Geometry

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Characterizing student mathematicsteachers’ levels of understanding inspherical geometry

a a

aFatih Faculty of Education, Secondary School Science and

Mathematics Education, Karadeniz Technical University, Trabzon61335, Turkey

Version of record first published: 21 Sep 2010

To cite this article: Bulent Guven & Adnan Baki (2010): Characterizing student mathematicsteachers’ levels of understanding in spherical geometry, International Journal of MathematicalEducation in Science and Technology, 41:8, 991-1013

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Bulent Guven & Adnan Baki

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International Journal of Mathematical Education inScience and Technology, Vol. 41, No. 8, 15 December 2010, 991–1013

Characterizing student mathematics teachers’ levels ofunderstanding in spherical geometry

Bulent Guven* and Adnan Baki

Fatih Faculty of Education, Secondary School Science and Mathematics Education,Karadeniz Technical University, Trabzon 61335, Turkey

(Received 10 November 2008)

This article presents an exploratory study aimed at the identification ofstudents’ levels of understanding in spherical geometry as van Hiele did forEuclidean geometry. To do this, we developed and implemented a sphericalgeometry course for student mathematics teachers. Six structured, task-based interviews were held with eight student mathematics teachers atparticular times through the course to determine the spherical geometrylearning levels. After identifying the properties of spherical geometry levels,we developed Understandings in Spherical Geometry Test to test whether ornot the levels form hierarchy, and 58 student mathematics teachers took thetest. The outcomes seemed to support our theoretical perspective that thereare some understanding levels in spherical geometry that progress througha hierarchical order as van Hiele levels in Euclidean geometry.

Keywords: levels of understanding; spherical geometry; student mathemat-ics teachers

1. Introduction

Levels of understanding, which can be defined as the thinking process of studentsthrough a number of distinct cognitive levels, have always been a concern ofeducators [1]. Dina and Pierre van Hiele developed a model to describe differentlevels of human geometric reasoning. According to this model, the learner, assistedby appropriate instructional experiences, passes through the following five levels,where the learner cannot achieve one level without having passed through theprevious levels [2]:

Level 1(Recognition): The student recognizes geometric figures by their global

appearance and identifies names of figures, but she/he does not explicitly identifytheir properties.

Level 2 (Analysis): The student analyses figures in terms of their components and

properties, discovers properties rules of a class of shapes empirically, but she/he doesnot explicitly interrelate figures or properties.

Level 3 (Pre-deductive): The student logically interrelates previously discovered

properties rules by giving or following informal arguments.

*Corresponding author. Email: [email protected]

ISSN 0020–739X print/ISSN 1464–5211 onlineß 2010 Taylor & Francis

DOI: 10.1080/0020739X.2010.500692

http://www.informaworld.com

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992 B. Guven and A. Baki

Level 4 (Deductive): The student proves theorems deductively and developssequences of statements to deduce one statement from another, but she/he doesnot yet recognize the need for rigour.

Level 5 (Rigour): The student establishes theorems in different axiomatic systems

and analyses/compares these systems.

Crowley [3, p. 4] described the distinctive characteristics of the five levels of the

van Hiele model as follows:

. The progress from one level to the next is not through biological

development but rather depends more on instruction.. The linguistic symbols of each level are unique. In other words, each level is

regarded as having its own language and learners on different levels cannotunderstand one another.

. The intrinsic characteristic of one level becomes the extrinsic objects of studyof the next. In other words, properties of a particular geometrical conceptare inherent in its existence but may not be studied as properties until a laterlevel.

. The mismatch between the level of instruction and the level at which astudent is functioning may restrict the desired progress.

Mayberry [4] tested the hierarchical nature of the levels by studying pre-service

elementary teachers. Questions designed to evaluate each concept at each of the vanHiele levels were given to the participants in a clinical interview setting, and theresults were scored according to a success criterion for each level. If the participants’understanding really was hierarchical, then success at any one level should not occurunless success also occurred at each lower level.

In a large-scale study, Senk [5] investigated the relationship between students’ vanHiele levels and their achievement in writing geometry proof. Research findingsshowed that achievement in writing geometry proofs is positively related to van Hielelevels of geometric understanding. In other words, Senk demonstrated that bydetermining a student’s van Hiele level at the beginning of a high school geometrycourse, one could very accurately predict the student’s proof-writing ability at theend of the course.

Research involving the van Hiele levels has generally focused on the lower levels.Level 5 understanding has not been addressed in research studies. This is partly dueto the fact that too few of the participants in the studies conducted during the last 20years have exhibited any characteristics of level 5 thinking [6]. Indeed, reflectingupon this, van Hiele remarked that level 5 was theoretical and not of particularconcern as he saw the goal of K-12 instruction to be the development of level 4thinking [7]. Another reason research concerning level 5 has been neglected lies in thedifficulty in assessing this level using multiple-choice items. Usiskin [8] stated thatthe fifth level either does not exist or is not testable. This conclusion, together withthe lack of advanced participants in the studies that were being conducted, seems tohave closed the door with regard to research at this level [6].

1.1. Spherical geometry

Spherical geometry can be said to be the first non-Euclidean geometry [9]. For atleast 2000 years humans have known that the earth is (almost) a sphere and that the

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International Journal of Mathematical Education in Science and Technology 993

shortest distance between two points on the earth is along great circles (theintersection of the sphere with a plane through the centre of the sphere).

