by Hugh Burkhardt
An international observer, even one as friendly to, and involved in, the U.S. scene as I am, has difficulty viewing integrated mathematics as an issue. Like creationism in science education, the dis-integrated high school mathematics curriculum appears to be a self-inflicted wound, peculiarly American. Nowhere else in the world would people contemplate the idea of a year of algebra, a year of geometry, another year of algebra, and so on. I believe that my remarks fairly represent the accepted view in other countries. However, all educational systems have a huge resistance to change. So having recorded my naive observation that the emperor's clothes, albeit old, are still invisible, I will summarize the arguments.
The main advantages of integrated curricula are that they build essential connections, help make mathematics more usable, avoid long gaps in learning, allow a balanced curriculum, and support equity. I know of no comparable disadvantages, provided that the "chunks" of learning are substantial and coherent.
Building a student's robust cognitive structure, one that can be used flexibly and effectively in solving problems, depends on linking new concepts and skills with the student's existing understanding. This happens through active processing over an extended period, first of weeks as the curriculum points out key links, ultimately over years as the concepts are used in solving problems across a variety of contexts.
Compartmentalizing mathematics inhibits building such connections. For example, the different functions that represent the scaling of lengths, areas, and volumes are a practical example of links between algebra and geometry and the real world. The profound fact that doubling all lengths multiplies all areas by 4 and volumes by 8 underlies home-heating calculations and accounts for upper limits on the size of insects.
Making mathematics usable
The usefulness of mathematics depends on making such links with practical contexts. One has to ask, "Which concepts and skills, tools in my mathematical toolkit, will give me power over this problem?" For most practical problems, the useful tools will come from more than one area of mathematics. Design and planning tasks usually involve both space and number. Evaluating practical alternatives often involves data as well, with algebraic modeling adding power. As with all learning, students develop such "applied power" over practical problems only by using their mathematics successfully in increasingly challenging problems, an activity that is possible only in a well-integrated curriculum.
The amnesia problem<<br /> Year-long chunks of one-flavored curriculum create continuity problems. It is a massive extension of the well-known "summer amnesia" problem—students return to school in the fall having forgotten much of the previous year's work. Curricula in other countries, like the integrated curricula in the United States, have coherent units that focus on a single aspect of mathematics and that typically last from three to eight weeks.
Balancing the curriculum
When and for how long should we teach geometry? What about such "new" areas as data, probability and statistics, or discrete mathematics? After a lively program in middle schools, should they disappear until twelfth grade? What about problem solving over several aspects of mathematics? These balance issues are discussed and disputed in all countries, but finding good solutions, whatever your criteria, needs flexibility in scheduling that dis-integrated curricula simply do not allow.
Various issues are at work here. Algebra and geometry favor different learning styles, so a whole year of one seems inequitable. In prosperous middle-class homes where parents value education, children's tolerance of abstraction, with its delayed gratification, can be built; but if all young people are to have a fair chance to succeed in mathematics, they have a need and a right to a curriculum where the practical payoff from mathematical power is clear to them, month by month. A year of Euclid is an elitist filter. It would be even more lethal if so many had not been turned off by the abstractions of first-year algebra. A balanced diet is surely as important in the mathematics curriculum as it is everywhere else.
How do we get there from here?
Since, in human affairs, rational argument is mainly used to justify emotionally driven decisions, the rationale above for an integrated curriculum may have slow and limited influence. The new curricula are there for the converted. For others, change may best be effected by avoiding confrontation. Choosing traditional titles for the textbooks but having contents that are much more integrated can be effective. Low cunning is a powerful tool in achieving—or resisting—educational change; I recommend it.
|Hugh Burkhardt is a theoretical physicist and an "educational engineer." Working at the Shell Centre at the University of Nottingham, England and often "on tour" in the United States, he leads the development of tools to support systemic improvement in mathematics education. MARS, the implementation phase of the Balanced Assessment project, is his main current initiative.