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A Dialectic Analysis of Generativity:
Issues of Network-Supported Design
in Mathematics and Science
Walter M. Stroup
Department of Curriculum and Instruction
The University of Texas at Austin
Nancy M. Ares
Department of Teaching and Curriculum
University of Rochester
Andrew C. Hurford
Department of Curriculum and Instruction
The University of Texas at Austin
New theoretical, methodological, and design frameworks for engaging classroom
learning are supported by the highly interactive and group-centered capabilities of a
new generation of classroom-based networks. In our analyses, networked teaching
and learning are organized relative to a dialectic of (a) seeing mathematical and sci-
entific structures as fully situated in sociocultural contexts and (b) seeing mathemat-
ics as a way of structuring our understanding of and design for group-situated teach-
ing and learning. An engagement with this dialectic is intended to open up new
possibilities for understanding the relations between content and social activity in
classrooms. Features are presented for what we call generative design in terms of the
respective “sides” of the dialectic. Our approach to generative design centers on the
notion that classrooms have multiple agents, interacting at various levels of participa-
tion, and looks to make the best possible use of the plurality of emergent ideas found
in classrooms. We close with an examination of how this dialectic framework also
can support constructive critique of both sides of the dialectic in terms of content and
pedagogy.
MATHEMATICAL THINKING AND LEARNING, 7(3), 181–206
Copyright © 2005, Lawrence Erlbaum Associates, Inc.
Requests for reprints should be sent to Walter M. Stroup, The University of Texas at Austin, Depart-
ment of Curriculum and Instruction, 1 University Station D5705, Austin, TX 78712–0382. E-mail:
wstroup@mail.utexas.edu
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The highly interactive and group-centered capabilities of a new generation of
classroom-based networks support the development of new theoretical, method-
ological, and design frameworks for engaging classroom learning. We are inter-
ested in how mathematical and scientific content can frame the design of class-
room activities and supporting technologies. Content in this sense is an organized
body of knowledge developing over time and enacted through activity. In explor-
ing the mutually constitutive relation between content and classroom activity, we
structure this analysis of network-supported teaching and learning in terms of a di-
alectic of (a) seeing mathematical and scientific structures as fully situated in
sociocultural contexts and (b) seeing mathematics and science as ways of structur-
ing our understanding of, and design for, group-situated teaching and learning.
This work is intended to have practical and theoretical value for researchers and
classroom educators working with these next-generation systems.
Practically, we hope to avoid the possibility that the promise of next-generation
network capabilities will be less than fully realized. Simply aggregating student re-
sponses to multiple-choice questions, although relatively easy to do in a networked
environment and certainly useful on some occasions, misses the many ways in
which day-to-day classroom practice can be improved by a more thorough engage-
ment with generative design as an approach to network-supported learning. Ac-
cordingly, in this article we attend both to some of the generative design principles
that can help guide classroom-focused activity design and to some of the potential
pitfalls that can compromise the realization of the mediating potential of these
next-generation systems. Theoretically, we believe the approach to generativity
discussed in this article—centering on the idea that classrooms have multiple
agents participating at richly interactive levels of engagement and agency—makes
better use of the diversity of identities, the plurality of conceptual structures, the
shifting of collective understandings, and the evolution of disciplinary concepts
found in classroom activity. Attending to this plurality of expressive activity serves
to deepen our insights into the emergence and development of ideas and, simulta-
neously, into the designs that can best support the advancement of mathematical
and scientific thinking for all our students. Our hope is that this article will serve as
an invitation to researchers, designers, and practitioners to participate in shaping
this evolution.
Critical to our engagement with this multiplicity of expressive forms is the
framing of our analyses in terms of a dialectic that sees content as both socially
structured and socially structuring. As we note later, it is not new to think of
mathematics as socially situated (cf. Vygotsky, 1978, 1987). What is new are the
ways in which content now can be seen to structure the social activity of the
classroom, and then, reciprocally, what social conceptions of knowing as partici-
pation can do to help situate and advance the notion of content as enacted in
classroom activity. To situate our analysis of the proposed dialect for exploring
the relation between content and social activity in next-generation networks, we
182 STROUP, ARES, HURFORD
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begin by offering a sketch of what teaching and learning looks like in these
highly interactive environments. We then provide an analysis both of the net-
work capabilities that are central to the new systems and of aspects of generative
design, framed in terms of the proposed dialectic, that we hope can prove useful
to teachers and researchers in realizing the full potential of these systems relative
to classroom-based teaching and learning.
WHAT LEARNING IN NEXT-GENERATION
NETWORKS LOOKS LIKE
These new systems are typically designed specifically for classrooms. Rather
than simply import traditional network capabilities associated with business or
other less learning-centered environments, these systems are specifically opti-
mized for places where learners and teachers come together as groups, in a phys-
ically contiguous space, with the goal of advancing meaningful learning. Instead
of constraining the learning experience to be narrowly individualistic, this tech-
nology supports socially situated interaction and investigation. Moreover, the
group itself owns the learning trajectories and the processes of knowledge con-
struction, rather than outside experts or programmers. Students arrive in class,
turn on their devices, and find themselves situated in an activity that is not just
about mathematics or science but situated in an activity that invites them to par-
ticipate in mathematically and scientifically defined ways. The network allows
artifacts, electronic gestures (e.g., pressing a key), and patterns of interaction to
situate and animate the evolving experiences of students. Local actions on an in-
dividual device—such as, for example, pressing a key to move a simulated char-
acter—interact and are coordinated by the network and are projected to a public,
collective space in front of the classroom.1
DIALECTICS AND GENERATIVE DESIGN 183
1The implementations of next-generation networked systems may vary, but the top-level design fea-
tures of the systems are remarkably similar and typically include individual devices or “nodes,” support
for a range of topologies for real-time or near-real time interaction (e.g., peer to peer, peer to group, in-
cluding whole-class, or group to group), wireless flexibility and portability, a core set of meaningful
functionality in each device (e.g., at least that of a graphing calculator), and a mixture of public and pri-
vate display spaces (e.g., the public space can be a computer projection system as with participatory
simulations [Wilensky & Stroup, 1999a] or a calculator ViewscreenTM with some of the SimCalc mate-
rials [Kaput, Roschelle, Tatar, & Hegedus, 2002], and the private space can be the students’own indi-
vidual displays on a calculator or laptop computer). The network experience is “author-able” in that it
allows teachers, students, or others to create new activities or change the flow of a given activity. Partic-
ipants can exchange both group and individual artifacts or data types, including text, strings, numeric
values, ordered pairs, lists, matrices, individual and whole-class graphs, images, and, in some cases,
sounds or video.