In ancient civilizations, geometry literally meant ‘the science of measuring theland’ [10]. The typical study of geometry in modern classrooms works on theassumption that ‘the land’ to be measured is flat [11]. We know, however, that ourland is on the earth, which is, basically, a sphere. Although Euclidean geometry islocally a good description of the physical world, it cannot apply to navigation on thesurface of the earth, to astronomy, and to surveying [12]. Thus, students’understanding of spherical geometry is important to understand and explain thephysical world around them. However, although it is thousands of years old,spherical geometry is not taught in most schools [12].

The study of spherical geometry is not abstract, because students are well-acquainted with spheres. If students are given the proper tools, this study can be veryinteresting. Students can easily consider many elementary theorems from planeEuclidean geometry and explore them on a sphere. For instance, they might ask suchquestions as, ‘Is the angle–angle similarity theorem for triangles valid for spheres?’[13]. This approach gives students an appreciation that Euclidean geometry is one ofmany geometries. This is suggested in the Curriculum and Evaluation Standards forSchool Mathematics by NCTM [14] as follows: College-intending students also shouldgain an appreciation of Euclidean geometry as one of many axiomatic systems. Thisgoal may be achieved by directing students to investigate properties of other geometriesto see how the basic axioms and definitions lead to quite different and oftencontradictory results. For example, great circles, which play the role of lines inspherical geometry, always meet. Thus, in spherical geometry, instead of having exactlyone line parallel to a given line through a point not on the line, there are no such lines[14, p. 160]. The NCTM’s report implies that it should integrate other geometriesinto its content, thus making students aware of other geometries.

1.2. Purpose of the study

van Hiele [7] only looked at the levels of understanding in Euclidean geometry andwas not interested in non-Euclidean geometries. He found non-Euclidean geometriestoo theoretical to include in the school mathematics curriculum. We now know,however, that there have been different attempts to integrate spherical geometry intoschool mathematics [15–17]. Also, dynamic geometry software, such as ‘Cinderella’and ‘Spherical Easel’ allow non-Euclidean concepts to be worked within a computer-based environment. As a result of these developments, we can look at levels ofunderstanding in non-Euclidean geometries. More specifically, we intend toelaborate these levels of understanding in spherical geometry.

This study was undertaken to:

. investigate the existence of levels of understanding in spherical geometry;

. characterize the levels of understanding in spherical geometry and

. determine whether these levels form a hierarchy.

2. Methodology

Through a similar methodology to van Hiele, we tried to determine whether studentshave particular understanding levels in spherical geometry. To achieve this goal,

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we designed a model course to introduce the topics of spherical geometry to studentmathematics teachers.

In developing this, we looked at several textbooks in this field. We extensivelyutilized the ‘Non-Euclidean Adventures on the Lenart Sphere’ [18], ‘ExperiencingGeometry in Euclidean, Spherical, and Hyperbolic Spaces’ [19], and ‘Solid Geometryand Spherical Trigonometry’ [20] during the course. The course curriculum includescomputer-based activities requiring the use of the Spherical Easel Program, adynamic geometry software developed for spherical geometry. Topics of sphericallines (great circles), polar point of a great circle, spherical triangles, polar triangles,spherical polygons, spherical circles and applications of spherical geometry arediscoursed in order during the course.

The model course took 2 months in one semester, 3 h in a week, and 58 studentsat the Department of Science and Mathematics of the Faculty of Education,Karadeniz Technical University, participated in the model course. Six structured,task-based interviews were held with eight participants at particular times through thecourse to determine the spherical geometry learning levels. To achieve a wide rangeof understanding levels in spherical geometry, we chose students at different vanHiele levels. Two students were selected from each level. After identifying theproperties of spherical geometry levels, we developed the Understandings in SphericalGeometry Test (USGT) to check the validity of the prediction and test whether or notthe levels form hierarchy, and 58 students took the test.

2.1. Software used in the course

At the beginning of the course, we had to decide which learning tool would be usedin the spherical geometry lessons, real spheres (e.g. the Lenart Sphere Kit) orcomputer programs (Cinderella or Spherical Easel). Because the activities designedfor the course were based on exploration and accuracy, and many measurementswere required to produce acceptable conjectures along the lessons, we decided to usea dynamic geometry software. We decided on Spherical Easel due to its ease of use.This Java-based program allows students to make drawings and explorations inspherical geometry like they do in plane geometry with programs, such as Cabri andSketchpad. Spherical Easel is designed to be easy to use and will often lead youthrough the construction of a diagram. A screenshot of the program can be seen inFigure 1.

2.2. Instrument used to select students for interviews: the van Hiele geometry test

Determining a student’s van Hiele level has been a source of controversy due to thecriterion of the model. The most valid method to determine a student’s van Hielelevel has been through one-on-one question and answer involving the researcher andthe student [21]. Where a large number of participants are involved, however, thismethod may not be feasible. Therefore, this study utilized the van Hiele geometrytest (VHGT), a quantitative instrument that was developed from research conductedfor the Cognitive Development and Achievement in Secondary School Project(CDASSG). The VHGT [8] consists of 25 multiple-choice geometry questions to beadministered in 35 min. Usiskin [8] used direct descriptions from the writings of thevan Hiele to construct the VHGT. Every five questions correspond to particular van

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Figure 1. A screenshot of Spherical Easel.

Hiele level. This test has been widely used by others, yet has fallen under somecriticism as well. One advantage of Usiskin’s test is the ease in its administration andgrading. This instrument has been used many times in research studies over the past25 years [22].

