Examining the conceptual organization of students in an integrated algebra and physical science class
Westbrook, Susan L
The cross-disciplinary context of density and slope was used to compare the conceptual organization of students in an integrated algebra and physical science class (SAM9) with that of students in a disciplinespecific physical science class (PSO). Analyses of students’ concept maps indicated that the SAM9 students used a greater number of procedural linkages to connect mathematics and science concepts on the SAM9 students’ maps than did the PSO students. The maps produced by SAM9 students also tended to show a more compartmentalized approach to thinking about the content of the two disciplines, a finding contrary to the researcher’s original assertion. Traditional teaching territories and conceptual complexity were examined as possible explanations for the discrepancy between the predicted and actual outcomes.
Attitudes and beliefs about teaching and learning in science and mathematics classrooms are rapidly being transformed through the work and publications of national and professional organizations. Rather than emphasizing computation and “drill and practice,” the goals of the National Council of Teachers’ Mathematics’ Curriculum and Evaluation Standards (1989) include valuing mathematics, communicating mathematically, and reasoning mathematically. Moving beyond the traditional notion of science as information, the National Science Education Standards (National Research Council, 1996) identify science as an “active process” (p. 20) and inquiry as “the central strategy for teaching science” (p. 31). Making classroom mathematics a valuable enterprise, however, may be difficult to accomplish when students do not grasp the underlying disciplinary contexts within which problems are framed. Science teachers may also find inquiry-based science instruction hindered by the students’ lack of understanding of necessary mathematics concepts and representations. Integrated-or crossdisciplinary-mathematics and science curricula could provide a means through which science and mathematics teachers engage students in meaningful inquiries with valuable outcomes.
Although calls to study the potential effects of integrated mathematics and science curricula have come from several sectors of the mathematics and science education communities (Berlin, 1989; Good, 1991; Rutherford & Ahlgren, 1990), little research data are available to provide theoretical support for pedagogical models or the development of suitable integrated learning environments. In A Bibliography of Integrated Science and Mathematics Teaching and Learning Literature, Berlin (1991) reported that only 7% of 555 articles, books, and manuscripts written about the integration of mathematics and science content represented research on the subject. Further examination of that “research” literature, however, revealed that the studies listed rarely applied directly to classroom environments where mathematics and science were conceptually integrated. Since Berlin’s 1991 bibliography, editorials (e.g., Underhill,1995), “howto’s” (Lonning & DeFranco, 1994; McBride & Silverman, 1991), models (Berlin & White, 1994; Davison, Miller, & Metheny, 1995), curricular evaluations (Deal, 1994), university programs (Lonning & DeFranco, 1994; Stuessy, 1993), and numerous examples of activities proposed to integrate mathematics and science content have been published. There is no deficiency in the dialogue about approaches to take to integrate mathematics and science curricula. There does appear, however, to be a lack of systematic, prolonged inquiry into the processes and products of integrating mathematics and science in actual classroom settings.
In an attempt to promote more investigation of learning in integrated mathematics and science contexts, Williamson, Westbrook, Wright, and Fischer (1997) suggested that research be conducted on two levels: (a) pragmatic inquiries into the effect that curricular programs have on student understanding, attitudes, and process skills and (b) theoretical studies to examine the nature of the learner and the learning process within the integrated context. In order to explore the pragmatic and theoretical aspects of learning in an integrated mathematics and science classroom, the perceived outcomes of that integration must first be determined. What benefits do students and teachers gain from the integration of mathematics and science content? The assets of curricular integration can be perceived in three areas: context, authenticity, and conceptual complexity.
According to Science for All Americans (Rutherford & Ahlgren, 1990), “Science provides mathematics with interesting problems to investigate, and mathematics provides science with powerful tools to use in analyzing data (p. 16).” An integrated curriculum has the potential to foster an environment in which a variety of contexts can be used-logically and meaningfully. Content integration helps the mathematics teacher develop what Ball (1993) calls representational contexts: dynamic, rich models and problems that provide opportunities for students to think about and do mathematics. The content of science can be used to facilitate students’ understanding of mathematics as a conceptual tool available for daily application in real-world problem solving.
