I HAVE DEFINED MATHETICS AS BEING TO LEARNING AS HEURISTICS IS TO problem solving: Principles of mathetics are ideas that illuminate and facilitate the process of learning. In this chapter we focus on two important mathetic principles that are part of most people’s commonsense knowledge about what to do when confronted with a new gadget, a new dance step, a new idea, or a new word. First, relate what is new and to be learned to something you already know. Second, take what is new and make it your own: Make something new with it, play with it, build with it. So for example, to learn a new word, we first look for a familiar “root” and then practice by using the word in a sentence of our own construction.
We find this two-step dictum about how to learn in popular, commonsense theories of learning: The procedure described for learning a new word has been given to generations of elementary schoolchildren by generations of parents and teachers. And it also corresponds to the strategies used in the earliest processes of learning. Piaget has studied the spontaneous learning of children and found both steps at work—the child absorbs the new into the old in a process that Piaget calls assimilation, and the child constructs his knowledge in the course of actively working with it.
But there are often roadblocks in the process. New knowledge often contradicts the old, and effective learning requires strategies to deal with such conflict. Sometimes the conflicting pieces of knowledge can be reconciled, sometimes one or the other must be abandoned, and sometimes the two can both be “kept around” if safely maintained in separate mental compartments. We shall look at these learning strategies by examining a particular case in which a formal theory of physics enters into sharp conflict with commonsense, intuitive ideas about physics.
One of the simplest of such conflicts is raised by the fundamental tenet of Newton’s physics: A body in motion will, if left alone, continue to move forever at a constant speed and in a straight line. This principle of “perpetual motion” contradicts common experience and, indeed, older theories of physics such as Aristotle’s.
Suppose we want to move a table. We apply a force, set the table in motion, and keep on applying the force until the table reaches the desired position. When we stop pushing, the table stops. To our superficial gaze, the table does not behave like a Newtonian object. If it did, textbooks tell us, one push would set it in motion forever and a counteracting force would be needed to stop it at the desired place.
This conflict of ideal theory and everyday observation is only one of several roadblocks to the learning of Newtonian physics. Others derive from difficulties in applying the two mathetic principles. According to the first, people who want to learn Newtonian physics should find ways to relate it to something they already know. But they may not possess any knowledge to which it can be effectively related. According to the second, a good strategy for learning would be to work with the Newtonian laws of motion, to use them in a personal and playful fashion. But this too is not so simple. One cannot do anything with Newton’s laws unless one has some way to grab hold of them and some familiar material to which they can be applied.
The theme of this chapter is how computational ideas can serve as material for thinking about Newton’s laws. The key idea has already been anticipated. We saw how formal geometry becomes more accessible when the Turtle instead of the point is taken as the building block. Here we do for Newton what we did for Euclid. Newton’s laws are stated using the concept of “a particle,” a mathematically abstract entity that is similar to a point in having no size but that does have some other properties besides position: It has mass and velocity or, if one prefers to merge these two, it has momentum. In this chapter we enlarge our concept of Turtle to include entities that behave like Newton’s particles as well as those we have already met that resemble Euclid’s points. These new Turtles, which we call Dynaturtles, are more dynamic in the sense that their state is taken to include two velocity components in addition to the two geometric components, position and heading, of the previously discussed geometry Turtles. And having more parts to the state leads to requiring a slightly richer command language: TURTLE TALK is extended to allow us to tell the Turtle to set itself moving with a given velocity. This richer TURTLE TALK immediately opens up many perspectives besides the understanding of physics. Dynaturtles can be put into patterns of motion for aesthetic, fanciful, or playful purposes in addition to simulating real or invented physical laws. The too narrowly focused physics teacher might see all this as a waste of time: The real job is to understand physics. But I wish to argue for a different philosophy of physics education. It is my belief that learning physics consists of bringing physics knowledge into contact with very diverse personal knowledge. And to do this we should allow the learner to construct and work with transitional systems that the physicist may refuse to recognize as physics.1
Most physics curricula are similar to the math curriculum in that they force the learner into dissociated learning patterns and defer the “interesting” material past the point where most students can remain motivated enough to learn it. The powerful ideas and the intellectual aesthetic of physics is lost in the perpetual learning of “prerequisites.” The learning of Newtonian physics can be taken as an example of how mathetic strategies can become blocked and unblocked. We shall describe a new “learning path” to Newton that gets around the block: a computer-based interactive learning environment where the prerequisites are built into the system and where learners can become the active, constructing architects of their own learning.