Usiskin presents two options for scoring the van Hiele Test: the ‘3 of 5’ criterionand the ‘4 of 5’ criterion. The ‘3 of 5’ criterion indicates that a student has mastered agiven level if he/she correctly answers three or more of the five questions for thatlevel. The ‘4 of 5’ criterion has an analogous interpretation. We decided to use the ‘4of 5’ criterion for the following reasons: it takes a more conservative stance in theerror analysis since type I errors (some students might be at a lower van Hiele levelthan the researcher assigns to them) are generally considered to be more serious thantype II errors (some students are actually at a van Hiele level higher than that theresearcher assigns to them). It also reduces the number of students who do not the fitthe model, which is important for studies of small sample size such as ours [8].

2.3. Interviews conducted to characterize spherical geometry understanding levels:structured, task-based interviews

In this study, by analysing student mathematics teachers’ interpretations during thetask-based interviews [23], we hoped to make inferences about the characteristics ofthe students’ geometric thinking in spherical geometry. These interviews weredesigned to explore the student’s development of the geometric thinking processlongitudinally.

Participants were asked to complete activity sheets during the interviews. Toanalyse the participants’ thinking process, we asked them questions in the form ofinformal conversations while they were completing the activity sheets. We used threetypes of questions: descriptive, structural and contrast. For example, ‘Could youdescribe what you did when . . . .?’ (descriptive), ‘What are the steps you generally liketo follow in . . . .? ‘(structural),’What difference do you see between showing this inEuclidean geometry and spherical geometry? ‘(contrast). Talking on activities during

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Figure 2. A sample activity sheet.

interviews appeared to give access to the participants’ thinking process and level ofunderstanding in spherical geometry. Figure 2 presents a sample activity sheet thatwas given to the students.

In addition to task-based interviews, classroom observations were made to seethe students’ interactions as a whole in the environment. We participated indiscussions during small group work and whole classroom discussions. Thisparticipation took the form of dialogue with the students during the activities andtranscribing it soon afterwards while it was still fresh in our minds. This approachhelped us describe the context and investigate more extensively the participants’perceptions and understandings through their interactions with activities.

The large amount of qualitative data collected from the participants wascontinuously analysed throughout the course, involving an iterative process of datareduction and further data collection. For example, the interview transcripts wereread to identify concepts that summarized the students’ ways of thinking and toidentify disconfirming and confirming evidence. This process can be called contentanalysis [24]. In this process, ‘Hyperresearch’ software was used for codingqualitative data through a deductive way. In the next step, we tried to identify theproperties of levels of understanding by using themes arisen from the coding. Afterthe properties of levels were identified, special cases from the activity sheetscompleted by the participants were descriptively compared with the properties oflevels in spherical geometry.

2.4. Understandings in spherical geometry test

The interviews and observation helped us to identify properties leading to thedescription of four levels in spherical geometry. To check the validity of theprediction and test whether or not the levels form a hierarchy, we developedthe USGT. The test includes 20 open-ended questions (5 questions for each level).While developing the test, we used direct descriptions of spherical geometryunderstanding levels developed in the scope of this study. Having developed the test,we sent it and also a description of the spherical geometry understanding levels tothree mathematics education researchers who are capable in the field of learning

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levels in mathematics to determine that the questions in the test reflect the features ofthe levels – in other words, to determine that the test is valid. From their suggestions,we made some changes in the test. The test was administered in 45-minute sessions.One question from each level of the test is shown in Table 1 with its characteristic.

Scoring the responses to the test questions was also used to determine whether thelevels identified in spherical geometry form a hierarchy. We used the same patternfor spherical geometry that Usiskin [8] used to assign levels in Euclidean geometry.This study used the ‘3 of 5’ criterion. To obtain comprehensive data about levels ofunderstanding in spherical geometry, task-based interviews were carried out witheight selected student teachers. Data obtained from interviews were compared withdata obtained from USGT. As a result of this comparison, when we used the ‘4 of 5’criterion, we observed that six of eight student mathematics teachers’ levelsdetermined through task-based interviews coincided with their levels determinedfrom the USGT. When we used the ‘3 of 5’ criterion, we observed that the levels ofeight students obtained through task-based interviews coincided with their levelsdetermined from the USGT.

2.5. Scalogram analysis

Responses are coded as 1 and 0 (correct answer ¼ 1, wrong answer ¼ 0) as done byMayberry [4,25]. If a student scores at least three ‘1’ from the first five questions, itmeans that he/she has attained the first level. If this student has at least three ‘1’ fromthe second five questions, then he/she has attained the second level. If this studentfails to get at least three correct answers from the third and fourth five questions, thescore for this student can be represented as 1100. If the score is represented as 1101,this means that the student has met the criterion on levels 1, 2 and 4 but not level 3.Only 5 of the 16 possible response patterns should appear if the hierarchy is valid:0000, 1000, 1100, 1110 and 1111.

Guttman scalogram analysis was employed to reveal that van Hiele levels astested form a hierarchy. Guttman [26,27] argued for scales where items can be rankedin difficulty such that if a person responds positively to a given item, that personmust respond positively to all easier items. Thus, theoretically a given score on aGuttman scale can only be reached with one pattern of response, and if we know aperson’s score, we know how that person responded to all items in the scale.Guttman scaling, or scalogram analysis, then, is the estimation of reproducibilitygiven knowledge of a person’s scores, that is, the extent to which item responses fitthe ideal patterns [28].

The measure of errors for the entire scale is the coefficient of reproducibility(Rep) and is given by the formula [29]:

Coefficient of reproducibility ¼ Rep: ¼ 1 À

Total errors

Total responses,

where total responses can be calculated by multiplying item and sample size. Thus,

Rep gives the fraction of the scores that correctly placed in the response patterns.Therefore, in this study:

Rep: ¼ 1 ÀTotal number of errors

Number of levels Â Number of students

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Table 1. Sample questions for each level.

