2. The number of molecules is so large that statistical methods can be applied

When studying simulation as an educational method, a simulation program’s features (e.g., technology, visualization methods, level of interactivity) might more easily capture our attention, but the mental processes involved in student’s interaction with them must also be considered. These processes depend on many factors related to the learning context, e.g., learning task, prior knowledge, interest, instructional method, degree of instructional support, type of assessment. The role of some these factors in simulation-based instruction will be examined in Sects. 5.6 and 5.7. The present Section focuses on the more general issue of the ways in which simulation- and cognitive processes are related.
Consider the case of a middle school student, Mary, who is using a simulation to study the relation between temperature and particle motion in a gas. Let us assume that: Mary can change the gas temperature by moving a slider on the screen; the gas molecule motion is visualized via animation; and that a graph represents the statistical distribution of molecular velocities (Fig. 5.5).
The conceptual model underlying the simulation is the kinetic theory of gases, which is based on the following hypotheses:
1. The gas consists of molecules that have the same mass and are in a constant and random motion. The molecules collide with each other and with the walls of the container.
2. The number of molecules is so large that statistical methods can be applied.
3. The total volume of the gas molecules is negligible compared to the container volume.
4. All the collisions are perfectly elastic, and the interactions between molecules are negligible, except during collisions.
5. The molecules are considered to be perfectly spherical in shape.
6. Relativistic and quantum-mechanics effects are negligible.
Kinetic gas theory was first established in 1738 by the mathematician Daniel Bernoulli, who assumed a gas to consist of ‘‘very minute corpuscles, which are driven hither and thither with a very rapid motion’’ (in Newman 1956, p. 774).
Bernoulli was also the first to recognize that pressure is caused by particle collisions with a container’s walls, and that particle speed increases with increasing temperature. This model is also called the ‘‘Billiard ball model’’ of a gas, because the molecules in it are considered to be rigid spheres, such as the balls used in billiard games. A possible learning goal for this type of simulation is to understand the statistical mechanics’ explanation of temperature, and in particular, the Maxwell–Boltzmann distribution of molecular velocities, which express the percentages of molecules with velocities differing from the average velocity. How might Mary achieve this goal? By moving the temperature slider back and forth, she observes the corresponding changes in the motion of the molecules. A hypothetical dialogue between Mary and her teacher follows here below:
TEACHER What did you notice in this activity?
MARY Oh, that’s very simple; the molecules move faster as the temperature increases and slower when the temperature decreases.
TEACHER Great! At the highest temperatures, are there still a few slow particles?
MARY Obviously not, by increasing the temperature, all the molecules are moving faster—that is what temperature is all about.
Previously, when introducing simulation use, the teacher had explained the meaning of the graph on the right side of the simulation panel, by clarifying that the curve represents the probability of a particle moving at the velocity shown on the x-axis of the graph. The higher the curve, the greater the probability of finding a particle moving at that velocity. She then asks Mary the following question:
TEACHER And what kind of changes did you notice in the graph?
MARY If I increase the temperature, the peak of the curve moves to the right, and this confirms the fact that molecular speed increases with temperature.
TEACHER What about the shape of the curve?
MARY (thinking) If I increase the temperature, it becomes flatter… but to be honest I don’t know why this happens, I suppose it has something to do with the movement of the molecules.
Shortly afterwards, the teacher assigns another activity, telling the students to focus on the motion of individual particles as shown in the animation. The students’ task is to verify whether all the molecules move faster (or slower), by increasing (or decreasing) the temperature, and imagining how this may be related to the changes observed in the shape of the curve. While carrying out the task, Mary notices, to her great surprise, that some molecules move more slowly than the others, even at the highest temperatures:
MARY I can’t believe I’d never noticed this before! The speed may actually vary a lot from one molecule to the other, and the widest variety of particle speeds occurs at the highest temperature, although the speeds are more similar to each other at a lower temperature.
TEACHER Great Mary! And you’ve probably also noticed how this correlates with the shape of the curve…
MARY (moving her hands to show the changes in the curve) Of course! The pointier the curve becomes, the more the speeds concentrate around their mean value; and the flatter the curve, the more dispersed they are.
TEACHER That’s right! And you’d be surprised to discover how many other things you can learn about the gases just from studying this curve.
We shall now analyse what presumably occurred in this fictionalized account of simulation-based learning. Although our reconstruction is hypothetical, it may help us gain a better understanding of the ways in which simulation can support learning: While using the simulator, Mary constructed her own internal representation of the system. This internal representation is constructed ad hoc to make inferences on the causal relation between the change in temperature and the observed phenomena. It may therefore be interpreted as a mental model, in the sense of it being a temporary structure in working memory (WM). From the perspective of embodied cognition theories, this initial mental model is presumably grounded in the sensorimotor experience of setting the temperature value by moving the slider indicator while watching the animation, and in the correlated introspective state. At the same time, Mary is probably retrieving prior knowledge about temperature and gases from long-term memory. She already knows that gases can vary in temperature, as with hot or cold air. She also remembers an educational cartoon she saw in elementary school, which represented molecules as tiny coloured balls moving in all directions. Later, Mary expresses her mental model to her classmates in the form of a verbal rule:
MARY The molecules move faster as temperature increases, and slower as temperature decreases.
This rule is easy to remember and communicate, but fails to captures a basic feature of the statistical account temperature, and namely, that the molecules of any gas will move at a variety of velocities. In statistical mechanics, the temperature of a gas is a measure of the average kinetic energy of its particles.
Moreover, the relative proportions of molecules moving at different velocities is yielded by the Maxwell–Boltzmann distribution, which is the equation underlying the curves shown in the graph of molecular velocities. By focusing on the motion of individual particles, Mary realizes that the molecules move more quickly with a temperature increase, but that even at extremely high temperatures, a few molecules still move slowly; this insight changes her mental model of the phenomenon thereby. The new mental model also allows her to better understand the concept underlying the Maxwell–Boltzmann distribution. Interestingly, Mary’s own initial mental model is not unknown in the history of science; even Rudolf Clausius, who made a great contribution to the kinetic theory of gases, assumed that all molecules move with the same speed. It was only with Maxwell that the notion of a statistical distribution of velocities was introduced into the physics of gases.
As an additional activity, the teacher asked Mary if she can express her mental model visually, so as to communicate her idea to the other students. After several attempts, she proudly showed the class her picture (Fig. 5.6), explaining that the arrows represent molecular velocities. In brief, Mary did not use her mental models to draw logical inferences only, but actually ran a mental simulation of these models. She then expressed the results of her mental simulations as external models (e.g., verbal explanations for both the teacher and the class, gestures accompanying these explanations, and a picture drawn at the teacher’s request), so as to share her ideas with others.
We conclude this section by assuming that simulation-based learning can involve an epistemically rich interplay among different kinds of models even when students do not build simulation themselves, but use existing ones—as long as the learning activities involved thereby are sufficiently structured. This type of interplay is not exclusive to simulation-based learning in educational contexts, but may also occur in other contexts.