2
Similarity of Form and Substance: Modeling Material Kind
This chapter describes one study within a multiyear investigation of the emergence and development of model-based reasoning. Our emphasis on models follows from the widespread observation that, regardless of their domain or specialization, scientists' work involves building and refining models (Giere, 1993; Stewart & Golubitsky, 1992). Although models are central to the everyday work of scientists, they are nearly invisible in school science, especially in the elementary grades. Moreover, modeling is a form of thinking that is difficult to study, because it coalesces only over years of instruction in contexts where it is consistently valued and supported.
Accordingly, to conduct our research program, we have been collaborating for the last several years with elementary school teachers in a local district to reorient mathematics and science instruction around the construction, evaluation, and revision of models. We work closely with participating teachers to plan instruction and then study together the forms of student thinking that emerge. The emphasis is on cross-grade collaboration with an eye toward understanding the development of student thinking across the 5 years of elementary schooling. Because we seek to study the development of model-based reasoning, we focus especially on promising precursors to this form of reasoning in young school students, including physical models, maps, diagrams, and other forms of inscription (Lehrer, Jacobson, Kemeny, & Strom, 1999; Lehrer & Schauble, 2000). We are particularly interested in how young students come to mathematize the world, an activity that modern science has been pursuing since the time of Newton (Kline, 1980; Olson, 1994).
For example, in the first and second grades, students were provided with baskets of hardware and invited to construct models that showed how their elbows work (Penner, Giles, Lehrer, & Schauble, 1997). Children's first models reflected a preoccupation with resemblance. That is, these models primarily captured features that children found perceptually salient, like styrofoam balls to represent the “bump where your elbow goes” and popsicle sticks to represent fingers. As they reviewed these first models in their classrooms, children decided that resemblance was not a sufficient criterion for an acceptable model because, by and large, these models did not “work like”—for example, bend like—elbows. The children next generated a variety of new design solutions; springs, bending straws, and rubber balloons were used to construct “elbows” that could bend. During this second round of model construction, concern for perceptual salience faded and children focused instead on function. All the groups of students succeeded in getting their elbow models to bend, but initially, no one seemed concerned with the fact that they all bent a full 360°! However, as the class evaluated these models, one child objected, “Your real elbow doesn't bend that way…it gets stuck!” In the final round of model revision, student attention turned to modifying the models so that they reflected both potential sources of motion and constraints on motion.
In the third grade, other students carried these explorations further, and as they did so, their models showed an increasing concern for mathematical description (Penner, Lehrer, & Schauble, 1998). First, the third-graders conducted a series of investigations with a device constructed of two dowels connected with a hinge. The dowels could be connected at varying positions so that students could investigate the function of “elbows” with different points of tendon attachment. Students developed graphical and functional descriptions of the relationships between the position of a load and the point of attachment of the tendon—in effect, modeling the elbow as a third-class lever. They eventually concluded that the elbow trades off lift for reach, that is, although it takes more effort to lift a load at a distance from one's body, there are, nevertheless, advantages in mobility to this kind of “design.”
This brief example reflects some of the principles of the overall program of research. These principles include (1) collaborative design of instruction with teachers to focus on “big ideas” in mathematics and science that are amenable to modeling approaches; (2) intensive professional development to help teachers understand both content and student thinking about content; (3) emphasis on classroom tasks that make student thinking visible via inscriptions, maps, diagrams, models, and related mathematical descriptions; (4) expansion of mathematics beyond arithmetic, for example, to data, space, and geometry, measure, and uncertainty; and (5) recruitment of these mathematical resources for reasoning about science. We are interested in the development of student knowledge about mathematics and science and, equally important, as we explain later, about students coming to understand the epistemologies of mathematics and science. In this chapter, space precludes discussion of our work on professional development; this topic and the design of the overarching research program are explained in Lehrer and Schauble (2000).
Over the years our research, we have studied student thinking both within and across concentrated units of modeling instruction. Here, we report results of a study of one fifth grade classroom where students recruited a variety of mathematical resources to construct and evaluate models of material kind—that is, density. Before describing the instruction, however, we first discuss a particular commitment in this research that contrasts sharply with prevailing approaches to model-based reasoning— that is, its reliance on mathematical resources.
“QUALITATIVE” VERSUS “QUANTITATIVE” MODELS
The pioneering work on expertise performed in the 1980s by researchers like Larkin, McDermott, Simon, & Simon (1980) and Chi, Feltovich, and Glaser (1981) focused educators and psychologists on the important role that qualitative analysis plays in the reasoning of practicing scientists. Today, researchers in education (e.g., White & Frederiksen, 1998; Wiser, Grosslight, & Unger, 1989), psychology (e.g., Clement, 1989), and philosophy (e.g., Nercessian, 1993) continue research in this tradition, uncovering new knowledge about the contribution of conceptual, causal, or “mental” models, analogies, and images to thinking and problem solution. This research has been both influential and instructionally fruitful. However, the work has also spawned a by-product that, in our view, has been less salutary. Specifically, the dichotomy between quantitative and qualitative reasoning (sometimes referred to as formal and informal reasoning) has been overgeneralized and reified. Many now believe that qualitative reasoning, which builds on students' existing conceptual structures, is a better starting point for instruction than is quantitative reasoning, which is widely assumed to be mere calculation—the proceduralized manipulation of algorithms. For example, it is taken for granted within the field of science education that students should learn to engage in qualitative forms of reasoning before they reason mathematically or quantitatively about problems.
This position can result in two kinds of distortions. First, it trivializes mathematics itself. Although school instruction places undue focus on the manipulation of arithmetic algorithms, we later argue that mathematics is not equivalent to computation (National Council of Teachers of Mathematics, 2000). Second, a commitment to the priority of qualitative reasoning cuts students off from one of the fundamental trends in modern science, particularly modeling activity in modern science—the progressive mathematization of nature (Kline, 1980; Olson, 1994).
Although mathematics is often viewed as the discovery of preexisting structure, an alternative perspective is that it develops within a history of collective argument and inscription (Davis & Hersch, 1981; Kline, 1980; Lakatos, 1976). An implication for mathematics education is that students, even at an early age, should have opportunities to develop their understanding of the nature of mathematical argument and the relationship between mathematical procedures and concepts (Strom, Kemeny, Lehrer, & Forman, 2000). More broadly, learning mathematics includes developing an epistemology in which mathematics is experienced as purposeful and meaningful. School instruction often caricatures mathematics by reducing it to procedures, framed within an epistemology of “answer.” Yet, studies of mathematical learning in reform classrooms paint a strikingly different picture, one in which mathematical cognition is a joint product of individuals' attempts to achieve meaning and collective “norms” (standards for evidence and ways of thinking) about the nature of mathematics (Cobb, Gravemeijer, Yackel, McClain, & Whitenack, 1997).
