Assessing Logo programming among Jordanian seventh grade students through turtle geometry

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Assessing Logo programming amongJordanian seventh grade studentsthrough turtle geometryAmal A. Khasawneh aa Department of Curriculum and Instruction , School of Education,Yarmouk University , Irbid, JordanPublished online: 16 Jun 2009.

To cite this article: Amal A. Khasawneh (2009) Assessing Logo programming among Jordanianseventh grade students through turtle geometry, International Journal of Mathematical Education inScience and Technology, 40:5, 619-639, DOI: 10.1080/00207390902912845

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International Journal of Mathematical Education inScience and Technology, Vol. 40, No. 5, 15 July 2009, 619–639

Assessing Logo programming among Jordanian seventh

grade students through turtle geometry

Amal A. Khasawneh*

Department of Curriculum and Instruction, School of Education,Yarmouk University, Irbid, Jordan

(Received 11 February 2008)

The present study is concerned with assessing Logo programmingexperiences among seventh grade students. A formal multiple-choice testand five performance tasks were used to collect data. The results providedthat students’ performance was better than the expected score by theprobabilistic laws, and a very low correlation between their Logoprogramming performance and school mathematics achievement wasrevealed. Most of the made misconceptions were due to geometricalaspects rather than Logo primitives, and were concentrated on the angleof rotation, the angle of complete rotation and the angle of regularpolygon. In addition, students’ problem-solving ability was limited whileconducting some Logo programming tasks, and acceptable in others.In regard to the results, it is recommended that teaching Logoprogramming should be used in different contexts that enhance students’learning, and develop problem-solving processes.

Keywords: assessment; Logo programming; mathematics achievement;misconceptions; problem-solving; turtle geometry

1. Introduction

Developments in information technology (IT) and its impact on mathematicslearning have been many and fast in the last two decades. To support an IT initiativein Jordan, research projects should be conducted in order to understand the impactof IT on enhancing mathematics learning and problem-solving processes. Althoughcomputer programming should have a place in school curriculum, it is underpremature debate. In this article, the researcher takes into account students’performance in Logo programming, and their problem-solving abilities in thecontext of turtle geometry.

Compared with the abundant accomplishments in theWest, Jordan, in theMiddleEast, has hardly made any initial investigation in the context of Logo programmingand its impact on students’ learning. An IT initiative in Jordan’s schools reflects someof the overseas experiences. However, the application of computer software, such asprogramming languages in schools of different teaching and learning environment andless financial support, needs to be developed gradually. In the next section,

*Email: [email protected]

ISSN 0020–739X print/ISSN 1464–5211 online

� 2009 Taylor & Francis

DOI: 10.1080/00207390902912845

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illumination, based on work with Logo, from overseas research is presented as

a background of this study.

2. Background

In the vision of Papert [1], ‘the child programmes the computer, and in doing so,

this acquires a sense of mastery over a piece of a powerful technology and establishes

an intimate contact with some ideas from science, from mathematics and from the

art of intellectual model building’ [1, p. 5]. To achieve this vision, Papert established

a cognitive culture of the Logo computer language that initiates communication with

the turtle. Logo intellectual roots are in artificial intelligence, which extended thecapacity of computer to act intelligently in mathematics logic and in developmental

psychology, where the three disciplines shaped the strength of Logo language

through turtle geometry, integrated with understandable primitives [1].Logo was originally designed as an interactive tool for constructivist learning,

and much more than a procedural programming language. It is a philosophy of

education that takes two main visions: Reform and revolutionary [2]. For the reform

vision, Logo should support traditional schooling, while the revolutionary vision,

which was adopted by Papert, was against the traditional form of school. Overall,reformers viewed Logo as a tool that should be introduced into schools to assist

educational change. In the meanwhile, the revolutionary approach tended to view

Logo as a computer environment where children can do things in new ways [2].

Moreover, developers of Logo geometry have assumed three strands of curriculum:

paths, shapes and motions. These strands are the basis of relational understanding,

where paths draw straight or non-straight lines and closed or non-closed paths;

shapes do procedures to draw polygons; motions represent the transformationalgeometry [3]. Needless to say, through Logo’s evolution, it was utilized across the

world in many different contexts and in many different classrooms with different

results [4]. Papert [5] concludes that the slow pace of the Logo turtle’s progress

into the school world may only reflect the sluggish transition of school to the turtle

microworld.During the last 20 years, different research approaches were being used and

integrated with 130 different implementations of Logo, each of which has its own

strength. Many of these implementations concern teaching and learning geometryas an important branch of mathematics. Examples of these implementations are

ucbLogo.mswLogo, comenius Logo, experLogo, lego/Logo and arLogo which is an

Arabic port of ucbLogo [6]. Enkenberg [7] believes that most of the graphics

software are good tools for learning geometry. He states ‘the world seems out to be

geometrical to the computer users when looking at it through the lenses of computer

screen’ [7, p.106]. He argues that Logo emphasizes what to know and how to know,

and its power is based upon procedurality, recursivity and possibility of extending

the amount of words the turtle knows [7, p.110].Among Logo’s evolution, beginning with the early uses of Logo to the uses

of Logo mature, Logo provided children with a graphical environment rich in

problem-solving processes. Using the turtle and writing procedures to draw different

shapes, children were able to develop systematic strategies for problem-solving.