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The capabilities of these systems go well beyond simple aggregation of student
responses to multiple-choice questions. In moving beyond aggregating answers,
two broad classes of activities are supported—iconic and noniconic—and can be
discussed with particular attention paid to either the students (Roschelle, 1990) or
the role of the teacher (Mack, 2002), or to the interactions between teacher and stu-
dent (Ares, Stroup, & Schademan, 2004).
In one class of network-supported activity, students can assume iconic roles in a
simulation. That is, they can control the movement of a rabbit or a wolf in a preda-
tor–prey simulation or control a light in a simulated traffic grid (see Figure 1) and
work toward the goal of improving traffic flow (Wilensky & Stroup, 1999b, 2000).
These network-supported activities are iconic because students assume roles
that are “like the thing.” The students can—almost literally—behave like a rabbit
of wolf or like a traffic light, and the relevant behaviors of the system emerges from
the interactions (e.g., the relative populations of rabbits and wolves oscillate in
ways consistent with fundamental ideas in population dynamics or the traffic be-
comes more or less congested in ways related to the traffic-related strategies the
students implement). Student participation is also represented by icons that share
overt features of the roles they assume: The icons can look and move like predators
or prey, or the icons can look like a light positioned at an intersection and turn red
184 STROUP, ARES, HURFORD
FIGURE 1 Students use interactive classroom network to control lights in a simulated traffic
grid.
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or green (Wilensky & Stroup, 1999a).2These participatory forms of network-sup-
ported interactivity significantly enhance the kinds of role-playing activities that
have been used for decades in education either for teaching concepts such as the
spread of a disease in a population (cf. Stor & Briggs, 1998) or for systems dynam-
ics proper (cf. the inventory management game developed by Jay Forester at Mas-
sachusetts Institute of Technology in the 1960s and recently revived in Peter
Senge’s, 1994, widely read Fifth Discipline). Some recent work has situated iconic
role-playing activities relative to agent-based forms of systems modeling that in-
clude the use of powerful new tools for agent-based programming (cf. StarLogoT
and NetLogo as developed by Wilensky, 1999; StarLogo by Resnick, 1994, 2004;
or AgentSheets by Repenning, 1993). Network architectures for implementing
real-time role-playing activities with computers or graphing calculators are now
available (cf. HubNet; Wilensky & Stroup, 1999b, 2000) and significantly enhance
the interactivity, replay, and real-time data-collection capabilities associated with
iconic role-playing activities.
A second broad class of activities is noniconic: Student participation takes
place in terms of representations, symbols, and gestures that are less like a particu-
lar natural object and are more structured by the notations and tools of the domains
of mathematics or science themselves. A simple example is having students submit
functions that are the “same” or equivalent to the function f(x)=4x. On individual
graphing calculators, students can manipulate the symbolic representation of func-
tions (e.g., Y1=4x;Y2=2x+2x;Y3=40x/10) as well as the graphing and table
features to find functions they think are the same as f(x)=4x. Using the network,
they can submit their favorite example and the collection of functions then can be
displayed together at the front of the class or sent back to the individual devices for
further consideration. Similarly, in a statistics-oriented activity, students could use
their local devices to adjust the bin sizes of a histogram or adjust the size of the
samples they take from a given population and see what, if any, effects these ac-
DIALECTICS AND GENERATIVE DESIGN 185
2As is noted in this publication and others (e.g., Wilensky & Stroup, 2000), possibly the first major
instance of a participatory simulation used in the context of systems dynamics and systems learning
was The Beer Game, as developed by Jay Forrester and his systems dynamics group in the early 1960s.
These participatory simulations were called “flight simulators” in a way that alluded to the military use
of simulator environments in World War II. There is a significant literature related to The Beer Game,
and interest in this participatory simulation has been recently revitalized as a result of its appearance in
Senge’s (1994) The Fifth Discipline. Diehl (1990) appears to have been the first to use the phrase “par-
ticipatory simulations” to describe these activities. Over the years, different technologies have been
used to implement participatory simulations. These implementations range from the use of simple pa-
per and pencil (e.g., Senge, 1994; Stor & Briggs, 1998) to the use of electronic badges (so-called
Thinking Tags; see Borovoy et al., 1998; Borovoy, McDonald, Martin, & Resnick, 1996; Colella,
Borovoy, & Resnick, 1998), handheld technologies (Stroup, 1997b, [a TI-83 graphing calculator];
Solowayet al., 2001 [using a Palm OS device]), and a new, network-based HubNet architecture
(Wilensky & Stroup, 1999b). Our focus here is on participatory simulations as implemented in
next-generation classroom networks.
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tions have on the public display of the data (Finzer & Stroup, 2000). One instance
of a real-time networked architecture that tends to embody noniconic forms of par-
ticipation is the Navigator 2.0TM system recently released by Texas Instruments
(2005).
Some activities integrate aspects of iconic and noniconic interactions. For ex-
ample, using network-supported capabilities, the SimCalc Project (Mathworld;
Kaput & Roschelle, 2000; Kaput et al., 2002) has students manipulate piece-wise
constant velocity graphs that control the motion of a simulated character (e.g., an
elevator) on their local devices. The graphs and respective motions can be submit-
ted and displayed simultaneously in the upfront space, and questions about what
the graphs and motions have in common can be discussed. A number of the pro-
jects currently funded by the National Science Foundation in the United States and
some commercial entities are collaborating in developing these new forms of net-
work-supported interactivity, and it looks as if the fusing of iconic and noniconic
interactivity will become increasingly the norm relative to the functionality sup-
ported by these new systems.
The point we make in this article is that new forms of mediated activity and
participation are supported by these next-generation systems. In this article, we
take up the question: What’s new about these systems as mediating environ-
ments? Networking of various forms has been available for decades in class-
rooms, so it is important that we distinguish the capabilities of the new systems.
Then, moving beyond the systems themselves as mediating tools, we can ask:
How do mathematics and science, as domains of human inquiry, interact with
the social context and pedagogy of learning in a group space like a classroom?