Accordingly, as we considered ways to develop mathematics that could conceivably be employed to make sense of material kind (i.e., density), we were careful to ensure that students would first have the opportunity to develop an understanding of the “big ideas” in mathematics that could serve as effective resources for this purpose. Practically, this meant that students learned mathematics before attempting to employ mathematical reasoning as a tool for understanding nature. Our rationale for separating rather than “integrating” strands of instruction for mathematics and science was that these two disciplines have contrasting epistemic roots. Mathematics often relies on a logic of certainty, illustrated canonically by proof as a form of explanation. In contrast, science relies on a logic of reasoning about uncertainty, moderated through models (and the related idea of residual between the model and the modeled world) or other means of inscribing the world.
Too early integration of mathematics and science runs the risk of shortchanging one at the expense of the other. For example, students could learn about the mathematics of ratio by employing ratio to calculate the density of a material, but this emphasis would foreshorten other mathematical senses of ratio that might profitably be developed (we later discuss some of these). Similarly, density could be used merely as a “context” for teaching students about ratio, but only with a corresponding foreshortening of related ideas about material kind. Given these shortcomings of traditional forms of integration, we decided to separate mathematics and science instruction. Specifically, we decided to lead with mathematics instruction and to follow with instruction about density in which students were encouraged to use mathematical ideas to model material kind. As we later describe, we did not adhere rigidly to distinguishing mathematics from science when it made more sense to consider them jointly. For instance, precision of measure arises whenever attributes of the world are considered, so students grappled with this aspect of measure during instruction about density and material kind.
THE TEACHING STUDY
Participants in this study were 20 students and their teacher in a fifth-grade classroom in a suburban public school in south central Wisconsin. Many, but not all, of the students had been instructed in previous grades by teachers participating in our research program. Students were otherwise representative of children in the community. They were White and of mixed socioeconomic status. A few were classified as having special needs, such as cognitive or learning disabilities. One was autistic and is not represented in this study, but participated in class activities.
The study was conducted in the spring semester, so most of the students were 11 years old. Our general approach was to work intensively with the teacher to plan instruction and then to study the forms of student thinking that emerged, both by tracking discussion, small group activity, and student work in the classroom, and by administering one-on-one interviews with the participating students. Data sources included daily video of classroom instruction supplemented by our field notes, collection of student work, and tape recordings of student interviews.
Over the spring semester, researchers worked closely with the teacher, Mark Rohlfing, to plan a series of lessons organized around two topics. During February and March, Mr. Rohlfing worked toward developing in his students a firm understanding of measure and ratio. As described later, the emphasis was on understanding, not on the calculation of rote procedures. To that end, the researchers and Mr. Rohlfing emphasized the accessible idea of similarity as the foundation from which to build an increasingly mathematical understanding of ratio and proportion (Lehrer, Strom, & Confrey, 2000). Students explored ratios of lengths of two-dimensional and three-dimensional geometric figures and expressed these ratios in various mathematical forms. They also explored volume measure with an eye toward developing mathematical descriptions of the structure of the space occupied by an object. This unit took approximately 8 weeks and included a total of 29 classes, usually of about 11/2 hours in duration.
During the late spring (April and May), Mr. Rohlfing's students began to explore the properties of various kinds of materials. As we later explain, modeling approaches were emphasized throughout this unit. Of interest to the researchers were whether and how students would use the mathematics of measure and similarity as a means for modeling ideas about material kind—that is, density. Approximately 20 classes were devoted to investigation of material kind. Most of these classes lasted about 11/2 hours.
We first describe the mathematical ideas that Mr. Rohlfing's students investigated and provide some highlights of the varieties of student thinking that these classroom activities elicited. We then follow by outlining the classroom instruction on material kind, again providing examples of student thinking. Finally, we report the results of final interviews in which students were asked to reason about complex problems in both of these two domains.
DEVELOPING MATHEMATICAL RESOURCES
We developed and posed a series of tasks in volume measure and similarity, many of which were drawn from work the previous year in a third-grade classroom. In each strand of mathematics, we began with a careful description of everyday perception and then helped students develop mathematical explanations for their perceptions.
Volume
We first focused on helping students develop a theory of measure, rather than merely achieving simple procedural competence. Because many of these students had had extensive prior experience in earlier grades with core ideas about length and area measure (such as the need for standard units and zero point; see Lehrer, Jacobson et al., 1998; Lehrer et al., 1999), we extended consideration of these fundamental ideas to volume measure, using activities and ideas derived from previous research (Lehrer, et al., 2000). To forestall students considering volume as a product of a simple algorithm, we first emphasized qualities of volume as space occupied. Students began by exploring different ways to partition three rectangular prisms (“fish tanks”) composed of cubes to decide which tank could hold the most water. Despite differences in appearance, all three rectangular prisms had equal volume measure, and students arrived at this conclusion by considering various ways of structuring the space (i.e., by counting individual cubes or by partitioning the prisms into arrays and then counting groups of cubes). These counting strategies for given units helped students recognize that measure of volume often requires consideration of “hidden” units (some of the units in the prisms were not visible from the surface).
From these initial investigations, we posed a series of increasingly complex tasks that involved structuring three dimensions into units and arrays of units. For example, students designed different rectangular prisms given a constraint of constant volume, or estimated the cubic volume of different boxes (also rectangular prisms) given only the lengths (as edges of cubes). Most students in the class solved problems like these by forming rectangular arrays of cubes organized into layers. Some multiplied the number of cubes in each layer by the number of layers, whereas others relied on skip counting the number of cubes in each layer by the total number of layers (see Battista & Clements, 1996). In short, students' counting strategies became progressively more efficient, and their structuring of the space occupied nearly always accounted for hidden units.
To develop ideas about volume when a simple counting strategy is apt to be inefficient or impossible, students also attempted to find the volume of forms that were not rectangular prisms. One student's solution to a problem of this kind is displayed in Fig. 2.1. Here, the area of the base of the cylinder was approximated; then the student multiplied the area by the height to arrive at the volume. Note the use of color-coding (simulated in gray tone in our figure) to delineate fractions of units that could be combined to yield a whole.
