Talk of integration is, for most educated liberals, a rhetorical engine that can be coupled to any trainload of goods that needs to be pulled over the mountain pass of public opinion. Surely no one can object to the ideal of integration. The alternative--disintegration, segregation, isolation--leaves good intellectual produce stranded on a siding with no hope of forward motion.

The goal in education is to get our train--our students--over the mountain pass. In our haste to reach the summit, we should not ignore the key question of strategy: can we climb the long grade better with one giant train led by the powerful engine of integration or with several smaller trains progressing along different tracks and at different speeds?

A strategic analysis of "integration" requires what economists call "disaggregation." We begin by examining three basic issues--philosophy, coherence, and instruction--and then conclude with some cautious recommendations.

Philosophy

It is easy to imagine several possibilities for how the proposed change might come about:

• By employing mathematical methods thoroughly in science instruction, taking the necessary steps to coordinate the curricula of the two subjects. This would have great benefits both for mathematics and for science. For mathematics, it would ensure that students see the mathematics they study actually being used, and it would reinforce their learning. For science, it would help advance instruction from the present descriptive (almost pre-scientific) stage to a form that provides an honest introduction to modern scientific method. Adding modest amounts of mathematics will help make school science scientific.
• By employing scientific examples and methods thoroughly in mathematics instruction, taking necessary steps to coordinate the curricula of the two subjects. This too would have great benefit both for mathematics and for science, but in uncommon ways. For mathematics, it would reinforce the perspective of investigation, exploration, and experimentation that is so important to all contemporary curricular recommendations. For science, it would help underscore the importance of careful data analysis, logical thinking, and modeling as part of the scientific method.
• By teaching mathematics entirely as part of science--as its language and ubiquitous tool--in order to better motivate and reinforce the study of mathematics while at the same time strengthening the quantitative component of science instruction. I suspect that this would be a disaster both for science and for mathematics. Few teachers are equipped to give both subjects the emphasis they deserve. Mathematics teachers are likely to slight (and distort) science in their haste to cover traditional mathematics uncontaminated with the vagaries of real data; science teachers are likely to select only a part of mathematics that is necessary and useful for the science they are teaching, leaving untaught vast parts of the subject that are necessary for later work but not directly applicable to school science.
• By teaching science entirely as part of mathematics--as its principle application. This too, I fear, would produce a disaster for the same reason: no teachers are prepared to do equal justice to both subjects. (I must add that there are more likely to be positive exceptions in elementary school than in middle school or high school, since at the elementary level the same teacher normally teaches both subjects. Of course, relatively few elementary teachers are really well prepared in either science or mathematics. The limited number who are capable of delivering a properly integrated curriculum would not make a significant dent in the problems of U. S. education.)
• By employing mathematical methods thoroughly in science, and scientific methods thoroughly in mathematics, coordinating both subjects sufficiently to make this feasible. This is, I submit, an ideal situation. Each discipline, science and mathematics, would accrue benefits from an infusion of methods of the other, but neither would lose its identity or distinguishing features in an artificial effort at union. There are, after all, important differences between science and mathematics, both philosophical, methodological, and historical. These should not be lost in a misguided effort at homogenization.

Students gathering data in a science or social studies course can be asked to use elementary tools of data analysis to organize these data and to formulate conjectures for further testing. By using real data, students will encounter all the anomalies of authentic problems--inconsistencies, outliers, errors. Children can, of course, practice arithmetic on any data they gather. Examples of ratios and proportional reasoning abound in science, providing extensive experience in a very important topic that students often do not master. Older students can gain good experience in routine mathematical tools of graphing, calculation, and simple algebra (in curve fitting). Beginning algebra students can fit lines by eye, then figure out their equations. More advanced students can investigate transformations to linearize non-linear data.

In a similar fashion, students studying mathematics can be asked to apply what they learn to situations in the world around them. Children can be introduced to scientific strategies of observation, recording, and calculation with numeric data such as records of rainfall or temperature. Collections of leaves can be used to develop non-numeric habits of classification and pattern-recognition. Geometry students can be asked to lay out the foundations of a building with string and stakes, and discover just how important the word "plane" is in plane geometry--and how difficult it is to achieve in the real three dimensional world. Algebra students can be asked to gather data and make conjectures about patterns in algebraic structures, and then try to prove them.

Clearly there are many possibilities of such integrating activities. My point in citing these examples is not to catalogue what might be done, but to illustrate a very important difference in perspective. In teaching science, one would want to employ whichever mathematical techniques are suitable to the task, whereas in teaching mathematics one would seek whichever science domains help illustrate and apply the mathematics. These differences in perspective are part of the instinct of the professionals who teach science and mathematics, and they are, for the most part, entirely justifiable.

These differences are essential to proper understanding of a fundamental difference between mathematics and science:

Science seeks to understand nature.
Mathematics reveals order and pattern.

So too are the crafts of mathematics education and science education. Good mathematics teachers develop various strategies to help students construct for themselves the mental structures of mathematics that will become a unique part of their own working intelligence. Similarly, science teachers lead students to develop the habits and instincts of a working scientist, to make the scientific method part of their own repertoire of approaches to the world.

Effective education, therefore, must not only teach students about science and mathematics, but teach them in what respects they are similar and in what respects they are different. There is no intrinsic value to an educational program of integration whose primary purpose (or effect) is to diminish student understanding of these essential differences.

