Connecting School Science

INTEGRATION MODEL

The Berlin-White Integrated Science and Mathematics (BWISM) Model identifies the following six aspects: (1) learning, (2) ways of knowing, (3) process and thinking skills, (4) conceptual knowledge, (5) attitudes and perceptions, and (6) teaching. We do not intend to imply that these aspects are isolated or exclusive of one another. The identification and elaboration of these as separate aspects is meant to provide some clarification of the characteristics that are in constant interplay as we seek to define, implement, and evaluate integration. These aspects, in various combinations, can serve as a basis to generate operational definitions and comparable research.

Learning

Integration can be viewed from the perspective of the learner and how scientific and mathematical concepts, processes, skills, and attitudes are processed and organized in the cognitive structure of the learner. Both science and mathematics educators value a constructivist view of learning (Piaget 1970; Vygotsky 1978) and the need to construct meaningful knowledge (Ausubel 1963; Novak and Gowin 1993). On the basis of cognitive research, the following general propositions may serve as the infrastructure for the development of integrated school science and mathematics:

• Knowledge is built on previous knowledge.

• Knowledge is organized around big ideas, concepts, or themes.

• Knowledge involves the interrelationship of concepts and processes.

• Knowledge is situation- or context-specific.

• Knowledge is advanced through social discourse.

• Knowledge is socially constructed over time.

Ways of Knowing

Science seeks to advance knowledge through the observation and manipulation of phenomena in order to explore the nature of the environment and human existence in that environment. Science searches for consistent and verifiable patterns to build a knowledge base and explain the real world. This way of knowing may be characterized as one of induction whereby a pattern or generalization emerges or is revealed from the data.

In mathematics, the quest for knowledge more frequently involves modeling in search of patterns and relationships that are not bound by the observable world and its relationships. The phenomena may not be readily observed or even envisioned because of extreme conditions such as size (e.g., very small or very large), time (e.g., past or future), or degree of abstraction. Mathematics often involves modeling and the use of logical, symbolic systems to describe patterns and relationships. These symbols can be manipulated without the constraint of reality or the need for concrete representation.

Figure 3.1 illustrates how the processes of induction and deduction are cyclically interrelated. The observation of the environment, collection of data, and search for relationships (induction) can deliver a basis for quantitative description. Mathematics can be used to express and communicate the relationships in the data as models (graphic, symbolic numeric, geometric, or functional). These models may be subjected to mathematical transformations that can then be used to make predictions (deduction). Often the mathematical models can predict real-world phenomena that have not as yet been observed or even contemplated. For example, a mathematical model could predict the existence of planets in the solar system or subatomic particles not yet discovered. These predictions can guide further exploration of the system within which the prediction was made, leading to additional observations and manipulations to verify or refute the predictions. If the predictions are verified, new knowledge and theories may result. If refuted, the results may be reconsidered mathematically and a revised model constructed.

Fig. 3.1. Ways of knowing: induction and deduction (reproduced from Barnes et al. [1992], with permission from the National Center for Science Teaching and Learning, A. L. White, Director.

The most significant feature of this "ways of knowing" cycle may be the importance of the symbiotic relationship between the inductive and deductive processes. It is a reasonable goal to provide ample opportunity for all students to acquire an understanding of the connections between these ways of knowing. If students are to benefit from the symbiotic relationship and power of these ways of knowing, the increased use of mathematical modeling in science classes and the use of student-generated scientific data in mathematics classes are recommended. Attention to this "ways of knowing" cycle aggregates the processes of induction and deduction to promote an integrated, holistic view of the generation of knowledge.

Process and Thinking Skills

Integration can be viewed from the perspective of process and thinking skills valued in both science and mathematics. Although the terms may differ, the mathematics curriculum standards recognizing problem solving (inquiry), reasoning, communication, and connections (integration) emerge as processes central to school science as well as mathematics (National Council of Teachers of Mathematics [NCTM] 1989; National Research Council 1993b, 1993c).