One of the characteristics Question related to characteristic

Level 1: Recognizing basicgeometric figures inspherical geometry

Which triangle(s) shown above is spherical triangle? Explainyour answer. (Question 2)

Level 2: Determining andexplaining whether ornot plane figures can bedrawn on the sphericalsurface

Which quadrilateral shown above can be drawn on thespherical surface? Explain your answer? (Question 8)

Level 3: Making logicalinference (not formalproof) by using his/herknowledge of spherical

You know that if the sides of two triangles are proportionalthen these triangles are similar. Is the same theorem validfor spherical geometry? Explain your result with examples.(Question 12)

and Euclidean geometry

Level 4: Starting directly tofollow deductive wayinstead of using specialcases and inductive stepsin the process of theproof

Triangle ABC and DFH are polar triangles. Prove that thesides and angles of triangle DFH are respectively thesupplements of the angles and sides of triangle ABC triangle.(Question 18)

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and

Rep: ¼ 1 ÀTotal number of

errors

Number of levels Â

58

:

A coefficient of reproducibility greater than 0.9 is commonly assumed to ‘indicatea scability’ [30,31]. This means that if the value of Rep is greater than 0.90, it isaccepted that the levels form a hierarchy.

3. Findings

3.1. Construction of levels in spherical geometry

The student mathematics teachers’ levels of understanding in spherical geometrywere identified through interviews and classroom observations. As in Euclideangeometry, students possess different understanding levels in spherical geometry. Thelevels constructed for spherical geometry are:

(1)

(2)(3)(4)

Transition

Definition–comparisonPre-deductiveDeductive.

The characteristics and indications of the levels are described as follows.

3.1.1. Level 1: transition

At this level, figures in spherical geometry are recognized by their appearance, andthe student at this level considers that there should be various differences between theproperties of plane figures and their equivalents on sphere (e.g. plane triangle andspherical triangle). However, he/she does not know those differences. The student isready to study spherical geometry, but the previously learned concepts of Euclideangeometry seem to dominate his/her perceptions while working on new tasks inspherical geometry. For that reason, this level is labelled as a transition to sphericalgeometry. At this level, the student is also aware of the existence of differentgeometries other than Euclidean.

Students were asked to draw a spherical line that passes through points A and Bon the sphere by using the definition of line on the sphere. As seen in Figure 3, thestudent could identify and name the given figures in the spherical surface, but wasnot aware of their definitions and properties. This can be easily seen from thespherical line drawn by the student, because a spherical line (great circles) is definedas follows:

. The great circle is a circle whose centre is the centre of the sphere and whose

radius is equal to the radius of the sphere, or. the spherical line is the intersection circle of a sphere and a plane that passes

from the centre of sphere.

It can be seen in Figure 3, however, that while the student knows that spherical

lines are circles on the sphere, he did not use the antipodal points (points that lie atthe intersection of a great circle and a line through the centre of the circle on the

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Figure 3. A great circle drawn by a student at level 1.

sphere) and centre of the sphere. He only used his images about spherical lines gainedwhile studying with the Spherical Easel Program.

At the transition level, the student is inclined to accept a new geometry with theprinciples and axioms of Euclidean geometry. Even if the student perceives a newgeometry, he/she is not aware of its basic principles and axioms, and does not realizehow these principles and axioms play a role in constructing a new geometry.Therefore, a different geometry for the student means a different plane or surface.Characteristics and their indications of transition level is shown Table 2.

3.1.2. Level 2: definition and comparison

At this stage, perceptions based on definitions and properties start to replaceperceptions based on the appearance of the geometric figures. The student learns thedefinitions and properties of spherical figures and he/she can compare the definitionsand properties of the same shape concepts in the two geometries. At this stage,knowledge based on definitions and logical inferences seems to replace intuitive andvisual knowledge. As seen in Figure 4, students can draw basic figures of thisgeometry by using the formal definitions and properties of figures. In Figure 4, twostudents drew great circles by using its definition (the intersection of the surface of asphere by plane that passes through the centre of the sphere, the centre of a great line isthe centre of sphere), and as seen in Figure 5 students can construct the polar pointof a great circle by using its definition and properties (all spherical lines connectingany spherical line and its polar point is perpendicular it, by using this procedure,students at this level can construct the polar triangle of any triangle).

The student starts to use the new terminology belonging to a new geometry, suchas great circles, polar point of a great circle and polar triangle. The student canfollow a proof in plane and explain why that proof is not valid in spherical geometry.

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Table 2. Characteristics and their indications of transition level.

Characteristics

1. Recognizing different geometries

2. Recognizing basic geometric figures inspherical geometry

3. Perceiving that appearance and proper-ties of a figure in plane is different fromits spherical equivalent

4. Predicting spherical equivalent of thefigure (given in plane) on sphericalsurface

5. Not being able to give reason why aparticular property or axiom is true inspherical geometry

6. Being not aware of that a new geometryis constructed when axioms in Euclideangeometry change

Indications

When a geometric shape is shown to thestudent, she/he asks in what geometry

The student identifies a spherical lineamong given geometric figures

The student knows that the sum ofinterior angles of a spherical trianglecannot be 180

The student predicts the image of thetriangle on spherical surface

The student does not know what thesum of interior angles of a sphericaltriangle is exactly, even though she/heknows that the sum of interior anglesof a spherical triangle cannot be 180

The student believes that axioms inEuclidean geometry are still valid inspherical geometry. For example, she/he thinks that even in spherical sur-face, only one line passes through twopoints

Figure 4. The great circles (spherical lines) drawn by students at level 2.