Within a framework of seeing learning as a form of participation in communities
of inquiry (whether localized in the classroom or in larger communities of math-
ematical, scientific, social, or neighborhood-based inquiry), we offer specific de-
sign principles related to a dialectic of mathematical and scientific ideas as both
(a) structuring of and (b) structured by the social activity of the classroom. We
use this dialectic framework to clarify what is meant by generative design, ex-
tending previously existing ideas of generativity up to group-level analyses of
teaching and learning. Next-generation capabilities support this shift to
group-level analyses and design. To further focus this group engagement, four
features of generative design are discussed in terms of respective sides of the di-
alectic. Relative to mathematics and science being used to overtly structure the
activity of the classroom, (a) space-creating play and (b) dynamic structure are
highlighted and discussed (see 2 in Figure 2). Relative to scientific and mathe-
matical participation being structured by social activity, (c) agency and (d) par-
ticipation (see 1 in Figure 2) are highlighted and analyzed with special attention
to supporting varied forms of participation in classrooms.
Our focus on design highlights the ways in which the dialectic between social
and mathematical or scientific structures illuminates new possibilities for genera-
186 STROUP, ARES, HURFORD
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tive classroom practice. The sense is that in these highly interactive forms of activ-
ity, the line between learning activity and content becomes so blurred and inter-
mingled that the mathematics and science actually become the foundation for
highly socially situated pedagogy. Throughout, we attend to results and issues
raised in relation to ongoing work with these next-generation systems.
DIALECTIC AS CREATIVE TENSION
Many researchers now embrace the idea that content and learning exist in, and
emerge from, social activity. These analyses have made significant contributions
to our understanding of teaching and learning and have heralded a new era of not
just greater subtlety in the ways we look at the classroom, but also in what it
means to teach and learn. Teaching and learning have come to be understood as
forms of participation in activities and processes much larger than solitary indi-
vidual comprehension or engagement (cf. Gutierrez & Rogoff, 2003; Lave &
Wenger, 1991; Moll, 1990). We propose that the next step in this analysis is to
see not only how mathematical and scientific ideas are organized by social activ-
ity (1 in Figure 2), but also how they can play a structuring role in the social
space of classrooms (2 in Figure 2) and thus become involved in organizing so-
cial activity.3
DIALECTICS AND GENERATIVE DESIGN 187
3Prawat (1996) advanced a similar sense of ideas participating or having what Dewey (1925, 1938)
called “sentiency.” Similarly, classroom interactions can be considered an evolving, memetic ecology
of ideas and expressive artifacts in a way that builds on Dawkin’s (1976) idea of a meme.
FIGURE 2 The development of ideas via the dialectic of mathematics and science as socially
structured (1) and socially structuring (2).
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At the center of this dialectic analysis is the emergence and interaction of ideas
as they evolve in relation to generative, network-supported, classroom activity (see
Figure 2). Our use of the term dialectic follows its use in ancient Greek thought.
Unlike the more recent Hegelian use that anticipates a synthesis of opposites, we
look to revitalize an earlier sense of dialectic that predates Plato and that views dia-
logue, discourse, and disputation themselves as deepening our understandings of
the world.4Dialectic is a kind of juxtaposition of ideas, often literally a debate,
rather than a resolution or synthesis. Understanding emerges via the activity of
holding in creative tension even ideas that seem paradoxical. We believe that care-
ful examination of the proposed dialectic (Figure 2) can have implications for
evolving notions of generative design supported by next-generation network tech-
nology. As we discuss more fully later, our use of the term generative refers to or-
chestrating classroom activity in ways that occasion productive and expressive en-
gagement by participants, characterized by increased personal and collective
agency.
We see in this dialectic a potentially useful tension and interdependency be-
tween the structuring role of mathematics and science and the structuring role of
social activity. Designing with this dialectic in mind moves the focus away from
having to decide between the two and toward productively leveraging the dialect’s
generative potential. In addition, attending to this dialectic presents us with a novel
way to address the question raised by Shulman’s (1987) analysis of pedagogical
content knowledge: What is the relation between content and the activities of
teaching? Content, in the proposed framework, is seen as structuring dynamically
the network-supported learning activity itself. In a significant sense, then, the con-
tent becomes the pedagogy. Reciprocally, pedagogy, understood as emergent ways
of coming to participate in communities of mathematical and scientific practice,
develops content. By seeing content and pedagogoy as coconstructive, we can be-
gin to explore how learning activity becomes dynamic, enacted, or lived content.
We can also use the dialectic to address the question of the relations between the
classroom community’s developing insights and our situated participation in
larger communities of mathematical and scientific practice (Lave, Smith, & Butler,
1988; Newman, Secada, & Wehlage, 1995; Resnick & Rusk, 1996).
The dialectic is offered as a way of engaging these issues as well as more practi-
cal design challenges related to working with next-generation classroom networks.
The overall sense is that, too often, social constructivists ignore the role of mathe-
matical and scientific structure, and, too often, content specialists ignore the struc-
turing role of social activity. Rather than continuing to talk past each other, our
188 STROUP, ARES, HURFORD
4Parmenides’ (510 BC) foundational poem is seen as a starting point for the ongoing development
of the idea of dialectic: “There is need for you to learn all things … both the unshaken heart of persua-
sive Truth and the opinions of mortals, in which there is no true reliance … that the things that appear
must genuinely be, being always, indeed, all things” (Diels & Kranz, 1951/1962, p. 246).
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hope is that this dialectic analysis can help engage us all with the ways in which
content and social activity mutually constitute meaningful teaching and learning
(see also Greeno, 1997).
NEXT-GENERATION NETWORKS
AS NEW MEDIATING TOOLS
Due to the group-focused interactivity and data-collection capabilities of
next-generation networking, we now have a new mediating tool to explore the dia-
lectic relation between content and activity in designing for the dynamics of class-
room teaching and learning.
As Vygotsky (1978) noted, “The use of artificial means, the transition to medi-
ated activity, fundamentally changes all psychological operations, just as the use of
tools limitlessly broadens the range of activities within which the psychological
functions may operate” (p. 55). Mediated activity changes psychological opera-
tions in ways that are closely associated with the sense in which tool use broadens
the forms of activity within which psychological functions operate. This article,
then, is not so much about technology per se, but rather about a specific instance of
the interaction and co-evolution of design, technological affordance, and theory
(Dewey, 1938; Hickman, 1990). New forms of activity—generative design—car-
ried out in relation to new tools of interaction and participation—the network—of-
fer the potential of changing and broadening learning in classrooms. Because a sig-
nificant number of these networks are about to become widely available and are
poised to become a major presence in classroom learning, it is important to under-
stand the role these next-generation designs and forms of mediating activity can
have in advancing the evolution of what it means to teach and learn.
