- UM Repository

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PRESERVICE SECONDARY SCHOOL MATHEMATICS TEACHERS'
KNOWLEDGE OF PERIMETER
Wun Thiam Yew, *Sharifah Norul Akmar Syed Zamri & Lim Hooi Lian
School of Educational Studies, Universiti Sains Malaysia
*Faculty of Education, University of Malaya
Abstract: This purpose of this article is to examine preservice secondary school
mathematics teachers‟ knowledge of perimeter. Clinical interview technique was
employed to collect the data. Interview sessions were recorded using digital video camera
and tape recorder. Subjects of this study consisted of eight preservice secondary school
mathematics teachers enrolled in a Mathematics Teaching Methods course at a public
university in Peninsula Malaysia. They were selected based on their majors
(mathematics, biology, chemistry, physics) and minors (mathematics, biology, chemistry,
physics). This article presents the analysis of the responses of the subjects related to a
particular task. The finding suggests that half of the preservice secondary school
mathematics teachers in this study had the correct notion of perimeter that simple closed
curves, closed but not simple curves, and 3-dimensional shapes have a perimeter.
Preservice secondary school mathematics teachers‟ linguistic knowledge and ethical
knowledge of perimeter were discussed. The implication of this finding was also
discussed.
Keywords: Preservice secondary school mathematics teachers, knowledge of perimeter,
case study, clinical interview
Abstrak: Artikel ini bertujuan untuk menyelidiki pengetahuan guru praperkhidmatan
matematik sekolah menengah bagi perimeter. Data kajian ini dikumpul melalui teknik
temu duga klinikal. Sesi temu duga telah dirakamkan dengan perakam video digital dan
perakam audio. Subjek kajian ini terdiri daripada lapan orang guru praperkhidmatan
matematik sekolah menengah yang sedang mengikuti kursus Kaedah Mengajar
Matematik di sebuah universiti awam di Semenanjung Malaysia. Mereka dipilih
berasaskan bidang pengkhususan major (matematik, biologi, kimia, fizik) dan minor
(matematik, biologi, kimia, fizik). Artikel ini membentangkan analisis respon subjek ke
atas satu tugasan tertentu. Dapatan kajian menunjukkan bahawa separuh daripada guru
praperkhidmatan matematik sekolah menengah dalam kajian ini mempunyai idea
perimeter yang betul iaitu lengkung tertutup ringkas, tertutup tetapi bukan lengkung
ringkas, dan bentuk 3-dimensi mempunyai perimeter. Pengtahuan linguistik dan
pengetahuan etika guru praperkhidmatan matematik sekolah menengah dibincangkan.
Implikasi dapatan kajian ini turut dibincangkan.
Kata kunci: Guru praperkhidmatan matematik sekolah menengah, pengetahuan
perimeter, kajian kes, temu duga klinikal
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INTRODUCTION
One cannot teach what one does not know. Teachers must have in-depth knowledge of
mathematics they are going to teach. Therefore, it is important that teachers need to have
a comprehensive knowledge of mathematics to enable them to organize teaching so that
students can learn mathematics meaningfully. Fennema and Franke (1992) advocated that
"No one questions the idea that what a teacher know is one of the most important
influences on what is done in classroom and ultimately on what students learn" (p. 147).
Furthermore, “Teachers who do not themselves know a subject well are not likely to help
students learn this content.” (Ball, Thames, & Phelps, 2008, p. 404). This applies also to
mathematics teacher.
Literally, perimeter means the measurement around. However, numerous definitions of
perimeter were provided by the researchers or mathematics educators. Table 1 shows
some of these definitions. According to Beaumont, Curtis, & Smart (1986), “the study of
the concept of perimeter is unified by considering the simple closed curve” (p. 5). A
closed but not simple curve has more than one interior. For all simple closed curves, “the
perimeter is the total length of the curve - the total distance around it.” (Beaumont et al.,
1986, p. 5). Perimeter is an extension from the concept of length. Beaumont et al. (1986)
emphasized that perimeter is the distance around and measured in linear units (e.g., mm,
cm, m, km). Review of research literature had shown that some students and preservice
teachers encountered difficulty in distinguishing between the attributes of perimeter, area,
and volume (Baturo & Nason, 1996; Beaumont, Curtis, & Smart, 1986; Ramakrishnan,
1998; Reinke, 1997).
PURPOSE OF THE STUDY
The purpose of this study was to investigate preservice secondary school mathematics
teachers' knowledge of perimeter. Specifically, this study aimed to investigate preservice
secondary school mathematics teachers' conceptual knowledge, linguistic knowledge, and
ethical knowledge of perimeter.
