1 Introduction

Because we work with elementary school-aged children and their teachers, we are particularly concerned with how to support the development of modeling (Lehrer & Schauble, 2003). That is, we seek to understand what it takes to initiate youngsters into the practices that comprise what Hestenes (1992) refers to as “the modeling game,” so that over time, children come to appreciate what models are intended for, how models are constructed, that they entail decisions about what and how to represent, and that multiple models of the same natural system are possible and even desirable (Chap. 1). It is especially important for students to grasp that any modeling choice could potentially have been made in a different way and that decisions about models, therefore, involve tradeoffs in utility, efficiency, precision, and message in relation to a question. The framework for modeling competence (FMC; Chap. 1) frames these ideas as meta-modeling knowledge, but as we will describe, young children seem to display tacit understanding of these issues before they are prepared to defend them as explicit criteria.

To provide contact with productive ideas like these, in the design research that we pursue with students and teachers we aim to support youngsters as they begin to participate in modeling practices (Lehrer & Schauble, 2006). This entails a deliberate departure from the more common emphasis on learning canonical scientific models and applying them to solve problems. We often encourage students to struggle with inventing models before we introduce ready-made or conventional models, because we find that problematizing the modeling process enhances students’ opportunities to understand the nature and status of models, which are not transparent to children (Grosslight, Unger, Jay, & Smith, 1991), and to learn about characteristics and functions of the natural systems represented by models (Lehrer, Schauble, Carpenter, & Penner, 2000).

2 Modeling as a Scientific Practice

Like all scientific practices, modeling makes sense only when one understands the goal structure within which it is embedded (Sandoval, 2014). Scientific practices rely for their vitality on their epistemic function within knowledge-making communities (Rouse, 2007). Describing their superficial structures to students is not usually an effective way of teaching them, because practices are both more variable and more context sensitive than they are often portrayed in school science (Chinn & Malhotra, 2002; Gooding, 1990). Yet it is common for educators to focus primarily on teaching those structures in the form of strategies, skills, inquiry cycles, and conventional models, but often with insufficient concern for first generating and sustaining the epistemic context within which scientific practices meaningfully function. As a result, students may learn to reproduce the structures they are taught, but fail to understand the conditions under which those strategies or procedures are useful, or how to adapt them when it is appropriate to do so. Students who are taught general rules of reasoning tend to interpret them as recipes, focus unduly on duplicating the rule or strategy transmitted by the teacher, and hence, develop a distorted picture of scientific thinking (Berland & Reiser, 2011; Manz, 2015). For example, they may worry about producing the teacher-required three pieces of evidence to support every claim but think little about how an observation or event legitimately assumes the status of evidence. They may become preoccupied with designing controlled experiments but fail to consider the more problematic elements of experiment, such as whether their proposed variables and measures are trustworthy stand-ins for their still-emerging constructs of interest. There is nothing inherently wrong with teaching skills of scientific reasoning, but we should not be surprised that if skills are taught as domain-general solutions to problems that youngsters have not yet sufficiently conceived, students respond by “doing school” rather than “doing science” (Berland & Reiser, 2011). To foster the development of scientific practices and to promote the emergence of what Ford (2015) calls a “grasp of practice,” a productive first goal is to engender classroom communities whose members share the task of working collaboratively to develop, revise, and critique knowledge about the natural world (Lucas, Broderick, Lehrer, & Bohanan, 2005). Scientific reasoning, at least as experienced by young children, should be tied tightly to local forms of meaning making. In schools, local means at the classroom level; accordingly, this goal is accomplished by organizing the activity of classroom communities around extended knowledge-building experience within specific content domains.

3 Bringing Modeling to the Early Grades

We conduct our research as extended design studies in which our aim, working with collaborating teachers, is to create classrooms that reflect these priorities and then longitudinally study the thinking that develops as students are promoted across elementary grades (Cobb, Confrey, diSessa, Lehrer, & Schauble, 2003; Sandoval, 2004). As researchers learn about the resources and difficulties that students bring to this enterprise, we work with participating teachers to identify and test curricular and pedagogical means of adjusting instruction in response.

