Programming to Learn in Other Content Areas
The relationship between programming and other content areas may not be intuitively obvious. Or, perhaps, the programming examples in this chapter may already have involved concepts you recognize from other content areas. For example, you may relate the concept of variables to an algebra course. Papert (1980) saw the relationship as potentially much more general. To understand his view, it is useful to appreciate his vision of a computer microworld. A computer microworld provides an environment representing some discipline. The student can explore and manipulate this environment and experience systematic consequences as a result of actions taken. These consequences allow the student to construct an understanding of the environment through processes of assimilation and accommodation. Papert adopted the ideas of assimilation and accommodation from Jean Piaget. In assimilation, external experiences are interpreted as fitting with existing mental structures. In accommodation, mental structures are changed to fit experiences. These interacting processes describe the development of personal understanding by construction (see discussion of constructional theory in Chapter 2).
Turtle Geometry
The LOGO turtle provides access to the microworld of turtle geometry. Papert (1980) argued that geometry could be understood through action and that turtle geometry is just another style of "doing geometry." Euclid’s style was logical, Descartes’ style was algebraic, and the turtle’s style is computational. If you find this comparison a bit abstract, think about your own understanding of the concept "circle." You may represent a circle as a definition: a closed plane figure with all points equidistant from a common point. As an algebraic expression, a circle can be represented as x2 + y2 = r2 (assuming the Cartesian center is 0, 0), with r representing the radius of the circle. If you fix the radius as 1, the unit circle can be defined as x2 + y2 = 1. A circle can also be represented as the product of a LOGO program. Most students willing to experiment with LOGO eventually come across the simple program REPEAT 360 FD 1 RT 1. For a more sophisticated method for generating a circle, consider the following procedure (based on Yoder, 1992):
to CIRCLE :RADIUS
REPEAT 360
FD :RADIUS
WAIT 10
PD
FD 1
PU
BK 1
BK :RADIUS
WAIT 10
RT 1
END
If you have an opportunity, enter these commands and run the program. The program draws a circle with the radius you specify. The turtle moves forward the distance you input as the radius, puts the pen down, and moves forward one step. This creates a point. The turtle then puts the pen up and moves back to the center of the circle. Finally, the turtle turns one degree to the right. This sequence is repeated 360 times. If you watch the turtle repeatedly run out, make a mark, and then run back as it generates the circle (the WAIT commands have been inserted so that program execution slows down), you may understand what a circle is in a different way than you would if you just thought about "a closed plane figure with all points equidistant from a common point."
Certainly the computational method of defining shapes is the only method among those briefly presented here that defines geometry in terms of action (movements of the turtle). Young children intuitively understand spatial notions in terms of action and the LOGO approach (Battista & Clements, 1988). One experience cited in support of this claim is the frequently observed tendency of children (or adults) to "become the turtle" to solve LOGO problems. You can watch children moving their bodies as they think about a problem. Teachers often suggest that students become the turtle when students want help. Try it. Can you walk in a circle? Translate what you are doing into LOGO commands. Would your program read, "Move ahead a little, turn a little, ahead a little, turn a little, and so on"?
Although it is most frequently used with elementary or middle school students, LOGO is not limited to understanding only the most basic features of geometry. Turtle geometry can be used to explore advanced topics in geometry as well (Abelson & diSessa, 1981; Yusuf, 1995).