###### October 1, 2008

# Teaching Density to Middle School Students

By Dawkins, Karen R Dickerson, Daniel L; McKinney, Sueanne E; Butler, Susan

Abstract: Content knowledge and pedagogical practices are of particular concern to middle school science instructors teaching density. First introduced in elementary grades with the ideas of floating and sinking, density taught in middle school is geared toward understanding through the use of mathematical formulas. Using a lesson-plan study design, the authors identified preservice teachers' content understandings regarding density and how those understandings influenced their teaching practices. The results indicated that, although teacher candidates had developed reasonable, intuitive understandings of density, many experienced difficulty connecting those understandings to the mathematical relationships involved. Therefore, university science educators must carefully examine the levels at which their students understand density so that these future educators have the necessary conceptual knowledge to teach this concept effectively. Keywords: content knowledge, density, middle school, pedagogical practices, preservice teachers

The National Science Teachers Association (Coble and Koballa 1996) calls for all teachers to possess an in-depth and conceptual understanding of scientific concepts. Content knowledge and pedagogical practices are of particular concern to middle school science instructors teaching density. First introduced in elementary grades with the ideas of floating and sinking, density taught in middle school is geared toward understanding through the use of mathematical formulas. The depth of understanding a teacher requires to successfully teach this concept poses difficulties. Therefore, examining the content understandings and pedagogical practices of preservice teachers is worthwhile. We address these issues with the following questions:

1. What understandings do preservice science teachers have regarding density?

2. How do their conceptions influence their teaching practice regarding density?

An investigation of these questions is an exploration of teachers' pedagogical content knowledge. This investigation is necessary because although many contextual constraints inform the instructional practice of teachers, "the public knowledge portrayed to students is filtered through the lens of teacher knowledge" (Gess- Newsome 1999, 88).

Literature Review

The concept of density is complex because it is not a direct measurement, but rather the expression of a relationship (ratio) between two measures-mass (or weight) and volume. Piaget (1971) addressed this difficulty in his study of children's ideas about movement and speed when they are unable to discriminate between the value of speed and the value of distance or time. If provided with rich learning experiences, children may develop some sense of the properties of matter such as mass, volume, and density prior to formal instruction (Klopfer, Champagne, and Chaiklin 1992); however, those intuitions rarely develop into sophisticated scientific understandings by the time they graduate from high school (Roach 2001). Continuing difficulties observed in students may be due to their lack of conceptual or procedural knowledge (Heyworth 1999).

The difficulty with conceptual knowledge involves the idea of ratios as specifically applied in scientific concepts (Bar 1987) and the mathematical concepts of ratio and proportions (Streefland 1985). Students do not usually intuitively understand the ratio aspect of science concepts (Kariotoglou and Psillos 1993). In addition to problems with conceptual knowledge of the science concepts themselves, there are also common difficulties in procedural knowledge related to proportional reasoning. The problems occur in a mathematical context (Tourniaire and Pulos 1985) as well as in a scientific context (Akatugba and Wallace 1999). Students tend to avoid proportional reasoning when solving science problems. For example, Anamuah-Mensah (1986) found that secondary school students preferred to use algorithms when solving acid-base titration problems, often choosing the wrong mathematical operations because they did not understand the conceptual relationship. Frazer and Servant (1987) showed that students attempted to calculate the concentration of a solution by using a formula that multiplied the amount of solute by the volume of the solution. Their answers provided no evidence of reasoning regarding the meaning of concentration in terms of the given measurements. It is especially important for educators to understand that, even when students can cite definitions and formulas of science ratio concepts, they do not necessarily understand the meanings or know how to handle calculations involving such relationships.