Cross-disciplinary environments also allow students to participate in what Brown, Collins, and Duguid (1989) have called “authentic” inquiries. Integration of mathematics and science content and processes in combination with inquiry-oriented methods provide the student with more realistic experiences (Roth, 1992, 1994). Data collection is seen as a necessary and integral part of asking and answering questions. Graphing becomes a conceptual and analytical tool applicable to a real problem, rather than an item in a problem set at the end of the chapter. Development of equations and other representations follows naturally from instances where data are generated and sense-making is attempted.
Information from general educational research such as Gardner’s (1983) theory of multiple intelligences and current interpretations of brain research (Cohen, 1995) lend additional support for integrating, rather than separating, the curriculum. Complex, contextrich experiences provide opportunities to disequilibrate learners and encourage conceptual development and organization. Integrated curricular programs have the potential to increase students’ abilities to access and use the information they are learning, as they draw on experiences from across multiple disciplinary boundaries.
The benefits of curricular integration, from the teacher’s perspective, are fairly well defined. A learning environment with built-in contextual relationships, authenticity, and conceptual complexity provides the mathematics teacher with a platform for contextually based development of mathematics concepts and skills, meaningful situations to facilitate the use of mathematics as a conceptual problem-solving tool, and practical applications of mathematics tools, skills, and concepts. The science teacher benefits from access to numerical systems, representations, and concepts needed to assist the students’ analysis and interpretation of data and subsequent understanding of scientific phenomena. The impact of cross-disciplinary efforts on the student’s understanding of science and mathematics is not as easy to grasp. How are concepts and processes from the two disciplines conceptually organized by students participating in integrated mathematics and science courses? Do students in such courses develop a “big picture” view of the interrelationships among crossdisciplinary concepts? Are the cross-disciplinary linkages readily apparent? These questions take on added relevance when the integration occurs in high school classrooms. Are such ventures productive in terms of acquisition and development of more abstract mathematics concepts included in algebra and geometry?
The purpose of this study was to answer the question, “How does participating in an integrated algebra and physical science course affect the students’ conceptual organization of the concepts examined in the context of the course?” If curricular integration provides a conceptual context for doing mathematics and processing tools for contemplating science, students in an integrated course could, hypothetically, develop a more “integrated” notion about science and mathematics concepts. The result would be that students in integrated courses would develop a more cohesive, big-picture perspective of the interrelationships among concepts relevant to both disciplines. This inquiry examines that assertion.
Design and Methodology
Curriculum
The study was part of a year-long evaluation of an integrated science and mathematics curriculum called SAM9. The comprehensive, integrated algebra and physical science curriculum was developed for students in ninth grade. The course materials, written by the teachers and the researcher, were in the first year of implementation. The three-phase learning cycle (exploration, invention, and expansion) outlined by Renner and Marek (1990) served as the instructional and philosophical model for the curriculum. Implementation of the SAM9 curriculum required extensive collaboration and planning between the mathematics and science teachers; the course was team taught during a two-period block to allow sufficient time for laboratory explorations and student interactions. Emphasis in the course was placed on concept development, application of mathematical and scientific content to everyday problem solving, conceptual connections across disciplines, and authentic assessment techniques. (More specific descriptions of the SAM9 curriculum have been explicated in Williamson, et al., 1997, including a complete description of the philosophical framework for the curriculum.)
Setting
The project site was a rural high school in the southeastern United States. The school served approximately 825 students in grades 9 through 12. The project teachers included a physical science teacher and a mathematics teacher who had participated in an 8month project to develop the SAM9 curriculum. Both teachers were experienced professionals who had taught several different courses in their disciplines during their tenure as teachers.