Prior forms of classroom networking have served primarily in two ways: (a) as a
portal to sources of information or interactivity centered outside the classroom
(e.g., visiting the Cable News Network web site or filling out a web-based ques-
tionnaire) or (b) to implement computer-assisted instruction (CAI) or tutoring en-
vironments. Two features characterize these uses of networking in these prior sys-
tems. First, the experience is fundamentally individual: Most of the activity could
be carried out at home or in a local library as little or no use is made of the social
space of the classroom. Second, the knowledge or the trajectory of learning is
owned by a distant expert for web content or the computer programmers or design-
ers for CAI or tutoring environments. Especially for low socioeconomic status stu-
dents who have had their experience of school-based technology skewed heavily
toward the use of drill-and-practice CAI environments (e.g., see Wenglinsky,
1998), a fundamental tenet of mathematics and science reform—that the class-
room should be a community of inquiry characterized by joint ownership and con-
struction of understanding—is undermined.
DIALECTICS AND GENERATIVE DESIGN 189
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Of course, the network alone does not determine the nature of the learning ex-
perience (Papert, 1990). However, a technological resonance can exist between
technological affordances and the activities of teaching and learning. The sense is
that some technologies more readily support generative group explorations (i.e.,
are the “right tool” for the job). New mediating activities supported by new tools
support new possibilities for learning.
GENERATIVE DESIGN FEATURES:
PLAY AND DYNAMIC STRUCTURE
A number of approaches to generative teaching and learning have been devel-
oped (e.g., Freire, Freire, & Macedo, 1998; Learning Technology Center, 1992;
Senge, 1994; Wittrock, 1991),5and these do focus on forms of activity that sup-
port the continuous improvement of individual and group functioning. With this
article we are working to reframe some of these ideas in terms of the proposed
dialectic. In this section we specifically attend to the ways in which mathemati-
cal and scientific concepts can be used to structure the social activity of class-
rooms (2 in Figure 2). We also are explicit about designing technologies and ac-
tivities for highly interactive group spaces in ways that move beyond simply
scaling individual models of learning to the group. We explore Part 2 of the dia-
lectic shown in Figure 2 through the use of the scientifically and mathematically
structured ideas of space-creating play and dynamic structure. This exploration
extends previous understandings of generative teaching and learning in ways that
are well supported by next-generation network capabilities. Some aspects of this
approach to generative design share features with aspects of thought-revealing
activity as discussed by some researchers within a “models and modeling”
framework (e.g., Kelly, 2003; Lesh, Carmona, & Post, 2002; Lesh, Hoover,
Hole, Kelly, & Post, 2000). Content-related expressivity is highlighted in all
these approaches to generativity.
190 STROUP, ARES, HURFORD
5Generative teaching, as discussed by Wittrock (1991), involves students’ ability to create arti-
facts that embody their constructed understandings. In a closely related way, researchers from the
Learning Technology Center (1992) at Vanderbilt emphasized aspects of creating “shared environ-
ments that permit sustained exploration by students and teachers” (p. 78) in a manner that mirrors
the kinds of problems, opportunities, and tools engaged by experts. Senge (1994) claimed “for a
learning organization, ‘adaptive learning’ must be joined by ‘generative learning,’ learning that en-
hances our capacity to create” (p. 14). Perhaps most significant in its connections to the sociocultural
and critical analyses taken up more directly later in this article is Paulo Freire et al.’s (1998) use of
“generative words” in ways that explore the “creative play of combinations” (p. 87) to create new
words as part of developing literacy with adult learners in Brazil (see also Callahan, 1999; Stroup,
1997a).
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Space-Creating Play
If students are asked to create and display functions that are the “same as 4x,” they
are generating a space or collection of related mathematical objects. In the net-
worked version of this activity, the space of related objects is the result of students’
“play-full” explorations of possibilities, using their individual computing devices,
and the use of the network to share their examples and to have students learn from
the examples of others.
In considering how play actually works for children, it is important to empha-
size that play is not simply an “anything goes” state of affairs. Instead, play is an
organized form of activity:
The premise that Durkheim, Vygotsky, and Piaget share … is that thinking and cog-
nitive development involves participating in forms of social activity constituted by
systems of shared rules that have to be grasped and voluntarily accepted.… The sys-
tem of rules serves, in fact, to constitute the play situation itself. In turn, these rules
derive their force from the child’s enjoyment of, and commitment to, the shared ac-
tivity of the play-world. (Nicolopoulou, 1993, p. 14)
Thinking and cognitive development related to play is structured by a system of
shared rules that need to be understood and accepted. The power of this form of ac-
tivity comes precisely from the children’s dedicated engagement related to being
part of the “play-world.” Generative teaching and learning as discussed in the prior
literature have had some aspects of play associated with them, but they
underutilized the emergent space of behaviors and artifacts for classroom-based
(group) learning and teaching. The network-based activities discussed earlier in
this article are more group focused. They are also more overtly engaged with the
sense in which learners are seen to be “playing” in mathematically and scientifi-
cally structured spaces. This play then creates a space of objects or emergent be-
haviors that embody students’ understandings of the mathematical or scientific
content and that can serve as the objects of attention and analysis for the class.
For example, a generative activity would ask students to find functions that are
the same as 4x, whereas an overscripted activity would ask students to create pairs
of single-digit, positive integer terms involving xthat sum to 4x. The sense is that
the structure of the activity should emerge from what students create in a way that
they find expressive, that they can make use of, and that is not overly constrained
from outside. In the former case, students may feel as if the conversation can con-
tinue or that there is more they can do and explore, whereas in the latter case, the
sense is of having found the very limited set of correct examples. The generative
use of mathematical and scientific ideas to structure classroom activity does not
collapse the space of possibility but instead opens up ways of instantiating and
constructing understandings.
DIALECTICS AND GENERATIVE DESIGN 191
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In generative participatory play, willing and doing are unified. Technology and
teachers assume a convening role, not a controlling role, in which students are cen-
tral to what expressive artifacts (e.g., a graph or a kind of motion for a simulated
agent) and insights are produced. Students move to assume a convening role as
well, not just deciding to play along, but also to play a part in what the activity it is
about, what matters, and where the classroom activity should go next. The separa-
tion of willing and doing in most classroom activity contributes to student alien-
ation in the classroom. Our belief is that the integration of willing and doing will
meaningfully advance students’ sense of ownership and that this ownership will
scaffold engagement in authentic activity outside the classroom and outside
school.