Because both intuitive understandings and mathematically based scientific understandings support each other in the development of profound and useful conceptions about density, it is appropriate to incorporate both into the process of teaching (Smith, Maclin, Grosslight, and Davis 1997). Intuitions about density may involve concepts about quantities, although not necessarily in a formal mathematical relationship; therefore, when we use the term intuitive understanding in this investigation, it refers to ideas that children may develop through their experiences with materials apart from a formal mathematical relationship (as in, D = m/V). References to mathematically based scientific understandings refer to applications of precise numerical relationships. In practice, instruction of children often focuses on the notions of floating and sinking or heaviness without attention to the finer points of equal masses and different volumes or vice versa, ideas that could easily extend children's inferences about density without involving calculations (Kohn 1993). In older students, the intuitive aspects are often ignored, giving way to a focus on two factors: (a) memorizing a definition and (b) solving problems using the algorithm D = m/V, which students can usually perform even if they have little conceptual understanding of density. Teachers may ignore the connections between the experience-based understandings developed in elementary school and the mathematical relationships presented through formulas in middle and high school, creating a cognitive gap that prevents students' rich and thorough understanding of complex concepts such as density.

Method

We chose a method-lesson-plan study-that would allow us to examine preservice teachers' ideas about density and related, appropriate pedagogical practices. Education researchers have previously used this research tool as an effective interview technique (De Jong 2000). We explored the prospective teachers' initial knowledge base in the context of preparing lessons because this content appears to be teachers' most important issue in planning new lessons (Sanchez and Valcarcel 1999). Following an instrumental case study design, preservice middle school science teachers served as the participants for this two-year investigation. They participated in a lesson-planning task and an interview when they enrolled in their first science education course and again in the following year when they became involved more intensely in lesson planning and field-based experiences. Because the sample is small, we make no claims to generalize beyond this study; however, findings cited in this article are consistent with aspects of other studies, reinforcing ideas about difficulties with conceptual understandings regarding complex concepts (e.g., density; Akatugba and Wallace 1999).

The participants involved in Year 1 included seven students preparing to teach middle-level science, who grouped into three clusters (two or three people per cluster) to accomplish the lesson- planning task. They prepared plans for two lessons during a regular onehour class. Within a week of the lesson-planning session, all seven participants met to participate in a videotaped postinterview to explain their lessons. The data sources included lesson plan documents the participants prepared in small groups and transcripts of videotaped interviews conducted posttask.

During Year 2, a subset of Year 1 participants, including one person from each of the lesson-planning groups, worked as a single team to accomplish the task. We selected these participants based on convenience. The process was similar in both years; however, during Year 2, we asked probing questions in the preinterview session about the relationship of density to the idea of a ratio and the possibility of using a graph to illustrate the ratio. The information gathered from Year 1 informed the Year 2 process. We identified areas of confusion and planned probing questions prompt subjects to think about density as a ratio. During Year 1, lesson plans focused little on density apart from the formula. Because we had a special interest in exploring the participants' intuitive understandings as well as their application of formal mathematical relationships, the lesson-planning task required that they plan a lesson (Day 1) addressing density without any reference to the formula, D = m/V. For the second lesson (Day 2), the preservice teachers were to introduce the formula and include problems for students to solve. The analysis of the data focused primarily on the two facets of understanding density (intuitive and mathematical). It was an iterative process of examining the written lesson plans and interview transcripts regarding the participants' conceptual understandings and their choices of pedagogies to facilitate their students' understandings through the lesson designs. An informal system we used to organize information is shown in table 1. Although the grid was helpful in organizing information, the categories were not necessarily so clean. For example, evidence found in the lesson plan documents also related to the participants' conceptual understandings.

Findings and Evidence

Using both the lesson plan documents and the interview transcripts from both years, we identified several areas of interest.

Intuitive Understandings

The preservice teachers relied heavily on the idea of floating and sinking, or layering of liquids of different densities. When asked how floating and sinking relate to density, one comment was typical: "Objects less dense than water will float on water; objects more dense will sink." On further questioning, none of the participants could explain buoyancy:

Interviewer: "What are boats made of?"

Participant: "I think some are made of steel."

Interviewer: "Is steel less dense than water?"

Participant (smiling): "Hmm, that's a problem, isn't it?"

