz|1990}}; {{Cite|Remillard|2005}}), and ''accommodate'' the needs of their students, making continual adjustments and providing feedback ({{Cite|Stein|1996}}; {{Cite|Wilson & Lloyd|2000}}). Depending on the perceived success of the plan, teachers may ''add'', ''modify'', or ''omit'' parts of the curriculum, either due to interest or ability ({{Cite|Remillard|1992}}; {{Cite|Tarr, Reys, Reys, Chávez, Shih, & Osterlind|2008}}).


Cobb et al. describe the social perspective as "concerned with ways of acting, reasoning, and arguing that are normative in a classroom community" (p. 118). This means that individual student reasoning is described in the ways they participate in normative classroom activities. In contrast, the psychological perspective "focuses squarely on the nature of individual students' reasoning or, in other words, on his or her particular ways of participating in communal activities" (p. 119). The key difference is that the social perspective focuses on normative activities, and the psychological perspective highlights differences in individual student participation. Cobb et al. build their social perspective from sociocultural theory (e.g., [[Cole (1996)|Cole, 1996]]; [[Lave (1988)|Lave, 1988]]; [[Rogoff (1997)|Rogoff, 1997]]) and ethnomethodology and symbolic interactionism ([[Blumer (1969)|Blumer, 1969]]), and base their psychological perspective on constructivism ([[Piaget (1970)|Piaget, 1970]]; [[Steffe & Kieren (1994)|Steffe & Kieren, 1994]]; [[Thompson (1991)|Thompson, 1991]]) and distributed accounts of intelligence (e.g., [[Hutchins (1995)|Hutchins, 1995]]; [[Pea (1993)|Pea, 1993]]). Concerning their use of a psychological perspective, Cobb et al. clarify:
This perspective on teachers' curriculum use leads Brown to conclude that teaching is a form of design, a process that "is about crafting something in order to solve a human problem, to change the state of a particular situation from a current condition to a desired one, and to accomplish a goal" (p. 23). Understanding teaching as design highlights the dynamic between teachers and their materials, and Brown provides three constructs for understanding this relationship. The first contrasts ''offloading'', ''adapting'', and ''improvising'' and gives us a way to think about the degree to which teachers appropriate instructional materials. Second, Brown describes a framework for examining interactions between teachers and features of their materials. Lastly, he describes ''pedagogical design capacity'', described as a teacher's "ability to perceive and mobilize existing resources in order to craft instructional contexts" (p. 24).


<blockquote>
=== Types of Curriculum Use: Offloading, Adapting, and Improvising ===
The goal of analyses conducted from this psychological perspective is therefore not to specify cognitive mechanisms inside students' heads. Instead, it is to infer the quality of individual students' reasoning in, with, and about the world, and to account for developments in their reasoning in terms of the reorganization of activity and the world acted in. (p. 121)
</blockquote>


Together, Cobb et al. did not seek to describe the social and psychological perspectives together as some grand theory; rather, each perspective compliments the other, just as individual student activity shapes classroom activity and vice versa.
From his dissertation work ([[Brown (2002)|2002]]), Brown identified three ways teachers appropriate their materials as they design instruction. ''Offloading'' occurs when teachers follow curriculum materials closely, giving agency to the materials for guiding instruction. On the other end of the scale is ''improvising'', when teachers craft instruction spontaneously and/or without specific guidance from their materials, thus shifting agency to themselves. In between, ''adapting'' occurs when teachers modify their materials to support instructional goals, thus sharing the agency between themselves and the materials.


==== Aspects of the Classroom Microculture and Individual Students' Reasoning ====
Brown emphasizes that this scale does not necessarily correlate to teacher expertise and is not intended to measure the fidelity with which materials are used. It is designed to describe the nature of teachers' interactions with resources, not the outcomes of those interactions. Each position on the scale can serve a strategic purpose and be used with varying degrees of effectiveness. Brown explains,


Cobb et al. use the following interpretive framework for studying both classroom and individual mathematical activity and learning (p. 119):
<blockquote>
"Just as a novice teacher might offload instructional responsibility to a scripted lesson due to limited understanding of the subject matter, so might an expert teacher offload instructional responsibility to a worksheet that supports her goals, freeing her to roam the room and respond to student needs as they arise" (p. 25).
</blockquote>