Relative to space creation, what distinguishes playing along from playing a
meaningful part includes, in some sense, the size of the space the students can ex-
plore. Playing along invokes a sense of constraint and limited possibility. Activ-
ities such as “solve this linear system by adding the same thing to both sides” and
“simplify the expression 2x+2xby combining like terms”are highly constrained.
Playing a part, on the other hand, involves one’s own explorations being juxta-
posed to others’, to the group’s evolving notion of the domain, and to the more for-
malized insights of the dynamic communities of science and mathematics. The dif-
ference can be as simple as asking students to submit expressions, using a network,
that are the “same as 4x” or as complex as finding ways to improve traffic flow in a
participatory simulation.
Traditional models of tutoring, including their embodiment in some forms of
networked design, typically center on playing along. Even in more recent “cogni-
tive” tutoring environments that are a clear improvement over traditional CAI envi-
ronments, a relatively small space of correct expressions is evaluated as acceptable
(Carnegie Learning, 2003; Heffernan, 2001, 2003). One of these cognitive tutoring
environments begins by asking the student to find a linear expression describing
the motion of a boat, rowed at a constant speed, crossing a river with a current hav-
ing a uniformly constant rate of flow (Heffernan, 2001, 2003). Not only are stu-
dents constrained to a small set of possible responses, they are not allowed to raise
questions about the nature of the problem or its significance in rowing a real boat
across a real river such as the ones they might find in their world outside of school.
In essence, the student is actively coached by these environments to converge to
one of the possibilities in a small solution space.
Similarly, sending out to students a “family” of nearly identical tasks, ones
varying only in terms of the highly constrained randomization of one or more pa-
rameters, is not generative even if, in principle, the space of possible tasks and so-
lutions is large. For example, some recent activities mediated by new connected
technologies allow one student to evaluate a peer’s attempt to describe the motion
of a simulated elevator using either piece-wise constant velocity graphs or
piece-wise linear position graphs (Roschelle & Vahey, 2003). Without going into
192 STROUP, ARES, HURFORD
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detail, powerful ideas in calculus are intended to be, and can be, part of the match-
ing activity. Each pair of students gets a unique matching task in which each task
has a single correct solution. Although this work has focused on important social
dimensions of peer-to-peer interactions (cf. Roschelle, 1990), variation is accom-
plished by having the technology select random pairs of graph segments for each
duo of students to work on. According to the designers, a large space of possibili-
ties can be explored because each student pair is working with a unique but related
task.
Peer-to-peer dialogue notwithstanding, from our perspective,this kind of activity
is not generative because the technology itself is responsible for generating the
space, not the students’ own thinking.6Moreover, once the task is assigned, there is
only one right answer per group. And finally, the sense in which these pairs may be
seen to be exploring a family of related functions is all but inaccessible to anyonebut
the programmer and, possibly, the teacher. With this design, larger insights about
families of functions are unlikely to be visible to, or generated by, the students.
Dynamic Structure
When students in a networked space create and then display functions that are “the
same as 4x,”the emergent structure of the activity is itself lived or brought into being
by what the learners do and come to understand. Unfortunately, the idea of structure
has come to be understood in a relatively static way in mathematics research. Forex-
ample, the process–object analyses of mathematics learning have tended to collapse
structure to a relatively static kind of object (Dubinsky, 1991; Sfard, 1991). Sfard
stated “[T]he structural conception is static, instantaneous, and integrative,the oper-
ational is dynamic, sequential, and detailed”(cited in Kieran, 1993, p. 193). From
within this process–object stance, the “static” structural aspects of mathematical in-
sight are to contrast with the “dynamic” operational aspects.
Sociocultural researchers have tended to respond critically to this static or in-
trinsic sense of structure:
When Soviet psychologists speak of the “structure of an activity,” they have in mind
something very different from what has come to be known as “structuralism” in
Western psychology [and mathematics education]. The units are defined in terms of
the function they fulfill rather than of any intrinsic properties they possess. (Wertsch,
1981, p. 19)
DIALECTICS AND GENERATIVE DESIGN 193
6Recent iterations of this activity now give students the ability to choose the initial challenge and
highlight the after-activity discussion wherein the students are asked to describe what clues were most
useful in successfully completing the activity. This emergent space of clues is captured in a shared pub-
lic space by having the teacher write down the list of useful clues as students call them out. In these
ways, the designers have increased the generativity of the activity(Roschelle, personal communication,
November 6, 2003).
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Dynamic structure is intended to point to a functional or operational sense of struc-
ture, not fixed or intrinsic attributes. A space of functions is created by the 4xactiv-
ity, and in a significant sense this space is brought into being by learners’ own un-
derstandings of the mathematical ideas of equivalence. Their mathematical ideas
are what dynamically structure the learning activity. This socially animated under-
standing of structure fits well with the ideas Soviet psychologists—and especially
Vygotsky (1987)—bring to learning.
What may come as a surprise to some, however, is the sense in which this lived
and larger-than-the-individual meaning of dynamic structure may be exactly what
Piaget (1970) was pointing to in suggesting that formal mathematical constructs
(e.g., the algebraic group) could serve as the prototype of what he meant by emer-
gent cognitive structure.7Structures emerge in relation to the activity of individu-
als as well in the development of mathematics and science as domains (Piaget,
1970; Piaget & Garcia, 1989). Individuals and larger communities of mathemati-
cal and scientific inquiry are different levels at which we can attend to creative
agency. Structure, understood in this dynamic way, is a patterning or coordination
in the kinds of operations on elements of the system. It is this larger dynamic sense
of structure that allows Piaget to talk about group-situated learning as “co-opera-
tion” (Montangero & Maurice-Naville, 1997, p. 140). Students are seen to be ex-
changing, adding on to, and transforming understandings in ways that mirror for-
mal operations in mathematics. This dynamic, emergent sense of structure found
in both Vygotsky’s (1978, 1987) and Piaget’s work is consistent with, and ex-
tended by, the sense of generativity that follows from the notion of mathematics
and science structuring social activity. Viewed this way, there are striking parallels
and forms of complementarity between Vygotsky’s analyses of language and
Piaget’s analyses of operational thought that can be brought to the task of thinking
about, and designing for, activity in classrooms supported by next-generation net-
worked functionality. In summary, emphasizing this side of the dialectic, genera-
tive learning and teaching come to be understood as organized by space-creating
play and dynamic structure.
