## Modeling Nature

## Rich Lehrer and Leona Schauble

University of Wisconsin-Madison

Rather than submitting a published paper, we decided to share the piece of text that we judged most pertinent to the goals of that meeting, namely, the opening to a proposal submitted two years ago to the National Science Foundation. Accordingly, enclosed is the opening statement of that proposal, which describes our view of a worthwhile focus for science education in the elementary and middle grades and sketches a vision of how this goal might be achieved. The proposal takes seriously the idea of working toward overarching themes or "big ideas" in science, such as those described by 2061, across the years of a child's education. It then goes on to tackle the question, what would it really take to achieve this goal? We have omitted the description of planned research that follows this proposal opening, since it does not really bear on our upcoming discussion.

### Summary of Proposed Activities

The purpose of this research is to study the conceptual development of elementary and middle school students as they move across grades of instruction designed coherently and cumulatively around three of the central themes described in the science standards: growth and diversity, behavior, structure and form. National standards concur that science literacy involves coming to understand such central themes in a deep way. However, as yet there is little research to guide educators in the task of orchestrating instruction over years of schooling toward the long-term development of these core ideas.

Understanding the organizing themes of science does not emerge over a few months, so this work ideally will be conducted within contexts where educators are already committed to this objective. Accordingly, the research site is a local school district (five participating schools) where the Principal Investigators have worked for several years—most recently, to establish reform mathematics practices and concepts (such as geometry, data, measurement, functions, and uncertainty) that can support a modeling approach to science. With this foundation, the PIs now seek to develop a cumulative approach to science, emphasizing model-based reasoning as the cornerstone of scientific practice and the most promising route toward the themes outlined in the standards. Researchers and district teachers will work to put these forms of instruction in place, and over a three-year period, to study transitions in student thinking. Participants include nine teachers and their students: six at elementary school level (three in primary grades and three in upper elementary) and three in the sixth grade (first year of middle school). Administrators agree that students will "graduate" across grades within the participating classrooms during the project, affording the opportunity to conduct longitudinal study to learn how students' understanding of these "big ideas" emerges and takes hold.

The investigation is a form of design experiment. Starting from previous work and theory on the development of model-based reasoning, researchers will work with teachers to design an environment in which students learn about central themes in science via the invention and revision of models, and then systematically study the student thinking that emerges as the design unfolds. An existing, long-term professional development collaboration is the primary means for realizing the design. Research methods include: yearly administration of science achievement benchmark items (with comparison to both local and national samples), an experimental study of the development of students' representational repertoires, analysis of videotapes of daily classroom interactions during targeted instructional units along with follow up interviews, analysis of student reasoning and justification of their models during classroom "critique" sessions, and analysis of student performance on transfer "model-eliciting tasks."

The standards paint a picture of the kind of scientific literacy that can emerge over years of well-planned education supported by thoughtful curricula and excellent pedagogy. To create these conditions for all children, research is needed to illuminate the outcomes that can reasonably be expected (both products and processes of student learning) and to enhance understanding of what it takes to achieve them-not over a few weeks or months, but over years-the time span required for students to grasp profound ideas.

#### Tackling the Challenges of Standards-Based Instruction

National science education standards vary somewhat in their selective emphasis, but concur that science literacy involves grappling with central themes, like equilibrium or change, in ways that foster students' capabilities and propensities to conduct increasingly substantive inquiry (e.g., American Association for the Advancement of Science, 1993; National Research Council, 1996). Yet there is little research, especially at the pre-secondary level, to guide teachers in the complex task of orchestrating instruction on discrete topics and domains of science toward the long-term development of these central themes (Collins, 1998).

To construct such a knowledge base, researchers need to systematically study the conceptual development of students as they move through years of instruction designed to emphasize consistency and coherence with regard to central themes or "big ideas" (Bardenn & Lederman, 1998). Since understanding of these organizing themes does not emerge over a few months or even a year, this work needs to be conducted within test bed contexts in which teachers and administrators are committed to this approach. We are fortunate to be working in such a context. In previous work funded by the National Science Foundation, the James S. McDonnell Foundation, and OERI, we have been collaborating with all four public elementary schools in a local school district to evolve a research-based approach to reforming elementary school mathematics and science instruction. In these schools, mathematics has expanded to include geometry and space, data, measure, and functions, as recommended by the National Council of Teachers of Mathematics (Lehrer, Jacobson et al., 1998; Lehrer and Schauble, 1998). Moreover, during our past four years of work, we have observed the power that these mathematical resources provide for student reasoning about science.

