URI WILENSKY Center for Connected Learning Northwestern University Annenberg Hall 311 2120 Campus Drive Evanston, IL 60208 uriw@media.mit.edu 847-647-3818 |
Epistemology & Learning Group Learning & Common Sense Section The Media Laboratory Massachussets Institute of Technology 20 Ames Street Room E15-315 Cambridge, MA 02139 uriw@media.mit.edu |

Proceedings of the Nineteenth International Conference on the Psychology of Mathematics Education. Recife, Brazil, July 1995.

As part of the Connected Probability project, we have extended the StarLogo parallel modeling language (Resnick, 1992; Wilensky, 1993) and tailored it for building probabilistic models. The StarLogo language is an extension of the computer language Logo that allows learners to control thousands of screen "turtles". These turtles or computational agents have local state and can be manipulated as concrete objects. Through assigning thousands of such turtles probabilistic rules, learners pursue both forwards and backwards modeling. Forwards modeling involves exploring the effects of various sets of local rules to see what global pattern emerges, while in backwards modeling learners try to find an adequate set of local rules to produce a particular global effect. In this report, two case studies of probabilistic modeling projects are presented.

Connected Mathematics is situated in the constructionist learning paradigm (Papert, 1991). The Constructionist position advances the claim that a particularly felicitous way to build strong mental models is to produce physical or computational constructs which can be manipulated and debugged by the learner. As described by Wilensky (1993), Connected Mathematics also draws from many sources in the mathematics reform movement (e.g., Confrey, 1993; Dubinsky & Leron, 1993; Feurzeig, 1989; Hoyles & Noss, 1992; Lampert, 1990; Schwartz, 1989; Thurston, 1994).

A Connected Mathematics learning environment focuses on learner-owned investigative activities followed by reflection. Thus, there is a rejection of the mathematical "litany" of definition-theorem-proof and an eschewal of mathematical concepts given by formal definitions. Mathematical concepts are multiply represented (Kaput, 1989; Mason, 1987; von Glaserfeld, 1987) and the focus is on learners designing their own representations. Learners are supported in developing their mathematical intuitions (Wilensky, 1993) and building concrete relationships (Wilensky, 1991) with mathematical objects. Mathematics is seen to be a kind of sense-making (e.g., Schoenfeld, 1991) both individually and as a social negotiation (e.g., Ball, 1990; Lampert, 1990). In contrast to the isolation of mathematics in the traditional curriculum, it calls for many more connections between mathematics and the world at large as well as between different mathematical domains (e.g., Cuoco & Goldenberg, 1992; Wilensky, 1993).

Because, when learners build computational models, they articulate their conceptual models through their design, researchers can gain access to these conceptual models (see e.g., Collins & Brown, 1985; Pea, 1985). The researcher is given insight into the thinking of the learner at two levels: as model builder and as model consumer.

The challenge for such an approach is to design the right middle level of primitives so that they are neither (a) too low-level, so that the extensible model becomes identical to its underlying modeling language, nor (b) too high-level, so that the application turns into an exercise of running a small set of pre-conceived experiments.

- There is a considerable literature attesting to the difficulty people have with understanding probability (e.g., Kahneman & Tversky, 1982, Nisbett et al, 1983, Konold, 1991). Standard instruction has been shown to provide little remedy. Educators have responded to this research by advising students not to trust their intuitions when it comes to probability and to rely solely on the manipulation of formalisms. As a result, learners construct brittle formal models of the core probabilistic concepts and fail to link them to everyday knowledge. Connected Mathematics provides an alternative to this formalistic stance. It asserts that powerful probabilistic intuitions can be constructed by learners (Wilensky, 1993; 1994; forthcoming). By taking up such a challenging domain, a strong proof of the value of Connected Mathematics can be demonstrated.
- Computational environments can open doors to new ways of thinking about probability. Computational environments allow users to construct stable products (e.g., normal distributions, see below) using random components. This construction would be very difficult to do without computational environments. From a constructionist perspective, this ability to build meaningful products from random components is a prerequisite for making sense of the core notion of randomness.
- Particularly in the area of probability and statistics, the educational goal should emphasize interpreting (and designing) statistics from science and life rather than mastery of curricular materials. In order to make sense of scientific studies, it is not sufficient to be able to verify the stated model; one needs to see why those models are superior to alternative models. In order to understand a newspaper statistic, one must be able to reason about the underlying model used to create that statistic and evaluate its plausibility. For these purposes, building probabilistic and statistical models is essential.
- Many everyday phenomena exhibit emergent behavior: the growth of a snowflake crystal, the perimeter pattern of a maple leaf, the advent of a summer squall, the dynamics of the Dow Jones or of a fourth grade classroom. These are all systems which can be modeled as composed of many distributed but interacting parts. They all exhibit non-linear or emergent qualities which place them well beyond the scope of current K-12 mathematics curricula. Yet, through computational modeling, especially with parallel languages such as Starlogo, pre-college learners can gain mathematical purchase on these phenomena. Modeling these everyday complex systems can therefore be a motivating and engaging entry point into the world of probability and statistics.