Subjects
Approximately 100 ninth-grade students (average age 15 years 4 months) were involved in the study. Twenty-eight students were randomly assigned by the school computer to back-to-back Algebra I and Physical Science classes and subsequently participated in the SAM9 curriculum. The remaining students, representing a range of mathematical backgrounds (Algebra I, Algebra Ia, Algebra II, and Geometry), were enrolled in one of three Physical Science courses taught by the science teacher of the SAM9 team. Since it would be difficult to understand how the SAM9 algebra experiences were being constructed by the students if data were compared with those derived from students enrolled in other mathematics courses, the data set was modified to contain only the students in the Physical Science classes who were enrolled in an Algebra I course. The final samples consisted of 26 SAM9 students and 22 Physical Science-only (PSO) students. Equivalence between the groups was determined from the students ‘ scores on the standardized mathematics test administered at the end of the eighthgrade year. No significant between-group differences were found (one-factor ANOVA; p = .1765); for the purposes of this study, the mathematical abilities of the SAM9 and PSO groups were considered to be equivalent.
The Investigation
This study focused on the first learning cycle investigation of the school year. Density was the target science concept; slope was added to the learning cycle developed for SAM9 curriculum. Students in the SAM9 and PSO classes began the investigation by exploring the relationships between the masses and volumes of several solid and liquid substances. The concept that density is an intrinsic comparison of a substance’s mass to its volume was invented by the students during the subsequent class discussion of the data.
At the conclusion of the invention discussion, the SAM9 and PSO students embarked on different paths. The SAM9 students began the expansion phase with a graphical analysis of the data collected during the exploration. The graphical representations then became the focus of a class discussion, as the students made sense of the way the lines representing each substance looked on the graphs. The mathematics teacher led the students to the notion that the slope of each line resulted from proportional changes in the masses and volumes for each substance. The focus of the ensuing class discussion was on the idea that the slope of the line could be found by comparing the changes in the y-coordinates and the x-coordinates. The SAM9 students then engaged in activities to measure and calculate the slopes of several common landmarks (stairs, ramps, etc.) in and around the school. They also solved traditional and alternative problems related to slope and density. The culminating activity for the investigation was an exploration of flotation characteristics of various materials and the relation of flotation to density. Students were required to design and conduct an experiment to test a hypothesis about the concept of flotation.
The flotation inquiry was the only expansion activity in which the students in the PSO classes participated. It is important to note that the physical science teacher did require the students to graph the mass and volume data and make the same sorts of comparisons as in the SAM9 class. Formal instruction concerning slope, ratios, and proportions was not included in the classroom experiences of the PSO students, however.
Instrumentation
Concept mapping (Novak & Gowin, 1984) was employed to explore the connections the students made between traditionally segregated mathematics and science concepts. A concept map is a structural representation of a person’s understanding of a particular concept and the concept’s linkage to other ideas and examples. The maps are usually drawn using circles or boxes bearing the concept names. Lines are drawn between the concepts that the mapmaker views as related; words are written on the lines to show the nature of the relationship between the adjoined concepts. Concept maps are usually drawn to show hierarchical arrangements among the concepts; major concepts appear at the top of the map, while examples occur at the bottom of the map.
Concept maps were selected for use in the study because they have been previously reported to be useful in ascertaining students’ conceptual organization in science classrooms (Rogers, 1993) and in representing changes in students’ constructions of science concepts (Novak, 1990). Westbrook and Rogers (1996) used concept mapping to explore changes in students’ understanding of the concept of flotation and found the students’ mapping structures to be both stable with respect to persistent notions about the phenomenon and sensitive to changes in the students’ understanding of the concept. (See also Ruiz-Primo & Shavelson, 1996.)
The concept maps were completed by the students at three points in the learning cycle sequence: prior to the investigation (pre-exploration), after the invention discussion (postinvention), and at the conclusion of the expansion activities (postexpansion). Development of concept maps by the students occurred in two phases. Initially, the students were given a list of 20 words related to density, measurement, and graphing and were instructed to group the terms into five labeled categories of their choosing. After grouping the terms, the students were instructed to use the word sorts as a guide to constructing concept maps to show their ideas about the relationships among the concepts. The postexpansion word list included terms concerning motion (e.g., acceleration, deceleration, time, speed); the postexpansion map served as a preassessment for the unit to follow. Students’ map constructions of the motion terms were not considered within the scope of this study but will appear on the sample map included in Figure 1.
