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286Effectiveness of Logo

The Effectiveness of Using Logo toTeach Geometry to PreserviceElementary TeachersMichael T. BattistaTeacher Dev. & Curr. StudiesKent State UniversityKent, Ohio 44242

According to Hatfield, "mathematicslearning is primarily a person-centered,constructive process; students buildand modify their knowledge from ex-periences with task-oriented situationscharacteristic of mathematics. Studentsmust experience opportunities and de-velop feelings of responsibility for re-vising, refining, and extending theirideas as the ideas are being con-structed" (1979, p. 53). One approachto developing students mathematicsconcepts and problem solving skillsthat is consistent with this "construc-

tive" view of mathematics learning is to engage them in active interactionwith computer-based environments. In this approach, students understandingof a mathematical concept is developed or enhanced by asking them toinstruct the computer to do some type of mathematical task. For example,students can use predefined programs that slide, turn, and flip a figure toexplore ideas in transformational geometry; they can decide how to direct agraphics program to draw a rectangle; or they can write a computer programto find the greatest common divisor of two numbers.A natural choice for a computer environment in which students can be

actively engaged in constructing and exploring mathematics is Logo. Accord-ing to Feurzeig and Lukas (1972), Logo "provides an operational universewithin which students can define a mathematical process and then see itseffects unfold" (p. 39). With its turtle graphics capability. Logo is especiallyuseful in geometry because geometric figures can be created by directing themovement of a cybernetic "turtle" on the computer screen. Because thisscreen is an excellent representation of a small portion of a mathematicalplane, many educators have suggested that having students investigate andexplore the world of Logo turtle graphics is an excellent way to have themlearn about school geometry (Billings, 1983; Billstein, 1982). As Abelson anddiSessa have noted, with Logo, students can "regard plane-geometric figures

School Science and MathematicsVolume 87 (4) April 1987

Effectiveness of Logo 287not as static entities but as tracings made on a display screen by acomputer-controlled "turtle" (1980, xiv).

Thus, it can be hypothesized that having students draw and investigategeometric shapes in a Logo environment is one way for them to learngeometric concepts constructively. That is, if a student is attempting to directthe turtle to draw a rectangle, for example, he or she must do far more thanconjure up a mental image of a rectangle, as would be done with a freehanddrawing. The student must construct a conceptualization of rectangle that isexplicit enough to develop a set of move commands that can be entered intothe computer. This activity forces the student "to externalize intuitiveexpectations. When the intuition is translated into a program it becomesmore obtrusive and more accessible to reflection" (Papert, 1980, p. 145).Since the program becomes a tangible representation of the students conceptof rectangle, the validity of the conceptualization can be tested by runningthe program on the computer. In this way, the students conceptualization isexpressed in a form that allows it to be actively manipulated, modified, andstudied. Alternately, if students are given a Logo procedure that will quicklydraw regular star polygons when the proper number of sides and angle ofturn are input, they will be able to investigate the mathematical relationshipthat exists between these inputs more easily than if the figures were drawn byhand. Furthermore, the computer seems to have a motivational effect onstudents. Often, students can develop geometric concepts by investigatinginherently "neat" phenomena such as Logo procedures that draw spirals ormany-pointed stars (Papert, 1980).However, despite anecdotal reports of students developing and using

mathematical notions in a Logo domain and several research studies reportingLogo programmers gains in learning certain mathematical topics (Ross &Howe, 1981), there is a lack of convincing empirical evidence supporting thenotion of increased geometry achievement as a result of the use of Logo. Themajor goal of the present study, therefore, was to compare the effectivenessof having students learn specific geometric ideas by conducting explorationsin a Logo environment to conducting these same explorations with paper andpencil.