Coherence

• Planning appropriate coordination of mathematics and science. Effective instruction will depend on a curriculum that introduces important topics in both science and mathematics in a suitable order with appropriate linkages between them. This will not be too difficult in the elementary grades, but as mathematics and science curricula proliferate and diversify in the middle and upper grades, it will be increasingly difficult to develop a workable model even on paper, much less in the classroom.
• Implementing coordination whenever student electives begin to affect curricular choices. Coherence in this context seems to be impossible. Each chemistry class, for instance, will have students whose mathematical preparation spreads out over three or four years of high school mathematics. Similarly, advanced mathematics classes will have many students who do not elect to take advanced science courses. In such a context, the liabilities of an integrated curriculum quickly overwhelm the advantages.
• Coordination within science itself. Science is not really homogeneous; there are many sciences with different traditions and methodologies. Coordination among these sciences will be at least as difficult, if not more so, than coordination between science and mathematics. While it may be possible, albeit difficult, for some scientists and teachers to agree on a coordinated multidisciplinary pattern for biology, chemistry, earth science, and physics, such a plan will do little good so long as other scientists and teachers argue vigorously for interdisciplinary science following themes such as environment, health, and energy. Surely there is little prospect for thorough integration with mathematics if the science curriculum itself is up for grabs. The NCTM Standards provide a contemporary fixed point in curriculum planning for mathematics. Where is there a comparable document for science that enjoys as widespread support?

Instruction

Science teachers do not have sufficient breadth across the sciences to teach all of science in an integrated fashion. Very few school science teachers, for example, are sufficiently comfortable with physics to introduce students in a suitable fashion to fundamental physical concepts such as mass, force, momentum, and energy, much less to angular momentum or action at a distance. Teachers who teach out of their zone of comfort too often teach only vocabulary and terms, since that is all they really know.

Except for those prepared in chemistry and physics, most science teachers do not have sufficient preparation in quantitative methods to successfully integrate mathematics and science, much less to teach mathematics beyond the level of the middle school curriculum. Teaching high school mathematics really does require as preparation a full undergraduate major in mathematics. Current science teachers do not have this background, and prospective science teachers could not be expected to acquire it unless they undertook a six year program of teacher preparation.

Finally, to complete my litany of liabilities, few mathematics teachers are well prepared in even one science; practically none are competent in all. What's worse, the predominant emphasis of school science has strongest links to the biological and life sciences, which is the area in which mathematics majors are typically least prepared. Moreover, unless mathematics teachers have studied a lot of science, they are not likely to understand or empathize sufficiently with the observation-rich, hands-on, laboratory-intensive aspects of the scientific method.

Silver Linings

There are, however, some silver linings in this thundercloud. Elementary school is an obvious exception to many of my concerns about philosophy, coherence, and instruction. I believe it should be possible to develop a good coherent joint curriculum in science and mathematics in the first 4-6 grades, doing justice to both fields while also laying sound foundations for future study. The major impediment would concern the number of teachers who are capable of teaching such a curriculum. It is clearly small. But it is also possible, and important for the nation, to educate a cadre of elementary school mathematics-science specialist teachers who would be both enthusiastic about science and also capable of teaching a coordinated curriculum.

My second silver lining is actually the beginnings of a sunburst: instead of worrying about integrating content, let's think instead about integrating instructional methodologies. Exploratory, investigative, discovery learning that is typical of the best science instruction is one of the features of the new NCTM standards. Children learn by doing, their actions helping construct their personal knowledge. Involvement in learning increases, as does long-term retention. Active, exploratory learning works as well in mathematics as it does in science.

Similarly, the compelling logic of inference and deduction can help students experience the special power of science. Absent the rigorous logic of inference that is typical of mathematics, science instruction can easily degenerate into description, demonstration, and memorization. Without the intrinsic authority of inference, the epistemology of school science becomes extrinsic, hence heretical: students believe what teachers tell them, not what they have logically demonstrated from evidence. If the methodology of science is to be faithfully expressed, the essence of mathematics must be taught as part of science.

So learning theory suggests many plausible benefits to blending (or "integrating," if you insist) the ways in which mathematics and science are taught. These benefits are important for students, but unfortunately that argument is insufficient to bring about significant change. We all know that what really matters is politics.

It may just be that blending methods is better politics than blending content. To infuse methods of science and mathematics into each other's instruction, one avoids the absolute need for careful coordination, with the inevitable controversy and painful compromise. Methodological issues such as exploration, group work, data collection, discussion, argument are activities that apply to all ages. The issues of symmetry or of dominant style also vanish: science teachers can stress the methods that suit science, using more quantitative and mathematical approaches as supplements whenever appropriate and to the extent that they prove effective. Similarly, mathematics teachers can use a blend of presentation and discovery method in whatever balance they find useful. Neither must feel threatened by domination from the other.

Finally, about instruction. By blending methodologies instead of content, one preserves the established traditions of essentially separate science and mathematics teacher preparation where scientific and mathematical content predominate. In school, teachers could engage in paired teaching, or team teaching, with the science teacher helping the mathematics teacher learn how to make productive use of exploratory assignments while the mathematics teacher helps the science teacher see how to introduce quantitative, logical methods into science teaching. The system as a whole then builds on the strengths of the corps of teachers as a whole, rather than floundering on the inevitable weaknesses of individual teachers when confronted with the task of teaching an integrated science and mathematics curriculum.

So, in conclusion, the proposition I put before you for discussion is really quite simple: integrate how you teach before worrying so much about integrating what you teach.