This experience would give an intuitive understanding of geometrical concepts and

Logo programming. In addition, the socially contextualized uses of Logo would

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improve positive attitudes towards learning. Some researchers have supported suchclaims. Rieber [8], in an experimental group programming with Logo’s turtle, foundthat Logo programming had made significant differences in thought patternsof children and their learning of geometric concepts. Clements and Battista [9], foundthat Logo groups showed improvement in their ability to recognize angles ofdifferent sizes, rotate angles, estimate the size of various angles and understandgeometric shapes and properties. In the same context, Noss [10] revealed thatstudents learning Logo typically develop better conceptions of some geometricconcepts than students taught without Logo. Lehrer et al. [11] concluded thatLogo is an effective means for elaborating children’s procedural interpretations ofsome pre-proof geometry concepts compared with the declarative specifications.These results were emphasized by other researches that showed the positive effectsof Logo environment on students’ understanding of geometrical concepts, andproposed that a key task for mathematics educators is the creation of mathematicalmicroworlds, which must take account of the learner, the teacher and theprogramming activity [12–14]. Although a review [15] of around 80 major studiesof the use of Logo in mathematics teaching and learning highlights the conflictingnature of the results, it suggests that the use of Logo may afford opportunitiesfor children to explore mathematical ideas in a meaningful context.

Using Logo Mathematics Tutorial (LMT2), Yusuf [16] concluded that middleand high school students had deeper conceptualization of quadrilaterals, polygons,angles, rotations and circles. Moreover, in a national Logo project in Costa Rica,Edwards [17] investigated a mathematical microworld written in Logo within threephases: Open exploration of the microworld, group discussion and problem-solving.It was found, through observation and written work, that students were successfulin exploring the function of the microworld. Their hypotheses were improved afterdiscussion, and they were successful in the problem-solving tasks. In addition,Kapa [18] investigated changes in students’ problem-solving control, planningstrategy and sharing processes during Logo tasks. Based on Polya’s problem-solvingprocesses (understanding the problem, creating the solution, implementing the planand evaluating the solution), high achievers were systematically and constantlyengaged in Polya’s four processes, while low achievers needed teachers’ help andtried to be engaged in the four processes [19].

In order to understand students’ difficulties to learn the Logo programminglanguage, Norte et al. [20] conducted a study at a high school in Portugal, who neverlearned a programming language before, and that exhibited many learningdifficulties. The results of the study supports the notion that the possibility tolearn by exploration must be given to the students. In addition, it encouragesprospective teachers to adopt Logo under a constructive learning environment.In the same context, Wulf [21] adopted a constructivist pedagogical approach toteaching Logo programming in high school and undergraduate courses. Wulf ’sapproach focuses on higher cognitive levels and addresses multiple student learningstyles and intelligences.

Logo programming, in Mexican public schools, is a main tool of a nationalproject, whose objective is to implement technological tools into the mathematicsclassrooms. As part of an evaluative research associated to this project, many of theteachers had initial difficulties in adapting to the proposed pedagogical modelassociated to Logo, but they were motivated to learn and gradually changed theirpractice. However, students were able to carry out Logo activities on their own quite

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well, and their mathematics performance was improved [22]. Moreover, Logointeractive activities, that were designed to be used in the lower secondarymathematics programmes [23], presented many technical and didactic challenges,the foremost of which was the difficulty of preserving the spirit of Logo, and itsbenefits as a programming and constructive environment.

Through its turtle geometry, Logo programming language facilitates theteaching and learning of analytic geometry and calculus from the notion ofcurvature [24]. Moreover, Logo turtle offers intrinsic representations of functions.These representations are complementary to standard representations, and areexpressed as procedures in Logo. These turtle procedures link mathematical topicswith turtle geometry, and offers opportunities for exploration and discovery [25,26].In the meanwhile, Logo’s turtle has been used to introduce many of the fundamentalconcepts that underlie computer graphics and computer-aided design to under-graduate and graduate students in colleges and universities [27].

In order to enhance mathematical problem-solving at the Advanced LevelCurriculum reform of the British National Curriculum, a problem-solving andpowerful mathematical thinking tool, which supports Logo’s turtle graphics, wasdeveloped [28]. The researchers argue that the new problem-solving tool can makea significant contribution to the enhancement of pupils’ problem-solving skills at alllevels in school. While Logo is restricted to real plane, the new tool user can explorea wide range of activities on the complex plane.

According to the previous background, the main focus of Logo projects has beenon Logo as a programming language and an environment for exploration. Moreover,it is a tool for learning the basic geometric concepts through understanding anddebugging, and a tool for developing problem-solving processes described by Polya,rather than a programming language by itself. In the meanwhile, if the purposeof learning a programming language at the advanced levels is to develop the higherthinking abilities, such as problem-solving and designing Logo programs oralgorithms, students should have knowledge about the structure of this language,and students’ Logo programming performance must be assessed formally. ManyLogo teachers endorse formal assessment by giving tests and grades, while othersrecommend individualized and open-ended environments [29]. In addition, manystress the importance of assessing Logo learning in order to track students’ progressand to improve instruction through diagnostic teaching. In this context, Watt [30]put on more emphasis on the product of Logo work than on the thinking processes,such as Logo programming knowledge, ideas essential to an effective use of Logofor understanding geometry concept and producing Logo programs that drawspecial figures in problem-solving environment.