During the ensuing discussion, however, one participant stated, "Even if the boat is made of steel, the steel is spread out over the water-not in a big solid glob." There was an intuitive notion about the relationship between density and buoyancy, but the interviewer did not take the conversation in the direction of a more sophisticated understanding involving displacement of water and other factors. One participant said she would probably "hold off on water displacement until after they [students] have a good grasp on density."

Density as a Property of Substances and Focus on Direct Measurements

During Year 1, participants made no references to density as a property of substances. Most of the examples and demonstrations cited in their lesson plans involved solid objects floating in water or layered liquids. Materials cited either in discussion or in lab instructions included the following: bricks, feathers, wood blocks, salad dressing, oil, and colored water. If you exclude the water present in the sinking and floating examples, no elements or compounds were mentioned in any lesson. The absence of such a reference during Year 1 prompted the researchers to introduce the idea to the preservice teachers during Year 2 for their consideration. We address their responses in one of the following sections.

The problems participants posed during Year 1 focused primarily on the measurment of mass and volume of solid objects and then on the substitution of those values into the density formula. Participants paid a great deal of attention to measuring the volume of irregularly shaped objects by means of water displacement. All three groups of preservice teachers proposed labs in which students would measure mass and volume before calculating density. As seen in the literature regarding children's inability to understand a complex concept such as speed, these participants seemed much more comfortable focusing on the quantities that could be directly measured, rather than the relationship between those quantities.

Dependence on Textbooks

The participants involved in the Year 1 project indicated, both directly and indirectly, their reliance on textbook examples and on examples their teachers provided. During the postinterview, one participant ended an explanation with the phrase "like in the textbook." In discussing buoyancy, this same participant stated, "I had to constantly refer back to the book." During the postinterview, we posed the question: "Where did you get the idea for this activity?" Two out of three groups cited demonstrations they had observed during their middle or high school years. Asked where they thought their teachers got their ideas, one participant said, "Probably from the textbook just like us." Gess-Newsome (1999) cited a prospective science teacher who, after presenting a microlesson in class, explained:

You know, I'm a biology major. I took all the required course work for my degree, and did quite well. But no one has ever explained to me what it is that I am expected to teach about biology. In micro-teaching, I selected lessons that I had seen in workshops or that other instructors had taught. I wasn't trying to be unique. I just didn't know what else to do. (51)

At this point in the participants' program, they had an in-depth understanding of the public schools' science curriculum and standards; it was reasonable for them to default to what they had read or seen-unless an intervention provided better ideas.

Influence of Prompts on Developing Understanding and on Lesson Plan Development

After analyzing Year 1 data, we identified two areas to address with questions during the preinterview of Year 2: density as a property of substances and the use of a graph to represent the relationship between mass and volume of substances. In the Year 2 preinterview (with 3 participants), the interviewer asked: "How could you represent the density of a material on a graph?" One participant drew horizontal and vertical axes and after a brief discussion among the three, she labeled the horizontal axis volume and the vertical axis mass. It took only a little prompting to suggest that they graph a material whose density they already knew- water. At least one participant stated that water's density was one gram per milliliter. They used those units to label intervals on the graph and plotted the first point. It took no prompting for them to plot additional points (proportional reasoning) and to draw a line representing the density of water. On questioning, they concluded that a direct mathematical relationship existed between mass and volume for each substance. Further discussion led them to find a chemistry book that had a table with the densities of common substances. They chose one less dense and one more dense than water and plotted points on the same graph. Figure 1 shows the graph created by the participants, with one line representing the density of water, one representing aluminum (denser), and one representing ethyl alcohol (less dense). Their spontaneous discussion included the observation that the slopes of the lines showed a comparison among the densities and that steeper slopes represented greater densities. The interviewer decided not to make explicit the idea that the chemistry table listed substances (not objects with nonuniform densities), but rather to wait until later to determine if the participants incorporated that concept into their plans.