Solution strategies like these afforded students an opportunity to think of volume as the iteration of the area of “slices” (which were conceptualized by some students as vanishingly small). Although volume formulae typically assume this understanding, we found it necessary to provide students with experiences of structuring and restructuring three-dimensional space in a variety of ways. Not all of the student inventions proved useful, but exploration of the strengths and weaknesses of students' proposed strategies elaborated the class' understanding of the structure of “the space occupied.” For example, when comparing the volumes of an open one-liter cylinder and an open one-liter rectangular prism (students did not know either volume, or that the two containers were equivalent, but did establish that they were the same height), students proposed seven different strategies within the span of the lesson. Several of these proposed strategies partitioned and rearranged parts of each base to compare the areas. The argument was that equal heights implied that comparative volumes could be established by comparative areas of the bases, taking a view of volume as a product of area and height. Another student suggested that these area-based strategies could be circumvented by using string to measure the perimeter of each base. He claimed (incorrectly) that the larger perimeter would imply the greater volume. Although this conjecture proved untenable, counterclaims made by students to examine it helped them further differentiate among perimeter, area, and volume. For example, students compared the volumes and circumferences of two 8.5 × 11 inch sheets of paper, one folded in half and the other rolled to create a cylinder. (Use of extreme cases like these is a good mathematical heuristic or “habit of mind.”) In summary, class invention and investigation of measurement strategies related conceptions of the structure of space occupied to ways of finding its measure. These served as an important preamble to considering other aspects of measure, like precision, that arose during the unit on material kind that we describe later.
Similarity
We anchored similarity to students' experiences with magnification. Our aim was to help students develop a mathematical explanation for their perception of “same, but bigger.” In our view, this perceptual grounding served as a resource for making sense in ways that typical word problems about recipes or relative sweetness of solutions do not. That is, in the scale of geometric figures, students can readily perceive both constant ratio and violations of constant ratio.
Students first picked out “families” of magnified rectangles from collections of rectangles with different dimensions. Then they described what was the same about the rectangles in each family. We told students that similar rectangles (those within a common family) are described as those with the same relationship between the sides. Students next sorted another dozen rectangles into three different groups of similar rectangles. For each group, students found a “rule” relating the long side of each rectangle in that group to the short side, for example, “2 × short side = long side.” Follow-up tasks included finding other rules that described different groups of rectangles and using these simple algebraic expressions to generate new instances of similar rectangles. Although ratios of sides were expressed as simple algebraic relationships, not all relationships were integers. For example, a 2:3 ratio was expressed as “ss = ⅔ ls” and “ls = ss + 1/2 ss.” After students generated these algebraic expressions of ratio, they plotted bivariate graphs of short versus long sides of their families of rectangles, as displayed in Fig. 2.2.
All similar rectangles on these graphs were described by lines through the origin, and different families were characterized by different lines. Constructing these graphs afforded students the opportunity of thinking about the meaning of lines of different steepness when the long side of the rectangle was represented on the ordinate and the short side on the abscissa. Students expressed an interest in characterizing the extent of steepness, so we introduced notions of “stairs” (right triangles) and slopes. Students “tried on” ratios of both legs of the triangles that characterized the slope of each line, finding that changes in x per unit y resulted in quantities that decreased with steepness. Consequently, they represented steepness numbers as changes in y per unit x, so that higher quantities were associated with greater steepness (although some students suggested simply switching the axes). Students also constructed and compared lines with and without “shift numbers” (their terminology for expressing the distance of a line from the origin at the x-intercept). These comparisons distinguished between multiplicative (e.g., ls = 2 × ss) and additive
relationships (e.g., ls = 2 × ss + 3), and also, between similar and nonsimilar figures.
Following these experiences with similarity in two dimensions, we moved to a comparatively simple case in three dimensions. Students worked with similar cylinders and developed graphical displays of similar solids by jointly considering the circumference of the base and height of the cylinder. Steepness numbers and shift numbers again figured prominently in representing families of similar forms. In addition, students
FIG. 2.3. Katie's report about three formulae relating the circumference of a cylinder to its height.
reasoned about the equivalence of different algebraic expressions of similarity, as depicted in Fig. 2.3 and Fig. 2.4.
In summary, similarity lent itself to initial perceptual grounding, followed by development of multiple ways of representing ratio. One form of expression used by the students was the quantitative relationship, a/b, interpreted as “a units per unit of b.” For example, students noted: “For every one of the short side, there's two of the long,” or, “The long side is 31/2 times the short side.” Another form of expression that we cultivated was graphical. This graphical description of similar forms extended the sense of ratio by making its generality explicit. A line described any similar figure of a particular ratio, even figures that had not been plotted, or often even conceived. The quantitative expression of the ratio was readily recovered as the slope of the line, a concept that itself relied on similarity (i.e., similar right triangles).
In addition to these mathematical considerations of the sensibility and extension of ratio, similarity also provided a ready analogy for understanding the densities of materials. Just as different similar forms are characterized by different ratios of lengths or angles, material kinds (e.g., brass, aluminum, wood, water) are characterized by unique ratios of mass and volume. Working from the assumption that models are basically analogies (Hesse, 1965), we intended to work with students on the mathematics of similar form and then use these concepts to serve as a base domain for helping students understand the target domain of density.
The analogical mappings between base and target domains were, in principle, sustained by mathematical descriptions of the two cases that were nearly identical. For example, substituting mass and volume for short and long sides in Fig. 2.2 makes the analogy evident. However, central to the differing epistemologies in mathematics and science is the kind of relationship that the line expresses in each of these cases. In the case of mathematical similarity, the line describes a relation of necessity. The two-dimensional figures are ideal cases, and by definition, every similar figure will fall exactly on the same line. In contrast, in the case of material kind, the points plotted represent the measured mass and volume of actual three-dimensional figures. Here, cases may or may not fall exactly on the line, so the line serves as a potential model. The difference between mathematical similarity and material kind highlights a central aspect in the epistemology of modeling that we wanted students to consider explicitly—that is, the idea of mismatch, or residual between the model and the world.
MODELING MATERIAL KIND
Fundamental properties of material are foundational concepts in science, but many of these properties—volume, mass, and especially density—are notoriously difficult for students to grasp. Many science curricula and texts present volume and mass as mere definitions (one gets the impression that these are regarded as the formal definitions for ideas that students should already find self-evident) and then define density as a mathematical formula expressing the ratio of mass per unit volume. Smith, Snir, and Grosslight (1992) and Smith, Maclin, Grosslight, and Davis (1997) suggest that the problem with these approaches is that they fail to connect with students' intuitive theories about material and substance. According to the research conducted by Smith and her colleagues, children do not necessarily regard matter as continuous, as occupying volume, and as having the property of mass. Instead, they define matter by its readily perceptual qualities; for example, one can see it, one can feel it, it has “felt weight.” The implication is that what one cannot see or feel (e.g., very small particles) is not counted as substance. Moreover, preadolescent children typically fail to differentiate weight and density, and instead, conflate both of these concepts into a general sense of “heaviness.” Smith and her colleagues provide compelling evidence of these aspects of students' theories concerning weight and density, and we begin from the premise that their insights are largely on target. One means for addressing these alternative conceptions is through analogy (e.g., to sweetness) and qualitative modeling, a route that Smith and her colleagues extensively explored.