{| class="wikitable"
=== Facets of the Teacher&ndash;Tool Relationship: The Design Capacity for Enactment Framework ===
|-
!Social Perspective
!Psychological Perspective
|-
|Classroom social norms
|Beliefs about own role, others' roles, and the general nature of mathematical activity in school
|-
|Sociomathematical norms
|Mathematical beliefs and values
|-
|Classroom mathematical practices
|Mathematical interpretations and reasoning
|}


Social norms are features of the classroom participation structure ([[Erickson (1986)|Erickson, 1986]]; [[Lampert (1990)|Lampert, 1990]]) that include things like how a class explains and justifies solutions, makes sense of explanations, and questions alternative interpretations. For the research, Cobb et al. looked for occasions when norms might be violated to inform their understanding of the classroom's social norms. Because this is a social perspective, this view of social norms avoids singling out student behavior and instead views participation in norms as a joint activity. For the corresponding psychological perspective, Cobb et al. sought to understand how students saw their role and the role of others in mathematical activity. The social and psychological are related as social norms affect how individuals see their own roles and vice versa, with neither more important than the other.
Brown's ''Design Capacity for Enactment'' (DCE) framework ([[Brown (2002)|2002]]) attempts to describe the aspects of both curriculum resources and teachers that influence why teachers interact with curriculum materials in different ways. The DCE framework includes three basic aspects of curriculum: physical objects, representations of tasks (procedures), and representations of concepts (domain representations). Physical objects include the material nature of the curriculum materials, including supplementary and recommended materials. Representations of tasks refers to the instructions and procedures given to teachers and students in using the materials, such as lesson guides or recommended homework problem sets. Domain representations consist of the ways materials organize concepts using diagrams, models, analogies, and the sequencing of topics.


Because social norms are not unique to mathematics classrooms, Cobb et al. looked for math-specific norms ([[Lampert (1990)|Lampert, 1990]]; [[Simon & Blume (1996)|Simon & Blume, 1996]]; [[Voigt (1995)|Voigt, 1995]]; [[Yackel & Cobb (1996)|Yackel & Cobb, 1996]]) they referred to as ''sociomathematical norms'', including things such as agreement on what counts as a mathematical solution, a sophisticated solution, an elegant solution, and a mathematically acceptable explanation. For students to engage in these sociomathematical norms, Cobb et al. believed students must co-develop beliefs about mathematics that encourage them to act autonomously. Cobb et al. described this development of autonomy as moving from relatively peripheral participation towards more substantial participation, instead of an individual act (cf. [[Forman (1996)|Forman, 1996]]; [[Lave & Wegner (1991)|Lave & Wegner, 1991]]).
Brown identified three teacher resources in the DCE. The first, subject matter knowledge, consists of knowledge about the domain ({{Cite|Ball|1991}}; {{Cite|Stodolsky & Grossman|1995}}), while pedagogical content knowledge ({{Cite|Shulman|1986}}) combines general knowledge about teaching with domain-specific teaching knowledge. Lastly, teachers' goals and beliefs ({{Cite|Ball & Cohen|1999}}) describe teachers' motivations to teach particular content in particular ways. Sometimes goals and beliefs conflict with instructional approaches ({{Cite|Spillane|1999}}; {{Cite|Wilson|1990}}) and can be a barrier to the implementation of reforms (Cohen, [[Cohen (1988) Tech|1988]], [[Cohen (1988) Practice|1988]]).


As instructional designers, Cobb et al. also wished to define ''classroom mathematical practices'', arguing that "it is feasible to view a conjectured learning trajectory as consisting of an envisioned sequence of classroom mathematical practices together with conjectures about the means of supporting their evolution from prior practices" (p. 125). Unlike sociomathematical norms, which are only specific to a mathematical classroom community, classroom mathematical practices are analyzed in relation to specific mathematical tasks or ideas. Cobb et al. see these practices from a social perspective, but reflexively relate them to the mathematical interpretations and reasoning of individual students.
Together, the three curriculum factors and three teacher factors provide a basis for understanding teachers' offloading, adapting, or improvising with curriculum resources. The factors are not all-inclusive, as other researchers have studied characteristics such as the "voice" of curriculum ({{Cite|Herbel-Eisenmann|2007}}), contextual factors ({{Cite|Grossman|1990}}), cultural teaching norms ({{Cite|Stigler & Hiebert|1998}}), professional identity ({{Cite|McClain, Zhao, Visnovska, & Bowen|2009}}; {{Cite|Smith|1996}}), or teachers' orientation toward curriculum materials ({{Cite|Remillard & Bryans|2004}}).