A review of the literature reveals widespread support for the inclusion of process (or inquiry) skills in science curricula. Science is a dynamic, ongoing, tentative search, and the process skills reflect the nature of science and the typical activity of scientists; they are readily transferred to other problem situations, and they generalize to real-life problems (American Association for the Advancement of Science 1963; National Research Council 1993b, 1993c; Padilla 1986). The National Science Teachers Association and numerous state and local curriculum specialists have emphasized process skills as specific, planned outcomes of science programs. The development of basic and integrated science process skills has been a major emphasis of science education programs since the 1950s (Tobin and Capie 1980). The basic process skills include observing, inferring, measuring, communicating, classifying, and predicting. The integrated process skills include controlling variables, defining operationally, formulating hypotheses, interpreting data, experimenting, and formulating models. (See Padilla [1986] for definitions of these skills.) These same science process skills are integral to the development of problem-solving ability, which has been strongly endorsed as the primary focus of school mathematics (NCTM 1980, 1989).

Champagne (1992) has developed a comprehensive taxonomy of science and mathematics thinking skills that includes reasoning, metacognitive, information-manipulating, information-management, formal reasoning, and symbolic-representation skills. Integrated science and mathematics activities have the potential to engage students in challenging, authentic, relevant problem-solving tasks that provide for the application and practice of higher-order thinking skills.

Conceptual Knowledge

Integration can be viewed from the perspective of the overlap of conceptual knowledge of science and mathematics. This requires an examination of the concepts, principles, and theories of science and mathematics to determine which ideas are unique to science or to mathematics and which ideas may overlap. An examination of current (NCTM 1989; Rutherford and Ahlgren 1990) and emerging (National Research Council 1993b, 1993c) documents on curriculum standards reveals a shared commitment to the development of a limited number of fundamental concepts. Topics common to both science and mathematics curricula include the study of measurement, patterns and relationships, probability and statistics, spatial relationships, and variables and function. Examples of ways to interrelate science and mathematics concepts could include (a) natural selection with sampling, randomization, probability, ratio, proportionality, and area; (b) activation energy and reaction rates with the area under the curve of a distribution; (c) population growth and genetics with sampling and probability; and (d) reflection and refraction with angular geometry.

Often concepts previously identified as "science" or "mathematics" can be related through analogous embodiments. For example, the variables that describe the properties of a lever and those that describe the characteristics of a frequency distribution are analogous and can link school science and mathematics. The balance point of a lever is the fulcrum. This is the point around which the clockwise and counterclockwise forces on the lever system are equal, or "balanced." Similarly, the mean of a distribution is the point in a frequency distribution around which the positive and negative deviation scores are equal and result in a "balance" point for the distribution. There are times when the concepts of science and the concepts of mathematics have enough in common for it to be more meaningful to the student and more efficient to connect the learning of these concepts (Penafiel and White 1989; White and Berlin 1989).

Attitudes and Perceptions

Integration can be viewed from the perspective of attitudes and perceptions related to the development of scientific and mathematical literacy. The beliefs and perceptions one holds about science and mathematics as disciplines may also be similar. Finally, creating an environment and providing opportunities to enhance one's scientific and mathematical self-efficacy may serve as another affective connection.

Current science- and mathematics-education reform documents are quite explicit about valued scientific and mathematical attitudes (NCTM 1989; National Research Council 1993b, 1993c). The following list (Loucks-Horsley et al. 1990, p. 41) has been modified to embody shared attitudes:

• Desiring knowledge: viewing science [and mathematics] as a way of knowing and understanding

• Being skeptical: recognizing the appropriate time and place to question authoritarian statements and "self-evident truths"

• Relying on data: explaining natural occurrences by collecting and ordering information, testing ideas, and respecting the facts that are revealed

• Accepting ambiguity: recognizing that data are rarely clear and compelling, and appreciating the new questions and problems that arise

• Being willing to modify explanations: seeing new possibilities in the data

• Cooperating in answering questions and solving problems: working together to pool ideas, explanations, and solutions

• Respecting reason: valuing patterns of thought that lead from data to conclusions and, eventually, to the construction of theories

• Being honest: viewing information objectively, without bias

Another shared affective consideration is to promote science and mathematics as human enterprises. As such, they are approximations of truth and subject to change. Integrated experiences in science and mathematics may serve to dispel the notion of these subjects as unchangeable, irrefutable, proved bodies of knowledge that can furnish correct answers to all questions.