In Jordan, Ministry of Education launched an instructional computing programfor tenth grade since 1990. Then, a course of IT has been adopted in schools,including grades 7–12, with the beginning of the twenty-first century. The mainpurposes of this major topic of school curriculum are to improve students’ abilities tolearn through the computer environment, as well as their cognitive skills, especiallyproblem-solving processes. The IT curriculum is taught in isolation of other schoolcurricula, such as mathematics. Logo programming is one of the major modulesin the seventh grade’s textbook of the IT curriculum. This module aims at givingstudents the experience needed to programme the computer using Logo language,and to improve students’ problem-solving abilities. Logo programming moduleincludes Logo language primitives and Logo procedures. The types of activities

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practiced by students are using the turtle in order to draw paths and geometricalshapes, and construct more complex shapes that need many procedures. Logomodule lasts 15 sessions, each of 45min.

3. Objective of study

The major goal of this study is to assess Logo programming experiences of seventhgrade students in the context of geometric concepts, such as angle, angle size, circle,polygon (regular and irregular), properties of quadrilateral shapes and the interiorangle of regular polygon. In addition, the study investigates the correlation betweenstudents’ performance in Logo programming and their school mathematicsachievement, and identifies misconceptions in Logo programming aspects andgeometric aspects. Moreover, this study is to investigate students’ ability to writepaper and pencil Logo programmes, and to solve problems using Logo’s turtle.In particular, this study tries to answer the following questions:

(1) To what extent do seventh grade students understand written Logoprograms, and can predict geometric shapes represented by the writtenprogrammes?

(2) Is there a correlation relationship between seventh grade students’ achieve-ment in Logo programming and their school mathematics achievement?

(3) What are the misconceptions that seventh grade students made in differentaspects of Logo programming regarding different aspects of geometry?

(4) Are seventh grade students able to solve problems using Logo programming?

4. Sample of study

Four large schools were selected randomly among schools that contained at least sixsessions of seventh grade students, and are located in the north of Jordan. A sampleof (228) seventh grade students was selected using the randomized cluster method;126 females and 102 males. Students were of different socioeconomic status andof different mathematics achievement. All of them studied a module of Logoprogramming within a credit of 1 hour semester course of IT as a part of their schoolcurriculum. This module lasted for 3 weeks, and produced an introduction of Logolanguage through turtle geometry. Often, students work in groups in order to coverthe turtle activities in the textbook. In the meanwhile, they work individually forthe purpose of assessment. Teacher’s role is to guide and be a source of learning.A PC LOGO for windows, version 1.03, 1994, Harvard Associates, Inc. is utilizedin the Jordanian schools.

5. Instruments

A written test of 18 multiple-choice items and one question of the completionformat (Appendix A) were used in order to answer the first three questions of thestudy. The researcher conducted the test after the analysis of the Logo programmingmodule in regard to its purposes and content. The Logo programming aspects were:the primitives, the repeat command, procedures, using constructed proceduresto construct more complex ones and recognizing incorrect commands in a givenprocedure. On the other hand, the geometric aspects were: rotation, angle of


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rotation, circle, quadrilaterals (square, rectangle, rhombus and trapezoid) andpolygons (regular and irregular). In Appendix A, questions 1, 2, 5, 8, 18 assessangle of rotation; questions 10, 14, 15, 17 assess rotation of geometrical shapes;questions 3, 4, 9, in addition to question 19 of the completion format and questions6, 7, 11, 12, 13, 16 assess polygons and circle. The internal consistency coefficientof the test was (0.86).

To answer the fourth question of the study, five open problems (Appendix B)were administered to the sample in order to suggest paper and pencil Logoprogrammes of given shapes, and to construct the given shapes using the turtle.

6. Data collection

This assessment study was conducted at the end of the school year 2006/2007 afterseventh grade students studied the Logo programming module. Data were collectedin three phases. The first phase was conducted through a written multiple-choice testand a completion item, where the whole sample participated in responding to thistest individually. The researcher and the regular computer teacher administered thetest during a 50min session in their regular classroom. In the second phase, a set offive tasks (Appendix B) was administered to five equivalent groups of the sampleof the study, taking into consideration their mathematics achievement andsocioeconomic status. Group 1 was asked to solve tasks 1 and 2; group 2, tasks 2and 3; group 3, tasks 3 and 4; group 4, tasks 4 and 5 and group 5, tasks 1 and 5.The five groups included 46, 46, 46, 45 and 45 students, respectively. Data from thesecond phase of the assessment were collected through a written worksheet, wherethe groups were asked to write the hypothesized Logo programmes that could drawthe shape. At the third phase, five groups, each of 10 students, were selectedrandomly from those who did not master their tasks from each of the five groupsof the second phase. Each student in each group was asked to work individuallyusing the computer in order to construct a Logo programme (text screen) and thecorresponding graphics screen of the tasks he tried before. Students’ works wereassessed through observation, the printout of the text and the graphics screens.