=== Methodological Considerations ===
=== Pedagogical Design Capacity ===


Cobb et al. focused their analysis on the classroom mathematical practices, as they believed this to be the least developed aspect of their interpretive framework. The data used for analysis included video recordings of all classes during teaching teaching experiments, all students' written work, and video recordings of student interviews collected before and after the teaching experiment. Cobb et al. took [[Glazer & Strauss (1967)|Glazer and Strauss's (1967)]] constant comparison method because it had been adapted for design research ([[Cobb & Whitenack (1996)|Cobb & Whitenack, 1996]]). Acting as participant observers, Cobb et al. documented classroom activity in ways that exposed patterns or themes which could then be tested with each cycle of new data. With this method, Cobb et al. treated mathematical learning "as a process of coming to use conventional tools and symbols in socially accepted ways" (p. 127).
Brown describes ''pedagogical design capacity'' (PDC) as a teacher's "skill in perceiving the affordances of the materials and making decisions about how to use them to craft instructional episodes that achieve her goals" (p. 29) and more succinctly defines it as "a teacher's capacity to perceive and mobilize existing resources in order to craft instructional episodes" (p. 29). Whereas the DCE framework referred mostly to the resources, which Brown refers to as the nouns of the interaction between teachers and their materials, PDC applies to the verbs of the interaction and teachers' ability to accomplish new things with their knowledge ({{Cite|Ball & Cohen|1999}}). Clarifying, Brown adds:


Data analysis was approached chronologically, and with each mathematical concept the researchers conjectured about normative reasoning and communication, as well as individual students' reasoning. Separate records were kept for the social and psychological perspectives to make them distinct and to highlight the reflexive relationships between them. Conjectures about mathematical norms took three forms: (a) a taken-as-shared purpose, (b) taken-as-shared ways of reasoning with tools and symbols, and (c) taken-as-shared forms of mathematical argumentation (p. 129). For this paper, these conjectures were made in the context of a measurement activity, where students used tools to assign numerical values to objects and were required to defend both their process and results.
<blockquote>
 
PDC represents a teacher's skill in perceiving affordances, making decisions, and following through on plans. Whether such design decisions manifest as offloads, adaptations, or improvisations is a separate matter. It is the skill in weaving various modes of use together and in arranging the various pieces of the classroom setting that is the mark of a teacher with high PDC, not whether they happen to be offloading, adapting, or improvising at any given moment. Rather, PDC describes the manner and degree to which teachers create deliberate, productive designs that help accomplish their instructional goals. (p. 29)
=== Measurement Practices ===
</blockquote>
 
This paper was based on a 14-week teaching experiment with 16 first-grade students, with the first half of the experiment focused on linear measuring and the second half focused on mental computation with numbers up to 100.
 
==== Background to the Teaching Experiment ====
 
The classroom teacher was a member of the research team and allowed the researchers to intervene during classroom work. Two cameras were used to record every lesson while three researchers took field notes. Copies of work was collected from all students and interview data was collected from students before and after the experiment. The research team also recorded their weekly meetings. Five students were interviewed midway through the experiment, selected due to the diversity in their reasoning shown in the initial set of interviews. These five students were watched more closely during the experiment and researchers made detailed notes of the students' reasoning in their field notes after each class.
 
Initially, Cobb et al. envisioned that by iterating a tool along an object, students would give meaning to the accumulation of distance (cf. [[Piaget, Inhelder, & Szeminska (1960)|Piaget, Inhelder, & Szeminska, 1960]]; [[Thompson & Thompson (1996)|Thompson & Thompson, 1996]]). This way, students should understand that if measuring by pacing from heel to toe, for example, the number word said with each step represented the accumulated distance and not a label for that particular step. This understanding could be extended to other measuring tools representing multiple units, like a strip of paper that was five steps long. Relating it to their interpretive framework, Cobb et al. claimed "our instructional intent was therefore that a taken-as-shared spatial environment would become established in which distances are quantities of length whose numerical measures can be specified by actually measuring ([[Greeno (1991)|Greeno, 1991]])" (p. 132).
 