Here, we suggest an alternative, largely unexplored route—through deep understanding of the relevant mathematics. We are not advocating the application of what Smith et al. (1997) referred to as “quantitative reasoning” (e.g., “formal knowledge of a density calculation procedure,” p. 350). This type of procedural approach to mathematical pedagogy has not proven a fruitful foundation for understanding density. For example, in earlier research, when high school students were taught procedures to plot coordinate graphs representing mass-volume relationships as lines, they failed to understand that the line signaled an invariant ratio and thus, did not understand the line as a representation of density (Rowell & Dawson, 1977a). Moreover, despite accurate calculations, many students cannot flexibly partition volumes or understand the equivalencies among various algebraic expressions of mass, volume, and density (Hewson, 1986; Hewson & Hewson, 1983; Leoni & Mullet, 1993; Rowell & Dawson, 1977b; Smith et al., 1997). Mathematical understanding, as opposed to cranking out computations prescribed by a mathematical formula, has its own intuitive bases and typical trajectories (Fennema & Romberg, 1999). We conjecture that when these ideas are cultivated and supported, they can serve as invaluable resources for reasoning about science. Moreover, these resources are essential for helping students capitalize on modeling approaches that fully acknowledge the epistemological foundations of modeling.
Design of classroom pedagogy about material kind was guided by consideration of the mathematical resources students had developed. We designed instruction to capitalize on students' general understandings of the mathematics of measure, as well as their specific experiences with volume measure. We also counted on students' ability to draw on similarity as a model for substance. Yet, the instruction did not ignore the physical grounding afforded by material kinds. Consequently, students hefted, submerged, and compared objects and generally acted on them in ways consistent with building physical intuitions. Classroom instruction was oriented toward using mathematics as explanations and extensions of these emerging intuitions. Our hypothesized developmental trajectory for material kind began with comparisons and orderings of weight, volume, and kind, and then proceeded to quantification of differences (e.g., how much heavier?) and invariants like density.
Weight
In the classroom, we opened the discussion of material kind by asking students to rank by weight 24 small, three-dimensional figures composed of wood, Teflon, aluminum, and brass (six of each kind). Discussions about the relative weight of these objects, which began with the simple idea of rank-order comparison, eventually expanded into consideration of how much heavier and related issues about precision.
Mr. Rohlfing presented the collection of 24 objects. In order of increasing volume, each of the four sets of materials included six identical figures; a small sphere, a small cube, a short cylinder, a short rectangular prism, a tall cylinder (same circumference as the short cylinder but twice the height), and a tall rectangular prism (same perimeter as the short prism, but twice the height). Mr. Rohlfing arrayed the objects on a table at the front of the class and challenged students to find an efficient way to rank-order the 24 objects by weight: “How can we order them by weight without comparing every object to every other one?”
As previous research would predict, this challenge motivated some debate about what it means for objects to be “heavy.” However, compared with middle school participants in previous studies, these students readily differentiated volume from material kind as independent variables that determined the weight of any object in the set. For example, Craig asserted that the “gold” objects were the heaviest materials, but then was uncertain about whether the tall rectangular prism or the tall cylinder had the greater volume. He went on to note that whichever had the greater volume would be the heaviest object of all. We conjecture that students' strong foundational understanding of volume was important in helping them achieve this differentiation. In fact, after these initial discussions, students generated an algorithm for identifying the minimum number of pairwise comparisons that could still ensure a correct rank order of objects; the form of this algorithm was essentially a bubble sort. In subsequent classes, students used a pan balance to execute the algorithm (e.g., comparing pairs of objects on the balance to conclude which was heavier). Within short order, the objects (labeled by letter) were listed on a master chart by rank order of weight, heaviest to lightest.
Mr. Rohlfing next “upped the ante” by posing the question, “Can we tell how much heavier one object is than another?” This is a question that was deliberately sidestepped by Smith et al. (1997), who decided to rely on qualitative forms of modeling partly because, “Students find it extremely difficult to measure the mass and volume of different samples of the same kind of material with enough accuracy to determine that different sample sizes have the same density” (p. 368). As one might expect, given our commitment to mathematical understanding, we took the opposite tact. That is, we regarded the issue of precision as an opportunity for making a fundamental point about modeling—that modeling entails not only mapping, but also making decisions about how to interpret mismap-ping or residual.
Groups of students worked to weigh and then double-check the weight of the objects (in fact, they found the mass of objects, but in this instruction, we deliberately did not differentiate between weight and mass). The tools that they used were chosen to foreground the same concepts about measure that they had earlier encountered while measuring volume (and, moreover, in their earlier years of instruction while measuring length and area). First, students needed to come to regard the very idea of measure as comparison (as opposed, for example, to simply reading numbers off a scale). Ideas about standard units, appropriate units, and zero point (i.e., origin of the scale) emerged once again in the context of weight. For example, only a few sets of standard masses were available for use on the balance. Even using all the available masses was insufficient to balance the large, brass rectangular prism (the heaviest object in the set). To surmount this difficulty, students spontaneously invented the strategy of finding the weight of one of the lighter objects (for example, the Teflon prism), and then using that object (now of known weight) as one of the standards of comparison for finding the weight of heavier objects. Discussions in the classroom turned to the problem of propagation of error, as students noted that weight measurements conducted with fewer standard weights tended to result in less variable measures across groups of students than measurements made with many weights.
As groups of three students weighed and then reweighed the objects, it became progressively clear, to the consternation of all, that in spite of meticulous care, different groups were recording somewhat different measures. Students identified contributions to variability (e.g., calibration of the scales, placement of objects in the pans, eye level when the scale was read, propagation of error), and were eventually able to reduce, but not to obliterate, the variability of measures.
For several days, students discussed ways of interpreting the distributions of measures that they recorded. These discussions centered around identifying a procedure that the class as a whole could accept for deciding on the true value of the measures. Several students argued for the mode, claiming that if two or more students “got the same answer,” that solution must be “right.” However, once students observed a case in which the mode was nowhere near the center of the distribution of measures, this reasoning was called into question. As one student argued, “The mode is good, but what if there's like, three 25s and then a 24, 23, 19, and 17? You could take all the 25s because they're the same, but what if they're wrong?” Other students initially preferred averages, but reconsidered when they discovered a case in which a clear outlier pulled the average toward one tail of the distribution. Some students advocated for the median, but others were dissatisfied with a median that did not fall on an observed case (e.g., when there were an even number of measures, and the median fell between two of the observed measures). After several class periods devoted to these discussions of central tendency (students referred to these as “typical numbers”), the class achieved an agreement to use trim means of the observed measures as the true value of weight for each of the objects. The use of the trimmed mean represented a consensus not only about a measure of center, but also about the role that distribution of measurements should play in the selection of a useful indicator of typicality.