One such example is Alan, a student with a strong mathematical background who nevertheless felt that he "just didn't get" normal distributions. Using a version of the parallel modeling language Starlogo which was enhanced for focusing on probability investigations (Wilensky, 1993; 1994), Alan developed a model for explaining his question,"Why is height (in men) normally distributed?" Alan's they was that perhaps "Adam" had children which were either taller or shorter than him with a certain probability. If this process was repeated with the children, then a distribution of heights would emerge. To explore what kinds of distributions were possible from this model, Alan built a "rabbit jumping" microworld. Taking advantage of the parallel modeling environment, Alan placed sixteen thousand rabbits in the middle of a computer screen. He then gave each rabbit a probabilistic jumping rule. (In Alan's model, the location of the rabbit corresponds to a person's height and a jump corresponds to a set deviation in height). The first such rule he explored was to tell each rabbit to jump left one step or right one step each with probability 1/2. After a number of steps, the classic symmetric binomial distribution became apparent. Alan was pleased with that outcome but then asked himself the question: what rule should I give the rabbits in order to get a non-symmetric distribution? His first attempt was to have the rabbits jump two steps to the right or one step to the left with equal probability. He reasoned that the rabbits would then be jumping more to the right so the distribution should be skewed right. His surprise was evident when the distribution stayed symmetric while moving to the right. It didn't take too long though before he realized that it was the different sized probabilities not the different sized steps that made the distribution asymmetric. This example, while seemingly elementary, captures many facets of the model building approach to learning about complexity:

- The question was owned by the learner
- Theories were instantiable and testable
- Buggy theories could be successively refined
- The modeling environment did not limit the directions of inquiry

The model Harry built is initialized to display a box with a specified number of gas "molecules" randomly distributed inside it. The user can then perform "experiments" with the molecules.

The molecules are initialized to be of equal mass and start at the same speed (that is distance traveled in one clock tick) but at random headings. Using simple collision relations, Harry was able to model elastic collisions between gas molecules, (i.e., no energy is "lost" from the system). The model can be run for as many ticks as wanted.

By using several output displays such as color coding particles by their speed or providing dynamic histograms of particle speeds/energies, Harry was able to gain an intuitive understanding of the stability and asymmetry of the Boltzman distribution.

Harry's story is told in greater detail elsewhere (Wilensky, forthcoming). {Originally, Harry had thought that because gas particles collided with each other randomly, they would be just as likely to speed up as to slow down. But now, Harry saw things from the perspective of the whole ensemble of particles. He saw that high velocity particles would "steal lots of energy" from the ensemble. The amount they stole would be proportional to the square of their speed. It then followed that, since the energy had to stay constant, there had to be many more slow particles to balance the fast ones.

This new insight gave Harry a new perspective on his original question. He understood why the Boltzman distribution he had memorized in school had to be asymmetric. But it had answered his question only at the level of the ensemble. What was going on at the level of individual collisions? Why were collisions more likely to lead to slow particles than fast ones? This led Harry to conduct further productive investigations into the connection between the micro- and macro- views of the particle ensemble.

As a result, many more learners used the GPCEE model and, since it was an extensible model, they extended it. Among the extensions that users built were tools for measuring pressure in the box and viscosity of the gas, pistons to compress the gas, different shapes for the container, different dimensional spaces for the box, diffusion of two gases, different geometries for the molecules (e.g., diatomic molecules with rotational and vibrational freedom), and sound wave propagation in the gas. In order to build the computational tools for these extensions, users had to build conceptual models. They came to ask such questions as: What kind of thing is pressure? How would you build a tool to measure it?