When teachers and students pursue this work, they inevitably confront a series of decisions that are typically “black boxed” in school science. Scientists must figure out ways to navigate the many sites of contingency that are an inherent part of scientific practice (Manz, 2015; Metz, 2004). Among others, these include how we determine the scientific relevance of questions being proposed for study (Lehrer, Schauble, & Lucas, 2008) and how to invent conditions for seeing, which include measures, but also material configurations, observational protocols, instruments, experimental or naturalistic comparisons and designs, and transformations and displays of data (Lehrer & Schauble, 2015). This is not to argue that students should reinvent all these things on every occasion, but taking a long-term view suggests that beginners should come to appreciate that people create these epistemic tools, which, for that reason, are subject to critique and challenge by others who may advocate for alternatives. Resolving these sites of indeterminacy raises both conceptual and material challenges – how to symbolize and talk about natural systems, but also, how to practically set up and maintain conditions in the material world that allow nature to be effectively studied (e.g., Latour, 1999; Pickering, 1995). Science rests on the understanding that nature is not always self-revealing (Shapin & Schaffer, 1985). Developing this appreciation requires experiencing the generation of questions, investigations, and data that are made, not given or found, and using these experiences to ground and critique questions and data generated by others. Unfortunately, much of school science experience asks students to think about artificial conditions in which Nature has already been silenced (Chinn & Malhotra, 2002). The indeterminacy that is central to scientists’ decisions and the contingencies of the material world are frequently obliterated, perhaps to achieve efficiency or to ensure that students will get the “correct” answer.

How and in what sequence an educator should expose these sources of indeterminacy (e.g., which questions are apt to be most productive) and contingency (e.g., how particular tools facilitate some outcomes but close off others) to students, and how students can best be supported in understanding and resolving them, are pressing pedagogical questions. Our studies focus on how students, with teacher assistance, grapple with these issues, and we find not only that they are accessible to youngsters, with appropriate pedagogical support, but also that working through them at some level seems to play a critical role in helping students to appreciate and understand both their own invented solutions, strategies, and heuristics, and more conventional ones.

4 Focus on Representational Competence

Although we have studied several of these sources of indeterminacy, we have focused especially on supporting children’s representational competence (Danish & Enyedy, 2007; diSessa, 2004; Lehrer & Schauble, 2013), which we regard as a foundational underpinning for modeling. Exploring and reflecting on the implications of a variety of ways of representing the natural world are key components of modeling and provide an accessible way to introduce youngsters to modeling practices. Symbolic representations of all kinds – physical models, drawings, diagrams, maps, mathematical descriptions, including computational simulations – rest on decisions about what the representation should include and what should be eliminated from the natural system, so that features considered theoretically central are highlighted and the potential distraction of irrelevant aspects is dampened (Latour, 1990). Repeatedly generating and interpreting symbolic representations fosters in students the gradual development of a repertoire of representational forms and design tradeoffs that can be deployed to support problem solving in new situations (Lehrer et al., 2000). Teachers encourage students to compare and evaluate alternative representations and continually provide press for revisions that enhance the precision, power, and validity of representations, and thereby increase the range of questions that students can pursue (Lehrer, Schauble, & Lucas, 2008).