For example, when one constructs a figure to prove the sum of interior angles ofspherical triangle is 180 , the student at this level can realize that one cannot draw aparallel line to another line on the spherical surface as seen Figure 6. He/she knowsthat great circles CE and AB cannot be parallels. They intersect at two points thatare antipodal points.

The student can identify whether or not there exist equivalents of plane figures onthe sphere by using experimental results within a computer-based environment (e.g.is there trapezoid on sphere?). The student also compares definitions and propertiesof the same shape concepts in the two geometries. For example, as in Figure 7, while

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Figure 5. Construction of the polar point of a given spherical line.

Figure 6. One cannot draw a parallel line to a given line on the spherical surface (CE cannotbe parallel to AB).

two lines intersect at one point in plane, two lines intersect at two points that areantipodal in spherical geometry. Characteristics and their indications of definitionand comparison level is shown in Table 3.

3.1.3. Level 3: pre-deductive

The student, at the level of transition, has intuitive perceptions about sphericalgeometry. The student, at the level of definition and comparison, supports his/herintuitive perceptions by theoretical definitions and properties. At the level of pre-deductive, the student can make informal logical inferences about the situations inspherical geometry (as seen in Figures 8 and 9) by using special cases of geometricfigure, but is not able to support his/her inferences with formal proofs. In other

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Figure 7. Two lines intersect on one point in plane, two lines intersect on two points that areantipodal in spherical geometry.

words, though the student at this level is able to draw conclusions from specificinstances, the student is not competent enough to justify his conclusions by providingformal proofs.

Researcher: What can you say about the sum of the exterior angles of a spherical triangle?Student: The sum of interior angles of spherical triangle is greater than 180 . Therefore,the sum of exterior angles should be less than 180 .Researcher: Can you prove that?Student: No I can’t.Researcher: In plane triangle the measure of an exterior angle is equal to the sum of themeasures of the remote interior angles. Do you think that it is true in a spherical triangle?Student: No it isn’t. We know that in spherical triangle we can draw a triangle whom allinterior angles are 90 . So the theorem is not valid for spherical triangle. I think themeasure of an exterior angle is less than the sum of the remote interior angles.Researcher: Can you prove that?Student: Is my proof not a proof?

As seen in the above explication of the student, the student inferred that in

consequence of the sum of interior angles of a spherical triangle being greater than180 , the sum of exterior angles of the triangle should be less than 360 (bycomparing it with the result in plane geometry). However, he could not support hisinference with formal proof. Similarly, he inferred by using a trirectangular sphericaltriangle (a triangle with three right angles) that the measure of an exterior angle of aspherical triangle is less than the sum of the measures of the remote interior angles.However, he could not formally demonstrate that.

Furthermore, students at this level can achieve a conjecture by planning theirown computer-based designs step by step. The students at this level can indepen-dently plan, carry out and bring the conclusion to their computer-based projects.However, they cannot formally explain why the result they experimentally reached istrue. During the course, a group of students investigated a relationship between theangles and sides of a spherical triangle by using the Spherical Easel program asshown in Figure 10.

As the result of thoughtful and well-designed observations, the students reachedthe following results but could not prove them. As considered in Figure 10:

. If A þ B 5 180 (m(1) þ m(3)) then a þ b 5 180 (m þ l)

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(a)

For

the s

tudent,

sph

eri

cal lin

es

are

gre

at

circ

les

whic

h t

heir

centr

es

are

th

e c

en

tre o

f th

e s

phere

(b)

The s

tudent

can d

raw

the lin

e p

ass

ing t

hro

ug

h t

wo p

oin

ts g

iven

on

the s

pheri

cal su

rface

(c)

Th

e s

tud

en

t ca

n e

xpla

in h

ow

tw

o lin

es

pass

thro

ug

h t

wo p

oin

ts g

iven

on

the s

pheri

cal su

rface

The s

tudent

can

follo

w f

or

exam

ple

, th

e p

roof

of

the s

um

of

inte

rior

an

gle

s is

18

0 a

nd

expla

in w

hy t

his

pro

of

is n

ot

valid

in t

he s

ph

eri

cal geom

etr

y (

reaso

n w

hy o

ne c

an

not

dra

w a

para

llel lin

e t

o a

noth

er

line

on t

he s

pheri

cal su

rface

)

(a)

Th

e s

tud

en

t kn

ow

s th

at

an e

quila

tera

l tr

ian

gle

or

rect

angle

giv

en

in p

lan

e c

an b

e d

raw

n o

n t

he

spheri

cal su

rface

, b

ut

a s

qu

are

cannot

be d

raw

n o

n t

he s

pheri

cal su

rface

beca

use

its

inte

rior

ang

le is

90

(b)

The s

tudent

class

ifies

pla

ne fi

gure

s in

term

s of

the p

oss

ibili

ties

of

their

dra

win

gs

on

the s

pheri

cal

surf

ace

. Fo

r exam

ple

, th

e s

et

of

delt

oid

s su

ch a

s d

elt

oid

, rh

om

bu

s ca

n b

e d

raw

n,

and

th

e s

et

of

trap

ezo

ids

such

as

trapezo

id, p

ara

llelo

gra

m, re

ctan

gle

and

sq

uare

cann

ot

be d

raw

nThe s

tudent

can

make

the f

ollo

win

g c

om

pari

son

s.