7. Data analysis

Data were analysed according to the questions of the study, where questions 1 and 2were analysed quantitatively by computing the means and the standard deviationsof students’ performance in the multiple-choice test. Then three hypotheses weretested statistically. In the meanwhile, data collected to answer questions 3 and 4 wereanalysed qualitatively depending on the percentages of students’ correct answers,and the quality of the text screen and draw printout.

8. Results

8.1. Results of question 1: seventh grade students’ understandingof written Logo programs

To answer question (1), the hypothesis: ‘there is no significant difference between themean score of seventh grade students on Logo programming test and the expectedscore according to the probability laws’, was tested. Means and standard deviations

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of the whole sample and of males and females students were computed as shownin Table 1. Two tailed t-test was used to test the previous hypothesis, where theexpected mean score is 4.5 (0.25 multiplied by 18). The results revealed that thecomputed t (20.10) was greater than the critical value (1.984) at (�¼ 0.05), with 227degrees of freedom, which means rejecting the null hypothesis, and the differenceis in favour of the sample performance on Logo programming test.

In addition, the null hypothesis: ‘there is no significant difference between themean scores of the seventh grade male and female students on Logo programmingtest’, was tested. The two tailed t-test of non-homogeneity of variance showed thatthe computed t(5.53) was greater than the critical t(1.984) at (�¼ 0.05), with 225degrees of freedom. This means rejecting the null hypothesis, and the difference isin favour of females.

Apart from supplying statistics, seventh grade students’ paper and pencilresponses to item 19 (write three properties of the geometric shape that the followingseries of Logo commands could draw: fd 50 lt 70 fd 50 lt 110 fd 50 lt 70 fd 50,and give a name to this shape) were analysed. The analysis revealed that 41% outof the whole sample were able to predict the correct shape that the given Logoprogramme could draw, which is a rhombus. While 31% predicted the shape to bea parallelogram, 15% predicted it to be a square, a trapezoid, or a quadrilateral and13% gave random responses. In the meanwhile, students recognized the majorproperties that reflect the critical attributes of the rhombus that could be recognizedfrom the given Logo programme; 52, 31, 28 and 23% addressed ‘all sides are equal’,‘opposite angles are equal’, ‘the shape has four sides’ and ‘opposite sides are parallel’,respectively. In addition, students addressed few incorrect responses, suchas: irregular, regular and the sum of the opposite angles is 180�, where they added70 to 110�.

8.2. Results of question 2: the correlation between Logo programmingperformance and school mathematics achievement

For 228 seventh grade students, it was found that Logo programming performancecorrelated 0.053 with their scores of school mathematics achievement. To test if thiscorrelation is significantly larger than zero; that is, Ho: �¼ 0 tenable, the t-test wasused. The critical t-value (1.972) at 226 degrees of freedom is greater than thecomputed t (0.798) at (�¼ 0.05) [31]. This means that the correlation is insignificantbetween seventh grade students’ performance on Logo programming test andtheir school mathematics achievement. This could be an indication that the Logoprogramming test needs more advanced skills, such as: analysing the given Logo

Table 1. Means and standard deviations of the sample.

Sex N Meansa SD

Females 126 10.33 4.18Males 102 7.97 2.12All 228 9.28 3.59

Note: aMax score¼ 18.


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programme, translating the program into a shape (using paper and pencil) and

searching for the angle of rotation. For example, in a school achievement test,

students were asked to find the angle measure of a regular octagon, theyautomatically needed to apply a rule. However, when students are asked, ‘what is

the angle measure of the regular polygon that the following repeat command draws:

repeat 8[fd 50 lt 45]?’ they need to discover the shape (regular octagon), the angle

of rotation and the angle measure of the regular octagon which is the complementary

of the angle of rotation. Needless to say, students made a couple of mistakesin regard to the angle of rotation as shown through the results of the third question

of this study.More items in school mathematics tests indicate different skills, such as: ‘what is

the measure of the third angle of an isosceles triangle if each of the base angles

is 30�’? Responding to this question, students needed to apply the relation that thesum of the triangle angles is 180�. While in the Logo programming test such as item

18 (to draw an isosceles triangle, a student printed the Logo commands: fd 70 lt 150

fd 40 lt 120 fd 40, he made a mistake in the fourth command, what is the best

correction of this command?). Responding to this item, students need to analysethe command lt 150 as the first angle of rotation, to find the angle of the base (30�),

to get the measure of the third angle (120�) and to correct the mistake that the angle

of rotation in the fourth command is 60� instead of 120�.Although a number of studies have revealed that there is a positive relationship

between success in Logo programming and some cognitive capabilities such as thegeneral academic achievement, Geva and Cohen [32] found that with the second and

fourth grade students, mathematical problem-solving skills are good predictors

of students’ performance at computer and paper and pencil programming skills.

Where as Webb [33] concluded that for 11- and 14-year-old students, mathematicsability is the best predictor of knowing syntax, interpreting graphics programs and

generating logical relations programmes. In the meanwhile, Wiebe [34] concluded

that mathematics achievement is a good predictor of total at-computer programming

success and success with the more complex programming skills. Wiebe’s conclusion

might support the low correlation revealed in the current study between Logoprogramming performance and school mathematics achievement. Where in the first

phase of this study, Logo programming ability was measured away from the

computer, where it might not reflect the real programming ability using the turtle

graphics in this case.