Models and other representations are analogies, in which components and their relations are mapped from a source (e.g., planetary motion) to a target domain (e.g. an atom). Bridging analogies (Clement, 2009) bootstrap hybrids of source and target domains (e.g., Maxwell’s paddle wheel mechanisms served as a bridge between Newtonian mechanics and electromagnetism, as described by Nersessian & Chandrasekharan, 2009). Related research on the development of analogical thinking (e.g., Gentner & Toupin, 1986) provides the insight, supported by our own observations, that youngsters find it easiest to begin by generating and interpreting representations whose similarity with the phenomena being represented is reasonably evident (Chap. 1; Lehrer & Schauble, 2000). The most basic kind of model in which similarity is especially salient is models constructed from remnants, that is, those in which actual components of the original phenomena are lifted out of their context and deliberately rearranged and displayed to call attention to key relationships. Remnant models are accessible even to the youngest students. For instance, first graders in one of our studies investigated the effects of soil composition and moisture on the growth of prairie plants (Lehrer & Schauble, 2012). They initially encountered these plants in visits to a restored prairie, and shortly thereafter participated in the construction of a rain garden near their classroom, intended as a model for studying how soils and moisture affect plant growth. The rain garden featured nine plots laid out in a grid, and within each plot students planted a standard set of prairie plants. The garden was intended as a model of conditions in the prairie, but those conditions were implemented so that variation in key variables could be more clearly identified (e.g., both soil and moisture were systematically varied across the grid, the latter by a gradient in elevation). The site was thus a simplification and standardization of the prairie and served as a context in which even young children could develop and explore conjectures about relationships between environmental factors and plant growth. Note that although the model was composed of elements of the prairie, it also deliberately omitted many species and brought into immediate contact variations in soil moisture that would occur at greater remove in the prairie. Such deliberate selectivity indicates that physical microcosms exemplify (Goodman, 1976) and do not simply copy, despite the origins of this form of model in resemblance.

In a second-level model the teacher gathered remnants from the garden—in this case, clippings from the plants—and glued them to a classroom display that highlighted the 3 × 3 design of the garden. This second-level remnant model served as an always-present mnemonic device that highlighted and reminded youngsters about the underlying structure of the garden outdoors, and thereby facilitated indoor planning for subsequent studies of plant growth and insect population in the outdoor garden.

Remnants, like the plant components in the indoor secondhand model, are literally parts of the original phenomena. But other forms of representations also preserve similarity with the modeled world. Drawings, diagrams, and maps do so as well, although not to the same degree. Creating scientific drawings introduces children to important principles of inscription in that drawings amplify some aspects of a system while simultaneously reducing others (Latour, 1999; Tytler, Prain, Hubber, & Waldrip, 2013). For example, children may use bubbles or other visual devices to magnify the structure of a leaf or to make visible a root structure that would ordinarily not be visible. More realistic drawings would not employ these devices. To emphasize important components and relationships, diagrams usually omit details that are found in drawings, and maps preserve some aspects of a space, such as relative distance and orientation, but leave out many others (e.g., lines represent highways, dots stand in for entire towns, and lesser details like telephone poles and stop signs are usually omitted altogether). For novices of all ages, and especially for youngsters, perceptual similarity seems to play a supportive role in initially establishing and then maintaining the mapping relationships between the source and the potential target. Reasoning with representations can be cognitively demanding, especially for children, and, as is the case in analogical reasoning, similarity seems to support the mapping process in modeling.

5 Provoking Change: What Develops?

To support developing representational competence, teachers position children in lines of inquiry where representations are re-used, extended, and expanded in ways that help make visible new aspects of a natural system. These extensions and innovations may rely less on similarity and more on relationships that are not immediately available to perception. The first graders in the prairie study were surprised to see a large, to-scale drawing of prairie plants, illustrating that much of the mass of the plants is underground, in their root systems. During a routine maintenance prairie burn conducted by a teacher, the children learned that the root mass prevents the native plants from being destroyed by fire by supporting the plant’s later regrowth. To pursue questions about root growth, the students subsequently grew a variety of seeds in root chambers filled with soils of different composition. Students debated over ways to record changes in the length of the roots as they grew. Their initial displays were made with string, which resembles roots and, moreover, can be hung below a baseline representing ground level. Because these displays looked like roots, children readily accepted them as representations. But as the roots grew and students tried to compare the lengths of string, a disadvantage became evident—the string tended to curl and was difficult to see, even from a moderate distance. As a result, it did not adequately support the comparison of root lengths across days of growth. To resolve this problem, students agreed to substitute strips of paper, which were easier to see and compare, but less directly resembled roots. Moreover, instead of dangling the strips below a baseline representing ground, students now arranged them above the line, consistent with an intent to communicate increase in length and to describe growth by the differences in length over time.