In p

lan

eThe len

gth

of

line is

infinit

eIf

tw

o d

iffere

nt

lines

do n

ot

have inte

rsect

ion,

they a

re p

ara

llel

There

is

only

on

e s

egm

ent

pass

ing

tw

o p

oin

ts

In s

phere

The len

gth

of

line is

finit

eThere

are

no p

ara

llel lin

es

in t

he s

pheri

cal

geom

etr

yThere

are

tw

o s

egm

ents

pass

ing t

hro

ug

h t

wo

poin

ts

4. C

om

pare

s definit

ion

s and

pro

per-

ties

of

the s

am

e s

hap

e c

once

pts

in

the t

wo g

eom

etr

ies


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5. U

sin

g a

sp

eci

al la

ngu

age t

o d

efine

spheri

cal fig

ure

s6

. D

raw

ing

sph

eri

cal figure

s th

at

are

not

dra

wn in

pla

ne

Tw

o p

oin

ts o

n t

he lin

e d

ivid

e t

he lin

e into

thre

eTw

o p

oin

ts o

n t

he lin

e d

ivid

e t

he lin

e into

tw

opart

spart

sThe s

hort

est

dis

tance

betw

een t

wo p

oin

ts is

the

Not

alw

ays

line c

on

nect

ing

these

poin

ts(a

) Th

e s

tud

en

t u

ses

new

word

s lik

e g

reat

circ

le,

pole

poin

t an

d s

pheri

cal tr

ian

gle

(b)

The s

tudent

defines

a s

ph

eri

cal tr

iang

le a

s a t

rian

gle

whic

h s

ides

are

on g

reat

circ

les

The s

tudent

can

dra

w t

riangle

s co

nst

ruct

ed

by t

hre

e g

iven

poin

ts in s

pheri

cal su

rface


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Figure 8. The sums of interior and exterior angles of a spherical triangle are 180 and 360 ,respectively.

Figure 9. Trirectangular spherical triangle.

Figure 10. Relationship between the angles and sides of spherical triangle.

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Table 4. Characteristics and their indications of pre-deductive level.

Characteristics

1. Making logical inference (notformal proof) by using his/herknowledge of spherical geometryand Euclidean geometry

2. Making decision about whether ornot a theorem of Euclidean geom-etry is valid in spherical geometry.She/he usually use special cases offigures during the decision process

3. Using the software to supportinductive inferences, but proofs arenot formal proofs

Indications

The student knows that in both plane andspherical triangle the sum of interior andexterior angles is 540. Therefore, she/heinfers that because the sum of interior anglesof plane geometry is 180 , its exterior anglesis 360 , then the sum of interior angles of aspherical triangle is bigger than 180 , andthe sum exterior angles of a sphericaltriangle should be less than 360

In Euclidean geometry, the measure of anexterior angle of a triangle is equal to thesum of the measures of the remote interiorangles. The student knows that this theoremis not valid for a spherical triangle

At the previous level, the student makesdecision by using visual results on thescreen. At this level, the student uses thesoftware to support his/her conjectures, butshe/he cannot use necessary and sufficientconditions in his/her proofs

. If A þ B ¼ 180 (m(1) þ m(3)) then a þ b 5 180 (m þ l)

. If A þ B 4 180 (m(1) þ m(3)) then a þ b 4 180 (m þ l).

Characteristics and their indications of pre-deductive level is shown in Table 4.

3.1.4. Level 4: deductive

At this level, deductive reasoning replaces inductive inferences based on:

. special cases within spherical geometry,

. reflections of Euclidean geometry results in spherical geometry and

. visual results appearing on the screen.

The characteristics of this level seem to be similar to the fourth level of van Hiele.

We present the characteristics and indications of this level by means of the followingtask related to the construction of a formula for the area of spherical triangle.

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Figure 11. The area of spherical triangle.

We asked the students to prove that the area of triangle ABC is equal toR2 ¼ ðA þ B þ C À Þ during the clinical interview session. One of the studentsexplained his proof deductively as in Figure 11.

As seen in Figure 11, the student drew triangle ABC and determined its anglesas , and . Then she stated the area between two great circles as4 R2, 4 R2 and 4 R2 by using the formula of area of line. The student whodetermined that the sum of areas between great circles is equal to the sum of the areaof spherical surface and four times the area of triangle ABC acquired the area oftriangle ABC. Despite the fact that we did not ask him, he concluded from thatformula that the sum of interior angles of triangle is greater than 180 . Similarformal thought can be seen in the following example of a student trying to prove thetheorem about the relationship between a spherical triangle and its polar trianglethat states, The sides and angles of polar triangle are respectively the supplements ofthe angles and sides of the primitive triangle (Figure 12).

Let A0 B0 C0 be polar triangle of triangle ABC. Let H and T be the intersectionpoints of [AB] and [AC] with [B0 C0 ]. Since A is the pole of B0 C 0 , the spherical angle Ais equal to arc HT (Definition given to students at the beginning of the course: greatcircles which pass through the poles of a great circle are called secondaries to thatcircle. The angle between any two great circles is measured by the arc they intercepton the great circle to which they are secondaries). And since point C0 is the pole of[AB] and B0 is the pole of [AC], then m(C0 H) ¼ m(B0 T) ¼ 90 . Because m(C0 H) þm(B0 T) ¼ 180 , (HT) þ m(C0 B0 )¼180 . In this way, A þ a0 ¼ 180 .

