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.
PhilosophyThe language of the conference theme sounds symmetrical, as if the changes in science education required to achieve curricular integration will be comparable to the changes in mathematics education required for the same objective. But since the relationship between science and mathematics is not itself symmetrical--mathematics is the language of science, not the other way around--it is not at all clear whether the goal of integration is best described by a word with symmetrical connotations.
It is easy to imagine several possibilities for how the proposed change might come about:
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:
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.
CoherenceI come now to my second key issue: how one could coordinate mathematics and science in a coherent curriculum. Regardless of which model for integration one selects, careful coordination will be required. Again, I foresee several hurdles, most of which, unfortunately, appear to be insurmountable:
InstructionFinally, to the crux of the matter: teaching and learning. Is it possible--in this real world, not some Platonic universe--to teach an entire curriculum that integrates science and mathematics? The answer, I believe, is very simple: No. Indeed, since we cannot seem to teach either discipline separately very well, why should we think we could succeed with the vastly more difficult task of an integrated curriculum?
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 LiningsI don't enjoy providing such a negative analysis, and rather hope that perhaps others will be able to prove me wrong. Nevertheless, I don't see any escape from the general conclusion that any broad-brush attempt to integrate curriculum and courses in science and mathematics is doomed to failure.
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.
|Copyright © 1999.||Contact: Lynn A. Steen||URL: http://www.stolaf.edu/people/steen/Papers/integrating.html|