Let us begin with a closer look at the problem of prerequisites. Someone who wanted to learn about aerodynamics might lose interest upon seeing the set of prerequisites, including mechanics and hydrodynamics, that follow an exciting course description in a college catalogue. If one wants to learn about Shakespeare, one finds no list of prerequisites. It seems fair to assume that a list of prerequisites is an expression of what educators believe to be a learning path into a domain of knowledge. The learning path into aerodynamics is mathematical, and, as we have seen in our culture, mathematical knowledge is bracketed, treated as “special”—spoken of only in special places reserved for such esoteric knowledge. The nonacademic learning environments of most children provide little impetus to that mathematical development. This means that schools and colleges must approach the knowledge of aerodynamics along exceedingly formal learning paths. The route into Shakespeare is no less complex, but its essential constitutive elements are part of our general culture: It is assumed that many people will be able to learn them informally. The physics microworld we shall develop, the physics analog of our computer-based Mathland, offers a Piagetian learning path into Newtonian laws of motion, a topic usually considered paradigmatic of the kind of knowledge that can only be reached by a long, formalized learning path. Newtonian thinking about motion is a complex and seemingly counterintuitive set of assumptions about the world. Historically, it was long to evolve. And in terms of individual development, the child’s interaction with his environment leads him to a very different set of personal beliefs about motion, beliefs that in many ways are closer to Aristotle’s than to Newton’s. After all, the Aristotelian idea of motion corresponds to the most common situation in our experience. Students trying to develop Newtonian thinking about motion encounter three kinds of problems that a computer microworld could help solve. First, students have had almost no direct experience of pure Newtonian motion. Of course, they have had some. For example, when a car skids on an icy road it becomes a Newtonian object: It will, only too well, continue in its state of motion without outside help. But the driver is not in a state of mind to benefit from the learning experience. In the absence of direct and physical experiences of Newtonian motion, the schools are forced to give the student indirect and highly mathematical experiences of Newtonian objects. There movement is learned by manipulating equations rather than by manipulating the objects themselves. The experience, lacking immediacy, is slow to change the student’s intuitions. And it itself requires other formal prerequisites. The student must first learn how to work with equations before using them to model a Newtonian world. The simplest way in which our computer microworld might help is by putting students in a simulated world where they have direct access to Newtonian motion. This can be done when they are young. It need not wait for their mastery of equations. Quite the contrary: Instead of making students wait for equations, it can motivate and facilitate their acquisition of equational skills by providing an intuitively well-understood context for their use.
Direct experience with Newtonian motion is a valuable asset for the learning of Newtonian physics. But more is needed to understand it than an intuitive, seat-of-the-pants experience. The student needs the means to conceptualize and “capture” this world. Indeed, a central part of Newton’s great contribution was the invention of a formalism, a mathematics suited to this end. He called it “fluxions”; present-day students call it “differential calculus.” The Dynaturtle on the computer screen allows the beginner to play with Newtonian objects. The concept of the Dynaturtle allows the student to think about them. And programs governing the behavior of Dynaturtles provide a formalism in which we can capture our otherwise too fleeting thoughts. In doing so it bypasses the long route (arithmetic, algebra, trigonometry, calculus) into the formalism that has passed with only superficial modification from Newton’s own writing to the modern textbook. And I believe it brings the student in closer touch with what Newton must have thought before he began writing equations.
The third prerequisite is somewhat more subtle. We shall soon look directly at statements of what is usually known as Newton’s laws of motion. As we do, many readers will no doubt recall a sense of unease evoked by the phrase “law of motion.” What kind of a thing is that? What other laws of motion are there besides Newton’s? Few students can answer these questions when they first encounter Newton, and I believe that this goes far toward explaining the difficulty of physics for most learners. Students cannot make a thing their own without knowing what kind of a thing it is. Therefore, the third prerequisite is that we must find ways to facilitate the personal appropriation not only of Newtonian motion and the laws that describe it but also of the general notion of laws that describe motion. We do this by designing a series of microworlds.