Analysis of Concept Maps
Concept maps generated by students participating in research studies are often compared by assigning numerical scores to various aspects of the mapping structures (Novak & Gowin, 1984; White & Gunstone, 1992). Students’ maps are generally examined for the presence and accuracy of one or all of the elements, propositions, hierarchy levels, and examples. In some cases, the students’ maps are compared to those developed by “experts.” In this study, however, the students’ concept maps were not used to determine right or wrong assumptions, the depth of knowledge about a particular concept, or the degree to which the maps paralleled those constructed by experts. The researcher sought to explore the nature and types of linkages the students’ made among concepts traditionally assigned to instruction in either the mathematics or science classroom. Consequently, analyses of the maps were not performed using a numerical rubric. The maps constructed by PSO and SAM9 students were initially examined for similarities and differences. Trends were delineated by selecting particular topics, phrases, or structures and investigating each student’s map. As patterns emerged, the researcher “tested” the trends through the complete set of maps. If a particular trend persisted across the set of maps-or if contradictory trends between the two groups were apparent-the trend became part of the results of the study. The researcher continued to “predict and test” trends in the students’ maps until all recognized leads were exhausted. This method, similar to that used in naturalistic research endeavors (Lincoln & Guba, 1985), was considered most applicable to the question posed by the researcher.
Findings
Beginnings
Analysis of the preinvestigation concept maps made by the students in both courses (SAM9 and PSO) indicated little understanding of the meaning of, or relationships among, the terms in the list. Eight students in each group (n = 22, SAM9; n = 17, PSO) linked the term ordered pair to some other term related to graphing. None of the students linked graph to density, but 3 SAM9 and 4 PSO students linked slope to graph. Overall, it appeared that the students in the two groups began the learning cycle with similar, and slight, understandings of the ideas that would be developed during the investigation.
Map Structures
The structures of the preinvestigation maps made by the students in the two groups were generally incomplete and simplistic and contained few linking words. The map structures of students in both groups developed significantly during the three-week investigation. The postinvestigation maps of the PSO students had little discernible hierarchical structure and were headed by terms like science or density. (See a recreation of “JH’s” map in Figure 1.)
Maps made by students in the SAM9 class typically had a more delineated structure. Common headings for the SAM9 postinvestigation maps included words and phrases like “science-math-graphing,” “science includes math,” and “algebra-science,” which were then used to divide the maps into separate sections related to each term in the heading. A postexpansion map drawn by “EG” is shown in Figure 2 as an example of a map “split” to separate science and mathematics concepts. Several students made separate maps to describe what they considered to be the major ideas. One student drew separate maps for the headings math, density, and slope. The term science was noticeably absent in her mapping strategy, but the two main concepts targeted (i.e., slope and density) in the learning cycle investigation were represented.
Linkages Among Key Ideas
One of the instructional goals for integrating the concepts of slope and density in the investigation was to assist the students’ constructions of the different representations available for analyzing and understanding data gathered in the classroom. The slope of graphed data provides the science student with a way to compare densities of substances and make predictions about an object’s flotation in those substances. The intention was that the students would build an understanding of slope that would later support construction of other concepts. As noted earlier, the physical science teacher required the PSO students to graph the mass and volume data generated from the exploration activities. No attempt was made to help the students develop an understanding of the concept of slope. The SAM9 students used the graphs of the density data to develop a concept of slope; other examples (not related to graphing) of slope were also explored.
