By Daniel H. Watt
Source: Watt, D. H. (1979). A Comparison of the Problem Solving Styles of Two Students Learning LOGO: A Computer Language for Children. Creative Computing.
Introduction
This paper is one of a series of reports resulting from an 18-month study of an elementary school computer laboratory. The study, funded by the National Science Foundation, and carried out by the MIT LOGO Group in collaboration with the Public Schools of Brookline, MA, provided fifty sixth-grade students with between 24 and 36 hours of computer time and instruction in LOGO, a computer language designed for use by elementary school children. The work of sixteen of these students, representing a range of interests, work styles and abilities was monitored in great detail. From the data collected we have prepared a series of reports outlining in detail the work of each child, and analyzing the learning of all sixteen children in the areas of computer programming, mathematics and problem solving.
The contrasting work of Donald and Deborah, two of the sixteen students is presented in this paper. The contrasts between the students give rise to a number of pedagogical questions and issues which will be considered in the concluding section.
The computer language, LOGO, was developed approximately eleven years ago as a vehicle by which children could learn computer programming, problem solving and mathematics in a learning environment that grew out of the cognitive psychology of Jean Piaget and an approach to the design of computer languages, based on ideas arising from work in Artificial Intelligence. Children are typically introduced to LOGO by using the computer to control a “TURTLE,” an imaginary creature which “lives” on a graphics display screen, the movement of which is controlled by commands typed at a keyboard; for example: FORWARD 100, BACK 50, RIGHT 90, LEFT 45, etc.
FORWARD 100 moves the TURTLE forward “100 TURTLE steps” and leaves a trace on the display screen, while LEFT 45 causes the TURTLE to rotate 45 degrees to the left. Drawing pictures with the TURTLE is an initial programming activity with an immense resonance for most children. Children can identify with the TURTLE, imagining themselves going through its motions, as it carries out a particular task. At the same time, controlling the TURTLE becomes a metaphor for controlling the computer itself. Like the TURTLE, the computer responds to an ordered series of commands, and to procedures, defined as series of commands.
The way in which the action of the TURTLE can lead to a geometric design, as well as the method used by children to define procedures is illustrated in the following simple examples. Having drawn a square by repeating the steps, FORWARD 100, RIGHT 90 four times, a student can define a procedure by choosing a procedure title (BOX, for example) and typing a series of commands in order. When the student types the new command, BOX, the shape shown in figure 1 will be drawn on the display screen. A similar procedure, TRI, could be defined as follows:
TO BOX 10
FORWARD 100 20 RIGHT 90 30
FORWARD 100 40 RIGHT 90 50
FORWARD 100 60 RIGHT 90 70
FORWARD 100 80 RIGHT 90
END
TO TRI 10
FORWARD 100 20 RIGHT 120 30
FORWARD 100 40 RIGHT 120 50
FORWARD 100 60 RIGHT 120
END
Having defined procedures BOX and TRI, the student can now use these as commands to the computer, in effect developing his own “private language” as he goes along. In particular, they can easily be used as subprocedures in creating the drawing, HOUSE, or the abstract design, FLOWER.
TO HOUSE 10
BOX 20 FORWARD 100 30
RIGHT 30 40 TRI
END
TO FLOWER 10
REPEAT [TRI RIGHT 30] 12
END
These TURTLE commands, when embedded in a suitable interactive, procedural computer language such as LOGO, provide the basis for a rich universe of activities called Turtle Geometry, which includes cartoon drawings, simple and complex geometric designs, mathematical theory building, and computer games and animation. Extensions of Turtle Geometry have proven fruitful when used with advanced high school students or MIT freshmen and sophomores. The universe of turtle geometry provides a conceptual framework for such aspects of mathematics as coordinate systems, positive and negative numbers, the use of variables, symmetry and similarity, the significance of special angles (30-60-90, 180, 360, etc.). The computer programming involved in beginning LOGO activities can include the notion of procedures and subprocedures, recursion and iteration, the naming of procedures and variables, the use of conditional logic in the form of simple “stop rules” for repetitive procedures, the understanding of procedural hierarchy, and the use of “debugging” strategies.
Within the “universe” of turtle geometry there is room for different students, working individually, to create their own sub-universes or “microworlds” with their own limited but expandable set of concepts, and related activities and projects. The pedagogy of a LOGO classroom can be seen as the task of helping each child to create, explore and extend his or her own “microworlds.” In this paper we describe the “microworlds” of two students, Deborah and Donald, who approached a very similar project, in very dissimilar ways. We feel that by focussing on these two individuals we can demonstrate how the computational environment of LOGO fosters the learning of children with very different developmental levels, learning styles and academic abilities.
A Comparison of Donald and Deborah’s Work
Deborah and Donald each carried out one major project as part of their LOGO experience. They each programmed the computer to draw a cartoon-like “head.” Both projects, and the superprocedures used to draw them are shown in figure 5.
Deborah is considered by her teachers to be a “slow learner.” In her regular classes she often appears to be withdrawn, unrelated to the subject matter or to her fellow students. When she began work in LOGO, she was totally dependent on the teacher requiring his reassurance on matters as routine as when to type a carriage return.