In related studies of plant growth in other classes (Lehrer et al., 2000), students coordinated drawings and silhouettes created by pressing plants at different points of growth to interpret change of plant height as a Cartesian graph. The changing ratios of height to elapsed time made visible an “S-shape” of growth that seemed to be common across all the plants. As students noted, the S-shape was consistent with drawings and silhouettes, indicating that plants grew “slowly at first, then much faster, and slowed down again at the end.” This shape was identified as a common pattern and in later grades was generalized to other contexts of growth, including growth of individual organisms (such as hornworms) and populations (e.g., bacteria). As this narrative illustrates, a representation that initially relies on similarity can be gradually transformed, via analysis, use, and feedback, into a more conventional display that no longer relies on direct resemblance to its referent. Students gradually augment their preference for similarity as they develop a perceived need for increased explanatory power and increased ability to look through other symbolic descriptions to see new aspects of a system. For example, conceiving, measuring, and representing intervals on a graph that represents plant growth was provoked by the question, “What is the same and what is different in the way all our plants are growing?” Note, too, that the phenomena in question, plant growth, circulates among the representations, so that children’s appreciation of the characteristics of growth originates in the network of relations described by these representational re-descriptions of it (Latour, 1999).

Perhaps because they have a history with representations of all kinds, adults sometimes underestimate the cognitive work required to grasp why a convention like a coordinate graph is a reasonable way of representing a concept like plant growth. Yet representations, even those explicitly taught in school, are not automatically connected in novices’ minds to a history of useful applications. Even when a representation is familiar, considerable domain-specific knowledge may be required to support its interpretation in novel contexts. Moreover, youngsters’ initial grasp of the overall purposes for representations may be vague. If one’s goal is to learn about the world, why attend to a representation as a source of information or as a site for investigation—that is, why rely on a data display, a diagram, or a model—rather than the actual phenomena? Until they have had repeated opportunities to observe the conceptual advantages that representations provide, it is little wonder that youngsters tend to confuse representations with copies or depictions (for purposes of illustration or artistic expression), as documented by Grosslight et al. (1991).

Youngsters seem especially resistant to omitting information in representations, even though simplification is required to achieve amplification of the features that are considered important (Latour, 1990). But this does not necessarily mean that youngsters cannot distinguish models from copies; what may seem to be a general preference for copy may actually be a signal that children have not yet grasped what the model or representation is intended to accomplish.

For example, we observed a different class of first graders using a tub of hardware equipment to construct models that “work like your elbow” (Penner, Giles, Lehrer, & Schauble, 1997). Although there was considerable variability in the constructions that pairs of children produced, none of the initial models actually functioned like an elbow. Instead, all of them were depictive, rather than functional. The models featured Styrofoam™ balls to depict “the bump where your elbow goes” (as one child said), along with elaborately constructed “hands” and fingers represented with Popsicle™ sticks. These initial models seemed to express a concern for reducing representational ambiguity by making a persuasive case to peers for the representational validity of the constructions. When the teacher reminded children that the goal was to produce models that “worked like” their elbows, a girl pointed out that elbows, unlike the models in the classroom, bend. This observation instigated a flurry of revision. The next round of models all included bending “elbows,” but the models were now constructed with springs or pipe cleaners and rotated through a full 360-degree range. When challenged, children insisted that their arms also could move freely. To problematize this solution, a co-teaching researcher duct-taped children’s upper arms to their bodies and invited them to try to move their forearms through a 360-degree trajectory, resulting in the discovery that “our real elbows get stuck right here!” (e.g., they do not move backward). The final models constructed by the children all bent, but also included ways to constrain the range of motion. In closing interviews, a researcher attempted to learn the extent to which children were aware of this distinction between duplicating and modeling. During the interview, as one boy proudly showed off his construction, the researcher provoked this distinction by objecting, “But it doesn’t look like an elbow to me.” The student explained gently to the researcher (who he clearly considered in need of enlightenment), “Well, it’s only a model.”