The implications of the intentional development versus the algorithmic use of the mathematics concept of slope can be seen in the students’ maps. Six (23%) of the SAM9 students (n = 26) connected density to slope; 10 of the students (39%) linked density to graphing. Only 1 (5%) of the 20 final concept maps made by the PSO students indicated a relationship between density and slope, while 3 students made reference to a connection between density and graphing. Notably, 16 (62%) of the SAM9 and 11 (55%) of the PSO students linked slope and graph on the final mapping activity. It appears that the PSO students understood the connection between slope and graph, but were not aware of the relationship between slope and density or density and graph.
Relationships and Procedures
The maps were also examined to ascertain the types of linkages the students made among the terms on the map. Analysis of the maps indicated that the students tended to link terms according to the procedures through which the concepts on the maps might be linked and through the basic relationship (i.e., they were connected in some way) between the terms. Procedural linkages were defined as those valid connections among terms that indicated some action. Students denoted these linkages by using action verbs to link two (or more) terms. Relational linkages, on the other hand, were defined as any valid linkage between two terms that did not denote any procedure or action. Figure 3 shows examples of procedural and relational linkages excerpted from student maps. Although not always the case, the procedural linkages used by the students were generally more explicit than those that denoted nonprocedural relationships.
The students in the two groups showed relatively equivalent use of relational linkages among the concepts on which the investigation was focused. Fourteen of the 26 (53%) SAM9 and 11 of the 20 (44%) PSO postinvestigation maps showed relational linkages among density and mass and volume. The incidence of the use of procedural linkages, however, was greater among SAM9 students than PSO students. Fourteen of the SAM9 students (n = 26) drew procedural links among terms related to graphing (slope, ordered pairs, linear, nonlinear) on the final concept map. Three students (n = 20) in the PSO group showed an understanding of the procedural linkages among terms related to graphing. Those links were limited to “graph your ordered pairs” and “graph an ordered pair.”
Alternative Ideas
The maps drawn by students in the SAM9 class indicated certain alternative notions that did not occur on the maps drawn by the PSO students. Six of the 26 students in the SAM9 class showed an understanding of the link between density and slope; 1 of the PSO students indicated a link between those terms. Five of the SAM9 students linking density and slope, however, used the words “same as” to link the concepts. The SAM9 class investigated the relationship between density and slope, but the notion that density is the “same as” slope is an incomplete one. A more complete relationship could be stated as follows: “Density can be found by calculating the slope of a line on a mass vs. volume graph.” “KR” from the SAM9 class exhibited such an understanding by linking the terms this way: density “shown on a” graph “as” slope.
Another idea surfacing only on the maps drawn by the SAM9 students was “linear goes through (0,0).” This conception that linear is somehow linked to a line going through (0,0) was also “caught” on other student work. The notion likely resulted from the fact that in this first investigation the students only graphed data in which (0,0) was a logical data point; all resulting plots were linear. As the students gained more facility with a variety of data sets and types of plots, the notion was abandoned by most of the students. The researcher’s classroom observation notes, however, indicated that the concept “linear goes through 0,0” persisted as a viable idea for some students well into the fourth quarter of the school year.
Discussion
Analysis of the concept maps indicated that the students who experienced mathematics and science concepts that were intentionally and conceptually integrated (i.e., SAM9) delineated more linkages among the mathematics and science terms on the map than did their peers in the discipline-specific physical science class. The SAM9 students seemed more aware of the procedural connections between terms that linked the mathematics and science content. The maps generated by the SAM9 students also exhibited a more compartmentalized view of the mathematics and science content than did those of the students in the PSO classes. Specifically, the terms related to mathematics and science were drawn in different locations on the maps of many SAM9 students. This finding countered the original assertion that the SAM9 students would be more able to integrate the concepts. What would cause the SAM9 students to perceive mathematics and science as distinctly separate entities, while their PSO counterparts tended to blend the content of the two disciplines?