The Turtle World was a microworld, a “place,” a “province of Mathland,” where certain kinds of mathematical thinking could hatch and grow with particular ease. The microworld was an incubator. Now we shall design a microworld to serve as an incubator for Newtonian physics. The design of the microworld makes it a “growing place” for a specific species of powerful ideas or intellectual structures. So, we design microworlds that exemplify not only the “correct” Newtonian ideas, but many others as well: the historically and psychologically important Aristotelian ones, the more complex Einsteinian ones, and even a “generalized law-of-motion world” that acts as a framework for an infinite variety of laws of motion that individuals can invent for themselves. Thus learners can progress from Aristotle to Newton and even to Einstein via as many intermediate worlds as they wish. In the descriptions that follow, the mathetic obstacles to Newton are overcome: The prerequisites are rooted in personal knowledge and the learner is involved in a creative exploration of the idea and the variety of laws of motion.
Let us begin to describe the microworld by starting with Newton’s three laws, stated here “formally” and in a form that readers do not have to understand in detail:
1. Every particle continues in a state of rest or motion with constant speed in a straight line unless compelled by a force to change that state.
2. The net unbalanced force (F) producing a change of motion is equal to the product of the mass (m) and the acceleration (a) of the particle: F = ma.
3. All forces arise from the interaction of particles, and whenever a particle acts on another there is an equal and opposite reaction on the first.
As we have noted, children’s access to these laws is blocked by more than the recondite language used to state them. We analyze these roadblocks in order to infer design criteria for our microworld. A first block is that children do not know anything else like these laws. Before being receptive to Newton’s laws of motion, they should know some other laws of motion. There must be a first example of laws of motion, but it certainly does not have to be as complex, subtle, and counterintuitive as Newton’s laws. More sensible is to let the learner acquire the concept of laws of motion by working with a very simple and accessible instance of a law of motion. This will be the first design criterion for our microworld. The second block is that the laws, as stated, offer no footholds for learners who want to manipulate them. There is no use they can put them to outside of end-of-chapter schoolbook exercises. And so, a second design criterion for our microworlds is the possibility of activities, games, art, and so on that make activity in the microworlds matter. A third block is the fact that the Newtonian laws use a number of concepts that are outside most people’s experience, the concept of “state,” for example. Our microworld will be designed so that all needed concepts can be defined within the experience of that world.
As in the case of the geometry Turtle, the physics Turtle is an interactive being that can be manipulated by the learner, providing an environment for active learning. But the learning is not “active” simply in the sense of interactive. Learners in a physics microworld are able to invent their own personal sets of assumptions about the microworld and its laws and are able to make them come true. They can shape the reality in which they will work for the day, they can modify it and build alternatives. This is an effective way to learn, paralleling the way in which each of us once did some of our most effective learning. Piaget has demonstrated that children learn fundamental mathematical ideas by first building their own, very much different (for example, preconservationist) mathematics. And children learn language by first learning their own (“baby-talk”) dialects. So, when we think of microworlds as incubators for powerful ideas, we are trying to draw upon this effective strategy: We allow learners to learn the “official” physics by allowing them the freedom to invent many that will work in as many invented worlds.
Following Polya’s principle of understanding the new by associating it with the old, let us reinterpret our microworld of Turtle geometry as a microworld of a special kind of physics. We recast the laws by which Turtles work in a form that parallels the Newtonian laws. This gives us the following “Turtle laws of motion.” Of course, in a world with only one Turtle, the third law, which deals with the interaction among particles, will not have an analog.
1. Every Turtle remains in its state of rest until compelled by a TURTLE COMMAND to change that state.
2. a. The input to the command FORWARD is equal to the Turtle’s change in the POSITION part of its state.
b. The input to the command RIGHT TURN is equal to the Turtle’s change of the HEADING part of its state.