Revisiting Volume
As students reviewed their completed table of measures, Mr. Rohlfing asked if they had any conclusions about what affects “how heavy something turns out to be.” One of the students summarized, “The gold [that is, the brass] is the heaviest. Silver [aluminum] is next, then white [Teflon], and last is wood.”
“That's not always right,” another student objected. “Not all the gold ones are heavier than all the silver. It also depends on how big it is.”
“What do you mean by how big?” asked Mr. Rohlfing. Students recalled their earlier work at estimating volume of figures and put these tools to work to find the volume of the 24 objects. The tools included centimeter rulers, centicubes (small plastic cubes that are 1 cm on each dimension), graph paper (e.g., for tracing and estimating the base of cylinders), and displacement buckets with graduated cylinders. Students noted that the precision of these forms of measure varied, and moreover, that it varied with the form of the objects. For example, with centicubes, different groups concurred on the volume of the rectangular prisms, but water displacement yielded measures that were far more variable. However, water displacement was the only means children could identify for estimating the volume of the spheres.
Once again, multiple means of measure resulted in varying measures, and the students again debated and came to final decisions about the value that they would provisionally accept as the true value for volume of each of the objects. When the values for both volume and weight were arrayed in a table for all the 24 objects, Mr. Rohlfing brought the students back to a question that had come up in an earlier class about water displacement:
| Mr. R: | We had two ideas that surfaced with that. One was that the heavier something is, the more water is moved out of the way. The other theory was that it's the amount of space that the object takes up. Were those questions answered? How many people think those questions were answered last time? |
| Craig: | It's the amount of space that's taken up. |
| Mr. R: | What convinced you? |
| Craig: | It was something Katie said. If you stand up in the bathtub, you still weigh the same, but the water doesn't go up as much [as if you lay down]. |
| Luke: | My group had a big, light object and someone else had a small, heavy object. And the big, light object displaced more water. |
Note that Mr. Rohlfing scaffolded discussion by juxtaposing competing claims about the contributions of volume and weight to water displacement. Moreover, the students' reasoning about volume was tightly connected to experiences, both those generated in the classroom and those from home (tubs). These forms of dialogue contrast sharply with the initiate-respond-evaluate cycle so often observed in studies of classroom discourse.
At this point, Mr. Rohlfing asked another group to measure the weight of different volumes of water with an electronic balance. When the group returned from this task, they recorded their measurements on the board for the class to review:
| ML | Weight [of water] in Gm |
| 5 | 4.9,4.8 |
| 10 | 9.8,9.7 |
| 20 | 19.8, 19.7, 19.8 |
| 50 | 49.2,48.7 |
| 100 | 98.7,99.8 |
Katie, one of the students in the group, remarked, “It was always pretty close, within a couple of grams or less. We measured the same amount of water twice and got a range of values, so we know that even the electronic scale isn't very accurate. We also measured out 50 ml in one container and then poured it into another container, and found it was a little over 50 ml in the second container. So we figured out that the water measuring tools [e.g., graduated cylinders] aren't that accurate either.”
Mr. Rohlfing asked, “So what conclusions do you make about the measurement of water?”
Katie replied, “Somewhere around one” [e.g., water weighs 1 gm per ml].
The volumes and respective weights of these amounts of water were then added to the chart. As students proceeded next to consider the possibility that there might be “families of materials,” as they put it, they had at their fingertips a relatively firm and flexible understanding of (including differentiation of) weight and volume. Understanding these ideas involved many discussions about students' intuitive qualitative conceptions of weight, volume, and material. Equally importantly, mathematizing these ideas provided a firm foundation to which students' intuitive conceptions could be anchored.
Families of Material
After establishing best estimates for the mass and volume of each object, students investigated their families of material hypothesis by taking turns representing each object as a point on a large graph displayed on the wall of the classroom. Mr. Rohlfing labeled the ordinate as G, scaled in units of 20, and the abscissa as ML (cm3), scaled in units of 10. After brief discussion of the scale and meaning of the labels, students worked in pairs to locate and label each object on the graph.
Reconsidering Error. As students plotted the gold (brass) objects, students noticed that one of the points seemed out of place. Ashley suggested that one of them “… wouldn't work. It wouldn't fit if you were trying to draw a line through it” [gestured sweeping motion with hands]. Mr. Rohlfing agreed with Ashley's identification of the out of place point and then asked how else they might determine that the point in question was out of place. Another student, Craig, noticed that the object being plotted was the “same shape” as some of the other objects plotted, but it was represented on the graph as having a different volume. Craig declared that all objects of the same shape should fall on the same “milliliter line,” and he demonstrated what he meant by tracing a vertical line. As they replotted the point, Mr. Rohlfing referred to the brass objects, asking: “Are they all going to be on the same line?”
Most students shook their heads, “No.” Adam suggested that the graph itself was a source of imprecision because “It's going up by 20s on the side and so we're not going to be able to graph 28.5 very accurately.” He suggested that the graph could be made much more accurate by changing the scale: “If you made each line be one.” Other classmates agreed, but pointed out that the graph would then be far too large; they would need a stepladder to plot the points representing some of the objects.
At this point, Rachel raised the possibility that “the numbers might not be accurate.” Katie agreed, and then various members of the class recalled particular episodes of imprecision of measure of weight or volume. Adam summarized the discussion up to this point: “… [M]aybe because of propagation of error. Those are probably not all the right numbers there, and then even the graph isn't very accurate. So, putting both of those together make it even worse.” As students continued to plot points, Drew proposed an alternative meaning for each point (i.e., in addition to a location):
Clearly, Drew was building here on the earlier work the class had done in constructing rules to describe families of two-dimensional figures. Other students also now proposed that shapes in the same families fall on the same line and share the same rule, as Mr. Rohlfing wrote each algebraic expression on an easel for consideration by the class (e.g., v × 8.5 = w). The class began to predict where other to-be-plotted points might lie by noting both their measurements and the rules. This discussion raised the collective awareness of a difference between line in this context and in the previous context of similarity. Mr. Rohlfing asked the students why all the points did not fall exactly on the line, and several immediately volunteered, “Error!”