8.3. Results of question 3: misconceptions that seventh grade students

made in different aspects of Logo programming regarding different

aspects of geometry

To answer the third question, students’ responses to each item of the multiple choice

test were analysed. Misconceptions that students made were defined under four

major topics: (1) the angle rotation of a path, (2) the angle rotation of a shape, (3) the

regular polygons and (4) the circle. Frequencies and percentages of students

who emerged these misconceptions were calculated. Table 2 shows the natureof misconceptions, the items that represent the source of misconception and the

percentages of each misconception frequency. The criterion for considering the

response a misconception is 25% and above.

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8.4. Results of question 4: seventh grade students’ ability to solveproblems using Logo programming

To answer this question, five tasks (Appendix B) were utilized, and students wereasked to write a paper and pencil Logo programme that draws a given shape.To achieve this, the sample was divided randomly into five equivalent groups,where each group was asked to work on only two tasks; as stated before. Table 3shows the percentages of students who mastered each of the tasks.

. Responses to task 1 (the fan task): About one-third of the 92 studentsmastered the fan task. Most of the correct responses used the repeatcommand, such as: lt 30 repeat 3[repeat 3[fd 70 rt 120] rt 120], while two ofthem defined a triangle procedure, and then retrieved the defined procedureas follows: lt 30 repeat 3[triangle rt 120]. Two other students wrote a series

Table 2. Misconceptions and their percentages.

No. Misconceptions Items Frequency %

1. Cannot recognize angle of rotation while drawingpath or shape, considering the interior angle

1 57 25.00

Instead of the exterior angle. 2 110 48.258 101 44.00

18 79 34.65

2a. Cannot recognize angle of rotation while rotatingan equilateral triangle, students equallyresponded: 60, 180 and 360�.

10 113 49.56

2b. Cannot recognize that the rotated shape is anequilateral triangle, instead, students rotatedan isosceles triangle to form a fan.

14 61 26.75

2c. Students failed in estimating the rotation angle,about one fourth responded 180�, and onethird responded 60�.

15 135 59.21

2d. Students did not recognize that the rotated shapeis a square, and they considered the numberof rotations to be four instead of six, wherethey multiplied four by 90 to get 360� as acomplete rotation.

17 78 34.21

3a. Students did not recognize the measure of theregular octagon angle, so they failed to get theangle of rotation (complementary of 108�) todraw this polygon.

11 131 57.46

3b. Students did not differentiate between the rotationangle and the measure of the angle of a regularpolygon.

13 117 51.32

4a. Students responded that the repeat command:repeat 45[fd 1 rt 1] is a circle of 45 parameter.

6 115 58.44

4b. Students responded that the command: repeat360[fd 5 lt 1] represents an ellipse shape or halfan ellipse.

16 135 59.21

4c. Students missed the complete rotation (360�) of acircle, and they considered that the command:repeat 90[fd 1 rt 1] construct a circle.

7 112 49.12


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of Logo commands: lt 30 fd 60 rt 120 fd 60 rt 120 fd 120 rt 120 fd 60 rt 120

fd 120 rt 120 fd 60 rt 120 fd 60. Few students wrote a procedure that could

draw a fan, but with different angles of rotation as follows: repeat 3[repeat 3

[fd 50 rt 120] rt 110]. The other responses were incorrect, and could mostly

draw a triangle such as: repeat 3[fd 50 rt 120] rt 90, and repeat 3[repeat 3[fd

50 rt 120].. Responses to task 2 (five-edges star): Only five students (5.4%) produced

correct paper and pencil Logo programmes that could draw a star with five

edges. The programmes were a series of Logo commands such as: fd 50 rt

144 fd 50 rt 144 fd 50 rt 144 fd 50 rt 144 fd 50. Although there are different

strategies to achieve this task, the critical point for the students was to

deduce the measure of the angle of rotation. Depending on their geometric

knowledge, the students practiced the deductive thinking; first, they drew

a five-edges star with five segments, using paper and pencil, then each of

them assumed that the segments are equal and realized that a regular

pentagon is formed with interior angle of 108�. They continued to prove

that the measure of the angle of rotation in order to draw a five-edges star

is 144�.. Responses to task 3 (space rocket): Although most of the 92 students tried

to write paper and pencil Logo programme that could draw a space rocket,

only five students (5.4%) mastered the task. Two of them wrote three

procedures that could draw an equilateral triangle, a rectangle and

a trapezoid, and they retrieved these procedures and connected them

correctly to draw the rocket. The rest wrote a series of Logo commands

and mastered the task correctly. The following Logo programmes represent

the correct responses:

Program 1:To triangle

Repeat 3[fd 20 rt 120]

END

To rectangle

Repeat 2[fd 50 rt 90 fd 20 rt 90]

END

To trapezoid

fd 12 rt 60 fd 20 rt 60 fd 12 rt 120 fd 33

Table 3. Percentages of students who mastered the tasks using paper andpencil.