What have we gained in our understanding of Newtonian physics by this exercise? How can students who know Turtle geometry (and can thus recognize its restatement in Turtle laws of motion) now look at Newton’s laws? They are in a position to formulate in a qualitative and intuitive form the substance of Newton’s first two laws by comparing them with something they already know. They know about states and state-change operators. In the Turtle world, there is a state-change operator for each of the two components of the state. The operator FORWARD changes the position. The operator TURN changes the heading. In physics, there is only one state-change operator, called force. The effect of force is to change velocity (or, more precisely, momentum). Position changes by itself.
These contrasts lead students to a qualitative understanding of Newton. Although there remains a gap between the Turtle laws and the Newtonian laws of motion, children can appreciate the second through an understanding of the first. Such children are already a big step ahead in learning physics. But we can do more to close the gap between Turtle and Newtonian worlds. We can design other Turtle microworlds in which the laws of motion move toward a closer approximation of the Newtonian situation.
To do this we create a class of Turtle microworlds that differs in the properties that constitute the state of the Turtle and in the operators that change these states. We have formally described the geometry Turtle by saying that its state consists of position and velocity and that its state-change operators act independently of these two components. But there is another way, perhaps a more powerful and intuitive way, to think about it. This is to see the Turtle as a being that “understands” certain kinds of communication and not others. So, the geometry Turtle understood the command to change its position while keeping its heading and to change its heading while keeping its position. In the same spirit, we could define a Newtonian Turtle as a being that can accept only one kind of order, one that will change its momentum. These kinds of description are in fact the ones we use in introducing children to microworlds. Now let us turn to two Turtle microworlds that can be said to lie between the geometry and Newtonian Turtles.
VELOCITY TURTLES
The state of a velocity Turtle is POSITION AND VELOCITY. Of course, since velocity is defined as a change in position, by definition the first component of this state is continuously changing (unless VELOCITY is zero). So, in order to control a velocity Turtle, we only have to tell it what velocity to adopt. We do this by one state-change operator, a command called SETVELOCITY.
ACCELERATION TURTLES
Another Turtle, intermediate between the geometry Turtle and the Turtle that could represent a Newtonian particle, is an acceleration Turtle. Here, too, the state of the Turtle is its position and velocity. But this time the Turtle cannot understand such a command as “Take on such-and-such a velocity”. It can only take instructions of the form “Change your velocity by x, no matter what your velocity happens to be.” This Turtle behaves like a Newtonian particle with an unchangeable mass.
Thus, the sequence of turtles—geometry Turtle to velocity Turtle to acceleration Turtle to Newtonian Turtle—constitutes a path into Newton that is resonant with our two mathetic principles. Each step builds on the one before in a clear and transparent way, satisfying the principle of prerequisites. As for our second mathetic principle—“use it, play with it”—the case is even more dramatic. Piaget showed us how the child constructs a preconservationist and then a conservationist world out of the materials (tactile, visual, and kinesthetic) in his environment. But until the advent of the computer, there were only very poor environmental materials for the construction of a Newtonian world. However, each of the microworlds we described can function as an explorable and manipulable environment.
In Turtle geometry, geometry was taught by way of computer graphics projects that produce effects like those shown in the designs illustrating this book. Each new idea in Turtle geometry opened new possibilities for action and could therefore be experienced as a source of personal power. With new commands such as SETVELOCITY and CHANGE VELOCITY, learners can set things in motion and produce designs of ever-changing shapes and sizes. They now have even more personal power and a sense of “owning” dynamics. They can do computer animation—there is a new, personal relationship to what they see on television or in a pinball gallery. The dynamic visual effects of a TV show, an animated cartoon, or a video game now encourage them to ask how they could make what they see. This is a different kind of question than the one students traditionally answer in their “science laboratory.” In the traditional laboratory pedagogy, the task posed to the children is to establish a given truth. At best, children learn that “this is the way the world works.” In these dynamic Turtle microworlds, they come to a different kind of understanding—a feel for why the world works as it does. By trying many different laws of motion, children will find that the Newtonian ones are indeed the most economical and elegant for moving objects around.