==== The Classroom Microculture ====
 
For this experiment, the teacher typically embedded a need to measure into a story that was read at the beginning of each class. Students would then work by themselves or in pairs to solve the measurement problem before discussing their solutions as a class. [[Stephan (1998)]] found that in addition to the teacher encouraging students to try to understand and stay involved at all times, the following were social norms for whole-class discussions:
 
# Students were obliged to explain and justify their reasoning.
# Students were obliged to listen to and attempt to understand others' explanations.
# Students were obliged to indicate nonunderstanding and, if possible, to ask the explainer clarifying questions.
# Students were obliged to indicate when they considered solutions invalid, and to explain the reasons for their judgment. (p. 133)
 
As for sociomathematical norms, Cobb et al. wanted discussions to focus on the conceptual, not the calculational ([[Cobb (1998)|Cobb, 1998]]; [[Lampert & Cobb (1998)|Lampert & Cobb, 1998]]; [[Thompson, Philipp, Thompson, & Boyd (1994)|Thompson, Philipp, Thompson, & Boyd, 1994]]). Explanations that simply describe using a measurement tool to get a numerical result would not be enough; for discussions to reflect the conceptual, the discourse would have to include justifications of how the measuring procedure structures the physical object into quantities of length.
 
==== The Emergence of the First Two Mathematical Practices ====
 
Cobb et al. describe the first two practices to create a backdrop for the emergence of the third practice, which they describe in greater detail. In Sessions 1 through 3 of the research, students participated in a narrative where a king measured his kingdom by stepping heel to toe. Some students began by placing their heel at the edge of a rug and only counting "one" when they took a second step, while others counted their first foot as "one" and their next step as "two." In the class discussion, some students correctly reasoned that each count should correspond to a physical part of the rug, and the first step should not be "missing." In subsequent sessions, students debated over different measurements due to different foot lengths and what to do when only part of a step was needed to reach the end of a measurement. This led to the first mathematical practice, ''measuring by pacing''.
 
Over the next seven sessions engaged students in the next phase of the story, where the king was too busy to be in all parts of his kingdom to measure things with his feet. The solution that students had was to create a "footstrip," a piece of paper representing five of the king's steps. Now students could place the footstrip and count by 5 steps, and students recognized that this "big step" was more efficient for longer distances. More discussion occurred when only part of the footstrip was needed to complete a measurement, and a student proposed cutting the footstrip. This led researchers to observe an advance in measuring practice as students began to represent measurement as a property of the object being measured, not a physical act of placing the strip. Still, for some students "measuring with the footstrip structured the physical extension of an object into a sequence or chain of individual paces much as had measuring by pacing," while other students saw measurement as "the physical extension of the object being measured constituted into what they called a 'whole space' that was partitioned into paces" (p. 138). Therefore, for a distance of 12, students in the first group were more likely to look at the 12th pace in a sequence of steps, while the second group saw 12 as the distance from the first step through the 12th step.
 
==== The Emergence of the Third Mathematical Practice ====
 
=== Methodological Reflections ===
 
=== Trustworthiness, Replicability, and Commensurability ===
 
=== Usefulness ===
 
=== Limitations ===
 
=== Conclusion ===


In his dissertation, Brown ([[Brown (2002)|2002]]) used PDC to describe how two teachers with similar resources described by the DCE framework nonetheless differed in the enactment of their curricula. Likewise, PDC was also used to describe teachers with similar enactment despite different resources described by the DCE framework. While Brown had evidence that PDC may emerge in teachers over time as they become more familiar with resources, more research is needed to understand how PDC is developed. There is also a need to develop measures for PDC and better understand its role in achieving outcomes.


== About ==
=== Implications ===


=== Mendeley ===
Better understanding of PDC would have implications for teacher preparation, curriculum design, and research methods. If curriculum materials were PDC-aware, they could offer support for different kinds of use which would be backed by aligned professional development. There is a natural tension between flexible, open-ended instructional designs and maintenance of coherence intended by curriculum authors. Brown et al. ([[Brown, Pellegrino, Goldman, Nacu, Julian, Tarnoff, et al. (2004)|2004]]) developed an online system called ''AIM'' (Adaptive Instructional Materials) that integrated a database of resources with a capability to adapt those resources into lesson and course plans. Similar research on "learning objects" ({{Cite|Wiley|2001}}) has also explored such modular resources, which contrasts with research of scripted and inflexible curriculum resisted by teachers (e.g., {{Cite|Ben-Peretz|1990}}).