Deborah was able to slowly build her confidence and understanding by limiting her choices of LOGO commands and inputs, limiting the goals of her work, and by working in a way that minimized the chances of error. While Deborah began her LOGO experience by asking for help at literally every turn, she had a deeply ingrained resistance to new ideas or concepts. For a long time she rejected the use of subprocedures, which could have greatly expanded her possibilities. It was as though she deliberately provided herself with a very definite and restricted “microworld” in which to operate.
Deborah used as few different commands as possible in her work. Basic turtle commands along with arc primitives RARC and LARC were almost the only commands she used. For inputs to TURTLE commands, she used only multiples of 10, up to 100. If a larger effect was needed, she would use two steps, as in FORWARD 90, FORWARD 30.
Deborah began by using only inputs of 30 for all TURTLE commands, and gradually expanded to include other numbers, while continuing to use 30, 60 and 90 as her favorites. This “microworld” which Deborah chose for herself, is very nearly as rich as all of Turtle Geometry. It includes squares, triangles, “circles” “stars” “men” “rabbits,” and a variety of abstract designs, as well as the mathematical concepts of perpendicularity, inverse operations, the total turtle trip theorem, symmetry, similarity, estimation of lengths and angles, planning and debugging, and procedure writing.
By limiting her inputs to numbers such as 30, 60 and 90, Deborah actually enhanced the possibility that random explorations would produce interesting results. At the same time, she seemed to have a high degree of visual intuition, often choosing precisely the correct input to produce a desired effect. Her patience in a one-step-at-a-time mode of operation was remarkable. Her format was quite stereotyped: (1) carry out one turtle step (turn, move or penup); (2) check to see if that looks right on the screen; (3) if so, write down the step and continue; (4) if not, clear the screen, retype all the steps previously written down and try another choice for the questionable one; (5) when a design looks good, choose a name for it, and copy all the steps to make a procedure.
As an example of Deborah’s success with this approach, we examine the way in which she drew a six-pointed star without making a single mistake. She began by turning the turtle RIGHT 30, and used a combination of FORWARD 70s and RIGHT 60s to complete the star. The actual rotations required to construct the star were RIGHT 120 at each point, and LEFT 60 at each inner vertex. The way Deborah accomplished these rotations was quite typical of her work. After each forward step, Deborah would turn the TURTLE RIGHT 60. She kept turning it RIGHT 60, until the TURTLE was headed in what seemed to her to be the right direction. This required two repeats of RIGHT 60 at each point, and five repeats of RIGHT 60 at each inner vertex. At one inner vertex she missed the correct orientation, and calmly repeated RIGHT 60 a total of eleven times until the TURTLE was aimed in the right direction. When she copied the steps in her notebook, she copied all eleven RIGHT 60s without any hesitation.
It was not quite late in the series of classes that Deborah was ready to undertake a major project. She drew a picture of a rabbit in her notebook, and asked the teacher if he thought that would make a good project. He suggested modifying the rabbit, to make use of straight rather than curved lines, and helped her redraw the picture in a more simplified form.
Although Deborah began by trying to draw the rabbit as a long series of commands, she soon accepted the suggestion that she break the problem into parts, and make each part a separate subprocedure. Although her work was directed toward an overall goal, and involved a certain amount of “top-down” planning, she constructed each part of the rabbit piece by piece, in an exploratory fashion.
Once again her choice of inputs to FORWARD and RIGHT commands were such that it was relatively easy for her to make the design come out the way she wanted. Without any apparent planning, she chose the length for the sides of the rabbit’s head (FORWARD 90 FORWARD 30) in a way that made it easy for her to locate the eyes and nose symmetrically. To locate the eyes she positioned the TURTLE at one side of the head, and moved it FORWARD 30. Deciding that that “looked like the right spot for an eye,” she drew a circle of radius 20. She then moved the TURTLE twice more in steps of FORWARD 30, decided that that looked about right for the second eye. Thus she was able to locate both eyes symmetrically. It was apparent that she had not decided that the rabbit’s head was 120 steps across, and that she could therefore locate the eyes symmetrically 30 steps in from each edge. More simply, she “knew” that 90 and 30 were “useful numbers” and that if she kept trying FORWARD 30 “it ought to come out right.”
With the completion of her rabbit project, Deborah had almost totally reversed her initial feelings of dependence and incompetence. She invited her parents, teachers and school principal to visit the computer lab, and in many ways, demonstrated to her visitors and classmates her new found sense of confidence, satisfaction and power.
Donald’s work, which provides a striking contrast to Deborah’s, was characterized by a strong component of advance planning, and the creating of structures within which problems could be solved. At the same time, Donald was quite ineffective at the visually-based, exploratory modes of problem solving which were so useful to Deborah. He had difficulty estimating angles, and making use of the visual feedback provided by his explorations to improve his next attempt.