Integrated science and mathematics experiences based on personal and social issues and interests may motivate students to achieve. Opportunities to be successful help to encourage, support, and nurture students' confidence in their abilities to do science and mathematics. The promotion of scientific and mathematical efficacy may serve to counter the perceptions of science and mathematics as difficult, accessible to only a select group of individuals (e.g., Anglo males), and not essential to the general populace.

Teaching

Integration can be viewed from the perspective of science and mathematics teaching methods and strategies that overlap and are supportive of one another. Teaching, as related to integrated science and mathematics education, includes four dimensions: the structure and organization of the learning environment, instructional strategies, assessment, and the changing role of the teacher. The goal of integrated science and mathematics teaching is to enable students to "acquire both scientific and mathematical knowledge of the world and scientific and mathematical habits of mind at the same time" (Rutherford and Ahlgren 1990, p.190). Such an environment would include a broad range of content, give time for inquiry-based learning, stimulate and support discourse, furnish opportunities to use laboratory instruments and other tools, provide appropriate and ongoing use of technology, encourage alternative assessment procedures, and maximize opportunities for successful experiences. Learning experiences should engage students in connecting qualitative and quantitative descriptions of their world. Modeling tools such as STELLA (Structural Thinking, Experimental Learning Laboratory with Animation, High-Performance Systems, Inc., Hanover, NH 03755) can help students to make the connections (Linn 1986).

Integrated teaching strategies should be based on cognitive research into how students learn in general and how they learn science and mathematics in particular. Initial activities should focus on interesting phenomena arising from, and applied to, real-world situations. There should be ample opportunity to develop thinking and reasoning skills; both basic and integrated process skills; and problem-solving, decision-making, and communication skills. Activities should involve students in doing, analyzing, and reflecting about science and mathematics and in acquiring scientific and mathematical power. A wide variety of school and community instructional resources involving multiple sensory modes (tactile, auditory, and visual) should be available. Cooperative-learning strategies are recommended to reinforce the collaborative nature of scientific and mathematical endeavors.

A variety of useful strategies to integrate across disciplines has been proposed by Fogarty (1991a, p. xv; 1991b, p. 63):

• Sequenced-- topics or units of study are rearranged and sequenced to coincide with one another. Similar ideas are taught in concert while remaining separate subjects.

• Shared-- shared planning and teaching take place in two disciplines in which overlapping concepts or ideas emerge as organizing elements.

• Webbed-- a fertile theme is webbed to curriculum contents and disciplines; subjects use the theme to sift out appropriate concepts, topics, and ideas.

• Threaded-- the metacurricular approach threads thinking skills social skills, multiple intelligences, technology, and study skills through the various disciplines.

• Integrated-- this interdisciplinary approach matches subjects for overlaps in topics and concepts with some team teaching in an authentic integrated model.

Alternative, authentic assessment procedures are suggested to meaningfully link integrated science and mathematics curriculum, instruction, and assessment. Embedded assessments may include performance assessments, projects, and portfolios. A number of current books related to alternative science (Hein 1990; Kulm and Malcolm 1991; Ostlund 1992; Raizen et al. 1989, 1990) and mathematics (Kulm 1991; Lesh and Lamon 1992; NCTM 1993; National Research Council 1993a; Romberg 1992; Stenmark 1989, 1991) assessments are available to assist classroom teachers in planning appropriate and meaningful experiences to assess what students know and to improve instruction.

Of significance to the teaching of integrated science and mathematics is the changing role of the teacher. Teachers and students become natural partners in problem solving and in developing their own capability for lifelong learning (NCTM 1989). The teacher may be viewed as an intellectual coach assuming such various roles as role model, consultant, moderator, interlocutor, and questioner (National Research Council 1991). Teachers as specialists and the use of teaching teams emerge in the integrated approach.

An Example: "Just Drop It!"