Which Line? Having identified a key difference in the two experiences of line, the class was confronted with the problem of exactly where to draw the line. Using a pair of points on the graph as anchors, two students held up a yardstick to locate a line to describe the family of brass objects. The proposed result was a line with a nonzero intercept. One student, Cathy, objected, stating that the line needed to start at zero.
| Mr. R: | We need a line up there to represent gold. Katie was concerned that your line didn't go through zero. |
| Luke: | But these two dots do [meaning that the line intercepted the dots]. |
| Mr. R: | How do you get a line that doesn't start at zero? |
| Rachel: | If it's an added number. [This is a reference to the class' earlier work with similar rectangles, in which children discovered that expressions like “w + 4 = l” yielded lines that did not go through the origin and did not define similar figures.] |
| David: | When we used addition, it didn't start at zero, but when it was multiplication it did. [e.g., the reference to “multiplication” means expressions like “4w = l.”] |
At this point, Mr. Rohlfing asked students to consider what “zero really means.” Tara suggested that it meant that the volume was zero. David then added that zero volume implied that the object “wouldn't even be there.” A third student proposed that it would weigh “nothing.” The class explored the consequences of this implication for a symbolic expression; v × 8.75 + 1 = w. They judged that the application of this expression was impossible (“You can't have nothing that weighs something!”), so that a line representing a family of material would have to go through the origin. Several students proposed trying to induce a rule from the table of measurements and then using the rule to manufacture points for a line, but most suggested that Drew's rule might not be more reliable than fitting a line on the graph. Because of the additional constraint that lines must go through the origin, the strategy of simply intercepting two or more cases receded. Instead, students began to draw lines that appeared to come “close” to the majority of the cases displayed, with a clear bias, however, to ensuring that the line intersected at least one case. One student, Luke, proposed that the class could evaluate the resulting line by considering the rule, “Check the step number.”
Lines and Rules: Checking the Steps
In the lessons that followed the initial conjectures about families of materials, students considered the relationships among graphical and algebraic expressions of relationships among mass and volume. Mr. Rohlfing asked about the meaning of steepness in this context. David proposed that they “make steps” (i.e., find the slope), and Melanie suggested that the steps (for the different lines) be compared. Students recalled their experiences with steepness numbers in the context of geometric similarity. They found that if they began at the origin for the line describing the brass objects, one set of stairs could be “up 40 gm” and “over 5 (ml),” resulting in a steepness number of 8.
Several students then responded to a question posed by their teacher about the meaning of this number by rewriting the algebraic expression for brass as v × 8 = 40. Mr. Rohlfing asked if there were another way to express this rule, and Jordan suggested, “Weight divided by volume = 8.” Drew compared the steepness number to its counterpart in similarity: “It's like the scale factor between them.” Much of the remainder of this lesson was devoted to interpreting each line's slope as a mass to volume ratio, with an emphasis on the slope as a ready means for thinking about grams per milliliter of volume. At the conclusion of the class, students noted that steepness had a material implication. For example, Katie suggested: “This is pretty obvious, but anything below the water line would have less grams than volume.” Luke agreed, adding: “Because for the water line, the volume and the weight are the same. So anything above that would have a higher slope. You go over less” (gestures).
Reasoning About Novel Cases
On the final day of instruction, the researchers provoked a discussion to explore how the class might use these ideas to reason about unfamiliar kinds of materials, in particular, novel liquids. One of the authors displayed a bottle of unidentified liquid colored with green food coloring.
| RL: | I have something in here (shakes bottle). It's greenish. |
| Natalie: | Oh, yuck! |
| Jordan: | It's oil! |
| Brittany: | Colored water! |
| Tara: | Cooking oil! |
| RL: | What I'd like to know is if it belongs with any of the families You've already created, or if it needs its own family. |
| Brittany: | It goes with water. |
| Luke: | I don't think it would. I think it has more molecules per square milliliter than water does. |
| RL: | You can tell this by looking at it? |
| Luke: | It's thicker. |
| RL: | Things that are thicker will have more molecules? |
| Luke: | (Hesitates) … Not necessarily. |
| Melanie: | Oil is thick, but it's less dense than water. |
| Danny: | If it's water, and you put food coloring in it, it could be just a little bit different. |
| RL: | How could you find out the rule? |
| Rachel: | We should take one ml of that stuff and then weigh it. |
Another student objected that if just one measurement were made, it wouldn't be very accurate. Eventually the class agreed to divide into teams, each receiving a different volume of the “mystery liquid” to weigh. On the basis of their measurements, students concluded that the green mystery liquid (glycerin) weighed 1.3 g/ml, and a mystery gold liquid that RL presented next (cooking oil) weighed .96 g/ml. Eventually, a family line was plotted on the graph for the each of the two mystery liquids.
Students next turned their attention to the graph and discussed which of the represented materials would float in water and which would sink. Students were confident that of the 24 solid objects investigated earlier, only those made of wood would float. David pointed out that wood floats because “it's underneath the water line.” In other words, he noted that the line representing the “family” of wooden objects was less steep (hence below) the line representing water. On this basis, students predicted that the mystery gold liquid would float in water and the mystery green liquid would sink.
| Natalie: | If you pour the gold stuff in the water, the water would stay on the bottom. |
| RL: | What would happen if I poured the green stuff into here (gold liquid)? |
| Angela: | The gold stuff will float on the water, because it's less dense than water. It doesn't float. So the gold would be on top of the green. |
| Luke: | I agree with Angela, because you can look at the graph to see that the gold line is above the green line. So the gold liquid will float on top of the green liquid. The gold is under the water line, and take the green, which is over the water ine. |
When RL poured water, green liquid, and gold liquid into a common container, these predictions were confirmed. As the layers separated, Isaac tried to explain: “Let's say you took the green stuff and you've got molecules per milliliter or something like that, and you did the same, like let's say, with the water. The green stuff would have more. It's more compact together. It pushes that stuff [water] away from it, it has more per cubic centimeter.”
In sum, over the second part of the semester, students developed a history of common experience and assumptions in measuring the weight and volume of both regularly and irregularly shaped three-dimensional objects made of different materials. These shared experiences provided opportunities to understand principles of measure, reason about precision, and consider sensible ways of resolving its opposite, variability in measure. Students discovered and reconfirmed in a variety of contexts that water is pushed out of the way when solid objects are immersed in it, and that the amount of water pushed aside is equivalent to (and an alternative measure of) volume.
Relying on their earlier experiences with similar forms, students investigated the possibility that kinds of materials might form families defined by a constant ratio of weight to volume. Their investigations capitalized on the algebraic and graphical notations that they had developed previously in the context of similarity. However, they now debated whether and how a line on a graph could be regarded as an appropriate model for a relationship between weight and volume of a particular kind of material. This debate highlighted an important component of scientific epistemology (the inherent uncertainty of fit between model and world) even as it contrasted with the mathematical generalization (and its certainty) of the line describing similar forms. Finally, students used these ideas—and, in our view, equally importantly, these now-familiar notations—to reason about novel kinds of materials, that is, unfamiliar liquids.