GENERATIVE DESIGN FEATURES:
AGENCY AND PARTICIPATION
The network-based projects described earlier in this article represent important ex-
amples because they highlight the structuring of the classroom social space by
194 STROUP, ARES, HURFORD
7Piaget (1970) takes the idea of the group, as it is found in abstract algebra, as the “prototype of
structures in general”(p. 19) and explicitly links it to his characterization of constructivist learning: “It
is because the group concept combines transformation and conservation that it has become the basic
constructivist tool” (p. 21).
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mathematics or science proper (2 in Figure 2). However, the roles and identities of
students in relation to content (Lave & Wenger, 1991; Moll, 1990; Rogoff, 1995)
and the function of tool-mediated activity (Vygotsky, 1987; Wertsch, 1991) are
underspecified in these projects. In this section, we explore the side of the dialectic
that emphasizes the structuring function of social activities in characterizing
generativity (1 in Figure 2). We focus on agency and participation as two socially
significant features of classroom activity closely related to the ideas of space-cre-
ating play and dynamic structure. The notion of learning here extends what was re-
ferred to earlier as “participating in forms of social activity constituted by systems
of shared rules that have to be grasped and voluntarily accepted” (Nicolopoulou,
1993, p. 14) to make explicit that participation involves not only understanding
rule systems, but also shaping and creating shared rules through social interactions
in classrooms. Learning in this sense is also transforming individual and collective
participation in important practices of communities (e.g., mathematics and sci-
ence; see Lave & Wenger, 1991; Rogoff, 1995). Thus, we explore agency and par-
ticipation in terms of how mathematics or science is constructed, and by whom,
along with the mutually constitutive relations among content, activity, and tools in
next-generation networks.
Agency
Students’ opportunities to assume agency at both the individual and collective lev-
els in network-mediated social spaces are markedly different from traditional
classrooms. To be a visible and necessary participant in real time, public construc-
tion of knowledge is a significant form of agency that is unique to next-generation
network design. To date, three design features seem to be particularly important
relative to agency: (a) anonymity, (b) authorability, and (c) opportunity to expand
the content and representations that are the focus of activity.
Anonymity.
Many networks allow students to submit contributions to the
emergent system to be considered by the class without their identities being associ-
ated publicly with that information. Davis (2002) showed that, freed from who sent
in a response, students are better able to explore what the mathematical activity
represents, whether it is sending in functions that are the same as 4xor controlling
a traffic light in a simulation. Thus, anonymity facilitated the group’s ability to ex-
plore mathematical and scientific concepts in a nonthreatening way. From the
teacher’s perspective, “It just promotes a lot of discussion and everybody’s free to
discuss it because kids can be criticizing an equation that they themselves wrote
and nobody would know” (Davis, 2002, p. 208). Also, students indicated that the
representation of self in relation to the group space gave them a sense of how they
were doing relative to the class as a whole. At the individual level, then, students
identify with their responses, icons, and data that show up in the group display, fo-
DIALECTICS AND GENERATIVE DESIGN 195
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cusing on themselves as part of the collective construction of mathematical and
scientific objects. At the group level, efforts center on the co-construction and
analysis of content. Thus, opportunities to reflect anonymously on their respective
insights contribute to students’ and groups’ increased sense of agency in engaging
the learning activities.
Authorability.
Generative design requires that what the activity is about
emerges from participation and is brought into being by the participants. More
than visibility and engagement, agency also includes a sense of being able to con-
tribute to and change the flow of the activity. At individual, small-group, or
whole-class levels, network-mediated activity invites students to change the nature
of their participation and influence the evolution of what the activity is about. For
example, a group of students responsible for one vertical column of intersections
in the traffic gridlock activity can decide to implement a column-specific strategy
to cascade the lights. This has implications both for what they do individually and
for their group’s relations to the class’s activity. Depending on what results, they
can then shape the strategy and the discourse of the larger group. Ideally, this sense
of agency would extend to being able to author or alter the network-supported
communication, forms of interactivity, and analyses. Students could, for example,
notice that the gridlock simulation has cars leaving one side of the grid and reap-
pearing on the opposite side (they “wrap”). Out of a desire to explore situations
like this simulations, students might decide to try to find examples of systems that
come close to working in this wrap-around way (e.g., some aspects of an open,
rectangular hallway system in their high school). Alternately, they could reach into
the relatively accessible computer code and try to make the cars’ behavior more
like what they know of the world around them (cf. Wilensky, 1999). What the ac-
tivity is about and how the students can investigate it are changed in fundamental
ways. The conversation and investigation can change directions as learning trajec-
tories are co-constructed by the participants.
Constructing content.
In addition to being invited to contribute to network
activities because anonymity reduces the risk of doing so, and to authoring both
examples and computer code, students have the opportunity to expand the content
and representations involved. For example, in one participatory simulation used by
a teacher to explore positive and negative integers, students also recognized and
explored concepts of slope and rate, as well as their representation in graphs (Ares
& Stroup, 2004). Here, students used the mathematics of the network-mediated ac-
tivity itself to expand the content and representations involved. This aspect of
agency is linked to Lave and Wenger’s (1991) notion of legitimacy of participation
in which individuals’ substantive contributions involve them in shaping practice,
or systems of shared rules. Students in this case gained legitimacy by exercising
the agency afforded them in the networked activity to construct mathematical prac-
196 STROUP, ARES, HURFORD
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tice, embodied in their expansion of the content of the activity to include concepts
of rate, slope, and representation. Thus, the space of mathematical objects was en-
larged through their play-full engagement in the generative activity.
Participation
If one of the aims of network-related projects is to open mathematics and science
to more students (e.g., Kaput, 1994), then questions about the nature of the activity
from which such learning emerges are essential. In generative classroom learning,
students and groups of students not only learn in ways that foster powerful, dy-
namic understandings of mathematics and science, they also learn in ways that
support all students’ developing knowledge and skills that foster their successful
participation in mathematical and scientific activity in the larger world.
Use of cultural and social practices.
