THE PHILOSOPHY AND PRACTICALITY OF MODELING INVOLVING COMPLEX SYSTEMS
Nicholas Mousoulides
University of Cyprus
n.mousoulides(at)ucy.ac.cy
Bharath Sriraman
The University of Montana
sriramanb(at)mso.umt.edu
Richard Lesh
Indiana University
ralesh(at)indiana.edu
Abstract
In this paper we outline the philosophy of the models and modeling perspectives of learning. The paper is based on the Taoist premise that the true structure of things is irreducible to formal statements, propositions or rules. Examples and paradoxes from the natural and physical sciences are used to support the models and modeling philosophy of learning. We also delve into the notion of operational definitions for researching learning that occurs in models-modeling involving complex systems and presents examples of several activities drawn from the engineering sciences that illustrates the thesis of this paper.
1. Introduction
Modern science, especially the progression of research in physics and biology reveals that learning is a complex phenomenon in which the classical separation of subject, object and situation is no longer possible. That is, reality is characterized by a “non-linear” totality in which the observer, the observed and the situation are in fact inseparable. Yet, at the dawn of the 21st century, learning scientists are still using theories and research methodologies grounded in the information-processing premise that learning is reducible to a list of condition-action rules. Complex systems are those which involve numerous elements, arranged in structure(s) which can exist on many scales. These go through processes of change that are not describable by a single rule nor are reducible to only one level of explanation.
Biologists have found that methodological reductionism, i.e. going to the parts to understand the whole, which was central to the classical physical sciences, is less applicable when dealing with living systems. Analogously, the challenge confronting learning scientists who hope to create models of the models (and/or underlying conceptual systems) that students, teachers and researchers develop to make sense of complex systems occurring in their lives is: the mismatch between learning science theories based on mechanistic information processing metaphors in which everything that students know is methodologically reduced to a list of condition-action rules, given that characteristics of complex systems cannot be explained (or modeled) using only a single function - or even a list of functions. As physicists and biologists have proposed characteristics of complex systems arise from the interactions among lower-order/rule-governed agents – which function simultaneously and continuously, and which are not simply inert objects waiting to be activated by some external source. Given the paradigm shifts that have occurred in the physical and natural sciences, we propose a view of learning and the modeling of learning analogous to the study of complex systems. The three themes we explore are:
(1) A phylogenetic approach to learning
(2) Understanding the dichotomy between theoretical terms and observational terms.
(3) The use of operational definitions
2. A Phylogenetic approach to modeling learning
Today’s students are more likely to be engaged in professions that calls for competencies related to understanding complex real world phenomena, team work, communication and technological skills. So, in essence there are three kinds of complex systems: (a) “real life” systems that occur (or are created) in everyday situations, (b) conceptual systems that humans develop in order to design, model, or make sense of the preceding “real life” systems, and (c) models that researchers develop to describe and explain students’ modeling abilities. These three types of systems correspond to three reasons why the study of complex systems should be especially productive for researchers who are attempting to advance theory development in the learning sciences. In mathematics and science, conceptual systems that humans develop to make sense of their experiences generally are referred to as models. A naive notion of models is that they are simply (familiar) systems that are being used to make sense of some other (less familiar) systems - for some purpose. For example, a single algebraic equation may be referred to as a model for some system of physical objects, forces, and motions. Or, a Cartesian Coordinate System may be referred to as a model of space - even though a Cartesian Coordinate System may be so large that it seems to be more like a language for creating models rather than being a single model in itself. In mathematics and science, modeling is primarily about purposeful description, explanation, or conceptualization (quantification, dimensionalization, coordinatization, or in general mathematization) - even though computation and deduction processes also are involved. Models for designing or making sense of such complex systems are, in themselves, important “pieces of knowledge” that should be emphasized in teaching and learning – especially for students preparing for success in future-oriented fields that are heavy users of mathematics, science, and technology. Therefore, the claim is that modeling students modeling abilities is the study of a complex living system with layers of emerging ideas, sense making and a continuous evolution of knowledge, which suggests we adopt a phylogenetic approach to modeling the growth of knowledge and learning. The field of economics is an interesting case study which reveals paradigmatic shifts in theories from archaic models for simple agricultural economies to more complicated industrial economies onto the modern day integration of game theory, evolutionary biology and ecology that characterize current economic theories.