Teaching Territories
Reflection on the roles the teachers took in the investigation provided one explanation for the difference between the predictions and the outcomes of the SAM9 students’ conceptual organization. The science teacher initially controlled all the classroom interactions related to inventing the concept of density, including problem solving, equation development, and examination of patterns in the data set. The mathematics teacher taught only the portion of the investigation related to slope and graphing. This division of labor along traditional, disciplinary lines may have intensified the students ‘ beliefs that the content areas were separate entities with distinct boundaries. Those boundaries may have been made less obvious to PSO students, who engaged in exploratory and representational activities that were not “sorted” by the teacher. It is important to remember that the density/slope investigation was the first integrated investigation the mathematics teacher and science teacher taught together. As the school year progressed, territorial teaching of designated content areas became less frequent. In a later investigation about electric circuits, for example, the mathematics teacher led the discussion of the students ‘ data to assist the students ‘ development of Ohm’s Law, a science concept.
Conceptual Complexity
The differences in the quality of responses given by the PSO and SAM9 students on the concept maps also indicates that there may be a conceptual difference between integrating content at a “doing” level and integrating at an understanding level. For the most part, the students in the two groups did the same kinds of activities. The students in the PSO classes constructed and discussed graphs of their data just as the students in the SAM9 class did. The concepts foundational to graphing and solving equations, however, were not examined. “Plug and chug” was the method the science teacher claimed to use to teach her science students to solve mathematical equations; the algorithm “rise over run” was used to work problems related to slope on worksheets. The students in the SAM9 class were not only trying to make sense of the science data generated in the laboratory, but were also attempting to construct an understanding of the concepts of graphing, slope, and solving equations. The resulting conceptual complexity may have led to a need on the part of the students to make conceptual distinctions about the mathematics and science content they were learning. If this case is true, then students in integrated mathematics and science courses may need to “sort” concepts before they develop a more integrated framework with which to think about and act on those concepts. Conceptual sorting would then be an expected outcome of learning in a cross-disciplinary context. Further research is needed to examine this claim.
The students in the SAM9 class did-in the long run-construct connections among the concepts explored in the class. The researcher was able to observe one student’s (“FS”) integration of the density/slope concepts during an investigation of circuit electricity several months later. FS retrieved her graph of the mass and volume data and used the graph to construct an understanding of the current and resistance graph she made from her work with circuits. FS used her understanding of the relationship between the slopes of the lines and the relative densities of substances to make sense of a graph representing data collected about current and resistance in parallel circuits. This integration of both the concepts and processes of the two disciplines was a primary goal of the SAM9 curriculum.
Implications
The findings and assertions presented in this report represent trends suggested by a small sample of data. The sample size and the open-ended nature of the concept map certainly preclude any inferences beyond the classrooms in which these data were collected. The data do, however, provide beginning points for other researchers who wish to investigate student knowledge constructions within the context of an integrated mathematics and science classroom. How do students’ organizations of mathematics and science concepts develop in classrooms where mathematics and science are intentionally integrated? Do teaching “territories” influence the way students in a content integrated course develop an understanding of relationships among discipline-specific concepts? Is compartmentalization or “sorting” of concepts an intentional strategy in complex conceptual environments? Does that compartmentalization give way to more fluid understanding if the content is integrated for a school year or across several school years? What alternate notions are students likely to develop as a result of their experiences with an integrated mathematics and science curricula? Answers to such questions will help us better understand the processes that occur when students learn in classrooms in which cross-disciplinary curricula are implemented.
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Susan L Westbrook
North Carolina State University
Author Note: This research was funded in part by a grant from Eisenhower Grants for Higher Education and a grant from the North Carolina Board of Governors’ Small Grants for School Based Research Program.
Special thanks to the teachers in the project, Donna Cauley and Beronica Johnson. Without their long suffering and patience, the research would not have been possible.
Correspondence concerning this article should be addressed to Susan L. Westbrook, Department of Mathematics, Science, and Technology Education, North Carolina State University, Box 7801, Raleigh, NC 27695-7801. Electronic mail may be sent via Internet to westbrk@poe.coe.ncsu.edu
Copyright School Science and Mathematics Association, Incorporated Feb 1998
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