A diversity of avenues of participa-
tion is available in network-mediated activity, including text, physical and elec-
tronic gestures, as well as verbal contributions to classroom dialogue (e.g., conjec-
ture, prediction, observation, and explanation). Moreover, the collaborative
character of participation in those modes of contribution invites multiple ways of
belonging, as students have access to a variety of representations of phenomena
(texts, graphs, visual displays of emergent systems, language) and engage in in-
quiry-oriented discussion and analyses (Ares et al., 2004). Together, the varied
modes of participation and joint construction of knowledge mean there is unique
potential in networked classroom technologies to draw on students’ cultural and
social practices to support learning in mathematics and science. Lee and Fradd
(1995) noted that communication patterns vary across cultural groups and that
“students from diverse language backgrounds often have different interpretations
of verbal communication and paralinguistic expression … alternative communica-
tion patterns can provide … students with powerful ways of demonstrating their
knowledge and understanding” (p. 17). For instance, during the participatory sim-
ulation mentioned earlier involving integers, rate, and slope, two Hispanics collab-
orated in Spanish throughout the simulation, sitting next to two European Ameri-
can boys who did their own work and then compared their results in English. Both
pairs’ interactions were important and appropriate, expanding the ways of partici-
pating seen in more conventional teaching. In addition to Spanish and English
serving as cultural resources, choosing to collaborate versus working independ-
ently may also have had gendered or cultural roots. The dynamic structuring of
their activity occurred not only through the mathematics involved, but also through
the students’ use of individual and collective social, cultural, and academic re-
sources. Thus, being able to draw on varied ways of participating made good use of
important resources the students brought to the task.
DIALECTICS AND GENERATIVE DESIGN 197
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Expanding classroom social space.
Figure 3 shows that two dimensions
of the social space of classrooms—content and representation (y-axis), and ways
of participating, including the use of social and cultural practices (x-axis)—are ex-
panded in generative, network-mediated learning. Students’ participation in the
construction of mathematical and scientific objects and content, as well as the
modes by which they do so, are mediated in unique, inclusive, and generally more
expansive ways by this use of the networked technology. Using the previous exam-
ple, the solid circles in Figure 3 represent working independently, in groups, and
with the whole class in Spanish and English on rate, slope, graphs, signed num-
bers, and functions. Generative design expands the social space of the classroom.
Less generative designs and activities tend to narrow the curriculum and the forms
of participation. This constriction is represented by the white circles that denote
working independently or working with the whole class, in English, on textbook
examples of rate and slope.
Continuing with this example, during whole-class discussion, both pairs of stu-
dents’ hypotheses, predictions, and explanations about jointly constructed mathe-
matical objects were important contributions that shaped the whole class’s content
understanding. Thus, the dual dimensions of (a) content and representation and (b)
the ways of participating were mediated by the technology in ways that provided
an opportunity for students to draw on their social and cultural resources in engag-
ing the full range of concepts and representations embodied in the participatory
simulation (see Figure 3).
198 STROUP, ARES, HURFORD
FIGURE 3 Expanded social space formed by the interaction of content and cultural dimen-
sions.
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Thus far, we have organized this article in terms of the dialectic that we use to
characterize the relation between content and social activity. The central attribute
of our approach to design is to encourage expressive diversity that promotes the
evolution of ideas through the interactions of people and tools. This plurality of
ideas can be generated, propagated, shared, contested, and advanced when there
are multiple ways for those ideas to emerge and develop. The dialectic allows us to
heighten attention to, and make the best use of, this plurality.
THE DIALECTIC ENABLES CONSTRUCTIVE CRITIQUE
Up until this point we have used the dialectic as a way of drawing out implications
for classroom activity as supported by next-generation network technologies. We
now explore its utility as a framework for supporting a critique of the respective
limitations of both sides of the dialective in terms of content and pedagogy. What is
at stake here is the ways in which content and pedagogy are also dependent on each
other. This dependency enables critique; critique itself helps to further the genera-
tive potential of the designs that attends to the dialectic. Relative to network-
supported design in particular, the sense of dialect introduced in this article sug-
gests that creative tension—including constructive critique—is what can serve to
open up, and to help to more fully realize, the generative potential of the dialectic
between the structuring role of math and science and the structuring role of social
activity.
A Social Critique of the Mathematical and the Scientific
As significant an improvement as all these network-supported forms of interaction
are, in which historical notions of teachers’ and students’relation to content are al-
tered, historical notions of content itself are still being maintained, tacitly, in some
of the network-mediated efforts reported herein. One might observe that an un-
der-examined assumption of some of this work is that mathematics and science are
homogeneous, monolithic, and unambiguous. Viewed in this way, even an updated
notion of generativity can serve to lead only to a predetermined outcome. The very
idea that mathematics and science are socially constructed—especially as situated
in the classroom’s concrete, localized, and diverse construction processes—
problematizes this homogeneity and lack of ambiguity. From the dialectic perspec-
tive, generative activity can produce insights and growth possibilities for both ac-
tors and notions of content. Content is not static. Generative design takes seriously
the sense in which mathematics and science are themselves evolving and structur-
ing forms of activity and insight, not fixed entities. This evolving and structuring
sense of content is a conspicuous feature of network-supported activity. Ambiguity
DIALECTICS AND GENERATIVE DESIGN 199
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and heterogeneity are inherent in the developing use of mathematical and scientific
language and the evolution in what it means to know.
In a related way, as we move to more culturally situated notions of participation,
a somewhat uncritical assumption in some designs for network activity seems to be
that mathematics and science are universal languages. Vygotsky’s (1987) idea of
scientific knowledge can be seen to share this assumption even as it also attends to
how this language comes to have meaning from social interaction and activity. In
contrast, treating language as a dynamic mediating tool through which group-level
and individual knowledge structures emerge poses interesting challenges to uni-
versality. The network-supported evolution of ideas and discourse in the classroom
serves to problematize any notion of a universal, fixed language. The mediating
tools of mathematical and scientific language change in both form and significance
through activity. Thus, generative designs need to engage this evolution to tap into
students’ developing understandings. Clinging to notions of universality under-
mines this engagement.
Additionally, cultural ways of knowing, as embodied in the variety of students’
languages and ways of participating, expands the possibilities for construction of
content and participation, as well as the exercise of agency. Generativity, espe-
cially as it relates to space-creating play, makes good use of important contribu-
tions this linguistic and cultural diversity can bring to network-supported activity.
As long as we hold on to notions of a “universal language” as a gold standard, there
is little or no room for linguistic diversity to matter in and of itself, or for the gen-
erative potential of varied expressive forms of language and participation to be
realized.