[http://www.mendeley.com/catalog/participating-classroom-mathematical-practices/ http://www.mendeley.com/catalog/participating-classroom-mathematical-practices/]
The approach used for AIM by Brown et al. applied three key principles. First, a range of teacher expertise with content and instruction was supported with multiple ways of accessing the resources. Experts could browse or search for resources to assemble into plans or use pre-authored lessons. For teachers with less expertise, pre-authored materials annotated by developers to highlight affordances and constraints supported teacher learning and decision-making. Second, materials were designed to be resource-centric; that is, materials were organized around concepts, encouraging use in different contexts and avoiding a procedure-focused approach to organization. When materials were used in pre-authored lessons, designers attempted to be transparent ({{Cite|Davis & Krajcik|2005}}) in their annotations by explaining the decisions made to use the materials in the chosen way. Lastly, to encourage reuse of resources and to support customization, designers attempted to balance the need to have materials dependent on context yet sufficiently generalized to be useful in different parts of the curriculum. This was done with descriptions of pedagogical affordances that avoided limiting teachers to the use of single strategies for implementation.


=== APA ===
PDC's implications for professional development indicate a need to support teachers' decision-making about the resources they choose and how they use them. Brown suggests teachers receive help in evaluating features and affordances of materials and necessary modifications to align materials with instructional goals. This kind of professional development would have the added benefit of providing a context in which teachers could improve their understanding of instruction and student learning.


Cobb, P., Stephan, M., McClain, K., & Gravemeijer, K. (2001). Participating in classroom mathematical practices. The Journal of the Learning Sciences, 10(1/2), 113–163. doi:10.1207/S15327809JLS10-1-2_6
== Corrolary ==
=== BibTeX ===


;APA
: Brown, M. W. (2009). The teacher-tool relationship: Theorizing the design and use of curriculum materials. In J. T. Remillard, B. A. Herbel-Eisenmann, & G. M. Lloyd (Eds.), ''Mathematics teachers at work: Connecting curriculum materials and classroom instruction'' (pp. 17–36). New York, NY: Routledge.
;BibTeX
<pre>
<pre>
@article{Cobb2001,
@incollection{Brown2009,
abstract = {In this article, we describe a methodology for analyzing the collective learning of the classroom community in terms of the evolution of classroom mathematical practices. To develop the rationale for this approach, we first ground the discussion in our work as mathematics educators who conduct classroom-based design research. We then present a sample analysis taken from a 1st-grade classroom teaching experiment that focused on linear measurement to illustrate how we coordinate a social perspective on communal practices with a psychological perspective on individual students' diverse ways of reasoning as they participate in those practices. In the concluding sections of the article, we frame the sample analysis as a paradigm case in which to clarify aspects of the methodology and consider its usefulness for design research.},
address = {New York, NY},
author = {Cobb, Paul and Stephan, Michelle and McClain, Kay and Gravemeijer, Koeno},
author = {Brown, Matthew W.},
doi = {10.1207/S15327809JLS10-1-2\_6},
booktitle = {Mathematics teachers at work: Connecting curriculum materials and classroom instruction},
journal = {The Journal of the Learning Sciences},
chapter = {2},
number = {1/2},
editor = {Remillard, Janine T. and Herbel-Eisenmann, Beth A. and Lloyd, Gwendolyn M.},
pages = {113--163},
pages = {17--36},
title = {{Participating in classroom mathematical practices}},
publisher = {Routledge},
url = {http://www.tandfonline.com/doi/abs/10.1207/S15327809JLS10-1-2\_6},
title = {{The teacher-tool relationship: Theorizing the design and use of curriculum materials}},
volume = {10},
year = {2009}
year = {2001}
}
}
</pre>
</pre>


[[Category:Articles]]
[[Category:Summaries]]
[[Category:The Journal of the Learning Sciences]]
[[Category:Book Chapters]]
[[Category:Design Research]]
[[Category:2009]]
[[Category:Learning Sciences]]
[[Category:Curriculum Use]]
[[Category:Measurement]]
[[Category:Teacher Knowledge]]
[[Category:Elementary]]
[[Category:Realistic Mathematics Education]]

Revision as of 14:50, 17 August 2014

The Teacher-Tool Relationship: Theorizing the Design and Use of Curriculum Materials