Although our reform encompasses both mathematics and science, continuity across grades in this work has to date mainly been governed by considerations about the development of mathematical ideas. As a result, the mathematical resources are now largely in place in our participating classrooms. Here we propose to shift our focus toward developing a cumulative approach to elementary and middle school science, emphasizing modeling and model-based reasoning as the cornerstone of scientific practice and the most promising route toward the core themes outlined in the standards. We propose both to put these forms of instruction in place, and, over a three-year period, study the resulting transitions in student thinking. Moreover, we plan to extend across the border of elementary school, where our current work resides, into the first level of middle school-in Wisconsin, the sixth grade.

To "cash in" the promise of the standards, we must build a research base to support instruction that is guided by knowledge, not guesses, about how student understanding typically unfolds. We must learn about the conceptual resources and barriers that students bring to this enterprise and identify the curricular and instructional strategies that are most effective in fostering conceptual development. The overarching agenda of this research is to help articulate how the changing content of science, mathematics, and technology described in the national education standards can actually take hold in public school classrooms.

We plan to begin by exploring three unifying themes, representing the life, physical, and behavioral sciences: (a) growth and diversity, (b) structure and form, and (c) human and animal behavior. Our efforts will be aimed at designing classrooms that promote the development of students' understanding of the models and central conceptual structures within each of these strands. Our current work suggests that big ideas in these strands are susceptible to multiple forms of modeling and representation. They afford easy points of access to primary-grade children, yet provide sufficient "lift" so that older students will be challenged. In addition, they meet a variety of pragmatic constraints, including sufficient robustness and ready availability of materials and activities to support their generation and observation. This is important because the effort we are proposing is mainly a research, not a curriculum development initiative. Although we expect to devote modest efforts to materials development, most of the instruction will rely on selection, modification, and coordination of pre-existing curricula and programs.

#### Approaches to Modeling That Are Aligned with Children's Development

Our emphasis on models follows from the widespread observation that, regardless of their domain or specialization, scientists' work involves building and refining models (Giere, 1992; Stewart and Golubitsky, 1992). Core ideas like "diversity" or "structure" derive their power from the models that instantiate them, so to fulfill the promise of the "big ideas" outlined in national standards, students must realize these ideas as models. Although models are central to the everyday work of scientists (and accordingly, prominent in the Standards), they are nearly invisible in school science, particularly in early grades. The first challenge, then, is to identify forms of modeling that are well aligned with children's development, a problem we have been exploring for the past several years. We have learned that it is advisable to begin with models that resemble their target systems (e.g., the phenomena being described or explained) in easily detected ways, partly because resemblance helps children draw analogies between models and target systems (Brown, 1990). For example, when we gave students springs, wood, and assorted stuff from a hardware store and asked them to construct a device that "works the way your elbow does," first-graders' initial models were guided by perceptually salient correspondences (Carey, Smith, Unger, etc.). In particular, many used round foam balls to simulate the "bumps" of the elbow joint (Penner, Giles, Lehrer, and Schauble, 1997). However, this beginning concern with "looks like" provided a substantive grounding for revisions that eventually came to focus on "works like"-that is, relations and functions among components of the target system (e.g., ways of constraining the motion of the elbow). We have been especially interested in the leverage that mathematics provides in children's modeling activity.

#### Mathematics for Sustaining Model-Based Inquiry

When students have space and geometry, measure, and data at their disposal, as well as the more traditional forms of number sense, the transition to mathematical modeling of natural phenomena becomes feasible and powerful, even in the early grades. For example, physical models of elbows can lead, in turn, to graphical and functional descriptions of the relationships between the position of a load and the point of attachment of the tendon. Thus, elbows can be modeled as third class levers, an idea we explored with third-grade students (Penner, Lehrer, and Schauble, 1998).