All of the preceding discussion has dealt with Newton’s first two laws. What analogs to Newton’s third law are possible in the world of Turtles? The third law is only meaningful in a microworld of interactive entities—particles for Newton, Turtles for us. So let us assume a microworld with many Turtles that we shall call TURTLE 1, TURTLE 2, and so on. We can use TURTLE TALK to communicate with multiple Turtles if we give each of them a name. So we can use commands such as: TELL TURTLE 4 SETVELOCITY 20 (meaning “Tell Turtle number 4 to take on a velocity of 20).
Newton’s third law expresses a model of the universe, a way to conceptualize the workings of physical reality as a self-perpetuating machine. In this vision of the universe, all actions are governed by particles exerting forces on one another, with no intervention by any outside agent. In order to model this in a Turtle microworld, we need many Turtles interacting with each other. Here we shall develop two models for thinking about interacting Turtles: linked Turtles and linked Dynaturtles.
In the first model we think of the Turtles as giving commands to one another rather than obeying commands from the outside. They are linked Turtles. Of course, Turtles can be linked in many ways. We can make Turtles that directly simulate Newtonian particles linked by simulated gravity. This is commonly done in LOGO laboratories, where topics usually considered difficult in college physics are translated into a form accessible to junior high school students. Such simulations can serve as a springboard from an elementary grasp of Newtonian mechanics to an understanding of the motion of planets and of the guidance of spacecraft. They do this by making working with the Newtonian principles an active and personally involving process. But to “own” the idea of interacting particles—or “linked Turtles”—the learner needs to do more. It is never enough to work within a given set of interactions. The learner needs to know more than one example of laws of interaction and should have experience inventing new ones. What are some other examples of linked Turtles?
A first is a microworld of linked Turtles called “mirror Turtles.” We begin with a “mirror Turtle” microworld containing two Turtles linked by the rules: Whenever either is given a FORWARD (or BACK) command, the other does the same; whenever either is given a RIGHT TURN (or LEFT TURN) command, the other does the opposite. This means that if the two Turtles start off facing one another, any Turtle program will cause their trips to be mirror images of one another’s. Once the learner understands this principle, attractive kaleidoscope designs can easily be made.
A second microworld of linked Turtles, and one that is closer to Newtonian physics, applies these mirror linkages to velocity Turtles. No static images printed on this page could convey the visual excitement of these dynamic kaleidoscopes in which brightly colored points of light dance in changing and rotating paths. The end product has the excitement of art, but the process of making it involves learning to think in terms of the actions and reactions of linked moving objects.
These linked Turtle microworlds consolidate the learner’s experience of the three laws of motion. But we have asserted that multiple microworlds also provide a platform for understanding the idea of a law of motion. A student who has mastered the general concept of a law of motion has a new, powerful tool for problem solving. Let’s illustrate with the Monkey Problem.
A monkey and a rock are attached to opposite ends of a rope that is hung over a pulley. The monkey and the rock are of equal weight and balance one another. The monkey begins to climb the rope. What happens to the rock?
I have presented this problem to several hundred MIT students, all of whom had successfully passed rigorous and comprehensive introductory physics courses. Over three quarters of those who had not seen the problem before gave incorrect answers or were unable to decide how to go about solving it. Some thought the position of the rock would not be affected by the monkey’s climbing because the monkey’s mass is the same whether he is climbing or not; some thought that the rock would descend either because of a conservation of energy or because of an analogy with levers; some guessed it would go up, but did not know why. The problem is clearly “hard.” But this does not mean that it is “complex.” I suggest that its difficulty is explicable by the lack of something quite simple. When they approach the problem, students ask themselves: “Is this a ‘conservation-of-energy’ problem?” “Is this a ‘lever-arm’ problem?” and so on. They do not ask themselves: “Is this a ‘law-of-motion’ problem?” They do not think in terms of such a category. In the mental worlds of most students, the concepts of conservation, energy, lever-arm, and so on, have become tools to think with. They are powerful ideas that organize thinking and problem solving. For a student who has had experience in a “laws-of-motion” microworld, this is true of “law of motion.” Thus this student will not be blocked from asking the right question about the monkey problem. It is a law-of-motion problem, but a student who sees laws of motion only in terms of algebraic formulas will not even ask the question. For those who pose the question, the answer comes easily. And once one thinks of the monkey and the rock as linked objects, similar to the ones we worked with in the Turtle microworld, it is obvious that they must both undergo the same changes in state. Since they start with the same velocity, namely zero, they must therefore always have the same velocity. Thus, if one goes up, the other goes up at the same speed.2
We have presented microworlds as a response to a pedagogical problem that arises from the structure of knowledge: the problem of prerequisites. But microworlds are a response to another sort of problem as well, one that is not embedded in knowledge but in the individual. The problem has to do with finding a context for the construction of “wrong” (or, rather, “transitional”) theories. All of us learn by constructing, exploring, and theory building, but most of the theory building on which we cut our teeth resulted in theories we would have to give up later. As preconservationist children, we learned how to build and use theories only because we were allowed to hold “deviant” views about quantities for many years. Children do not follow a learning path that goes from one “true position” to another, more advanced “true position.” Their natural learning paths include “false theories” that teach as much about theory building as true ones. But in school, false theories are no longer tolerated.