In this investigation, students will compare the heights of bounces that result from dropping different types of balls from different heights. The bounces resulting from dropping the balls on different surfaces can also be compared. From the collected data, a formula relating the height of the bounce to the height of the drop will be developed. Using the formula, students will be able to accurately predict how high a ball will bounce from any given height.

This activity from Math + Science: A Solution (Wiebe and Ecklund 1987) illustrates the interdependence of the aspects of the integrated science and mathematics model. Children are involved in observing, experimenting, measuring, recording data, searching for patterns, interpreting, applying, predicting, and generalizing. The concepts and principles of science and mathematics developed in the activity include averages, coeffficients, conservation, constants, elasticity, equations, friction, graphs, mathematical modeling, ratio, and slope.

The investigation can encourage and support instructional methods valued in both science and mathematics education. It is suggested that children work in cooperative groups of two to four while engaged in inquiry and problem solving. This methodology can promote the development of valued scientific and mathematical attitudes as well as positive habits of mind or dispositions. Othcr valued instructional methods embodied in this activity include hands-on manipulation of the environment; varied representational modes (e.g, drawings, diagrams, graphs, formulas); building on past experiences; the personalization of learning; the use of familiar and available resources (e.g., balls used in tennis, golf, basketball, baseball, soccer, pool, etc.); varied levels of questioning (ranging from those requiring low-level thinking to those requiring higher-order cognitive processing); the use of a simple, four-function calculator to average the data; and the use of a graphing calculator for data fitting and modeling. Consistent with the model, this activity furnishes numerous possibilities for making appropriate and effective connections between science and mathematics teaching and learning.

Resources

Since the 1970s, a number of notable curriculum programs or supplements have been produced. Among these are the Minnesota Mathematics and Science Teaching Project (MINNEMAST-- Minnemast Center, 720 Washington Avenue, S.E., Minneapolis, MN 55414; MINNEMAST materials are no longer in print, but they can be obtained from the ERIC Document Reproduction Service, CBIS Federal, 7420 Fullerton Road, Suite 100, Springfield, VA ), Unified Science and Mathematics for Elementary Schools (USMES-- Educational Development Center, Newton, MA 02160; USMES materials are no longer in print, but they can be obtained from the ERIC Document Reproduction Service, CBIS Federal, 7420 Fullerton Road, Suite 100 Springfield, VA ), Activities Integrating Math and Science (AIMS-- AIMS Educational Foundation, P.O. Box 8120, Fresno, CA ), Great Explorations in Math and Science (GEMS-- Lawrence Hall of Science, University of California at Berkeley, Berkeley, CA 94720), the Jasper Series (Optical Data Corporation, 30 Technology Drive, Warren, NJ 07059) Teaching Integrated Mathematics and Science (TIMS-- University of Illinois at Chicago, Box 4348, Chicago, IL 60680), The Voyage of the Mimi and The Second Voyage of the Mimi (Sunburst/Wings, 101 Castleton Street, P.O. Box 100, Pleasantville, NY ), and School Science and Mathematics Integrated Lessons (SSMILES--School Science and Mathematics Association, Bloomsburg University, 400 East Second Avenue, Bloomsburg, PA ). Two documents to be published by the National Center for Science Teaching and Learning will soon be available as a resource for classroom teachers interested in integrated science and mathematics concepts, skills, and processes. The documents include a database describing approximately three hundred integrated science and mathematics activities (Berlin 1994) and a compilation of SSMILES (School Science and Mathematics Integrated Lessons) previously published in the journal School Science and Mathematics (Berlin in press). Materials related to consumer education, energy education, environmental education, nutrition education, technical education, and vocational education are also fertile areas for integrated science and mathematics activities.

CONCLUSION

If the implementation of integrated school science and mathematics is to occur in a rational and realistic fashion, it will be crucial to have teachers involved in the exploration and amalgamation of the six aspects as well as in gathering information on the effects of the integration. The model outlined in this paper can serve to guide educators, administrators, policy makers, and researchers in the development, implementation, and evaluation of school science and mathematics integrated activities and programs.

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