No. Tasks

No. of studentswho triedthe task

No. of studentswho mastered

the task %

1 Fan task 92 31 33.72 Five-edges star 92 5 5.43 Space rocket 92 5 5.44 Chessboard 90 0 0.05 Five squares 90 58 64.4

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END

To space

Rt 30

Triangle

Lt 90

Rectangle

Lt 90

Fd 50

Rt 30

Rt 180

Trapezoid

END

Program 2:Rt 30 fd 12 rt 60 fd 20 rt 60 fd 12 rt 120 fd 33 rt 120 fd 12 lt 30

Repeat 2[fd 50 rt 90 fd 20 rt 90]

Fd 50 rt 30


Others, who tried the space rocket task, wrote a series of Logo commands

that could draw the space rocket, but they missed the correct angles of

rotation, and they made many mistakes in writing Logo program to draw

a trapezoid, and many considered the trapezoid a regular polygon. Also,

few students used the repeat command to draw the sub-procedures triangle

and rectangle, but they did not complete the task.

. Responses to task 4 (chessboard): All of the 90 students who tried this

task failed to write a paper and pencil Logo program that could draw

an 8� 8 chessboard. Nevertheless, most of the students (77%) got a correct

Logo procedure to draw a square (part (a) of the task), while four students

only mastered part (b) of the same task (the rod of eight squares).

The following Logo program is an example that could draw the rod of eight

squares:



Lt 90 pu fd 30 pd


Pu fd 30 pd repeat 4[fd 30 rt 90]





. Responses to task 5 (five squares of different lengths): Repeat command

was often used in the paper and pencil responses of the 90 students who tried

the task. They realized that there are six squares with different

lengths, where the length varies from one square to another. As shown

in Table 3, 64.4% of the students wrote six repeat commands to draw the


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given figure in this task. The following is an example of the correct

responses:







Few students of those who mastered the five-square task wrote a long series

of commands without using the repeat command, in the meanwhile, no

student tried to write more advanced Logo programmes. Regarding thosewho wrote misleading programs, they recognized six squares, but they did

not recognize that their dimensions should be different. The followings are

examples of such incorrect Logo programs to draw the given figure:

Repeat 6[repeat 4[fd 100 rt 90]]



Repeat 6[repeat 4[fd 50 rt 90]rt 60]

In the second phase of problem-solving, five equivalent groups, each of 10

students, were selected randomly from those who did not master the paper

and pencil Logo programs; that is to say, the five groups who tried the given

tasks using paper and pencil. Each of the 10 students were asked to use the

computer individually in order to work on the two tasks they tried before.

Table 4 shows the percentages of students who mastered the tasks usingcomputer.It is clear from Table 4 that students are more able to solve Logo

programming problems using the turtle than with paper and pencil Logo.

For example, among the students who had tried the task of the five-edges

star, three (30%) mastered the task using the computer, while 5.4%

mastered the same task using paper and pencil. The first student tried to

solve the problem deductively; he drew a regular pentagon where its interior

angle is 108�, then he drew five isosceles triangles on each edge of the

pentagon, after that he realized that the measure of each triangle angle was

72, 72 and 36� at the edge of the star. Therefore, five equal segments can be

Table 4. Percentages of students who mastered the tasks using computer.

No. Tasks

No. of studentswho tried the

task using computer

No. of studentswho mastered the

task using computer %

1 Fan task 10 4 40.02 Five-edges star 10 3 30.03 Space rocket 10 3 30.04 Chessboard 10 0 00.05 Five squares 10 5 50.0

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drawn using repeat command with a rotation angle of 144�: Repeat 5[fd 70

rt 144], as shown in Figure 1.Another student tried to combine a pentagon procedure with five equilateral

triangles. The procedure he used was: repeat 5[fd 60 rt 72]lt 60 repeat 5[fd 60

rt 120 fd 60 lt 48], and Figure 2 shows the output.The third student tried the following input: lt 125 fd 70 rt 140 fd 70 lt 70 fd

70 rt 140 fd 70 lt 70 fd 70 rt 140 fd 70 lt 70 fd 70 rt 140 fd 70 lt 70 fd

70 rt 140 fd 70, then he tried to summarize the previous series of commands

as follows: lt 125 repeat 4[fd 70 rt 140 fd 70 lt 70] fd 70 rt 140 fd 70

(Figure 3).Other students presented many trials, but the inputs were incomplete

to draw the star, and the sources of their mistakes were due to the low

problem-solving abilities and their lack of understanding geometrical

aspects, especially the confounding angle of rotation, with the measure

of the interior angle of regular polygon.

Figure 3. A star without a regular pentagon.

Figure 2. A star with regular pentagon and equilateral triangles.

Figure 1. A star with five equal segments.


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9. Discussion, conclusions and recommendations

The purpose of this descriptive study is to raise an issue emerged from the purposeof teaching Logo programming at seventh grade in Jordan as a part of the IT course,

where the major goal of this module, as mentioned in the outlines of the course, is to

improve students’ problem-solving skills. An examination of the literature onlearning to programme in Logo leads to the question: What should students master

in order to solve problems by turtle geometry?. Returning to the main purpose,

which is improving problem-solving abilities, Logo programming mastery should beachieved. Littlefield et al. [35] suggested that Logo language commands mastery,

predicting mastery of geometrical shapes of a given Logo programs, writing a paper

and pencil Logo program of a given shape and producing Logo programme using the

computer, should be documented.The results of this study highlight the importance of defining the purpose of any

instructional module related to Logo programming. It was clear from the nature of

students’ responses that they recognize the function of Logo commands, where mostof their misconceptions were due to the conceptual misunderstanding of geometric

aspects. Such aspects are represented by angle of rotation while drawing a path,

angle of rotation while rotating a geometrical shape, the nature of rotated shape,the angle measure of a regular polygon and the angle of complete rotation.