In moving from sociocultural to more critical theoretical perspectives, ques-
tions arise about whose community, culture, and history are the foci of the de-
sign activities. Students’ lives are situated in social, cultural, historical, and polit-
ical arenas. How are these connections to students’ lives included in the
network-supported activity of classrooms? For example, a tacit assumption of
the traffic gridlock activity discussed earlier is that students’ experiences in the
world are important to the development of their traffic strategies. But there are
many features of the simulation that are not like the world. Who decides if the
simulation does or does not capture the significant relations students might want
to attend to? Additionally, is traffic flow itself truly a compelling issue for all or
even most students? It is likely that for most students there are a host of more
pressing and socially relevant challenges to address. The move to address the
complexity in the world around them is a good one. But for mathematics and
science to be seen as socially significant and powerful, the question of what
complex phenomena are worth investigating must be negotiated with students.
For personal and collective agency to be advanced by a particular activity, design
must more overtly attend to the following kinds of questions: Does this activity
200 STROUP, ARES, HURFORD
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really matter, and for whom? How do the questions raised connect to students’
history, culture, and community? In what ways do the activities facilitate stu-
dents’ development of meaningful ideas and insights in a way that can allow
them to take action on their world? We believe these questions must be more
explicitly engaged in our design efforts.
A Domain-Centered Critique of the Social Analyses
To be fair, although critiques of mathematics education as acultural, ahistorical, or
apolitical are certainly reasonable, critiques of sociocultural theoretical frame-
works are also important. Socioculturalists have helped researchers and educators
focus on the reality of the social construction of mathematics and science; how-
ever, they have done significantly less well in attending to the dialectic proposed
here because of inattention to the structuring roles of mathematical and scientific
reasoning for learning (Lee, 2001). There are two important aspects of this inatten-
tion: (a) There is a blindness to content in the way that domain specialists are often
blind to culture, language, and social activity, and (b) there is a blindness to the
ways in which social critique and analyses often are themselves structured by
mathematical and scientific ideas.
There has been little attention to date to the fundamental connections between
activity and domain-specific content learning (but see Gauvain, 1998). Social anal-
yses often make claims across disciplines in ways that ignore the unique features
of domains (e.g., processes of learning in science are different from those in learn-
ing English). There is a sense of making universal claims, albeit about social di-
mensions of learning, that are not sufficiently situated in relation to what activity in
mathematics and science classrooms claims to be about. Not only is this true at the
level of the domain as a form of situated activity, but also at the level of the partici-
pants’ account of the centrality of content in their own representations of what
classroom activity is about. An English teacher would claim that his work is about
teaching English, and a science teacher would make similar claims about what her
work is about. The ways in which they orchestrate classroom activity involve both
domain-specific and highly contextualized forms of social engagement. In assum-
ing a near-universal voice, the claims of social critique may miss important aspects
of the nature of the activity itself.
This article also encourages educational researchers, theorists, and practitio-
ners to attend to the ways in which science and mathematical domains structure so-
cial analyses. For example, earlier we discussed using statistics as a way of struc-
turing the social activity of the classroom. But it is also the case that statistics has
been central to the development of social analyses and critiques of school-based
learning (cf. Bowles & Gintis, 1976). Sociology, for example, is heavily dependent
on quantitative analysis. Formal aspects of statistical reasoning structure these
DIALECTICS AND GENERATIVE DESIGN 201
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analyses. Statistics shapes the questions posed, how these questions get answered,
and, in part, what counts as knowledge. Critical approaches do not stand outside
what we have referred to as content, but are themselves deeply situated relative to
the dialectic attended to in this article.
OVERVIEW
Generativity centers on the diversity and interactivity of learners’ ideas. Through-
out this article, the dialect of seeing mathematics and science as both socially
structured (1 in Figure 2) and socially structuring (2 in Figure 2) has supported our
exploration and extension of notions of generativity, especially in relation to net-
worked classroom technology. We do not view this use of dialect as setting up a
Hegelian synthesis of content and pedagogy, but as a kind of holding in creative
tension. Generativity emerges in relation to this tension through making visible,
and fostering attention to, the mutually constituative relations between social ac-
tivity and domain-related learning and insight. We have made a case for viewing
such expressive activity and the ongoing development of ideas, framed in terms of
the dialectic, as productive relative to theory and design for classrooms. In its full-
est realization, next-generation networking can help to support this dynamic
vision.
Future efforts extending the productive use of generativity, situated in relation
to the dialectic, need to do more to examine the role of language in classroom
learning and in the development of domains. Within a generative framework, the
languages of mathematics and science and the diverse languages students speak
create a vibrant space of possibility. We can inquire as to whether the network-sup-
ported activity has a potentially powerful mediating role in making good use of the
academic and social resources available to teachers and students in culturally and
linguistically diverse classrooms. Closely related is the question of whether the
generative network activity design is effective in fostering diverse students’ sense
of participation and agency relative to mathematics and science learning. And fi-
nally, attention to the unique mediating role of new artifacts—for example,
real-time public displays of jointly constructed representations—in shaping class-
room discourse offers important glimpses into the development of mathematical
and scientific knowledge and reasoning.
Overall, we believe that the capabilities of next-generation classroom networks
can assume a significant mediating role in advancing the evolution of what it
means to teach and learn. Generative design, emerging in relation to the dialectic
analyses presented herein, has both practical and theoretical value for researchers
and teachers interested in the relation between social interaction and the construc-
tion of content.
202 STROUP, ARES, HURFORD
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ACKNOWLEDGMENTS
Susan Empson, Richard Lesh, Taylor Martin, and Jeremy Roschelle in particular de-
serve special thanks and appreciation for their careful reviews and suggestions. The
article has been greatly improved by their insights and commitments. A predecessor
of this article appears in the Proceedings of the Twenty-Fourth Annual Meeting of the
North American Chapter of the International Group for the Psychology of Mathe-
matics Education (Stroup et al., 2002) and contributions to that article, including au-
thorship beyond that associated with this article, were made by Sarah M. Davis, Ste-
phen Hegedus, James J. Kaput, Andre Mack, Jeremy Roschelle, and Uriel Wilensky.
Critical to the development of this line of research is ongoing funding from the Na-
tional Science Foundation: Grant 09093 titled CAREER: Learning Entropy and En-
ergy Project and Grant 126227 titled Integrated Simulation and Modeling Environ-
ment. Texas Instruments has also generously supported this work. We gratefully
acknowledge this support. The views expressed herein are those of the authors and
do not necessarily reflect those of the funding institutions.
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