An interview with Margaret Vickers, TERC, Cambridge, Massachusetts
In science observations lead to hypotheses, whereas in mathematics applications seem to flow from decontextualized theory. How can mathematics and science teachers best reconcile these rather different approaches to the mathematical and scientific education of students?
What you say may be true much of the time, but surely there are contrary examples. Some scientific explorations begin with an analysis of a theory, and some mathematical explorations presumably begin with the need to solve a practical problem. One path to reconciliation is to find and capitalize on such examples. I suspect the problem is partly sociological: science teachers seem much more comfortable with manipulatives than mathematics teachers. Maybe computers would help--spreadsheets to support algebra, the Geometric Supposer for geometry. Maybe we just haven't given mathematics teachers the right manipulatives.
Manipulatives (or experiments) suggest an inductive style, whereas many mathematics teachers love their subject precisely because it is deductive. How can teachers instill in students a proper appreciation of the special role that deduction plays in mathematics?
I am not sure. I taught mathematics at the grade 10 level some years ago and found in that context that only a very few students entered into the discourse of logical deduction. Only a few were excited by a system of axioms and definitions whose logical implications were clear, even elegant. Most students did not get "into" this subject--but they were prepared to use mathematics as a tool.
Some educators have argued that departmental boundaries in education may set up artificial barriers to collaborations on curriculum development and improving instructional strategies. Based on your interest in the school-to-work movement and the fact that disciplinary boundaries are not relevant in the workplace, what do you think is the proper balance of disciplinary and interdisciplinary curricula in education, especially in grades 10-14?
It is simplistic to say that disciplinary boundaries are not relevant in the workplace. There are boundaries, but they are quasi-disciplinary, quite different from the boundaries schools define. For example, heating, ventilating and air conditioning (HVAC) is based on thermodynamics, which involves both chemistry and physics.
I am a strong believer in projects that encourage students to use science, mathematics and technology concepts simultaneouly. Good projects often entail a communications element as well. But in my experience, projects in which kids are expected to deal with all aspects of a problem--the history, the politics, the economics, the science--often lack focus. In fact, employees don't usually do serious tasks in areas that are not their own. HVAC technicians don't try to do accounting, although they may communicate information about costs.
Does this mean that you do not support the "all aspects of the industry" approach to vocational curriculum? Advocates of this approach emphasize the value to students of working simultaneously on the economic, scientific, political, and human aspects of ordinary community problems.
What I like about the "all aspects" argument is that it eschews the practice of training a person very narrowly in, for example, woodworking, while excluding all discussions about how to make a living with these skills, or what we should do about the rapid depletion of world supplies of furniture timbers. I am, however, suspicious of attempts to do projects that pull in every imaginable angle. In my experience, these are often superficial.
Mathematics teachers often worry that the context-rich environment of an interdisciplinary course will make it harder for students to sort out the mathematics principles from the surrounding context. Do science teachers have similar concerns about learning science? What has been your experience in this regard? How do you think you, as a science educator, might respond to mathematics teachers about this issue?
If a scientific principle is tied to one context only, student learning may be limited. Faced with a second context, a student may very well not see how to use the concept. But removing the principle from the context is not the answer. Rather, we should contextualize and re-contextualize and then do it again and again. For example, look at how an increase in surface area plays a role in the radiation of heat--in car radiators, in the vanes of base-board heaters in homes, and even in the temperature control system of the Australian frill-necked lizard!
What you call "recontextualizing" is similar to what mathematics teachers call "generalizing"--inferring common patterns from multiple contexts.
I am not sure that I agree. The idea behind recontextualizing is not just to infer common patterns from multiple contexts. Rather, it is to learn and experience the same thing in multiple setting and to work with an expert mentor who helps students reflect on the isomorphisms. In recontextualizing, students are not left on their own.
Students with a knack for mathematics tend to generalize easily, but for many others the different contexts seem to remain forever different. Might these students be served better by the traditional system in which students first learn a key principle and than apply it in many different contexts?
I don't think so. Some kids simply don't learn "principles" if they cannot see what they are for.
Margaret Vickers is Senior Scientist and Director of the Working to Learn Projects at TERC in Cambridge, Massashusetts. She can be reached by e-mail at email@example.com.
Last Update: July 7, 1997