A second example of the modeling power of mathematics comes from our previous
work with third-graders posing questions about the growth of the canopy of
Wisconsin Fast Plants™ (plants bred for rapid growth cycles). Students
modeled plant growth at different points in time as the volume of a cylinder.
Drawing on their mathematical knowledge, they conjectured that plant volume
would change according to principles of geometric growth (e.g., "growing"
similar cylinders). They first debated whether their observed violations of
the conjectured pattern should be attributed to measurement error. However,
they eventually revised their models to capture the idea that the ratios of
measures (height and circumference) were changing over time, not constant,
as they initially expected (Lehrer, Schauble, Carpenter, and Penner, in press).
In this instance, children's reasoning was guided by mathematical descriptions
of geometry and space that were constructed as we worked with their teacher
on a previous NSF-sponsored project, *Teaching and Learning Geometry for
Understanding.* This example is a good illustration of the second challenge
we undertake in the current work, namely, to investigate the nature and development
of modeling approaches that can be pursued when adequate mathematical resources
are "at hand." As noted, our proposed work will be conducted in a context
where teachers have already formed a powerful collaboration to introduce and
sustain mathematical inquiry.

#### Developing Meta-Representational Skills

The third challenge posed by the national Standards, related in principle to the first, is how to foster development: How can science pedagogy be designed to capitalize on students' growing capacities for thought? How can instruction "push" development in the sense originally proposed by Vygotsky (1978)? With respect to model-based reasoning, it is fruitful to begin with and then promote development of children's meta-representational skills-their ability to generate and selectively compose representations of phenomena (Cobb, Gravemeiher, Yackel, McClain, and Whitenack, 1977; diSessa, Hammer, Sherin, and Kolpakowski, 1991). Following Latour (1990), we refer to these repertoires of representation—including diagrams, maps, drawings, graphs, text, and related examples-as "inscriptions." Inscriptional competence is developed over years of schooling that emphasize generating, using, and progressively revising representations, so that core ideas in science come to be inscribed in multiple ways (Olson, 1994). To illustrate this process, and to demonstrate how inscriptional competence interacts with conceptual development, we call upon a third example from our previously supported NSF research (Lehrer and Schauble, in press). As we developed beginning studies to explore how children might take a modeling approach to a "big idea" like growth, we observed that primary-grade students' initial representations of growth were typically focused on endpoints (e.g., "How tall do plants grow?"). Students' questions about "tallness" led to related considerations about the attributes of the plant that could best represent height and how those attributes should be measured. As one might expect, students' resolutions to these problems varied by grade.

**First-grade inscriptions**. First-graders represented the heights
of plants grown from flowering bulbs, using green paper strips to depict the
plant stems at different points in the growth cycle. Consistent with our claim
that young children seek to preserve resemblance, they first insisted that
the strips be adorned with "flowers." However, as the teacher repeatedly focused
students' attention on successive differences in the lengths of the strips,
students began to make the conceptual transition from thinking of the strips
as "presenting" height to "representing" height. Reasoning about changes in
the differences of the strips, children identified times when their plants
grew "faster" and "slower." As explained, this work needed to be firmly grounded
in the prior discussions about what counted as "tall" and how to measure it
reliably.

**Third-grade inscriptions**. In the third grade, children "mathematized"
change (in this case, of Wisconsin Fast Plants™) in a variety of ways.
They developed "pressed plant" silhouettes that recorded changes in plant
morphology over time, coordinate graphs that related plant height and time,
sequences of rectangles representing the relationship between plant height
and canopy "width," and various three-dimensional forms to capture changes
in plant volume. As the diversity of students' representations increased,
new cycles of inquiry emerged: Is the growth of roots and shoots the same
or different? Comparing the height and depth of shoots and roots, students
noticed that at any point in a plant's life cycle, the differences in measure
were apparent. However, they also noted that graphs displaying the growth
of roots and shoots were characterized by similar "shapes" (S-shaped logistic
curves). Finding similarities in the form, but not the measures of shoots
and roots, students began to wonder about the significance of the observed
similarity. Why might growth of different parts have the same form? When was
growth the fastest, and what might be the functional significance of these
periods of rapid growth? The variety of inscriptional forms they either invented
or used provided many opportunities for these third-graders to develop meta-representational
competence. For example, students noted that coordinate graphs of two different
plants looked similar (e.g., equally "steep"), yet actually represented different
rates of growth, because the children who generated the graphs used different
scales to represent the height of their plants. The discovery that graphs
might look "the same," yet represent different rates of growth tempered the
class's interpretations of coordinate graphs in other contexts throughout
the year.

