6
	Science and Mathematics Education: Finding Common Ground
	Jane Butler Kahle
	Miami University

The current reforms in both science and mathematics education have many commonalities; in fact, for the first time, the two disciplines are advancing with common goals and objectives, as evidenced in the mathematics standards published by the National Council of Teachers of Mathematics (NCTM, 1989, 1991, 1995), and the science standards published by the National Research Council (NRC, 1996). Both sets of standards are based on the premises that all children can learn challenging mathematics and science, that literacy in both disciplines is necessary for productive work in the future, that learners construct their own knowledge, and that there are many effective ways to promote knowledge construction.

     One way to elaborate such standards is to provide tasks that illustrate some of the ideas promoted by those standards. Tasks that fit with these standards have common characteristics: they require time, they allow multiple solution paths, they are open-ended, they may be revisited and extended, and they develop basic skills. Individually, each characteristic is applicable to both science and mathematics; and collectively, these characteristics delineate a practical route from the rhetoric of standards to the reality of student achievement. Many of the skills involved in doing these tasks are critical to success in the sciences, as well as in mathematics.

     Useful and compelling mathematics tasks illustrate both the logical and algorithmic nature of mathematics as well as its whimsy and beauty. These dual goals again make a connection with science. A national study of mine (Kahle, 1983) indicated that both ninth- and tenth-grade girls and boys were motivated to continue to study advanced science when their science teachers stressed both the basic skills of science (many of which may be learned in mathematics) and its more creative elements. I believe that problems that illustrate the whimsy and beauty of mathematics will encourage and excite many students who heretofore have been turned off to mathematics.

     Mathematics instruction that fits with the NCTM Standards includes tasks that actively engage students in making meaning of mathematics and in proposing several possible solution processes. Such activities provide sites in which the NCTM process standards--mathematics as problem solving, mathematics as communication, mathematics as reasoning, and mathematical connections--can be emphasized and developed.

     Science instruction consistent with the National Science Education Standards (NSES) (NRC, 1996) is characterized by similar parameters: students identify a problem from their observations of nature; they propose several solutions (hypotheses); in any one class, their investigations take multiple approaches; and they discuss and consider all reasonable solutions. The science as inquiry standards in particular promote the following processes:

. . . asking questions, planning and conducting investigations, using appropriate tools and techniques to gather data, thinking critically and logically about relationships between evidence and explanations, constructing and analyzing alternative explanations, and communicating scientific arguments. (NRC, 1996, p. 105)

     Practically speaking, it is possible to write tasks that are directly applicable to both mathematics and science lessons. In this volume, Drug Dosage (p. 80) is a task in which students may either use a mathematical model to understand a scientific context or use a scientific context to understand mathematical ideas in the model. Because of their availability and low cost, paper towels are a common context for tasks that integrate mathematics and science¹ . In order to compare brands for absorbency, strength when wet, or cost per sheet, or to investigate concepts such as the relationship between the number of water drops absorbed by the towel and the area of the wet spot, students design experiments and collect and interpret data.

     The NSES includes an excellent example of an activity, "The Solar System," for an integrated mathematics and science program (NRC, 1996, pp. 215-217). The goal of the activity is to have students construct a scale model of the sun, moon, and earth using techniques developed by early astronomers. Students observe the stars, discuss the patterns they observe, and use a particular pattern (the North Star, Polaris, doesn't appear to move) to suggest methods for estimating the circumference of the earth. In particular, if they know angle of the North Star at two locations on the same longitude and a known distance apart, they may estimate the circumference of the earth by using a two-dimensional diagram of the three-dimensional situation (see Figure 6-1), and by using geometric knowledge about circles, tangents, and angles. Through activities such as this, students can see not only that geometric understanding is necessary to understand the science problem, but also that science provides contexts for geometric and mathematical ideas.

Figure 6-1: A diagram from the activity "The Solar System"
Source: NRC, 1996, p. 216.

     These possibilities arise because of common skills that are needed to study both mathematics and science. For example, in both disciplines, students need to be able to estimate, to use mathematical models, to interpolate and extrapolate, to identify false negatives, to detect bias, to convert two-dimensional drawings to three-dimensional models and vice versa, to make and interpret graphs and other diagrams, and so on. Furthermore, when students use data gathered in science investigations in their mathematics courses, they encounter many of the anomalies of authentic data: inconsistencies, outliers, and errors. Indeed, tasks that build these kinds of skills are good examples of activities through which it should be possible to develop aspects of the scientific literacy stressed in the NSES (NRC, 1996), as well as the mathematical understandings promoted in the NCTM's Curriculum and Evaluation Standards for School Mathematics (NCTM, 1989).

     Because of these connections between mathematics and science, the NSES calls for coordinating the science and mathematics programs in schools. Such coordination results in opportunities to advance instruction in science beyond purely descriptive studies and to provide mathematics classes with authentic problems. The NCTM Standards documents (NCTM, 1989, 1991, 1995) also clearly encourage making connections between mathematics and the sciences, particularly in the sections on mathematical connections. The documents note the pervasiveness of the connections between mathematics and other disciplines and encourage such connections in the classroom as a means of enabling students to see the utility of mathematics and to find motivation for mathematics.

     We have been singularly unsuccessful in developing a long-term relationship, let alone a successful marriage, between science and mathematics as they are usually taught in schools. An often-promoted solution is simply using science examples in mathematical problems. This solution is too simple, obviously, and has not been successful in the past. However, both the NSES and this volume provide glimpses of what will work. First, students need to be able to integrate the scientific skills of observing, classifying, inferring, collecting, and interpreting data, using mathematical skills such as reasoning, computing, communicating, and making connections. For all students, the process of doing both good science and good mathematics holds the most promise of successful integration, an integration constructed by each learner as her or his skills and understandings develop. Mathematics at Work may help inspire a future in which value is placed upon the processes, not products, of learning, where science and mathematics are integrated through common skills, where relevant activities that integrate mathematics and science are readily available for each student, and where the common ground that is shared by those interested in high-quality mathematics and science education is explicit.

References

Kahle, J. B. (Ed.). (1983).

Girls in school: Women in science. Washington, DC: National Science Board, Commission on Precollege Education in Mathematics. (ERIC Document Reproduction Service No. ED 258 812).

National Council of Teachers of Mathematics. (1989).

Curriculum and evaluation standards for school mathematics. Reston, VA: Author.

National Council of Teachers of Mathematics. (1991).

Professional standards for teaching mathematics. Reston, VA: Author.

National Council of Teachers of Mathematics. (1995).

Assessment standards for school mathematics. Reston, VA: Author.

National Research Council. (1996).

National science education standards. Washington, DC: National Academy Press.

Sneider, C. I. & Barber, J. (1987).

Paper towel testing. Great Explorations in Math and Science (GEMS) Project. Berkeley, CA: Lawrence Hall of Science, University of California, Berkeley.

Teaching Integrated Mathematics and Science (TIMS) Project, Inc. (1992).

Spreading Out II. Chicago: University of Illinois at Chicago.

Note

¹ See, for example, Spreading Out II (Teaching Integrated Mathematics and Science (TIMS) Project, Inc. 1992) and Paper Towel Testing (Sneider & Barber, 1987).

Jane Butler Kahle is Conduit Professor of Science Education at Miami University. She is Chair of the Committee on Science Education, K-12. She has been President of the National Association of Biology Teachers and the National Association for Research in Science Teaching as well as Chair of National Science Foundation's Committee on Equal Opportunities in Science and Engineering and Biological Sciences Curriculum Studies' (BSCS) Board of Directors.