For example, 49% of the students consider that the procedure: Repeat 90[fd 1 rt 1]

represents a circle. Another reason for the observed misconceptions is that the Logo

programming module is not structured well with regard to the time given to completethe tasks, assuming that students control learning process under the guidance of the

teacher.As Olive concluded, success in Logo programming aspects is necessary but

not sufficient for success with geometric concepts involved in the tasks, and some

students are successful in Logo programming but not in geometry [36]. This

might explain the low correlation between the seventh grade students’ performancein Logo programming test used in this study and their school mathematics

achievement.Although the findings of the study showed that students’ performance was

significantly better than the expectations of the probability laws, seventh gradestudents’ performance was still low, but this performance does not reflect that their

produced Logo programmes are failures. The fault may rest on the misunderstanding

of the benefit of Logo programming, the misleading objectives of learning Logo,and unfamiliarity and difficulty with the discovery teaching approach of Logo

programming.Considering methods of instruction, few details were given concerning the kind

of instruction taking place in Logo programming in the Jordanian schools, wherethe outlines of the module and the teachers of IT discipline did not offer a sufficient

description of methods of instruction. Few teachers described the instructional

method as a discovery-oriented environment within cooperative work, while othersmentioned that Logo programming commands and related concepts are presented

and exemplified by the teacher, and then the students are given problems to solve.

In this context, many questions can be raised: How would the teacher respondif students encounter difficulties with a given task? Does the teacher tell the groups

to proceed step-by-step? What should be done to correct the graph or the Logo

program?

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This study showed that seventh grade students recognize and learn the basiccommands: Forward, Backward, Right, Left, and Repeat. This was obvious fromtheir responses to many test items that asked them to predict the name of the shaperepresented by a given Logo program. For example, when students were asked ’whatis the geometric shape that the Logo commands fd 40 rt 90 fd 20 rt 90 fd 40 rt 90 fd20 draw?’, most of the students realized that the shape is a rectangle of length 40 andwidth 20. In another item that asked students to write three properties of thegeometric shape that the following commands could draw: fd 50 lt 70 fd 50 lt 110 fd50 lt 70 fd 50, many students addressed correct responses according to the givencommands.

Although students in this study demonstrated reasonable ability to predict Logoprogrammes’ outcomes, they demonstrated limited ability to produce a paper andpencil Logo programmes of given shapes. In the meantime, turtle geometryenvironment motivated a number of students to master some of the given tasks byanalysing them and writing programming codes, then executing the programmeand identifying and correcting programming errors. This result supports the ideathat Logo creates an environment in which children learn from error, and to bea good programmer is to be highly skilled at correcting bugs [1]. Moreover, itsupports Wiebe’s conclusion that a student who cannot solve a computer problemaway from the computer may be quite successful at the computer, where studentreceives the feedback from the computer while entering and running the programmemay help him/her toward a solution [34].

Concerning the written work of students, many of them, especially females, wereactive in connecting Logo language with geometrical concepts, and to conceptualizeand generate logic on problems to be solved. Indeed, students tried their best to usePolya’s model of problem-solving while producing paper and pencil Logoprogrammes, and while using computer.

I should mention that seventh grade students in Jordan have never been exposedto Logo programming or to other programming languages prior to this assessmentstudy. However, they have experienced using computers. Although the activities thatcover the programming module are turtle geometry oriented, this was not enoughto master all the tasks such as: Predicting the geometrical shape from a given Logoprogram, producing a paper and pencil Logo program for a given geometrical shapeor producing computer Logo program as input and output. This means that a lotof work is still to be done in order to achieve the major goal of this module, whichis improving students’ problem-solving skills. Moreover, a specified philosophy ofteaching Logo programming should be re-established in Jordan’s schools, anddesigners of IT curriculum should benefit from the international projects in this field.Nevertheless, the revolutionary expectations of Logo were not achieved in themainstream schools [2]; Logo programming has to be used in different ways such as:An interactive tool for constructivist learning, a programming language thatsupport, in depth, problem-solving skills and enhancing mathematics learning.

In summary, the findings of the main research questions provided that students’performance on a Logo programming test was significantly better than theexpectations of the probability laws. Although from this result, students’ ability topredict the geometric shapes represented by the written Logo programmes is still low.A positive low correlation between students’ Logo programming performance andschool mathematics achievement was revealed. Most of the made misconceptionswere due to geometrical aspects rather than Logo primitives, and were concentrated


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on the angle of rotation. Moreover, students’ problem-solving ability was limited

while conducting some Logo programming tasks, and acceptable in others. In the

meanwhile, students were more able to solve Logo programming problems using the

turtle than with paper and pencil.Needless to say, the results of this study would be of value to Logo programming

teachers in deciding what types of Logo programming tasks they ask their students

to do. It would be also of use to IT curriculum developers and textbooks writers who

are developing programming activities to incorporate into mathematics and other

curricula.Finally, I would like to notify that today several versions of Logo are available

for free, for example, go to:

http://el.media.mit.edu/Logo-foundation/products/software.htmlhttp://www.cs.berkeley.edu/�bh/v2-toc2.htmlhttp://www.softronix.com/Logo.htmlhttp://en.kioskea.net/telecharger/telecharger-836-the-logo-creator

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Appendix A: LOGO programming test

Read the following multiple choice items and circle the right choice:

1. The following Logo commands draw the given shape, FD 50 RT 60 FD 50

Which of the following shapes represents the angle of rotation?