It is clear that GPCEE is both a physics simulation, one in which experiments difficult or impossible to do with real gases can be easily tried out, and an environment for strengthening intuitions about the statistical properties of ensembles of interacting elements. Through creating experiments in GPCEE, learners can get a feel for both the macro- level, the behavior of the ensemble as an aggregate, and its connections to the micro-level what is happening at the level of the individual gas molecule. In the GPCEE application, learners can visualize ensemble behavior all at once, sometimes obviating summary statistics. Furthermore, they can create their own statistical measures and see what results they give at both the micro- and the macro- level. They may, for example, observe that the average speed of the particles is not constant and search for a statistical measure which is invariant. In so doing, they may construct their own concept of energy. Their energy concept, for which they may develop a formula different than the formula common in physics texts, will not be an unmotivated formula the epistemological status of which is naggingly questioned in the background. Rather, it will be personally meaningful, having been purposefully designed by the learner.

The necessity of creating his own summary statistic led one learner to shift his view of the concept of "average". In the GPCEE context, he now saw "average" as just another method for summarizing the behavior of an ensemble. Different averages are convenient for different purposes. Each has certain advantages and disadvantages, certain features which it summarizes well and others that if doesn't. Which average we choose or construct depends on how we wish to make sense of the data.

Having shown the GPCEE environment to quite a few professional physicists, I can attest to the fact that although they knew that particle speeds fell into a Maxwell-Boltzman distribution, most were still surprised to see more blue particles than red -- they had formal knowledge of the distribution, but the knowledge was not well connected to their intuitive conceptions of the model In a typical physics classroom, learners have access either only to the micro level - through , say, exact calculation of the trajectories of two colliding particles, or only to the macro- level, but in terms of pre-defined summary statistics selected by the physics canon. Based on this example, it would seem that it is in the interplay of these two levels of description that powerful explanations and intuitions develop.

Ball, D. (1990). *With an eye on the Mathematical Horizon: Dilemmas of
Teaching*. Paper presented at the annual meeting of the American
Educational Research Association, Boston, MA.

Brandes, A., & Wilensky, U. (1991). *Treasureworld: A Computer
Environment for the Exploration and Study of Feedback*. In Harel, I. &
Papert, S. Constructionism. Norwood N.J. Ablex Publishing Corp. Chapter
20.

Chen, D., & Stroup, W. (1993).*General Systems Theory: Toward a
Conceptual Framework for Science and Technology Education for All*.
Journal for Science Education and Technology.

Collins, A. & Brown, J. S.
(1985).*The Computer as a Tool for Learning Through Reflection*. In
H. Mandl & A. Lesgold (Eds). Learning Issues for Intelligent Tutoring
Systems (pp. 1-18). New York: Springer Verlag.

Confrey, J. (1993).*A Constructivist Research Programme Towards the
Reform of Mathematics Education*. An introduction to a symposium for
the Annual Meeting of the American Educational Research Association, April
12-17, 1993 in Atlanta, Georgia.

Cuoco, A.& Goldenberg, E. P. (1992). *Reconnecting Geometry: A Role for
Technology*. Proceedings of Computers in the Geometry Classroom
conference. St. Olaf College, Northfield, MN, June 24-27, 1992.

Dubinsky, E. & Leron, U. (1994).*Learning abstract algebra with ISETL
New York* : Springer-Verlag.Edwards, L. (in press). Microworlds as
Representations. in Noss, R., Hoyles, C., diSessa A. and Edwards, L.
(eds.) Proceedings of the NATO Advanced Technology Workshop on Computer
Based Exploratory Learning Environments. Asilomar, Ca.

Eisenberg, M. (1991).*Programmable Applications: Interpreter Meets
Interface*. MIT AI Memo 1325. Cambridge, Ma., AI Lab, MIT

Feurzeig, W. (1989).*A Visual Programming Environment for Mathematics
Education*. Paper presented at the fourth international conference for
Logo and Mathematics Education. Jerusalem, Israel.

Hacking, I. (1990). *Was there a Probabilistic Revolution 1800-1930?*
In Kruger, L., Daston, L., & Heidelberger, M. (Eds.) The Probabilistic
Revolution. Vol. 1. Cambridge, Ma: MIT Press.

Harel, I. (1988).*Software Design for Learning: Children's Learning
Fractions and Logo Programming Through Instructional Software Design*.
Unpublished Ph.D. Dissertation. Cambridge, MA: Media Laboratory, MIT.

Hoyles, C. & Noss, R. (Eds.) (1992). *Learning Mathematics and LOGO*.
London: MIT Press.

Kaput, J. (1989).*Notations and Representations*. In Von Glaserfeld,
E. (Ed.) Radical Constructivism in Mathematics Education. Netherlands:
Kluwer Academic Press.