POSTINSTRUCTION INTERVIEW
At the end of the semester, all but three of the students (n = 17) participated individually in a clinical interview about material kind (the complete interview is included in the Appendix). The interview lasted approximately 45 minutes. We conducted the interviews not to establish claims that students in this classroom were learning more than those taught in other ways (an objective that would require an experimental design using classrooms as the units of analysis). Rather, the purpose was to learn whether our impressions of student understanding formed during classroom instruction would be confirmed by the performance of individuals, and to develop a finer grained picture of the range of student conceptions. It is one thing be impressed by the performance of a class of students. It is another to be convinced that a majority of the students are benefiting.
Conceptual Differentiation
One pair of interview items (Questions 1 and 6 in the Appendix) assessed whether students thought about density as the coordination of mass and volume (as one student put it, “the amount of stuff packed into the space”) or whether, as suggested by other researchers, (Carey, 1991; Smith, Carey, & Wiser, 1985; Smith et al., 1997), they confounded weight and density into a global, undifferentiated concept of “felt weight,” that is, how heavy an item feels when you heft it, as opposed to an idea of “heavy for its size.” The fifth-graders did reasonably well with the items in this category.
Weight-Volume. Students were shown two objects of identical size and shape, one made of bronze and one of aluminum. The interviewer pointed out that the objects were of equal volume, and students readily agreed, because these same two objects were among the 24 used in the classroom investigations. Two large graduated cylinders half filled with water were also presented, and the students were asked, “Now, if I put each of these objects into the water, the gold one into this cylinder and the silver into this cylinder, would the water rise higher in one of the cylinders? Or would it rise to the same level in both?”
Approximately 82% of the students explained that because the objects had the same volume, they would displace the same amount of water. Most explicitly rejected the idea that the differing weights of these objects would push out different amounts of water, for example, “Some people think that bronze would go heavier, but both are the same size, so they push out the same amount of water. Weight doesn't matter.” The remaining 18% of students were apparently still struggling to unconfound the roles of weight and volume. However, these students, too, provided justifications that seemed to suggest that they were trying to reconcile their convictions (e.g., that the heavier object would displace more water) with the classroom discussions of density, for example: “The gold has more molecules and it's denser, so it will push out more water.”
Weight-Density. The interviewer first dropped a dime into a bucket of water and then added a much larger wooden block, asking, “Can you explain why the wood floats and the dime sinks?” If a student simply answered, “Density,” the interviewer added, “What does density have to do with it?” On this item, too, students performed well, with 65% of their explanations appealing specifically to conceptions about density. One characteristic reply was, “If you had one milliliter, the dime has more stuff packed into it than one milliliter of water.” An additional one fourth of the students told us that wood is a “lighter kind of material” than water. Previous research (e.g., Smith, et al., 1997) suggests that such appeals to kind of material usually mark a transition between undifferentiated conceptions of felt weight and conceptions that fully differentiate weight and volume. The remaining 10% of students talked about either weight or volume, without coordinating the two ideas.
Relative Densities
All but one of the remaining items of the interview concerned relative densities of materials. Some items required interpretation of graphical representations, others, explanation of the observable behavior of physical materials.
Predicting Floating and Sinking by Interpreting a Graph. The first set of questions (Item 2 in Appendix) posed to students employed the graph displayed in Fig. 2.5, which shows lines representing five novel families of materials.
The interviewer explained, “Jeff made a graph with lines that show families of materials. Each line shows the weights for different volumes of one kind of material. Imagine I had 20 cubic centimeters of these five kinds of material, A, B, C, D, and E. And then, suppose that D is a liquid. Which of these materials will float in D? How do you know?”
About 82% of the students correctly distinguished the lines representing floating materials from those representing sinking materials and pro
FIG. 2.5. Graph representing five “families” of material.
vided justifications based on density; for example, “E is less dense. Density is like molecules per cubic centimeter. The lower the weight and the higher the volume, the less dense.” An additional 12% of the students (n = 2) appropriately identified the floating and sinking materials, but did so by referring only to the position of the lines representing those materials on the graph. That is, their justifications referred to whether the line in question was “above” or “below” the line for material D, but did not include conceptual interpretations of density. One student was unable to interpret the graph.
Finding Densities and Reasoning About the Physical Intelligibility of the Origin. Two additional items also asked students to reason about relative density from graphical representations. From the graph in Fig. 2.6 described as showing “two families of materials, X
FIG. 2.6. Graph for calculating densities of materials.
and Y,” students were asked to find the density (e.g., a value) of both materials. Seventy one percent of the students were able to find both densities. A final graph, displayed in Fig. 2.7, probed students' understandings of the physical intelligibility of the intercept. The first of these (Item 4 in Appendix, the last of the graph interpretation items) is displayed in Fig. 2.7. The instructions were, “Matt was making a graph to show two families of materials. But Susan told him there was a mistake in his graph. Can you find the mistake and explain why it's a problem?”
Although all the students identified the nonzero intercept as “the mistake,” only 47% spontaneously appealed to physical intelligibility to explain why lines representing families of materials must go through the origin. As one student explained, “It can't be nothing and weigh something. Not even air can do that.” The remaining students recalled that the
FIG. 2.7. Graph showing an “error” in representing families of materials.
lines must go through the origin, often citing this condition as necessary for comparing densities, but they did not offer physical interpretations to justify this claim. No student cited other features of the graph as problematic.
Reasoning About Composite Materials
The last interview item (Item 5 in the Appendix) required students to reason about the relative density of composite materials. We expected this item to be difficult because our instruction did not include transformations of materials that changed density (e.g., thermal expansion or popping corn). The interviewer adjusted two balls of clay until the participant agreed that they were of equal size and weight. Then, one ball of clay was reshaped into a bowl and both pieces were put into a bucket of water. The interviewer said, “When I put this ball of clay into a bucket of water, it sinks. But when I shape the same amount of clay into a bowl shape like this, it floats on top of the water. How would you explain that?”
Slightly fewer than half the students (47%) referred to the relative densities of the clay versus the clay-plus-air composite, or alternatively, offered appropriate arguments about the amount of water displaced by the two samples of clay. As one student explained, “Ball: Water displaced wasn't more than its weight. Bowl: Water displaced was more than its weight.” Another 29% of the students suggested that the differences in volume were important (e.g., “The bowl has greater volume, so it pushes out more water”) but failed to integrate these differences with relative densities. The remaining 24% offered vague appeals to “air” to explain why the bowl-shaped piece of clay floats. When asked what air had to do with it, these students either claimed that air is a substance that makes things float by means of some unspecified property or suggested that air floats because it has little weight, giving no evidence that they were regarding the clay-plus-air as a composite substance.