Outline of Headings

  • Teaching as Design
  • Theoretical Background
    • Artifacts can Extend Human Capacities
  • Curriculum Materials as Artifacts
  • How Curriculum Artifacts Influence Instruction
  • How Teachers Interpret and Use Curriculum Artifacts
  • How Teaching is Design
  • Analyzing Teacher Use of Curriculum Artifacts
  • Types of Curriculum Use: Offloading, Adapting, and Improvising
  • Facets of the Teacher-Tool Relationship: The Design Capacity for Enactment Framework
    • Applying the Design Capacity Enactment Framework
  • Pedagogical Design Capacity
  • Design Implications
  • The Design of Materials
    • Multiple Points of Access
    • Resource-Centric Material Design
    • Creating Reusable Resources and Supporting Customization
  • The Design of Professional Development

Summary

Brown begins this chapter comparing the relationship between teachers and curriculum materials to those of musicians and their music; just as the same song played by different musicians takes on its own character, teachers interpret and adapt curriculum materials in ways that make their practice unique, even if there are similarities across classrooms. Curriculum materials are often used to promote educational reforms and the results of such efforts have been mixed (Ball & Cohen, 1996; Cohen, 1988; Cuban 1992, 1993; Snyder, Bolin, & Zumwalt, 1992), the reasons of which have been attributed to practitioners (Cohen, 1990; Spillane, 1999), policies (Spillane, 1998), and professional development (Putnam & Borko, 2000; Wilson & Berne, 1999). Other studies have focused on how teachers interpret curriculum materials (Ben-Peretz, 1990; Brown, 2002; Brown & Edelson, 2003; Lloyd, 1999; Remillard 2000, 2005; Wiley, 2001) and how curriculum materials might be better designed to meet the need of teachers (Brown, 2002; Brown & Edelson, 2003; Davis & Krajcik, 2005; Davis & Varma, 2008; Schneider & Krajcik, 2002). Brown's purpose in writing this chapter is to describe a theoretical framework for the relationship between curriculum materials and teacher practice. In doing so, Brown conceives of teaching as a design activity in which teachers evaluate their resources and make decisions in an effort to achieve instructional goals. Tensions between teachers (the agent) and their curriculum materials (the tool) build on a well-established body of learning theory (Gibson, 1977; Hutchins, 1996; Norman 1988, 1991; Pea, 1993; Wertsch, 1991, 1998), which Brown uses to highlight three key points:

  1. Curriculum materials play an important role in affording and constraining teachers' actions.
  2. Teachers notice and use such artifacts differently given their experience, intentions, and abilities.
  3. "Teaching by design" is not so much a conscious choice as an inevitable reality (p. 19).

Wartofsky (1973) theorized artifacts as human-created tools that play a prominent role in our survival. Artifacts can be transmitted or preserved across place and time and our progress is inseparable from the artifacts we use (Wertsch, 1998). Artifacts mediate human activity (Vygotsky, 1978; Wertsch, 1991, 1998) in ways shaped by the artifact's affordances and constraints. Affordances might be attributed to the functional properties of the artifact (Gibson, 1977) or perceptual cues that suggest how they might be used (Norman, 1988). Constraints of artifacts restrict the ways in which we act and what we see as possible (Burke, 1966; Wertsch, 1998).

Norman (1988) suggested that humans often design artifacts to afford and constrain human activities in particular ways. In the case of curriculum, materials are designed to promote certain instructional, assessment, and student practices while discouraging the use of others. Often, curriculum materials allow teachers to engage in practices they could not have done alone, thus representing a shared capacity across teachers and their materials. Artifacts are also not limited to physical materials (Wartofsky, 1973); our cultural and historical ideas and practices also act to shape our activity. Brown enumerates six characteristics of curriculum materials that influence instruction:

  1. They are static representations of abstract concepts and dynamic activities — a means for transmitting and producing activity, not the activity itself.
  2. They are intended to convey rich ideas and dynamic practices, yet they do so through succinct shorthand that relies heavily on interpretation.
  3. They observe a number of culturally shared notational rules, norms, and conventions in their representations — although fewer consistently and conventions in their representations — although fewer consistently used conventions exist for curriculum materials than for sheet music.
  4. They may reflect common or existing practices and at the same time aim to shape innovative or new practices.
  5. They represent an interface between the knowledge, goals, and values of the author and the user.
  6. They require craft in their use; they are inert objects that come alive only through interpretation and use by a practitioner (pp. 21-22).