MethodSubjectsThe subjects of the study were 69 preservice elementary teachers enrolled in ageometry course required of and specifically designed for them. Althoughresults of studies with preservice elementary teachers should not be general-ized beyond this rather special population, such studies are of interest forseveral reasons. First, because these future teachers mathematical back-grounds usually exhibit many deficiencies, it is essential for them to betterunderstand the mathematics that they will eventually teach. Thus, it isimportant for teacher educators to learn how best to teach mathematics to

School Science and MathematicsVolume 87 (4) April 1987

288 Effectiveness of Logo

these students. Second, it has been suggested that preservice elementaryteachers effectiveness in teaching mathematics, especially problem solving, isdependent on their being exposed to proper mathematics instruction them-selves (Krulik and Rudnick, 1982). The National Council of Teachers ofMathematics recommends that teachers of mathematics provide opportunitiesfor their students to be actively involved in learning, experimenting with,exploring, and communicating about mathematics as part of an environmentthat encourages problem solving (NCTM, 1980); thus, in order for elementaryteaching students to become teachers who follow this recommendation theythemselves should be exposed to such instruction. That is, in order forpreservice elementary teachers to take a constructive approach in teachingmathematics, they themselves must have had some exposure to learningmathematics constructively. In addition, in order for preservice elementaryteachers to be able to utilize computers in their own teaching, they themselvesshould be exposed to such utilization (Battista and Krockover, 1982).

Effectiveness of Logo 289

periods on the computers to complete the activities from this unit. While theLogo group completed the introductory unit on the computers, students inthe NonLogo group did activities unrelated to the mathematics that was to becovered in the treatment.

"... students were to discover theorems about whichdistinct regular star polygons exist . . ."

After the introductory materials were completed, students in both groupsbegan the small group geometry activities. These activities either requiredstudents to apply their knowledge of geometry to draw geometric figures, orhad students draw figures (such as regular polygons, regular star polygons,and translation, rotation, and reflection images) and attempted to guide themto make some geometric discovery about the concepts they were exploring.During the first treatment period, students were asked to draw (either withthe turtle or with pencil, paper, ruler, protractor, and compass) such figuresas three concurrent line segments; parallel and perpendicular lines; obtuse,acute, right, and scalene triangles; and a parallelogram, rectangle, trapezoid,kite, rhombus, and pentagon. (The goal of this activity was to have studentsexplore the defining characteristics of these concepts.) In another activity,students were to draw first a square, then a regular pentagon, then a regularhexagon, and so on and, ultimately, were to discover the relationship betweenthe number of sides of a regular polygon and the measures of its vertex,central, and exterior angles. (The Logo group wrote procedures to draw thesefigures; the NonLogo group used compass, ruler, and protractor.) Anotheractivity had students using either tracing paper and protractor or Logoprocedures to determine the measures of the rotational symmetries of variousfigures. Students also constructed figures having given rotational andreflectional symmetries, and were introduced to the rule of 360. During thesecond treatment period, students were to discover theorems about whichdistinct regular star polygons exist by systematically constructing suchpolygons. They further investigated the rule of 360. And they investigated thetransformational concepts of translations, reflections, and rotations byutilizing tracing paper (NonLogo group) or Logo procedures that wouldtranslate, reflect, or rotate a given geometric figure. In particular, studentslearned how to do the following within the context of a rectangularcoordinate system: specify a translation, reflection, and rotation; find theimage and preimage of a given figure under a given transformation; andspecify, the translation, reflection, or rotation that mapped one figure ontoanother.During each treatment, the two halves of the class were in separate rooms

School Science and MathematicsVolume 87 (4) April 1987

290 Effectiveness of Logo

and the instructor circulated about both rooms. Every day each student wasgiven an activity sheet or booklet in which the activities for the day weredescribed in great detail. The activity sheets for the two groups were identicalexcept that when the NonLogo group was directed to do something withcompass, ruler, protractor, and paper and pencil, the Logo group wasdirected to do the activity with the Logo turtle. Each treatment during thefall semester lasted seven 50-minute class periods (three days per week); eachtreatment during the spring semester was slightly extended and lasted nineclass periods.At the end of each treatment period, a paper and pencil tes