**Fifth-grade inscriptions**. In the fifth grade, children again
investigated growth, this time of tobacco hornworms (*Manduca*),
but their mathematical resources now included ideas about distribution and
sample. Students explored relationships between growth factors (e.g., different
food sources) and the relative dispersion of characteristics in the population
at different points in the life cycle of the caterpillars. Questions posed
by the fifth-graders focused on the diversity of characteristics within populations
(e.g., length, circumference, weight, days to pupation), rather than simply
shifts in central tendencies of attributes. Thus, as children's representational
repertoires stretched, so, too, did their considerations about what might
be worthy of investigation.

In sum, over the span of the elementary school grades, we observed characteristic shifts from an early emphasis on literal depictional forms (Styrofoam balls in elbow models, paper "stems" in models of change in plant height) toward representations that were progressively more symbolic in character. Diversity in representation and meta-representational competence both accompanied and produced conceptual change. As children developed a variety of representational means for characterizing growth, they came to understand biological change in more dynamic ways.

**The role of computational media**. This work, and the work of
other researchers (diSessa, et al., 1991, Olson, 1994) suggests that there
is a tight coupling between representational forms and conceptual change.
Because representational forms are constrained by available media, we propose
to explore the capabilities of computational media for enabling fundamentally
new forms of production. For example, Kaput's SimCalc tools make concepts
of the calculus (e.g., dynamic area models of integration) accessible to young
students. Similarly, the work of diSessa (e.g., Abelson and diSessa, 1980)
is a classic example of how computational media can be used to represent growth
and form. DiSessa has agreed to work with us to develop a computer-based "modeling
kit" to support students' modeling investigations. Similarly, Kaput will work
with us to customize SimCalc tools for modeling growth and development. The
collaboration of both of these researchers will be supported through the National
Center for Improving Student Learning and Achievement in Mathematics and Science
(DiSessa, Kaput, and we are PIs in the Center ). In addition, Richard Maclin,
Assistant Professor of Computer Science at the University of Minnesota-Duluth,
has agreed to collaborate on the design and implementation of the toolkit.
With these tools, we aim to investigate how expanding students' repertoires
of representational forms to include dynamic, computer-based notations can
influence their reasoning in each of the three strands of investigation (growth
and diversity, structure and function, behavior).

#### Developing Scientific Literacy

Thus, we propose to investigate how inventing and revising models of central themes in science contribute to the development of scientific literacy in elementary and middle school grades. The context in which we propose to work is a community of approximately four dozen teachers in four elementary schools who have been working for the past several years to focus their professional development on the study of student thinking in mathematics and science (Lehrer and Schauble, in press). In this district, teachers collaborate to investigate student thinking, document significant transitions in student thinking, consider new curriculum approaches and tools, develop strategies for communicating with and engaging parents as partners, and develop new practices for sustaining student invention and revision of models. The Appendix contains the table of contents of a teacher-authored volume about children's thinking about data (currently under review for publication) that serves to illustrate one of the forums developed for communication and community-building in this network of teachers. These processes, which are already in place and have proven their effectiveness in earlier work, will serve as the "engine" that drives the current implementation effort.

We will work with a small cadre of six teachers in this community to create the conditions for a cumulative science education. The six elementary teachers, supplemented by three sixth-grade teachers who are new participants in the collaboration, will serve as catalysts for the larger community in which the work will be embedded. We have also invited three teacher coordinators from the larger community to join, so that communication and feedback can be sustained. We expect that in time, this approach to science will spread throughout the district (our reform in mathematics took this trajectory). With our teacher-collaborators, we aim to investigate the outcomes when "big ideas" in science are systematically revisited and deepened, and when students are encouraged to develop a steadily increasing repertoire of powerful models. To accomplish these aims, we propose to work with three teachers at the primary level, three at the upper elementary level, and three at the sixth grade (plus the three additional teacher coordinators). These teachers have been identified in our previous work and are eager to participate. Some of them "loop" with their students, spanning two grades every two years (e.g., grades 2, 3 and grades 4, 5), a practice that reinforces the continuity of instruction across grades. The middle school teachers (grade 6) either teach both mathematics and science or coordinate the teaching of mathematics and science within "houses."