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Figure 12. The sides and angles of polar triangle are respectively the supplements of the anglesand sides of the primitive triangle.

Similarly, B þ b0 ¼ 180 , C þ c0 ¼ 180 . Because triangle ABC is polar triangle ofA0 B0 C0 , A0 þ a ¼ 180 , B0 þ b ¼ 180 and C0 þ c ¼ 180 .

Characteristics and their indications of deductive level are shown in Table 5.

3.2. Levels form hierarchy

Having characterized the levels of understanding in spherical geometry, we tried todetermine whether or not the levels form a hierarchy by using Guttman scalogramanalysis [4]. The students’ scores obtained from the test including 20 questions areshown in Table 6.

As illustrated in the Table 6, the students’ scores seem to form a hierarchy,although the scores of S2, S4, S5, S11, S14, S21, S23, S28, S41, S50 and S54 haveerrors deforming hierarchy. Most of them have one error. Only the fourth students’score includes two errors. Different values of Rep were calculated using the fourlevels, the first three levels and the first two levels to show the power of hierarchy forthe sub- sequential levels.

Rep2 ¼ 1 À

Rep3 ¼ 1 À

Rep4 ¼ 1 À

2

11612

17413

232

¼ 0:982,

¼ 0:931,

¼ 0:943:

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Shows two errors.


Table 5. Characteristics and their indications of deductive level.

Characteristics

1. Determining and using what isbeing given and asked in a proof

2. Starting directly to follow adeductive way instead of usingspecial cases and inductive stepsin the process of the proof

3. Using definitions, known theo-rems, relationships and results

4. Finding the wanted result andgenerate new conclusions fromthe result

Indications

The student draws a triangle on the sphere and showsits sides and interior angles

The student uses the figure as a whole to constructthe formula giving the area of spherical triangle

The student uses the area of spherical line as 2 R2 to find the area of ABC spherical triangle

Finding the wanted result and generate new conclu-sions from the result. The student finds theformula giving the area of a spherical triangle.From the formula, the student can also reach aconclusion that inside of the parenthesis must bepositive and then the sum of the interior angles ofthe spherical triangle must be greater than 180

Table 6. Students’ scores on USGT.

Student

S1

S2

S3

S4

S5

S6

S7

S8

S9

S10

S11

S12

S13

S14

S15

S16

S17

S18

S19

S20

Score

11100100a

11000010b

1010a

111010001111110011101010a

110011101010a

111011101110110011101100

Student

S21

S22

S23

S24

S25

S26

S27

S28

S29

S30

S31

S32

S33

S34

S35

S36

S37

S38

S39

S40

Score

1010a

11101010a

11001110110011101101a

110011101100111010001110100011001111111011101111

Student

S41

S42

S43

S44

S45

S46

S47

S48

S49

S50

S51

S52

S53

S54

S55

S56

S57

S58

Score

1011a

111011111110111011111110111011101011a

11100110a

11111010a

1100111011101110

Notes: aShows one error.b

For three calculations, the scores of Rep are greater than 0.90. Therefore, thelevels of understanding in spherical geometry form a hierarchy.

4. Conclusions and suggestions

When we consider that the purpose of teaching geometry in schools is to providestudents with the knowledge to understand and explain the physical world around

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Table 7. The comparison of van Hiele geometry understanding levels andspherical geometry understanding levels.

Properties

Visual properties

Definitions and propertiesof geometric figures

Informal inferences

Formal proof

Geometry

SphericalEuclideanSphericalEuclideanSphericalEuclideanSphericalEuclidean

Levels

TransitionVisualDefinition and comparisonAnalysisPre-deductivePre-deductiveDeductiveDeductive

them, Euclidean geometry is limited to achieve this purpose because the earth onwhich we live is spherical. This indicates the need for a new integrated geometrycurriculum that includes the concepts of spherical geometry. In the development ofthis kind of curriculum, the first step should be to identify the students’ levels ofunderstanding in spherical geometry. We found out that levels of understanding inspherical geometry do exist and these levels form a hierarchy. According to thesefindings, the topics in the new curriculum should be constructed from concrete toabstract, and the topics should take place in the hierarchical order parallel to thelevels of understanding. The levels of understanding in spherical geometry are asfollows:

(1)

(2)(3)(4)

Transition

Definition–comparisonPre-deductiveDeductive

Generally there are some similarities between van Hiele levels and spherical

levels. If we look closely at the levels of spherical geometry, we will see that thecharacteristics of these levels are slightly different from van Hiele levels. Both levelsstart with visual perceptions and then continue with intuitive perception ofdefinitions and properties. Finally, in both levels, the students reach deductivereasoning and make logical inferences. These similarities can be summarized as inTable 7.

These similarities do not imply that students at both levels possess the samepattern of understanding. For example, the student at the first level of van Hiele isnot aware of the properties of plane figures, but the student at the transition levelknows the properties of plane figures and is aware that some of these properties arenot valid in spherical geometry. The student at the analysis level can list theproperties of a figure and compare its properties to another figure’s properties inplane geometry. On the other hand, the student at the definition and comparisonlevel is inclined to compare the properties and relationships of spherical geometry tothe properties and relationships of plane geometry. The student at the pre-deductivelevel of spherical usually makes logical inferences by using special cases and results inboth plane and spherical surfaces. On the other hand, the student at the pre-deductive level of van Hiele does not use special cases in Euclidean geometry.