2.1.What are reasonable assumptions to adopt (or avoid) about the nature of student’s models (and underlying conceptual systems)?
Models & modeling perspectives trace their lineage to modern descendents of Piaget and Vygotsky - but also (and just as significantly) to American Pragmatists such as William James, Charles Sanders Peirce, Oliver Wendell Holmes, George Herbert Mead, and John Dewey. For example: (a) Dewey and Meade emphasized that conceptual systems are human construct, and that they also are fundamentally social in nature. (b) Pierce emphasized that the meanings of these constructs tend to be distributed across a variety of representational media (ranging from spoken language, written language, to diagrams and graphs, to concrete models, to experience-based metaphors) – each of which emphasize and ignore somewhat different aspects of the constructs they are intended to express and/or the “real life” experiences they are intended to describe. (c) Dewey emphasized that knowledge is organized around experience at least as much as around abstractions – and that the ways of thinking which are needed to make sense of realistically complex decision making situations nearly always must integrate ideas from more than a single discipline, or textbook topic area, or grand theory. (d) James emphasized that the “worlds of experience” that humans need to understand and explain are not static. They are, in large part, products of human creativity. So, they are continually changing - and so are the knowledge needs of the humans who created them. (e) Dewey emphasized that, in a world filled with technological tools for expressing and communicating ideas, it is naïve to suppose that all “thinking” goes on inside the minds of isolated individuals (Lesh & Sriraman, 2005).
3. The dichotomy between theoretical terms and observational terms
One could say that theoretical terms are invariants of operations represented by physical measurement devices. Physicists have “learned” (pun-intended) that theoretical terms have to be defined operationally in terms of theories provided experimentation can back up notions occurring within the theories. (Dietrich, 2004). The question is how can this be adapted by learning scientists? That is, how can learning scientists operationally define observational terms (namely perceived regularities that we attempt to condense into theories, or as Piaget attempted – to phylogenetically evolved mental cognitive operators) (Dietrich, 2004). The purpose of theoretical terms is to clarify the meaning of concepts. On the other hand the purpose of observational terms is to delineate how the concepts/constructs have been measured. Ideally there should be a perfect match between theoretical terms and observational terms, i.e., observations should confirm theory irrespective of when or where the observations are made as long as the initial conditions of an experiment are somewhat the same. The implication of this fact for the learning sciences, especially those trying to describe modeling of students, is that it is okay if actual observations of students modeling processes vary slightly within a finite set of modeling behaviors. This means that if learning scientists consistently report similar observations of students modeling processes when confronted with the same complex situation across age groups and locations, then these observations can be used to develop a sound theoretical construct. Conversely when the theoretical construct is tested (or subject to experiment) at a new location, researchers should be able to predict the types of behaviors that will be observed as long as the integrity of the experiment (starting conditions are replicated).
3.1 The use of operational definitions
Operational definitions are routinely used in physics, biology and computer science. For instance in quantum mechanics, physicists are able to define philosophically intangible but observable sub-atomic phenomenon by making predictions about their probability distributions. It is important to note that physicists do not assign a definite value per se to the observable phenomenon but a probability distribution. The implication for learning scientists is that the notion of operational definitions can be adapted to the study of modeling by making predictions of the range of observable “things”/behaviors/processes/conceptual systems” that students will engage in when confronted by an authentic model eliciting situation. The goal (analogous to physics) is to operationally define tangible constructs relevant to the learning sciences, in terms of a distribution of clearly observable student processes with auditable trails of artifacts that are shareable and generalizable. In this respect we preserve the Tao (or the whole) by not attempting to measure each individual process or adhere to John Stuart Mill’s wise suggestion that we move away from the belief that anything that is nameable should refer to a “thing”
4. Some Concrete Examples
4.1.The Paper Airplane Problem is a middle school version of a case study first used in Purdue University’s graduate program for aeronautical engineering. The original problem involved a wind tunnel. The goal was for graduate students to develop an “operational definition” to deal quantitatively with the concept of “drag” for various shapes of planes and wings (Lesh & Doerr, 2003).