In addition to the influence of curriculum materials on teachers, Brown notes the importance of understanding the "dynamic and constructive ways" (p. 22) teachers interpret and use curriculum materials (Barab & Luehmann, 2003; Brown, 2002; Davis & Krajcik, 2005; Matese, 2005; Remillard, 2005). Even when curriculum materials are provided, teachers still select materials according to their beliefs, skills, knowledge, and goals (Freeman & Porter, 1989; Tarr, Reys, Reys, Chavez, Shih, & Osterlind, 2008) or resist scripted or otherwise inflexible materials (Cohen, 1990; Remillard, 1992; Wilson, 1990) based on their goals and beliefs (Cohen, 1988, 1990; Lloyd, 1999; Lloyd & Wilson, 1998; Wilson & Goldenberg, 1998). Next, teachers interpret materials in planning and instruction (Ben-Peretz, 1990; Stein, Remillard, & Smith, 2007). Then teachers reconcile those interpretations with their instructional goals (Ben-Peretz, 1990; Remillard, 2005), and accommodate the needs of their students, making continual adjustments and providing feedback (Stein, 1996; Wilson & Lloyd, 2000). Depending on the perceived success of the plan, teachers may add, modify, or omit parts of the curriculum, either due to interest or ability (Remillard, 1992; Tarr, Reys, Reys, Chávez, Shih, & Osterlind, 2008).

This perspective on teachers' curriculum use leads Brown to conclude that teaching is a form of design, a process that "is about crafting something in order to solve a human problem, to change the state of a particular situation from a current condition to a desired one, and to accomplish a goal" (p. 23). Understanding teaching as design highlights the dynamic between teachers and their materials, and Brown provides three constructs for understanding this relationship. The first contrasts offloading, adapting, and improvising and gives us a way to think about the degree to which teachers appropriate instructional materials. Second, Brown describes a framework for examining interactions between teachers and features of their materials. Lastly, he describes pedagogical design capacity, described as a teacher's "ability to perceive and mobilize existing resources in order to craft instructional contexts" (p. 24).

Types of Curriculum Use: Offloading, Adapting, and Improvising

From his dissertation work (2002), Brown identified three ways teachers appropriate their materials as they design instruction. Offloading occurs when teachers follow curriculum materials closely, giving agency to the materials for guiding instruction. On the other end of the scale is improvising, when teachers craft instruction spontaneously and/or without specific guidance from their materials, thus shifting agency to themselves. In between, adapting occurs when teachers modify their materials to support instructional goals, thus sharing the agency between themselves and the materials.

Brown emphasizes that this scale does not necessarily correlate to teacher expertise and is not intended to measure the fidelity with which materials are used. It is designed to describe the nature of teachers' interactions with resources, not the outcomes of those interactions. Each position on the scale can serve a strategic purpose and be used with varying degrees of effectiveness. Brown explains,

"Just as a novice teacher might offload instructional responsibility to a scripted lesson due to limited understanding of the subject matter, so might an expert teacher offload instructional responsibility to a worksheet that supports her goals, freeing her to roam the room and respond to student needs as they arise" (p. 25).

Facets of the Teacher–Tool Relationship: The Design Capacity for Enactment Framework

Brown's Design Capacity for Enactment (DCE) framework (2002) attempts to describe the aspects of both curriculum resources and teachers that influence why teachers interact with curriculum materials in different ways. The DCE framework includes three basic aspects of curriculum: physical objects, representations of tasks (procedures), and representations of concepts (domain representations). Physical objects include the material nature of the curriculum materials, including supplementary and recommended materials. Representations of tasks refers to the instructions and procedures given to teachers and students in using the materials, such as lesson guides or recommended homework problem sets. Domain representations consist of the ways materials organize concepts using diagrams, models, analogies, and the sequencing of topics.

Brown identified three teacher resources in the DCE. The first, subject matter knowledge, consists of knowledge about the domain (Ball, 1991; Stodolsky & Grossman, 1995), while pedagogical content knowledge (Shulman, 1986) combines general knowledge about teaching with domain-specific teaching knowledge. Lastly, teachers' goals and beliefs (Ball & Cohen, 1999) describe teachers' motivations to teach particular content in particular ways. Sometimes goals and beliefs conflict with instructional approaches (Spillane, 1999; Wilson, 1990) and can be a barrier to the implementation of reforms (Cohen, 1988, 1988).

Together, the three curriculum factors and three teacher factors provide a basis for understanding teachers' offloading, adapting, or improvising with curriculum resources. The factors are not all-inclusive, as other researchers have studied characteristics such as the "voice" of curriculum (Herbel-Eisenmann, 2007), contextual factors (Grossman, 1990), cultural teaching norms (Stigler & Hiebert, 1998), professional identity (McClain, Zhao, Visnovska, & Bowen, 2009; Smith, 1996), or teachers' orientation toward curriculum materials (Remillard & Bryans, 2004).