Throughout his work Donald was extremely receptive to suggestions from the teacher, often making use of new ideas before he fully understood them. In this way, he was able to incorporate into his way of working, strategies that would continue to prove useful, as he gradually came to understand them through use in more than one context. He seemed to have the confidence that he could make use of the teacher’s suggestions effectively and that he would eventually understand them, even if the concepts were a bit hazy at first.
As an example of both the effectiveness and the difficulties associated with Donald’s structured planning approach as well as the frustration he experienced with visual approaches to problem solving, we consider his construction of a “house” from a square and a triangle, a common LOGO task, tackled by many students at an early stage of their LOGO experience.
At first Donald attempted an exploratory approach to solving this problem. He began by having the TURTLE draw a triangle on the screen, making use of the TRI procedure described above. Having started by drawing a triangle, Donald established a “framework” for solving the problem corresponding to the initial orientation of the triangle. Asked to draw a picture of what he was trying to accomplish, he made the diagram shown in figure 9. Since he was now dealing with two disorientations, the gap between the TRI and BOX procedures, and the tilted orientation of the entire shape, Donald had more difficulty than he could handle, and in an hour of exploration, never succeeded in resolving the problem in this form.
At the next class, the teacher suggested that Donald draw the BOX first. This suggestion provided him with enough new insight to devise a plan for solving the problem. Donald’s plan allowed him to avoid the usual problem of finding the rotation needed to attach the triangle to the upper left hand corner of the box. Instead, he moved the turtle to the upper right hand corner of the box, reversed its direction, and then drew the triangle so that its first side was along the top of the box. In this way he resolved the problem analytically, eliminating the need for visual explorations.
Donald’s HEAD project occupied him for more than 12 class periods — more time than any other student devoted to a single project. This project also began with a plan, a cartoon-like drawing of a man’s head, which formed the basis of Donald’s work. After one session of exploratory work, Donald drew a revised plan, and worked with his teacher to create a superprocedure, designed to draw the entire figure.
Donald’s original superprocedure, TO HEAD, included the first six features of the head: the outside (BOX), EYES, NOSE, MOUTH, BEARD and HAIR. Once the superprocedure was written, each subprocedure became a mini-project, requiring one or two classes to resolve. Each feature of the head required its own “construction plan,” a combination of the analysis and exploration needed to carry it out. While working on the features of his head, Donald made use of a great deal of teacher assistance, especially in developing approaches to geometric analysis that were necessary to overcome his difficulties with visual problem solving.
When Donald came to his subprocedure, EYES, for example, he could not resolve the problem of how to locate the eyes symmetrically, by means of exploration. After several unsuccessful attempts, he asked for advice from the teacher. The teacher helped him set up a kind of “grid” with which he calculated the location of the eyes, and he was then able to solve this problem without any further difficulty.
Other, more complex problems that Donald encountered in the course of the project were resolved in similar fashion by creating “structures” which enabled him to solve the problem analytically and avoid visual explorations. In the course of his work Donald encountered estimation of distances and angles, the geometry of arcs and circles, the total turtle trip theorem, and the use of both grid-based and intrinsic coordinate systems. He learned to use subprocedures and sub-subprocedures, to use patterned procedures making use of a REPEAT command, to make use of variables to control the size and shape of his “hat” and “flower” and to use a procedure with a conditional stop rule. Although Donald only “learned” these approaches to the extent necessary to solve the particular problems inherent in his project, each succeeding use of the same concept, reinforced his exposure to it, deepening his sense of mastery.
Donald’s final figure, drawn by the superprocedure, HEAD, represents an almost literal mapping of his revised plan into a computer program.
Conclusion
In summary, we observe that Deborah’s and Donald’s problem solving styles were markedly different in several ways:
- Planning Donald was a “planner.” He made an overall plan for his project, and created separate plans for each subpart of his project. Deborah had an overall idea of what she was trying to accomplish, but her actual project was carried out in an exploratory manner, one step at a time. Donald’s superprocedure HEAD, was written to serve as a framework for his project; Deborah’s superprocedure, RABBIT, was written as the final step in completing her project.
- Analysis vs. Intuition Deborah relied on an accurate visual intuition and a very limited set of inputs to TURTLE commands; Donald’s visual intuition was poor. He relied on mathematical analysis to resolve each problem he faced.
- Comfort With New Ideas Donald absorbed a great deal of new information and tried a number of new approaches as he carried out his project confident that he could learn the skills he needed as he went along; Deborah resisted new learning — her learnings consisted largely of consolidating and reinforcing old skills and methods.
- Obtaining Assistance From A Teacher Donald made regular use of assistance from his teacher, while Deborah insisted on “doing things in her own way,” and resisted suggestions which she did not fully understand.
The observed differences give rise to pedagogical questions that are not limited to teaching children to use computers: How can teachers learn the learning styles of their students? How can teaching approaches in normal classes be flexible enough to support such a wide variation in learning styles? How can a teacher help Deborah acquire some of Donald’s analytic approach? Can Deborah’s intuitive understanding be learned by Donald?
The success of both Donald and Deborah in a LOGO environment leads us to believe that additional experience in such an environment may be productive in beginning to answer these questions.