Our educational system rejects the “false theories” of children, thereby rejecting the way children really learn. And it also rejects discoveries that point to the importance of the false-theory learning path. Piaget has shown that children hold false theories as a necessary part of the process of learning to think. The unorthodox theories of young children are not deficiencies or cognitive gaps, they serve as ways of flexing cognitive muscles, of developing and working through the necessary skills needed for more orthodox theorizing. Educators distort Piaget’s message by seeing his contribution as revealing that children hold false beliefs, which they, the educators, must overcome. This makes Piaget-in-the-schools a Piaget backward—backward because children are being force-fed “correct” theories before they are ready to invent them. And backward because Piaget’s work puts into question the idea that the “correct” theory is superior as a learning strategy.
Some readers may have difficulty seeing the child’s nonconservationist view of the world as a kind of theory building. Let’s take another example. Piaget asked preschool children, “What makes the wind?” Very few said, “I don’t know.” Most children gave their own personal theories, such as, “The trees made the wind by waving their branches.” This theory, although wrong, gives good evidence for highly developed skills in theory building. It can be tested against empirical fact. Indeed there is a strong correlation between the presence of wind and the waving of tree branches. And children can perform an experiment that makes their causal connection quite plausible. When they wave their hands near their faces, they make a very noticeable breeze. Children can imagine this effect multiplied when the waving object is not a small hand but a giant tree, and when not one but many giant trees are waving. So, the trees of a dense forest should be a truly powerful wind generator.
What do we say to a child who has made such a beautiful theory? “That’s great thinking, Johnny, but the theory is wrong” constitutes a put-down that will convince most children that making one’s own theories is futile. So, rather than stifling the children’s creativity, the solution is to create an intellectual environment less dominated than the school’s by the criteria of true and false.
We have seen that microworlds are such environments. Just as students who prefer to do their programming using Newtonian Turtles with third law interaction are making Newton their own, children making a spectacular spiral in a non-Newtonian microworld are no less firmly on the path toward understanding Newton. Both are learning what it is like to work with variables, to think in terms of ratios of dissimilar qualities, to make appropriate approximations, and so on. They are learning mathematics and science in an environment where true or false and right or wrong are not the decisive criteria.
As in a good art class, the child is learning technical knowledge as a means to get to a creative and personally defined end. There will be a product. And the teacher as well as the child can be genuinely excited by it. In the arithmetic class the pleasure that the teacher shows at the child’s achievement is genuine, but it is hard to imagine teacher and child showing delight over a product. In the LOGO environment it happens often. The spiral made in the Turtle microworld is a new and exciting creation by the child—he may even have “invented” the way of linking Turtles on which it is based.
The teacher’s genuine excitement about the product is communicated to children who know they are doing something consequential. And unlike in the arithmetic class, where they know that the sums they are doing are just exercises, here they can take their work seriously. If they have just produced a circle by commanding the Turtle to take a long series of short forward steps and small right turns, they are prepared to argue with a teacher that a circle is really a polygon. No one who has overheard such a discussion in fifth-grade LOGO classes walks away without being impressed by the idea that the truth or falsity of theory is secondary to what it contributes to learning.