## The power of models

Although each science in the curriculum has its own lens through which it looks at nature or society, they are united by a common approach. Modern science can be described as an ongoing process of developing and deploying models—not model airplanes or runway models, of course, but *simplified representations* of complicated objects or systems. Models help us understand things by focusing on their essential features and leaving out the incidental or less important details.

*Developing* a model involves applying inductive reasoning to things or phenomena to produce a representation of their nature that accounts for their properties and behavior. This process involves making real-world observations and measurements, identifying patterns or laws, and producing concepts and theories. *Deploying* a model, on the other hand, involves using deductive reasoning about the model to make predictions and form hypotheses that can be tested empirically, either to solve practical problems or to make observations necessary to improve the model. Here is a “visual model” of the process of scientific modeling:

## Types of models

We are all familiar with *visual* models, like flowcharts or a biology class cell diagram, and also with *physical* models, such as ball-and-stick molecules or the plastic skeleton in the lab that sophomoric students like to leave in peculiar poses. Less familiar are conceptual, mathematical, and computational models.

A *conceptual* model is a description of a thing or phenomenon that highlights its central characteristics in a simple way and assumes its other aspects are of negligible significance. The kinetic-molecular model of gases is an example: gases are treated as a collection of tiny particles flying around and colliding with things. The model’s simplifying assumptions are that the particles have negligible volume, are moving around in empty space, and experience no attractions or repulsions that would accelerate or slow them down. These assumptions are not completely true—fortunately, otherwise the stars and planets would not have formed!—but they enable us to get a roughly correct view of the difference between gases and other states of matter.

A *mathematical* model states the properties and behavior of an object or system in the form of equations that show the relationship between different variables.^{[1]} Mathematical equations are properly understood as models because they take a messy slice of reality and express it with a few uncluttered symbols. Additionally, like any model they are simplified approximations, because they disregard some features of the situation. For example, when using the equation to describe the acceleration a force causes an object to undergo, we ignore any deformation or rotation the force might cause in the object. We also assume that the object is large enough to disregard quantum effects, and is moving slowly enough to disregard relativistic effects. Even though the model is not accurate in every respect, it is tremendously useful to working scientists and engineers; it gives very good results in many situations and its simplicity makes calculations easier (or at least feasible). The simplifications or idealizations involved in the model are also pedagogically useful; since it is impossible for students to learn everything all at once, it is sensible to start with the main features of a phenomenon.

A *computational* model is a computer program that simulates the behavior of a system by applying a series of mathematical functions to variable data inputs to determine possible results. Complex systems—like the climate or a viral epidemic—are affected by a great many factors, and it’s impossible to include every possible variable in the simulation. But we can use observations of past behavior to determine a range over which relevant quantities vary, and run a computer program many times, inputting different possible values for those quantities. We can then express the results by saying that there is a ___% chance of such-and-such result, etc. A computational model, like any model, is a simplification and thus only an approximation of reality, but if its assumptions are valid, it produces impressive results.

## Using visual or physical models in the science class

Science textbooks make frequent use of diagrams, flow charts and other visual models. For example, in physics books, we see force diagrams, kinematic and other vectors, and energy graphs. Chemistry texts use diagrams or 3-D models to depict atoms, molecules, systems, reactions, and state changes. Biology books use diagrams or physical models of organs, cells, and organic processes.

Many students, however, try to save time by skipping the figures and just scanning the text looking for vocabulary definitions or answers to homework questions. It’s worthwhile to explicitly encourage students to study the diagrams and other figures set off from the text. Although ideally students would study both the text and the figures, if they are going to look at only one or the other, they will probably get a better conceptual understanding from the models presented by the figures.

Using models in the classroom can also help teachers improve their explanations. Instead of just standing at the front of the room explaining something verbally, it’s more effective to refer to a visual or physical model while giving explanations. A 3-D model with moving parts is the most effective, but it also works to draw models on the board or walk students through diagrams in a textbook or show them models on a computer.

By reducing a complex object, system or process to a symbolic visual or physical model, we can help students form a preliminary understanding idea of how things work. Such models also help students understand scientific theories about the various parts of things, their functions, and the way they interact.

## Conceptual modeling

Visual and physical models are an entry point for the more important educational goal of developing conceptual models of the natural world. Here is where students need to be introduced to scientific theories. These should not be presented as a *fait accompli*: reasoning through the development of theories with students “models” good scientific thinking, instead of just telling them to accept whatever the experts tell us.

One issue that arises is that students have difficulty transitioning from simple conceptual models to more sophisticated ones. Most veteran teachers, upon revisiting a topic to explain it more deeply, have had a smart-aleck student raise his hand and ask indignantly: “So what you taught us earlier wasn’t true?” A misconception that science students must overcome is that a model is the same as the reality it represents. Conflating the two makes it difficult for them to later develop a fuller understanding.

Intellectual capacity grows gradually—initially, the concepts and analogies we use in our fumbling attempts to understand the richness and complexity of reality tend to significantly over-simplify things. Later, the prior superficial notion must be split open so that understanding may put down deeper and broader roots.