Kahneman, D. & Tversky, A. (1982).*On the study of Statistical
Intuitions*. In D. Kahneman, A. Tversky, & D. Slovic (Eds.) Judgment
under Uncertainty: Heuristics and Biases. Cambridge, England: Cambridge
University Press.

Konold, C. (1991).*Understanding Students' beliefs about
Probability*. In Von Glaserfeld, E. (Ed.) Radical Constructivism in
Mathematics Education. Netherlands: Kluwer Academic Press.

Lampert, M. (1990).* When the problem is not the question and the
solution is not the answer: Mathematical knowing and teaching*. In
American Education Research Journal, spring, vol. 27, no. 1, pp. 29- 63.

Mason, J. (1987). *What do symbols represent?* In C. Janvier (Ed.)
Problems of representation in the Teaching and Learning of Mathematics.
Hillsdale, NJ: Lawrence Erlbaum Associates.

Minsky, M. (1987).*The Society of Mind.* New York: Simon & Schuster
Inc.

Nisbett, R., Krantz, D., Jepson, C., & Kunda, Z. (1983).*The Use of
Statistical Heuristics in Everyday Inductive Reasoning*. Psychological
Review, 90 (4), pp. 339-363.

Papert, S. (1980).*Mindstorms: Children, Computers, and Powerful
Ideas*. New York: Basic Books.

Papert, S. (1991). *Situating Constructionism*. In I. Harel & S.
Papert (Eds.) Constructionism. Norwood, N.J. Ablex Publishing Corp.
Chapter 1.

Repenning, A. (1993).*AgentSheets: A tool for building domain-oriented
dynamic, visual environments*. unpublished Ph.D. dissertation, Dept. of
Computer Science, University of Colorado, Boulder.

Resnick, M. (1992). *Beyond the Centralized Mindset: Explorations in
Massively Parallel Microworlds*. Doctoral dissertation, Dept. of
Computer Science, MIT.

Richmond, B. & Peterson, S. (1990). *Stella II*. Hanover, NH: High
Performance Systems, Inc.

Roberts, N. (1978).*Teaching dynamic feedback systems thinking: an
elementary view*. Management Science, 24(8), 836-843.

Rucker, Rudy (1993).*Artificial Life Lab*. Waite Group Press.

Schoenfeld, A. (1991).*On Mathematics as Sense-Making: An Informal
Attack on the Unfortunate Divorce of Formal and Informal Mathematics*.
In Perkins, Segal, & Voss (Eds.) Informal Reasoning and Education.

Schwartz, J. (1989). "Intellectual Mirrors: A Step in the Direction of Making Schools Knowledge-Making Places." Harvard Educational Review.

L. S. Shore, M. J. Erickson, P. Garik, P. Hickman, H. E. Stanley, E. F.
Taylor, P. Trunfio, (1992). *Learning Fractals by 'Doing Science':
Applying Cognitive Apprenticeship Strategies to Curriculum Design and
Instruction*. Interactive Learning Environments 2, 205--226.

Stanley, H.E. (1989). *Learning Concepts of Fractals and Probability by
`Doing Science'* Physica D 38, 330-340.

Thurston, W. (1994).*On Proof and Progress in Mathematics*. Bulletin
of the American Mathematical Society. Volume 30, Number 2, April, 1994.

von Glaserfeld, E. (1987). *Preliminaries to any Theory of
Representation*. In C. Janvier (eds.) Problems of Representation in
Mathematics Learning and Problem Solving, Hillsdale, NJ: Erlbaum.

Wilensky, U. (forthcoming).* GPCEE - an extensible modeling environment
for exploring micro- and macro- properties of gases*. Interactive
Learning Environments.

Wilensky, U. (1994). *Paradox, Programming and Learning
Probability*. In Y. Kafai & M. Resnick (Eds). Constructionism in
Practice: Rethinking the Roles of Technology in Learning. Presented at
the National Educational Computing Conference, Boston, MA, June 1994. The
MIT Media Laboratory, Cambridge, MA.

Wilensky, U. (1993).*Connected Mathematics: Building Concrete
Relationships with Mathematical Knowledge*. Doctoral dissertation,
Cambridge, MA: Media Laboratory, MIT.

Wilensky, U. (1991). *Abstract Meditations on the Concrete and Concrete
Implications for Mathematics Education*. In I. Harel & S. Papert (Eds.)
Constructionism. Norwood N.J.: Ablex Publishing Corp. (Chapter 10).

Wright, W. (1992a). SimCity. Orinda, CA: Maxis.

Wright, W. (1992b). SimEarth. Orinda, CA: Maxis.