The lesson plans that the Year 2 participants submitted reflected, in some respects, the preinterview discussion. They introduced their first lesson with the layering of liquids of different densities and then had students determine masses and volumes of various objects (no substances). Instead of asking students to determine the density of the objects (as they had done in Year 1 lessons), they asked students to show the mathematical relationship by using a graph. In the plans, the preservice teachers did not address the problem of having only one point to plot. Perhaps it can be inferred that they would lead their students to determine other points just as they had derived additional points in the interview. They did not address the idea of density as a characteristic property of substances. Although the graphing idea incorporated the table of substance densities in the preinterview, the participants did not explicitly discuss the idea about substances. The ideas participants developed fully in the interview were developed fully in the lessons. Those that were subtly introduced were not addressed in the lessons.

Conclusion and Implications

Although generalizability is not applicable because of the methods employed, the notion of transferability (Creswell 1998) makes the conclusions and implications germane for many preservice science teacher programs. First, it appears that preservice teachers may possess discrete understandings about science content that they have not yet connected in a way that makes sense for them. Such findings are consistent with those of Piaget (1971). Probing questions may be one tool used to enable them to construct more useful holistic concepts. They demonstrated an understanding of the disconnected components, and a masterful facilitator could effectively prompt the combining of the parts into a product that makes sense. With a concept such as density, they may have developed intuitive understandings that are reasonable but may not have been encouraged to connect those understandings to mathematical relationships explored at higher grade levels, a process standard the National Council of Teachers of Mathematics (2000) currently advocates and articulates in The Principles and Standards for School Mathematics. Introducing the idea of density as a ratio that can be represented graphically encourages preservice teachers to visualize the relationship between mass and volume and to make comparisons among different substances. In some cases, instructors of preservice teachers must express ideas explicitly. For example, the notion that density, in at least one sense, is a distinguishing property of substances was not reflected in the participants' plans despite their using a table that suggested such an idea. Purposeful and explicit attention to conceptual understanding of ratio and proportion and the visualization of that relationship in the context of density would address the concerns of Anamuah-Mensah (1986) and Frazer and Servant (1987; i.e., application of algorithms without conceptual understanding).

Second, the study methods we employed appear to constitute a successful way to begin to build preservice teachers' pedagogical content knowledge (Gess- Newsome 1999). The lesson-plan study method allows teacher educators the opportunity to examine preservice teachers' understandings and to facilitate deeper consideration of the science content and content-specific pedagogy. For example, lesson plans in this study showed the common use of floating and sinking demonstrations as well as the confusion often associated with those processes. They also showed the frequent use of objects in determining density, with little attention to the idea that such materials (e.g., shoe boxes) do not have a uniform density. If lesson plans are judged only on the use of strategies, such as inquiry activities or a prescribed format, problems with conceptual understandings may go unchallenged. However carefully constructed a lesson is, superficial attention to the underlying scientific concepts will perpetuate the incomplete understandings demonstrated in this article. If science teacher educators use lesson plans as a window to their preservice students' understanding of science, perhaps the cycle of misconceptions can be broken. In general, students do not conceptually understand scientific relationships. To bring about change, teacher educators in science and mathematics must (a) help their students recognize that children have difficulty with relationships in science, (b) attend to their students' understandings regarding mathematical models and what they mean conceptually, and (c) provide students with effective pedagogical strategies to address scientific relationships. When selecting content, science and mathematics educators should select topics that make use of ratios when modeling effective, inquirybased instruction. Conceptually understanding ratios is important to both science and mathematics teacher educators, but unless they purposefully include instruction regarding how to teach about ratios in a scientific context, the cycle of misunderstanding will continue.

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Karen R. Dawkins, EdD, is director of the Center for Science, Mathematics, and Technology Education at East Carolina University, Greenville, NC. Daniel L. Dickerson, PhD, is an assistant professor of science education at Old Dominion University, Norfolk, VA. Sueanne E. McKinney, PhD, is an assistant professor of mathematics education at Old Dominion University, Norfolk, VA. Susan Butler, PhD, is director of the Educator Preparation Institute at Gulf Coast Community College, Panama City, FL.

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