Pedagogical Design Capacity

Brown describes pedagogical design capacity (PDC) as a teacher's "skill in perceiving the affordances of the materials and making decisions about how to use them to craft instructional episodes that achieve her goals" (p. 29) and more succinctly defines it as "a teacher's capacity to perceive and mobilize existing resources in order to craft instructional episodes" (p. 29). Whereas the DCE framework referred mostly to the resources, which Brown refers to as the nouns of the interaction between teachers and their materials, PDC applies to the verbs of the interaction and teachers' ability to accomplish new things with their knowledge (Ball & Cohen, 1999). Clarifying, Brown adds:

PDC represents a teacher's skill in perceiving affordances, making decisions, and following through on plans. Whether such design decisions manifest as offloads, adaptations, or improvisations is a separate matter. It is the skill in weaving various modes of use together and in arranging the various pieces of the classroom setting that is the mark of a teacher with high PDC, not whether they happen to be offloading, adapting, or improvising at any given moment. Rather, PDC describes the manner and degree to which teachers create deliberate, productive designs that help accomplish their instructional goals. (p. 29)

In his dissertation, Brown (2002) used PDC to describe how two teachers with similar resources described by the DCE framework nonetheless differed in the enactment of their curricula. Likewise, PDC was also used to describe teachers with similar enactment despite different resources described by the DCE framework. While Brown had evidence that PDC may emerge in teachers over time as they become more familiar with resources, more research is needed to understand how PDC is developed. There is also a need to develop measures for PDC and better understand its role in achieving outcomes.

Implications

Better understanding of PDC would have implications for teacher preparation, curriculum design, and research methods. If curriculum materials were PDC-aware, they could offer support for different kinds of use which would be backed by aligned professional development. There is a natural tension between flexible, open-ended instructional designs and maintenance of coherence intended by curriculum authors. Brown et al. (2004) developed an online system called AIM (Adaptive Instructional Materials) that integrated a database of resources with a capability to adapt those resources into lesson and course plans. Similar research on "learning objects" (Wiley, 2001) has also explored such modular resources, which contrasts with research of scripted and inflexible curriculum resisted by teachers (e.g., Ben-Peretz, 1990).

The approach used for AIM by Brown et al. applied three key principles. First, a range of teacher expertise with content and instruction was supported with multiple ways of accessing the resources. Experts could browse or search for resources to assemble into plans or use pre-authored lessons. For teachers with less expertise, pre-authored materials annotated by developers to highlight affordances and constraints supported teacher learning and decision-making. Second, materials were designed to be resource-centric; that is, materials were organized around concepts, encouraging use in different contexts and avoiding a procedure-focused approach to organization. When materials were used in pre-authored lessons, designers attempted to be transparent (Davis & Krajcik, 2005) in their annotations by explaining the decisions made to use the materials in the chosen way. Lastly, to encourage reuse of resources and to support customization, designers attempted to balance the need to have materials dependent on context yet sufficiently generalized to be useful in different parts of the curriculum. This was done with descriptions of pedagogical affordances that avoided limiting teachers to the use of single strategies for implementation.

PDC's implications for professional development indicate a need to support teachers' decision-making about the resources they choose and how they use them. Brown suggests teachers receive help in evaluating features and affordances of materials and necessary modifications to align materials with instructional goals. This kind of professional development would have the added benefit of providing a context in which teachers could improve their understanding of instruction and student learning.

Corrolary

APA
Brown, M. W. (2009). The teacher-tool relationship: Theorizing the design and use of curriculum materials. In J. T. Remillard, B. A. Herbel-Eisenmann, & G. M. Lloyd (Eds.), Mathematics teachers at work: Connecting curriculum materials and classroom instruction (pp. 17–36). New York, NY: Routledge.
BibTeX
@incollection{Brown2009,
address = {New York, NY},
author = {Brown, Matthew W.},
booktitle = {Mathematics teachers at work: Connecting curriculum materials and classroom instruction},
chapter = {2},
editor = {Remillard, Janine T. and Herbel-Eisenmann, Beth A. and Lloyd, Gwendolyn M.},
pages = {17--36},
publisher = {Routledge},
title = {{The teacher-tool relationship: Theorizing the design and use of curriculum materials}},
year = {2009}
}