This example illustrates two points. First, the goal children initially adopted in their constructions was to communicate persuasively what their display was intended to represent. We feel that it would be a mistake to prematurely override that concern, because it addresses a critical aspect of representational competence, namely, sense of audience—that is, how others are likely to “read” what one’s representation communicates. This is an important objective for early modeling instruction, one that requires time and pedagogical attention. It is to be expected that youngsters might focus on the insight that the interpreter needs to understand what the representation stands for. Only subsequently does children’s concern shift to representing less immediately visible aspects, such as function. The second point is the critical role of model test and revision. Especially when young students’ sense of modeling is fragile, modeling instruction needs to be pursued within contexts that support the generation of clear feedback about model fit (and misfit). In this case, children could easily see whether their constructions adequately reproduced the motion of their forearms. Young students may not be accustomed to holding their explanations to account. The ability to test one’s model provides clear feedback about whether and the extent to which the model works. Opportunities and encouragement for students to compare solutions and to revise models in response to feedback are arguably as important as having opportunities to engage in the initial stage of model development.

In these examples, youngsters required sufficient time to apprehend the affordances of representations, to explore the tradeoffs of using different conventions for representing, and to begin to develop a sense of audience, that is, to appreciate the range of ways that peers may make meaning from one’s attempts to represent. These accomplishments can be challenging for any novice who has weak domain knowledge, not just for young children. A novice of any age is often unsure what to emphasize in a representation and what can be omitted; struggling with these questions is an integral part of modeling practice. Therefore, we feel that rather than hurrying students past these uncertainties, teachers should engage students in addressing them directly. Designing representations, using representations to support an argument to an audience, and revising representations in response to feedback from peers are processes that help students see what a particular representation is intended to accomplish with a particular audience, and also eventually cumulate to a more general sense of representational competence.

Developing representational competence is a lifelong task; although it may be an especially pertinent agenda for children, adult professionals continue to invent and interpret representational re-descriptions (e.g., Vertesi, 2014). As students develop their grasp of representational practice, they become increasingly prepared to confront more complex challenges that go beyond simply representing events and objects and center more explicitly on questions about model fit and misfit—especially, whether and to what extent a model legitimately stands for the phenomenon it is intended to represent.

6 Reflecting on the Nature and Status of Models

As suggested earlier, models represent natural systems that are materially modified in some way. Thus, they inherently entail indeterminacy in the sense that representations and material arrangements have an open texture (Hesse, 1962). Other representations and configurations are always possible, so both students and scientists sometimes struggle to understand whether and how representations and material arrangements can be taken to support inferences about the modeled phenomenon. In the following we review two occasions on which upper elementary grade students encountered uncertainty about the nature and status of models. Considered together, these events provoke reflection about the characteristics of modeling tasks that may make modeling more and less challenging.

The first example comes from third graders’ visits to a stream to tabulate its aquatic organisms. The students observed that crayfish were present in some parts of the stream but entirely absent in others. They conjectured that perhaps the crayfish were clustering in places where they could hide in the substrate from predators. However, they were unsure how to test this assumption, because the stream was deep and most of the streambed was inaccessible to observation.

To support investigation, the teacher brought a plastic wading pond to the classroom, and students used duct tape to mark off four equal-sized quadrants. They then installed different types of substrate within the quadrants to explore their hypotheses about crayfish preference. These included white rocks, mixed rocks, mixed rocks with plants, and no substrate at all. The wading pool was filled with stream water and ten small crayfish were installed. Over the next weeks, students took turns visiting the pool, once in the morning and once in the afternoon, to count and record the number of crayfish observed in each quadrant. They initially displayed the results of the counts by affixing stickers to a circular display sectioned into quadrants, like the pool. This initial data display was deliberately designed by the teacher to capitalize on its similarity to the wading pool model. Once students became familiar with the data and what it represented, the stickers were replaced by a data table that more effectively supported cumulating and displaying counts of crayfish by quadrant over time.

The counts confirmed children’s initial expectations; more crayfish were indeed observed among the mixed rocks with plants than in any of the other quadrants. Students took this finding as confirmation of their expectation that the crayfish were actively hiding in that quadrant because the plants and rocks provided protection. The teacher, however, asked children whether they could be sure that the crayfish were not just randomly wandering around, meaning that there was no choosing going on, but rather, that the results were simply due to chance. To pursue this alternative interpretation, the teacher (supported by researchers) introduced a model of chance, with the intention of persuading children to apply it to the crayfish context.