We have a close and long-standing relationship with the District (see letter of support from the Verona Area Schools). By agreement with administrators, over the proposed three years of the project the participating students will "graduate" from one experimental classroom to the next, affording the opportunity to conduct longitudinal analysis of transitions in students' thinking for approximately 60 students, supplemented by cross-sectional comparisons of the entire sample of approximately 250 participating students. As noted, all six of the elementary teachers are "old-timers" in our existing reform teacher community. The three new participating sixth-grade teachers provide a test of the curriculum "lift" of our efforts and also provide further potential to study longitudinal change across the boundaries between elementary and middle school. Our purpose is to generate knowledge and products that will be useful to an audience of teachers and educators that extends well beyond this district-in fact, to all those concerned with the challenges of teaching standards-based science in a historical context where expectations about science for young students have traditionally been much less ambitious. In this proposed work, we focus primarily on the task of learning what it takes to generate such practices, an objective that necessarily must precede concerns with wider implementation.

We next explain our instructional plans and describe their potential for generating variation of student representation and diversity of models. We must be brief, so we will focus on the two strands, Growth and Diversity and Behavior, because we have already conducted several pilot studies to investigate the potential of many of these instructional activities. Then, we sketch some possibilities for Structure and Form. In the final year of the research we will open exploration of this strand to the extent feasible with the resources at hand. Our efforts in all three strands will be assisted by collaboration with the Center for Biology Education (CBE), a University-wide organization that links biological scientists with practicing educators. The CBE has agreed to provide content expertise ranging from animal behavior to the uses of gardens and restored prairies as sites for study of growth and diversity. We also have commitments of expertise from scholars currently associated with the NCISLA (especially diSessa and Kaput, but also Charles Anderson, Richard Lesh, and Paul Cobb).

#### Instructional Strands

#### Growth and Diversity

Although "evolution is widely perceived and appreciated as the organizing principle at all levels of life" (Bull and Wichman, 1998, p. 1959), evolutionary principles and methods are seldom understood in a deep way by students (Bishop and Anderson, 1990; Demastes, Good, and Peebles, 1996; Rudolph and Stewart, in press; Samarapunghavan & Weir, 1997). There are conjectures about the reasons for these difficulties, but we believe that one obstacle has received less attention-prior to high school, students are not typically provided with experiences and concepts that are foundational to evolutionary thinking. Accordingly, we propose to study students' developing understanding of important underpinnings of evolutionary theory. We have selected growth and diversity because these themes afford the opportunity to develop central biological principles important to evolutionary thinking, such as:

- Organisms can be described as collections of attributes and can be distinguished (classified) by variation among these attributes.
- Change in selected attributes of organisms (e.g., plant height) can be modeled mathematically, so that comparative study of patterns of change can be conducted at the organismic level, a level with great initial appeal to students who grow their own plant or care for their own insect.
- The "natural histories" of organisms (e.g., life cycles, fossil records) can be described and compared.
- Growth can be aggregated at several levels (genotypic, phenotypic, population).
- Population growth can also be modeled mathematically. Heritability and selection transform distributions of selected attributes in populations, giving concrete meaning to differences in levels of analysis.

Primary grade studies will focus on organismic change. Beginning with readily
available organisms (e.g., Wisconsin Fast Plants™, bean plants, *Manduca*,
butterflies, frogs, etc.), children will engage in activities typical of elementary
school "natural sciences," drawing the organisms at different points in time,
observing their behavior, and developing related forms of inscription. This
work provides opportunities to view growth developmentally, as life cycles
of organisms, and affords comparisons among the life cycles of different organisms.
However, onto this descriptive emphasis we propose to add a modeling perspective
that encourages children to make the transition from viewing an organism as
an object, toward viewing an organism as a collection of attributes. Once
children have decided upon attributes to measure and reliable ways to measure
them, we intend to focus student inquiry on the nature of change with respect
to selected attributes (e.g., height, length, surface area, volume) and to
develop a variety of means to represent change. By posing problems that involve
comparative conditions (e.g., studying effects of diet, determining growth
factors), we will also investigate children's early understandings of distribution
and how these ideas influence their conceptions of growth and diversity. It
is difficult to imagine how children might think about diversity in the absence
of conceptual tools like distribution, so we will pay careful attention to
their thinking about precursors of distribution, such as spread, clumps, and
holes in data. (Here we have been informed by work developed by Russell and
others in the development of *Investigations in Number, Data, and Space*.)