In the Paper Airplane Problem, sensible conceptions of “good floaters” involve (a) going slowly for a long distance (or for a long time), and (b) ending up close to intended targets. Therefore, the following issues tend to arise during solution attempts.
Going slowly means that distance ¸ time is a small number. So, a trial way to think about “floating” might be to multiply speed x distance, or to multiple speed x time. But unfortunately, neither of these two options is sensible. To see why, consider the following.
If speed is a small number and distance (or time) is a large number, then what would it mean to multiply these two quantities?
If we multiply speed x time (or distance/time x time), then the result reduces to a pure distance when the times are canceled.
Alternatively, going slowly could be interpreted to mean that time ¸ distance is a large number. So:
A second trial way to think about “floating” might be to multiply (time ¸ distance) x distance. Unfortunately, this option doesn’t make sense because the formula reduces to a pure time when the distances are canceled.
A third trial way to think about “floating” might be to multiply (time ¸ distance) x time. This makes sense! It means that “floating” can be conceived (and measured) using the formula time2/distance.
If problem solvers employ the third option described above, then a sensible definition of a GOOD “floater” also probably needs to go beyond the preceding considerations to involve some way of combining measures of accuracy and floating ability. Regardless how this is done, however, the concept of a “good floater” inescapably involves a system of relationships among observable quantities. Like the concept of drag, it is not a concept that can be abstracted from “pieces of information” that are directly observable. In fact, most concepts in elementary mathematics have this property that, when we unpack what it means to “understand” them, what emerges are conceptual systems that involve relationships, operations, transformations, patterns, and regularities which cannot be abstracted from “pieces of information” that are given in the world. In other words, the relevant conceptual systems are imposed on experience rather than simply being derived from experience; and, these conceptual systems do not acquire many of their most important mathematical properties until they begin to function as well-organized systems-as-a-whole. For this reason, the process of developing well coordinated conceptual systems-as-a-whole is similar to the process of developing well-coordinated action schemes – such as those that are involved in playing tennis or ballroom dancing. When students first begin to engage in relevant activities, they’re unlikely to notice more than a subset of the information that is relevant; they are likely to impose inappropriate assumptions (and actions) due to prior expectations; and, they are likely to attend to surface-level information rather than to deeper patterns and regularities. The preceding observations are consistent with results of expert/novice studies, which consistently show that expertise is generally about “seeing” as much as it is about “doing.” Therefore, when “seeing” involves thinking that is based on a coordinated system of relationships, patterns, and operations, the development of such systems usually must involve sorting out, integrating, refining, revising, or rejecting existing conceptual systems at least as much as it involves constructing conceptual systems that are completely new. Until relevant conceptual systems begin to function holistically (in a well-coordinated manner), learners or problem solvers generally: (a) fail to notice more than a subset of relevant information, and (b) project unwarranted assumptions into the situation because their thinking in based on inappropriate patterns and regularities.
4.2. The Water Shortage Problem
This environmental engineering problem presented in the activity asked from students to help the local authorities in finding the best place for transferring water to Cyprus. Water shortage is one of the biggest problems Cyprus face these days. As a result, students are very familiar with the problem and almost everyday there are discussions on TV about the possible solutions to the problem. The problem presented in the activity asked from students to find the best possible place for transferring water to Cyprus, using oil tankers. Since there were no data about the price of the water, this variable was not taken into consideration. Additionally, all possible providers were willing to supply Cyprus with all necessary water.