The teacher began by asking class members to conduct repeated trials of ten spins (representing the ten crayfish) with equally partitioned, two-color spinners. This exercise helped children work through many of the naïve conceptions that frequently surface in investigations with chance devices (Lehrer, Horvath, & Schauble, 1994; Metz, 1998). For example, many students believed at first that qualities of the spin—to the left vs. to the right or fast vs. slow—would allow them to systematically control the outcome. By comparing trials run under different conditions, the students eventually concluded that it was impossible to predict the outcome of any particular spin. They also noted that with increasing numbers of trials, the outcomes came close to 50% in each half.

Once these conceptions were addressed, the teacher shifted to evenly partitioned, four-color spinners to reflect the four quadrants of the wading pool. Each student predicted the results of spinning ten trials, ran the spinner, and recorded the outcomes. The class accumulated the data, displayed it in a pie chart, and observed that about a fourth of the outcomes occurred in each of the quadrants. Next, the teacher asked each student to design a unique spinner, using anywhere from 2 to 4 colors, and to spin 20 times, record the results, and display them in a graph. A follow-up discussion focused on how the outcomes reflected, but did not precisely copy, the design of the spinners.

Finally, the teacher felt the class was ready to connect these ideas about chance back to the original question about the crayfish. To contextualize these ideas, the teacher sought to highlight the difference between choice and chance by placing a different snack in each corner of the classroom. She produced a spinner with four quadrants, each labeled by a small picture: of grapes, crackers, broccoli, and cookies. As each student spun the indicator on the spinner, he or she went to stand in the corner representing the outcome. The class observed that approximately equal numbers of people ended up in each corner. Then the teacher asked students to move to the corner where she had placed the snack they preferred. As she had planned, almost all of the children went to the corner representing cookies. These results were also graphed and displayed, and the two displays, considered together, supported a lively discussion about the differences between choice and chance.

Having navigated this extended development of ideas about chance and preference, the teacher urged students to apply this thinking to the original question about crayfish. As students inspected their crayfish data once again, the teacher asked them to reconsider their original conclusion: Did the crayfish end up in certain quadrants by chance, or were they really actively choosing to go to the mixed rocks with plants? To her surprise, most students refused to reconsider their initial interpretation. They argued that it was evident that the crayfish preferred mixed rocks with plants, and the plausibility of their conjectured explanations (e.g., to hide from predators or to avoid predation by “camouflage” within similarly colored rocks) seemed to cement those opinions in place and render them impervious to further review.

In retrospect, there are several features of the chance model that, we believe, made it difficult for children to accept as a satisfactory model of crayfish behavior. The first is the well-established difficulty that people of all ages experience when thinking about chance, including the counterintuitive notion that although no one outcome can be predicted, structure will nonetheless emerge across a large enough sample of outcomes. It is not surprising that students initially entertained many of the naïve conceptions about chance that have been documented in previous research (e.g., Konold & Kazak, 2008; Lehrer et al., 1994; Metz, 1998, 2004). However, their clear advancement in thinking about chance devices suggests that this is not the entire explanation. We suspect that another barrier to assuming a modeling perspective was that children had previously formulated and convinced themselves of a plausible reason to support their initial interpretation. There is considerable evidence that plausibility affects reasoning (e.g., Klayman & Ha, 1987; Schauble, 1990). Third, our follow-up interviews with children revealed that they conceived of animals as agentive and intentional, making it especially counterintuitive to apply a model of blind chance as an explanation of behavior. Doing so would require a form of counterfactual reasoning that is critical to experimentation, but challenging. Finally, the chance model, unlike the representations described earlier, bore an especially low level of similarity with its target. A spinner simply bears no resemblance to a crayfish. Hence, preserving the mapping between the model and the target might have been difficult even if it had not violated students’ conceptions about animal behavior. However, during interviews conducted with a sample (n = 10) of children later in the year, participants were asked to consider a hypothetical bird feeder loaded with both walnuts and peanuts. Children were told that a biologist wanted to learn which food the birds preferred, so she carefully counted the number of times crows picked walnuts and the number of times they picked peanuts. After watching 20 crows, she found that 12 picked walnuts and 8 picked peanuts. We asked children what they thought of the results, in light of what the biologist wanted to learn. When asked what they thought of the results, four of the ten students spontaneously expressed doubt about what could be inferred about preference on the basis of the data we described. As one said, “But that’s still not a very big difference. If it was just by chance, then, well, that was still pretty close.” Another remarked, “You still wouldn’t be sure, because it was a close tie. And 8 is just 4 less than 12. It would have to be a few more walnuts to make me sure that it was walnuts and not peanuts.” Another said: “It (the difference) could be real; it could be just by chance.” Hence, a minority of the children began to entertain difference in light of chance variability, perhaps because they did not have firm preconceptions about the food preferences of crows.