The focus in the later elementary grades will be on functions as descriptions
of change and on distribution as an emergent property of population (to distinguish
organismic and population levels of description). We will collaborate with
Kaput, diSessa, and Maclin to develop computer-based toolkits that will help
older elementary students to characterize change as mathematical functions,
including ideas of constant ratios (e.g., lines) and changing ratios. Students
will employ these mathematical descriptions to compare the growth of selected
organisms: "Do Fast Plants™ and *Manduca* grow in similar ways?"
Variation will be an explicit object of study. Accordingly, we propose to
focus on variation as a potential effect of experimental treatments (e.g.,
certain food sources decrease the natural variation in the length of *Manduca*)
and to emphasize comparison of distributions as a way to determine the effects
of various factors on growth in the design of experiments. The examination
of growth factors will provide students with opportunities to represent and
examine covariation, and we will investigate useful forms for displaying covariation,
ranging from cross-classified categorical data to scatterplots and lines.
(We will be informed by the ongoing work of Paul Cobb and his colleagues on
middle-school students' understanding of covariation.) To expand student conceptions
of diversity, we propose to investigate classification as a tool for characterizing
similarities and differences among organisms in the world. For example, students
might develop competing systems of classification and then examine the consequences
of these systems for characterizing diversity. We propose to take advantage
of ongoing school-based prairie restoration projects managed by the University's
Arboretum and garden projects conducted by the Center for Biology Education.
In these contexts, students might represent changes in diversity in prairie
populations of plants or insects and propose potential reasons for the transitions.

The major innovation at middle school will be ideas related to heritability and selection, characteristics of the population level of analysis. Beginning with different (known) genotypes, children will conduct selection experiments on successive generations of Fast Plants™. Drawing upon their knowledge of distribution, students will examine selection and inheritance in light of their effects on populations. Using their emerging knowledge of mathematical functions, students will compare population growth of different organisms, such as people, and in more controlled settings, plants, bacteria and daphnia, with an eye toward characterizing similarities and differences in cumulating quantities across successive generations. Population growth will afford opportunities to compare and contrast different models (e.g., linear, exponential, power, various mixtures) and their long-term consequences (e.g. human growth and global population). Much of this work will be conducted in collaboration with the Center for Biology Education and with collaborators Kaput and diSessa.

#### Animal and Human Behavior

Students are surrounded and fascinated by the behavior of organisms. Moreover, understanding behavior is central to both the social and the biological sciences. Understanding behavior entails grasping a set of inter-related concepts, including:

- Descriptions of behavior vary in their level of detail (e.g., micro to macro) and in their scope of application (e.g., behaviors of individuals, groups, populations, and species).
- All organisms have repertoires of behavior that are species-specific. One can often identify reliable patterns in behaviors. Some behaviors are automatic and relatively inflexible; others are under voluntary control and are relatively flexible.
- The form and/or functions of behaviors may change over the development of an organism. Sometimes a behavior maintains its form while its function changes; other times, organisms develop new behaviors to achieve a similar function.

Understanding these ideas requires students to "bracket" and reflect about their own mental life, opening the possibility of exploiting natural connections between these topics and related ideas such as perspective-taking, study skills, and meta-representational competence. The mathematical resources useful in modeling behavior include representations of frequency, covariation, distribution, function, and classification models. Domain-specific models of behavior that we propose to develop with students include rules, programs, ethograms, and information-processing models.

The emphasis in the primary grades will be on helping children grasp the separation
between what we know and how we know it, and on building simple rule descriptions
of behavior. We will adapt ideas from the NSF-sponsored *Sciences Makes
Sense* curriculum to encourage students to analyze relationships between
an organism's access to perception and the information detected (e.g., Massey
& Roth, 1997). Working from our previous NSF research on design, we will
extend these ideas toward designing systems that "sense" and act, first with
the simple "robots" suggested in the *Sciences Makes Sense* curriculum,
and eventually in the second grade, using Lego Logo™ to consider simple
contingencies with systems that involve feedback.

Older elementary students will investigate relationships of covariation, for example, between records of weather data and feeding behavior of chickadees at schoolyard birdfeeders, or interactions among different species of birds. Contingency and covariation can also be extended to human behavior. Our pilot work suggests that human memory and problem solving are fruitful topics for developing mathematical descriptions of behavior. Finally, we have conducted research on students' construction of classification models to describe and predict the development of people's strategies for representing spatial relationships in their drawings (Lehrer and Schauble, 1998). These classification schemes raise mathematical problems analogous to those that scientists address when they develop taxonomic classifications of species. Because Lego Logo™ now includes the possibility of designing robots with on-board computers, we will also examine older children's ideas about behaviors that emerge from the interactions of these robots (Resnick, 1994). We will be working with children who have already had experience in the design of robots and are familiar with notions of feedback and control, as expressed by simple rule-based contingencies.