Table 2.1 summarizes student performance across the items in the interview. The table suggests that students did a credible job of displaying differentiated concepts about weight, volume, and density, especially in light of the well-established finding that much older students typically fail to do so (e.g., Hewson & Hewson, 1983; Smith et al., 1992). Most of the fifth-graders were able to reason about density situations from graphical representations of the kind developed in their classroom. Even when physical interpretation of the representations was designed to challenge common misconceptions, half or more of the students performed well.
| Level of UnderStanding | |||
| Full | Transitional | Little | |
| Differentiation | |||
| Weight-Volume | 82a | 18 | 0 |
| Weight-Density | 65 | 25 | 10 |
| Graphs of Relative Densities | |||
| Floating/Sinking | 82 | 12 | 6 |
| Finding Densities | 71 | – | 29 |
| Origin/Intercepts | 47 | 53 | 0 |
| Composite Densities | 47 | 29 | 24 |
a Denotes percent of students demonstrating a particular level of understanding.
CONCLUSION
This class of fifth-graders made considerable progress in reasoning about the density of solid objects and liquids, as well as about the relative density of materials and composites. Instead of jumping directly into these difficult ideas through definitions and formulae, the classroom teacher first worked systematically to build student understanding of foundational mathematical ideas about volume, similarity, and distribution. It may seem surprising how much is involved in helping students develop a robust understanding of these concepts. Mr. Rohlfing spent several lessons eliciting different forms of student thinking about apparently simple concepts like volume and encouraged students to discover the contradictions and connections among their ideas. As mentioned, during one lesson that we observed, students proposed seven different ways of ascertaining whether or not two different three-dimensional solids had the identical volume, and spent 2 hours debating which methods were valid and would therefore lead to the same solution. In mathematics, as in science, it is vital to acknowledge the complexity of student thinking about these fundamental ideas. Only when students are urged to cut off their sense making and resort to procedure does the variability of student thinking disappear. When students are routinely encouraged to propose, justify, and evaluate their own algorithms and conjectures, they have reason to reconsider their own prior conceptions and embrace consensual means, via evolving classroom standards of argumentation, to decide which ways of thinking make sense.
In this case, we found that students' conceptual differentiation of weight and volume can be both facilitated and anchored by helping students achieve a firm grasp on mathematics of measure. In Mr. Rohlfing's class, the mathematics helped to anchor definitions and arguments so they did not endlessly drift in a sea of opinions that sounded equally valid to students. Students' well-grounded experience with volume of regular and irregular solids—including inventing various ways to measure volume— served to help students really understand what volume is, and made it correspondingly less likely that they would ignore the idea of per volume when considering the differing weights of kinds of material. Similarly, the class' evolving understanding of concepts about distribution, including typicality and variation, was fundamental to helping them decide what do about the different groups' measurements for weight and volume of the 24 objects. A common challenge for teachers in orchestrating productive classroom argumentation is that students tend to talk past each other when they use terms like heavy and big (e.g., what does one mean by big?) with no clear way to call anyone's claims into account. In contrast, when ideas are mathematized, they can be called into account in very precise ways. Agreeing on what big means is partly a matter of reaching consensus on how to operationalize it in measure. Smith et al. (1997) also found that middle school students' firm understanding of volume as “the total amount of space an object takes up” was related to their differentiation of weight from density, a finding consistent with the interpretation that you can't differentiate what you can't conceive.
A final point is that it makes little sense to regard conceptual and mathematical understanding as opposing and competing ways of approaching science. In our view, it would be preferable to learn more about how these forms of understanding can mutually bootstrap each other. As Smith and Unger (1997) suggested, such bootstrapping involves a dialectical interplay that leads to restructuring knowledge, not simply “abstracting” knowledge from one domain in service of another. Here we have argued that mathematical ideas (e.g., about similarity and measure) can serve as rich resources for reasoning about science (e.g., material kind). It is equally the case that conceptual questions about the world (e.g., How can we decide which objects are heavier than which?) also provoke and lend meaning to mathematical questions (e.g., what is the best typical number for representing a distribution of measures?). When this interplay occurs, mathematics can be recruited as a powerful resource for reasoning about science—a means for modeling nature. Models so constituted are a crucible for grounding and testing arguments and ideas about the material world.
ACKNOWLEDGMENTS
Richard Lehrer and Leona Schauble contributed equally to this research; their order of authorship reflects alphabetical convention.
This research was supported in part by a grant from the U.S. Department of Education, Office of Educational Research and Improvement, to the National Center for Improving Student Learning and Achievement in Mathematics and Science (R305A60007-98). The opinions expressed do not necessarily reflect the position, policy, or endorsement of the supporting agency.
We thank Mark Rohlfing and his students for their participation.
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APPENDIX: Density Interview Questions
1. Differentiation of weight/density
This item requires use of the short, bronze cylinder (#28) and the short aluminum cylinder (#20). Place each of the cylinders in front of a graduated cylinder. (Switch left-to-right positions of the cylinder from child to child).
If I put this solid (pick up bronze cylinder) into the water, what will happen to the level of the water? (Point to water line. If you need to, prompt until child notes that the water level should rise. Record whether child produces this idea spontaneously). Now if I put each of these two cylinders into the water, the gold one into this cylinder and the silver one into this cylinder (pick both up and hold them over their respective cylinders), would this water line rise higher (gesture to one cylinder)? Would this water line rise higher (gesture to the other)? Or would they both rise to the same level? Why? (Justification is important).
2. Relative density
Jeff made a graph with lines that show “families” of materials. Each line shows the weights for different volumes of one kind of material. (Note: These items refer to Fig. 2.5 in the text.)
2a. Imagine I had 20 cm3 of these five materials, A, B, C, D, and E. And then suppose that D is a liquid. Which of these materials will float in D? How do you know?
2b Imagine now that B is a liquid. Which of these materials will float in B? How do you know?
3. Finding Densities
This graph shows two families of materials, X and Y. (Note: This item refers to Fig. 2.6 in the text).
3a. What does the steepness of the lines tell you about the density of the materials?
3b. What is the density of X?
3c. What is the density of Y?
4. Interpreting Origin (Note: This item refers to Fig. 2.7 in the text.)
Matt was making a graph to show two families of materials, X and Y. But Susan told him that there was a mistake in his graph. Can you find the mistake and explain why it's a problem?
5. Composite Densities
This item requires use of two equal-sized pieces of clay, one rolled into a spherical ball and the other shaped into a bowl; also a small bucket of water.
When I put this lump of clay into a bucket of water, it sinks (gesture with lump of clay over bucket, but do not drop it inside). But when I take the same amount of clay and shape it into a bowlshape like this, it floats on top of the water. How would you explain that?
6. Differentiation of weight and density
For this item you need one bucket with water, one dime, and the wooden cube with the hook removed.
Drop the dime into the bucket and add the wooden block. The block is much heavier than the dime. So can you explain why the piece of wood floats and the dime sinks? (If the child says “density,” ask: What does that have to do with it?)