A second occasion when students questioned the status of models occurred as sixth graders were investigating seasonal change in a local retention pond and its surrounding shoreline (Lehrer & Schauble, 2017). On both fall and spring visits, students collected samples of plants and animals in both aquatic and terrestrial contexts and recorded their findings to support conclusions about the number and diversity of species living in these locations. Back in their classroom, the sixth graders developed ecosystem column models to support more controlled testing of factors that affect the interrelationships between the pond and the plants along the shore. The ecosystem models were constructed in two connected liter bottles, one modeling an aquatic system and the other, a terrestrial system. Pairs of students made decisions about the substrate, plants, and animals to include in each part of the system, connected the ecosystem columns, and collected data on outcome variables of their own selection over time.

Thirteen different models were produced across the classroom. We were interested in whether students regarded the ecosystem columns as models or instead, thought of them as simple attempts to duplicate the original pond, albeit in a limited way. For example, we were unsure whether they expected that a model would vary if they constructed and ran it again under precisely the same conditions. To find out, we asked students: “If you build your eco-column again exactly the way you made it the first time, do you think it would come out exactly the same?” We were surprised but pleased to learn that the sixth graders expected that the model instantiations would vary; indeed, they appeared untroubled by that possibility. For example, Carlen replied, “Probably not.” He went on to explain that in the next repetition, one of the organisms might eat more of the plants, which would produce changes in the eco-column that were not seen in the previous version. Another student noted that one could not expect an exact copy because “…it’s nature…you never know what seed will plant (germinate) faster…‘cause you can’t control that. You can only control, like, the number of plants when you first planted it.” Yet a third student summarized, “You have to do everything a couple of times to actually see what would really happen.” Only one of the seven students interviewed entertained the possibility that replicating an identical design could produce exactly the same outcomes, but his belief was expressed only as a hypothetical: “It could.”

When directly asked, “Why did your class make eco-columns?”, most of the students explained that the models were intended to represent ecological processes that were operating in the pond, such as, “The elodea was a producer for the fish,” and, “We, like, questioned interdependent and dependent relationships.” These students acknowledged that although the ecosystem model and the pond differed in significant ways, what bound the two systems were process and relation, not literal ingredients. Students also proposed that a virtue of the model was to make some aspect of these ecological processes more visible. “Like, so see how they (fishes) live on the plants without having to, like, go under the water.” Nonetheless, a minority (two of the seven) of students seemed to think of the models as copies of the pond. “We wanna have, like, a little section of the pond that we can’t bring into the school. So we made one.” These two students expressed concern that the differences between pond and model might be problematic for the model-status of the columns: “They kinda work differently, ‘cause it (the model) didn’t have a lot of animals.” Given their expectation that the models could not be expected to function like the pond, it is not surprising that they did not seem to grasp that this was the intention for constructing them. In fact, when asked about the purpose of constructing the models, these students seemed to find the question confusing. One of them conjectured that the goal might have been to compare the eco-column to the pond to “…see which one’s better (cleaner).”

Unlike the third graders, the sixth-grade students were more willing to accept models that violated expectations about perceptual similarity and focused instead on invisible processes and relationships that they considered more important. However, even after a year’s work in the actual pond and several weeks in a follow-up modeling exercise, a couple of students remained unsure about the viability of a model that did not include all the components found in the original source. These models made variability of outcomes especially visible, in that several columns were initially designed with similar components but looked very different by the end of the semester. This kind of natural variation is often difficult for students to understand (Lehrer & Schauble, 2004), and it seemed to influence a minority of students to question the model-status of the system.