2. Which of the following Logo commands draw the shape to the right?

a. FD 20 RT 60 FD20 b. FD 20 RT 120 FD 20

c. FD 20 LT 60 FD 20 d. FD 20 RT 300 FD20

3. What is the geometric shape that the following Logo commands draw?FD 40 RT 90 FD 20 RT 90 FD 40 RT 90 FD 20a. square of 20 units length b. rectangle of length 40 and width 20c. quadrilateral with one right angle d. triangle with 20, 40, 40 dimensions

4. Given the Logo commands: FD 50 LT 30 FD 80 LT 150 FD 50 LT 30 FD 80, whichgeometric shape the turtle will draw?

5. To draw equilateral triangle as in the given shape, a student printed the Logo commands:FD 30 LT 60 FD 30 LT 60 FD 30. He made a mistake, which statement represents themistake?

a. he did not start with the command to triangleb. he used the command left instead of right in the 2nd and 4th commandc. angle of rotation in the 2nd and 4th commands is 60 instead of 300d. angle of rotation in the 2nd and 4th commands is 60 instead of 120

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6. Given the procedure: repeat 45[FD 1 RT 2], what is the geometric shape that the turtle willdraw?a. half of a circle b. one fourth of a circlec. a circle with radious of one unit d. a circle of 45 unit perimeter

7. Which of the following repeat commands draw a circlea. Repeat 90[FD 1 RT 4] b. Repeat 90[FD 2 RT 2]c. Repeat 90[FD 4 RT 1] d. Repeat 90[FD 1 RT 1]

8. The turtle draws the shapes 1, 2, 3. What are the angles of rotation respectively?

a. 120, 90, 60 b. 240, 90, 300c. 60, 90, 60 d. 60, 90, 120

9. What is the series of Logo commands that represents a parallelogram of different measuresof adjacent sides?a. FD 60 RT 70 FD 30 RT 110 FD 60 RT 70 FD 30b. FD 60 LT 70 FD 60 LT 110 FD 60 LT 70 FD 60c. FD 60 RT 90 FD 80 RT 90 FD 60 RT 90 FD80d. answers a and c are correct

10. What is the angle of rotation for the turtle to draw the shape to the right using anequilateral triangle procedure?a. 60 b. 180 c. 120 d. 360

11. Which Repeat command draw a regular pentagon?a. Repeat 5[FD 40 RT 60] b. Repeat 5[FD 40 RT 45]c. Repeat5[FD 50 RT 90] d. Repeat5[FD 50 RT 90]

12. Name the polygon that the turtle draws according to the following Logo command series?FD 40 LT 60 FD 40 LT 60 FD 40 LT 60 FD 40 LT 60 FD 40 LT 60 FD 40.a. regular hexagon b. regular octagonc. irregular hexagon d. regular pentagon

13. What is the measure of the angle of the regular polygon that the following Repeatcommand draws: Repeat 8[FD 50 LT 45]?a. 45 b. 135 c. 50 d. 225

14. Given the procedure: To triangle: Repeat 3[FD 20 RT 120], what is the geometric shapethat the Logo command: Repeat 6[Triangle RT 60] draws?


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15. A student retrieves a procedure that represents shape 1 in order to construct shape 2, whatis the angle of rotation to achieve that?

a. 180 b. 150 c. 60 d. 90

16. What is the nearest geometric shape that the repeat command: Repeat 36[FD 5 LT 10]could draw?

17. Which of the following Repeat commands draws the given shape by the turtle?a. Repeat 4[Repeat 6[FD 30 RT 90] RT 60]b. Repeat 4[Repeat 6[FD 30 RT 90] RT 90]c. Repeat 6[Repeat 4[FD 30 RT 90] RT 60]d. Repeat 6[Repeat 4[FD 30 RT 60] RT 90]

18. To draw an isosceles triangle as shown in the given shape, a student printed the followingLogo commands: FD 70 LT 150 FD 40 LT 120 FD 40, he made a mistake in the fourthcommand, what is the best correction of this command?a. RT 30 b. LT 60c. LT 150 d. LT 180

19. Write three characteristics of the geometrical shape that the following Logo commandscould draw: fd 50 lt 70 fd 50 lt 110 fd 50 lt 70 fd 50, and give a name to this shape.

Appendix B

1. Write a Logo program to draw the given shape:

2. Write a Logo program that draws a five edges star.

3. Write a Logo program that draws the given shape:

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4. a. Write a Logo procedure to draw a square

b. Write a Logo program to draw the given shape

c. Write a Logo program to draw a chessboard

5. Write a Logo Program that draws the given shape:


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Documents

Assessing Logo programming among Jordanian seventh grade students through turtle geometry