The results of this study show that students progress through different levels ofunderstanding as they learn spherical geometry and also that these levels form a

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hierarchy of understanding. Hence, we suggest that the courses and books which aimto teach spherical geometry should be designed in the light of these levels. Otherwise,we think that the teaching process would fail to provide students with a convenientlearning environment and would not go beyond traditional teacher-centredapproach. We know that one of the major reasons behind the students’ failure inunderstanding Euclidean geometry is the type of instruction which does not take intoconsideration the students’ level of understanding regarding Euclidean geometry.Therefore the levels of understanding in spherical geometry should be taken intoaccount in designing teaching activities in order to avoid encountering the sameproblem in the teaching of spherical geometry.

References

[1] S. Yee-Ping, An investigation of van Hiele-like levels of learning in transformation geometryof secondary school students in Singapore, Ph.D. diss., Florida State University, 1989.

[2] D. Fuys, D. Geddes, and R. Tiskler, An investigation of the van Hiele levels of thinking ingeometry among adolescents, J. Res. Math. Educ. Monogr. No. 3. NCTM, Reston, 1988.

[3] M.L. Crowley, The van Hiele model of the development of geometric thought, in Learningand Teaching Geometry, K-12 Yearbook of the National Council of Teachers ofMathematics, M.M. Lindquist and A.P. Shulte, eds., NCTM, Reston, 1987, pp. 1–16.

[4] J. Mayberry, The van Hiele levels of geometric thought in undergraduate preserviceteachers, J. Res. Math. Educ. 4 (1983), pp. 58–69.

[5] S.L. Senk, van Hiele levels and achievement in writing geometry proofs, J. Res. Math. Educ.20 (1989), pp. 309–321.

[6] S. Blair, Describing undergraduates’ reasoning within and across Euclidean, taxicab, andspherical geometries, Ph.D. diss., Portland State University, 2004.

[7] P. van Hiele, Structure and Insight: A Theory of Mathematics Education, Academic Press,New York, 1986.

[8] Z. Usiskin, van Hiele levels and achievement in secondary school geometry, cognitivedevelopment and achievement in secondary school geometry project, Final Report, TheUniversity of Chicago, Chicago, 1982.

[9] D. Taimina and D.W. Henderson, (n.d.). How to use history to clarify common confusionsin geometry. Available at http://www.math.cornell.edu/$dtaimina/MAA/MAA.

[10] M.J. Greenberg, Euclidean and Non-Euclidean Geometries: Development and History,Freeman, New York, 1980.

[11] J.M. Sharp and H. Corrine, What happens to geometry on a sphere? MTMS 8(4) (2002),182–188.

[12] D.W. Henderson, Experiencing Geometry on Plane and Sphere, Prentice Hall, New Jersey,1996.

[13] E.H. Davis, Area of spherical triangles, Math. Teach. 92 (1999), pp. 150–153.[14] NCTM, Curriculum and Evaluation Standards For School Mathematics, NCTM,

Reston, 1989.[15] I. Lenart, Alternative models on the drawing ball, Educ. Stud. Math. 24 (1993),

pp. 277–312.[16] I. Lenart, Non-Euclidean Adventures on the Lenart Sphere, Key Curriculum Press,

Berkeley, CA, 1996.[17] I. Lenart, The plane-sphere project, Math. Teach. 187 (2004), pp. 22–27.[18] I. Lenart, Non-Euclidean Adventures on the Lenart Sphere, Key Curriculum Press,

Colorado, 1996.[19] D.W. Henderson, Experiencing Geometry in Euclidean, Spherical, and Hyperbolic Spaces,

Prentice Hall, Upper Saddle River, 2001.

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[20] H.L. Leighton, Solid Geometry and Spherical Trigonometry, D. Van Nostrand Company,New Jersey, 1943.

[21] J.A. Frykholm, External variables as predictors of van Hiele levels in algebra and geometrystudents, A report, The University of Wisconsin-Madison, ED372924, 1994.

[22] C.W. Faucett, Relationship between type of instruction and student learning in geometry,Ph.D. diss., Walden University, 2007.

[23] G.A. Goldin, A scientific perspective on structured, task-based interviews in mathematicseducation research, in Handbook of Research Design in Mathematics and ScienceEducation, A.E. Kelly and R.A. Lesh, eds., Lawrence Erlbaum Associates, Mahwah,NJ, 2000, pp. 517–545.

[24] A. Strauss and J.M. Corbin, Basics of Qualitative Research: Grounded Theory Proceduresand Techniques, Sage, London, 1990.

[25] J.W. Mayberry, An investigation of the van Hiele levels of geometric thought inundergraduate preservice teachers. Unpublished Dissertation, University of Georgia, 1981.

[26] L. Guttman, A basis of scaling qualitative data, Am. Sociol. Rev. 9 (1944), pp. 139–150.[27] L. Guttman, A basis of scalogram analysis, in Measurement and Prediction, L. Guttman,

A. Suchman, P.A. Lazarsfel, S.A. Star, and J.A. Clausen, eds., Princeton UniversityPress, Princeton, 1950, pp. 60–90.

[28] M. Henry and L. David, Scaling of theory-of-mind tasks, Child Dev. 75 (2004),pp. 523–541.

[29] M. Herbert, A new coefficient for scalogram analysis, Public Opin. Quarterly, 17 (1953),268–280.

[30] W.H. Goodenough, A technique for scale analysis, Educ. Psychol. Meas. 4 (1944),pp. 179–190.

[31] W.S. Torgerson, Theory and Methods of Scaling, Wiley, New York, 1962.

Geometry

Documents

undergraduate

sample activity

karadeniz

a0 b0 c0

making logical

key curriculum

van hiele

trirectangular