We are currently investigating the feasibility of adapting software simulations originally developed by Brian Reiser at Northwestern University and Brian Smith at M.I.T. to support middle-schoolers' inquiry into longer, more complex chains of behavioral contingencies. With these tools, students can search libraries of videotapes of animals in their natural settings (e.g., redwing blackbirds) and develop ethograms to describe, predict, and test typical patterns of behavior. For human behavior, more complex contingencies and their functional consequences can be explored, perhaps by conducting inquiry about memory for gist in text comprehension experiments, or by using protocol transcripts as data for attempting to develop classification models that can accurately distinguish the conversation of girls from that of boys.

#### Structure and Form

In this final strand, we suggest some general concepts that we may begin to explore in more detail toward the end of the three-year project. One aspect of structure follows from the geometry of transformation: (algebraic) group structure arises from the repetition and transformation of units of form (e.g., symmetry groups of triangles). In previous work, children designed quilts by transforming units ("core squares") following isometries of the plane (reflections, rotations, and translations). Children's sense of structure (e.g., the core squares and their relations in the quilt) changed as they came to understand relationships among symmetry, color "theorems" (the core squares were divided into regions of different color, so that different transformations resulted in differences in adjacent regions), and transformations of units. Over time, children's aesthetic sense was transformed from one dominated by "cool colors" to one informed by the language of mathematics. Quilt designs judged as "interesting" were characterized by appreciation of how transformations and design of core squares were used to produce multiple symmetries (Lehrer et al., 1998; Jacobson and Lehrer, in press).

Building upon children's experiences with two-dimensional transformation, we propose now to explore their conceptions of three-dimensional structure by considering transformations of three-dimensional units (cubes, stick-models of tetrahedra) that will be employed to generate "sculptures." Some of these three-dimensional designs will be explored in the third grade to provide presentational and representational models of crystalline structure. In the upper grades (4, 5, 6), the approach to structure via transformation could be expanded to include other forms, like the helix. We will work with collaborators to design software tools that make these explorations productive. DiSessa's prior work with spiral forms could be productively exploited, especially in light of recent descriptions of the evolutionary design space of spiral shells and their potential adaptive functions (Dawkins, 1996).

Engineering structures -and comparison to biological structures-constitute the second aspect of structure that we intend to consider. We will begin by exploring children's conceptions of stability as they place loads on simple foam structures such as rectangular prisms (e.g., beams) and cylinders (e.g., columns) (Middleton and Corbett, 1998). In collaboration with Robert Corbett, a local architect who specializes in restoration, we propose to create design challenges that encourage students to explore patterns of covariation between properties of these forms and load-bearing capacity (e.g., For a column, what are the roles played by diameter and height in compression?). Follow-up design challenges will focus on combinations of tensions and compressions to meet various criteria, such as different ways to create load-bearing bridges or to create simple structures, like tents. Having investigated engineering models, we will turn toward analogs in animal, plant, and human systems. For example, the vertebrae of some mammals constitute an engineering structure-a protruding truss (Vogel, 1998).

#### Relating the Strands

It is important that children come to view these themes as connected. Indeed,
viewing science as modeling offers the potential of seeing how models developed
in one context can be used or modified for others. One aim, of course, is
to help students develop and use a library of models within each strand. Hence,
we would expect that students using line segments to indicate ratios of growth
of Fast Plants™ would also think to use the same models (or close approximations)
for describing the growth of *Manduca*. However, would students who had
developed ideas about distribution and inference to compare the growth of
Fast Plants™ under varying conditions also deploy distributional models
to contrast the effects of different memory strategies on recall? Accordingly,
we propose to investigate the growth and use of children's models within each
strand, and also to note when models developed in one strand are employed
in another.

**Note from Rich and Leona**: At this point, the document describes
a research plan oriented toward achieving these objectives. We have not included
the plan because it does not seem useful for the current purposes.

#### References

Abelson, H., & diSessa, A. (1980). *Turtle geometry*. Cambridge, MA:
MIT Press.

American Association for the Advancement of Science (1993). *Benchmarks for
science literacy*. New York: Oxford University Press.

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