It is possible that the more sophisticated thinking about models that we observed in the sixth graders is primarily a result of their greater age and reasoning capability. However, it is also possible that the aquatic models explored by the sixth graders were better suited to provoke thoughtful consideration of the epistemic status of the model. The third graders readily dismissed the model of chance as applicable to crayfish behavior. In spite of their rich experience within both the model world and the natural world, they rejected the proposed mapping between these two worlds. Students did not regard as plausible the alternative to their initial conjecture about crayfishes’ intentional choice of substrate. Doing so would have required them to entertain as a model a spinner device that did not perceptually resemble its referent, to engage in counterfactual thinking, and to tolerate a violation of a core assumption that intention underlies behavior. In contrast, although the sixth graders began by regarding their aquatic models as small-scale copies of the pond, only a minority of students retained that perspective at the end of their investigations. This may be because the variation among the models designed (e.g., students selected different substrates, animals, plants) invited generalization across the particular “ingredients” and turned students’ attention to more general functional relationships.

7 Conclusion

Careful readers will have noticed a number of correspondences between the approach described here and the FMC introduced in Chap. 1. Interestingly, these correspondences focus on key features that signal development. For example, both approaches note the importance of the transition from an initial emphasis on similarity as a criterion for assessing model fit to a more nuanced concern with theoretically motivated relations and functions. Second, both programs acknowledge that as novices become more practiced in modeling, they begin to become aware of and eventually to apply more principled and general criteria for what counts as an adequate model, criteria that the FMC refers to as meta-modeling knowledge. Third, both approaches emphasize the importance of model test and model revision as important mechanisms for supporting learning.

At the same time, there are also some differences. For example, we do not draw a very clear distinction between representations (or model objects) and models. We use the term “representational competence” to describe our goals for young children because we are hesitant to claim that young children grasp the full panoply of understandings that a professional brings to modeling. On the other hand, we think of the relationship between representations and models as a fluid continuum, and there is no clear point on the continuum that delineates the border from one to the other. Indeed, whether a novice applies meta-modeling knowledge in a given situation may vary with domain, task, and support. Our collaborating teachers continually “push the envelope” so that representational competence is always expanding toward more elaborated forms of modeling that acknowledge multiple models of the phenomena under investigation. We have not found it productive to encourage them to worry about whether “we are there yet.”

Moreover, we have not found it helpful to worry about the distinction between models “in the head” and model objects “in the world.” We acknowledge, of course, that models are only models to someone. However, from a pedagogical perspective we emphasize that models derive their legitimacy not only from the conceptions of individuals, but from the collective engagement of a knowledge making and critiquing community. Moreover, although people’s cognitive representations certainly influence the external representations they produce, the influence goes both ways: representations often influence—sometimes in fundamental ways—the conceptions of the person who originally generated them. Representations are not merely explications; they are also sometimes the source of invention, especially when they are hybrids that combine elements from a number of sources (Gooding, 2006).

As we have been arguing, one of the virtues of modeling is that it serves as a centerpiece for inducting children into productive approximations of the practice of science. Developing understanding of modeling threads an ensemble of practices highlighted by the Next Generation Science Standards (NGSS) in the United States. These standards delineate eight core practices of science, including argumentation, explanation, and modeling (NGSS Lead States, 2013). Among them, we regard modeling as occupying a central role, because it involves a dual relation between representation and material, and between individual and collective—that is, models are evaluated in light of collective understandings, which hold students accountable to the quality of evidence (measures, data, nature of investigation) and to anchoring their individual pursuits to collective endeavor and meaning-making. One aspect of high quality research questions is that they spur related questions, often raised by others to whom they are communicated, and high quality evidence is not simply an overwhelming amount of data, but rather, data constructed in light of the model proposed. We believe that these understandings are important epistemic and epistemological outcomes—epistemic in the sense that students employ models to know, and epistemological in the sense that students learn about the mechanics of model invention and revision—the signature practice of science (Nersessian, 2008).