Not every student wants to develop a more nuanced understanding of topics, of course; *in nil sapiendo vita iucundissima est* (ignorance is bliss). But teachers themselves can be responsible for the ossification of unduly simplistic ideas in the student mind. This happens when models are taught as* true* rather than as *approximations* of reality.

One thinks, for example, of the atomic model that has electrons orbiting the nucleus. Teachers often present this to students as though it were literally true. When these students later need to learn about the wave properties of electrons, it is rather difficult to dislodge the earlier concept. Or take the case of cell diagrams, which make the cell appear mainly empty, with just a few organelles depicted to symbolically represent cellular processes—it can be difficult later to displace oversimplified notions if the teacher taught the diagrams as reality rather than as a simplified model of reality.

While some student misconceptions are inevitable due to intellectual laziness, they can be reduced if teachers make sure to explain that models are only an approximation of reality and have sacrificed full accuracy for the sake of simplicity. And, even better, expert teachers can help students understand the reasoning behind useful models, walking them through the thinking that went into the formation of the model, as if they are active participants “accompanying” the scientists who developed the models. Young students are capable of grasping this.

An important aid in helping students understand the connection between conceptual models and the reality they describe is to include hands-on activities in the curriculum. Actually experiencing things is the best foundation for learning; *nihil in intellectu quod non sit prius in sensu *(nothing is in the intellect that was not first in the senses). Students have some capacity to grasp abstractions without seeing things for themselves, but real experience is certainly a natural and effective foundation for understanding things intellectually.

## Mathematical problem-solving

Traditionally, physical science classes have emphasized mathematical problem-solving, in order to help students understand the laws and other equations they are learning—and thus the physical relationships those equations model—as well as to help them develop their analytical abilities.

Critics contend that focusing on math problems detracts from conceptual understanding, since some students merely memorize steps for solving problems rather than developing deep understanding. Conceptual understanding should be emphasized over rote memorization, but any significant reduction in a course’s math content is a disservice to students.

Students perennially have difficulty writing equations that will enable them to solve problems, because they don’t really understand the material. Some have developed a work-around of memorizing steps to follow, but they need to be weaned from this habit. The key is grasping that the various ways of describing a situation—words, diagrams, graphs, data tables, equations—are different models of the same thing; writing equations involves translating a verbal or other model into a mathematical model. If a teacher can convey to students an understanding of the modeling process, these sorts of translations get easier, which not only helps them solve assigned problems but also fosters deeper understanding.

## Experiments

Historically, scientific models were initially developed more from observations of nature than designed experiments, and this continues to be the case in fields such as astronomy or geology. Experiments, however, have played an indispensable role in the development of science and will always be an important element of scientific work.

It is worthwhile to teach students about historically important experiments, showing how scientific theories were developed rather than just presenting them as eternal truths that we have discovered. It shows students the evidence and arguments for our scientific models, and how they have evolved over time, which not only helps students have a proper understanding of the models, but also conveys some of the human drama and adventure of science.

For younger students, school science labs tend more often to be activities that illustrate principles than experiments to find unknowns. But eventually students need to learn experimental methodology, both so that they can understand experiments done by others, and to prepare them for doing scientific work themselves.

Advanced students, at least, should learn something about how to formulate testable scientific questions based on an observations, data or models; how to figure out suitable experimental procedures; how to formulate a hypothesis or predict the results of an experiment; how to predict sources of error (or at least assess them after an experiment); how to predict the effects of modifications to a procedure; and how to explain the connection between experimental results and scientific models.

## Computers in the classroom

Computers can be used in a variety of ways in the science classroom. In some cases, they are used to present video lectures, or the same sort of text and figures that are found in books. As substitutes for teachers and textbooks, they have no great merit and some disadvantages. The interactive educational tools made possible by computers, on the other hand, can be helpful if they are used as supplements.

Computers are particularly valuable for the computational models mentioned earlier, which are used for simulating complex systems. This sort of application is beyond the needs of most primary and secondary education, but laying the groundwork for higher-level work is valuable.

For example, problems involving data analysis should be incorporated into high school science classes. Students need to learn how to use data from graphs or tables to find unknown variables or predict behavior, detect patterns and relationships in data, understand how to handle outliers in a data set, use appropriate graphing techniques (including correct scale and units), and record data and make calculations with appropriate precision.

## Conclusion

Working scientists spend a considerable portion of their time constructing, testing and refining models, and scientific journals likewise dedicate significant space to articles about modeling. Science education in primary and secondary schools needs to convey this. Instead of teaching science as a body of information, it is helpful to teach it as a process of developing and deploying models. The modeling approach is not only useful but also more accurate, because it reflects both the objective and the creative aspects of scientific knowledge. This has the merit of avoiding skeptical idealism (our scientific ideas are mere mental constructs) and naïve realism (scientific ideas are discovered directly rather than produced by creative–and occasionally fallible–reasoning). In short, teaching students about scientific modeling provides an important foundation for a deep understanding of science.

[1] Some equations can show causal relationships, such as F=dpdt, which shows how applying a force causes an object’s momentum to change. State equations describe the relationships between various properties or parts of a system at equilibrium, such as PV=nRT, which describes the relationship between different properties of a gas sample. Path or process functions show how a system changes in a transition between states, such as dE=fdx, which shows how an object loses energy due to friction it encounters along its way.