The Effectiveness of Using Logo to Teach Geometry to Preservice Elementary Teachers

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<ul><li><p>286Effectiveness of Logo</p><p>The Effectiveness of Using Logo toTeach Geometry to PreserviceElementary TeachersMichael T. BattistaTeacher Dev. &amp; Curr. StudiesKent State UniversityKent, Ohio 44242</p><p>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-</p><p>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</p><p>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</p><p>School Science and MathematicsVolume 87 (4) April 1987</p></li><li><p>Effectiveness of Logo 287not as static entities but as tracings made on a display screen by acomputer-controlled "turtle" (1980, xiv).</p><p>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</p><p>mathematical notions in a Logo domain and several research studies reportingLogo programmers gains in learning certain mathematical topics (Ross &amp;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.</p><p>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</p><p>School Science and MathematicsVolume 87 (4) April 1987</p></li><li><p>288 Effectiveness of Logo</p><p>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).</p></li><li><p>Effectiveness of Logo 289</p><p>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.</p><p>"... students were to discover theorems about whichdistinct regular star polygons exist . . ."</p><p>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</p><p>School Science and MathematicsVolume 87 (4) April 1987</p></li><li><p>290 Effectiveness of Logo</p><p>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 test over the</p><p>geometric content covered during that period was administered to the wholeclass (Test 2 for the first treatment period and Test 3 for the second). Test 1(the pretest) covered identifying definitions for concepts such as parallel andperpendicular lines, regular polygon, quadrilateral, parallelogram, rectangle,square, and rhombus; constructing altitudes and perpendicular bisectors fortriangles; classifying triangles by angle and by length of sides; determiningthe number of reflectional and rotational symmetries of a figure; determiningthe measure of a vertex, central, and exterior angle for a regular octagongiven its drawing; determining the number of diagonals of a convex polygon;and drawing a regular polygon with compass and ruler. Test 2 coveredidentifying definitions for concepts such as parallel and perpendicular lines,regular polygon, quadrilateral, parallelogram, rectangle, square, and rhom-bus; identifying the vertex and exterior angles for a polygon; classifyingtriangles by angle and by length of sides; determining the number ofreflectional symmetries and the measure of each rotational symmetry for afigure; completing figures so that they would have specified symmetry;determining the measure of a vertex, central, and exterior angle for a regularpolygon with 50 or more sides (no drawing given); finding the sum of themeasures of the vertex and exterior angles of a polygon; determining how fara robot would walk if it repeatedly went forward a given distance and turnedright a given angle until it was facing its original position; describing therelationship between the vertex and exterior angles of a regular polygon asthe number of sides increases; and drawing a regular polygon with compassand ruler. Test 3 covered definitions of translations, rotations, andreflections; locating points in a rectangular coordinate system; drawing aregular star polygon given its vertices; listing all distinct regular star polygonswith a given number of sides; determining the measure of a vertex angle of aregular star polygon; drawing the translation, rotation, and reflection image(and finding preimages) of a polygon on a coordinate system that appearedon graph paper; specifying what transformation would map one figure ontoanother, and applying the rule of 360 to answer questions about the path ofa robot that repeatedly moves forward the same amount then turns right a</p><p>School Science and MathematicsVolume 87 (4) April 1987</p></li><li><p>Effectiveness of Logo 291fixed angle. The Cronbach alpha reliabilities for Test 1, Test 2, and Test 3were .74, .77, and .78, respectively.At the end of each semester, students were given a questionnaire that</p><p>assessed their feelings about mathematics, geometry, and using computers inthe geometry course. The questionnaire consisted of Likert-type items inwhich students were to express the degree to which they agreed or disagreedwith various statements.</p><p>Results</p><p>Table 1 gives the mean percent scores for the various groups of students onthe first three tests administered for each semester of the geometry course. Italso diagrams the design of the study by indicating when each treatment wasgiven to each group. For the first treatment period, Test 1 was used as ameasure of students present level of achievement in the geometry course andTest 2 as the measure of geometry learning during the treatment period. Forthe second treatment period, the sum of a students scores on Test 1 and Test2 was used as the measure of present level of achievement in the geometrycourse and Test 3 as the measure of geometry learning during the treatment.</p><p>TABLE 1 ,</p><p>Mean Percent Scores and Experimental Design</p><p>First SecondGroup n Test 1 treatment Test 2 treatment Test 3</p><p>Fall 1 18 85.11 --Logo-- 79.65 --NonL-- 75.24Fall 2 15 88.47 --NonL-- 84.42 --Logo-- 81.15Spring 1 17 85.63 --Logo-- 73.11 --NonL-- 85.94Spring 2 19 83.34 --NonL-- 81.01 --Logo-- 80.80</p><p>As can be seen from the design of the study, it was possible to comparethe geometry learning of a Logo group to that of a NonLogo group fourtimes. This was done using four separate analyses of covariance on studentsraw test scores, with either Test 2 used as the dependent measure and Test 1the covariate (for the first treatment period), or Test 3 used as the dependentmeasure and the sum of Test 1 and Test 2 as the covariate (for the secondtreatment period). See Table 2. Neither comparison was significant in the fallsemester. However, both comparisons in the spring semester were significantat the .067 level or beyond. The adjusted post-test mean of the NonLogogroup exceeded that of the Logo group in three out of four of theexperimental periods.</p><p>School Science and MathematicsVolume 87 (4) April 1987</p></li><li><p>292 Effectiveness of LogoTABLE 2</p><p>Analyses of Covariance and Adjusted Post-test Means for Geometry Achievement</p><p>Source ofvariation df MS F p</p><p>CovariateTest 1</p><p>Main EffectsGroup</p><p>Residual</p><p>Fall, First Treatment (Post-test = Test 2)Adjusted post-test means: Logo =80.95, NonLogo = 82.97</p><p>1 1195.95 35.64 .000</p><p>1 17.39 .52 .47730 33.56</p><p>Fall, Second Treatment (Post-test = Test 3)Adjusted post-test means: Logo =79.30, NonLogo =76.78</p><p>CovariateTest 1+Test 2 1 483.47 11.37 ....</p></li></ul>

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