CONCEPTUAL FRAMEWORK OF THE STUDY
Nik Azis (1996) suggested that there are five basic types of knowledge, namely
conceptual knowledge, procedural knowledge, linguistic knowledge, strategic
knowledge, and ethical knowledge. In the present study, the researchers have adapted Nik
Azis‟s (1996) categorization of knowledge to examine preservice secondary school
mathematics teachers‟ knowledge of perimeter.
METHODOLOGY
The discussion about the methodology of this study consisted of five sections: research
design, selection of subjects, instrumentation, data collection, and data analysis.
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Table 1
Some of the definitions of perimeter
Researchers or mathematics educators
Ball, 1988, p. 170.
Definition of perimeter
The perimeter of a figure or a region is the length of its boundary.
Beaumont, Curtis, & Smart, 1986, p. 5.
The perimeter is the total length of the curve - the total distance around
it.
Bennett & Nelson, 2001, p. 658.
The length of the boundary of a region is its perimeter.
Billstein, Liberskind, & Lott, 2006, p. 743.
The perimeter of a simple closed curve is the length of the curve.
Cathcart, Pothier, Vance, & Bezuk, 2006, p.
325.
Perimeter is the total distance around a closed figure.
Haylock, 2001, p. 268.
The perimeter is the length of the boundary.
Kennedy & Tipps, 2000, p. 512.
Perimeter is the measure of the distance around a closed figure.
Long & DeTemple, 2003, p. 771.
The length of a simple closed plane curve is called its perimeter.
O‟Daffer, Charles, Cooney, Dossay, &
Schielack, 2005, p. 676.
Perimeter is the distance around a figure.
Rickard, 1996, p. 306.
Perimeter is the number of (linear) units required to surround a shape.
Suggate, Davis, & Goulding, 1999, p. 129.
Perimeter is the distance around the edge of a shape.
Research Design
In this study, the researchers employed case study research design to examine, in-depth,
preservice secondary school mathematics teachers' knowledge of perimeter. “A case
study design is used to gain an in-depth understanding of the situation and meaning for
those involved” (Merriam, 1998, p. 19). Several researchers (e.g., Aida Suraya, 1996;
Chew, 2007; Lim, 2007; Rokiah, 1998; Seow, 1989; Sharifah Norul Akmar, 1997;
Sutriyono, 1997) employed case study research design to study Malaysian students,
preservice teachers, and lecturers.
Selection of Subjects
The researchers employed purposeful sampling to select the subjects (sample) for this
study. Eight subjects were selected for the purpose of this study. They were preservice
secondary school mathematics teachers from a public university in Peninsula Malaysia
enrolled in a 4-year Bachelor of Science with Education (B.Sc.Ed.) program, majored or
minored in mathematics. These subjects enrolled for a one-semester mathematics
teaching methods course during the data collection of this study. The mathematics
teaching methods course was offered to B.Sc.Ed. program students who intended to
major or minor in mathematics. The researchers had selected four B.Sc.Ed. program
students who majored in mathematics, and four B.Sc.Ed. program students who minored
in mathematics for the purpose of this study. They do not have any teaching experience
prior to this study. Each preservice secondary school mathematics teachers was given a
pseudonym, namely Beng, Liana, Mazlan, Patrick, Roslina, Suhana, Tan, and Usha, in
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order to protect the anonymity of all interviewees. The brief background information
about the subjects is shown in Table 2.
Table 2
Subjects’ Ethnicity, Gender, Age, Major, Minor, and CGPA
Subject
Ethnicity
Gender
Age
Major
Usha
Indian
Female
(21, 9) Mathematics
Minor
Biology
CGPA
2.92
Mazlan
Malay
Male
(21, 8)
Mathematics
Chemistry
2.70
Patrick
Bidayuh
Male
(21, 7)
Mathematics
Chemistry
3.04
Beng
Chinese
Female
(22, 9)
Mathematics
Physics
3.82
Roslina
Malay
Female
(21, 8)
Biology
Mathematics
3.15
Liana
Malay
Female
(21, 5)
Chemistry
Mathematics
2.77
Tan
Chinese
Male
(22, 7)
Chemistry
Mathematics
3.69
Suhana
Malay
Female
(20, 10)
Physics
Mathematics
2.52
Instrumentation
Eight types of tasks were devised for this study. A sample of the tasks, Task 1.1, is given
in Appendix A. This article reports only the responses of the subjects on Task 1.1. This
task was adapted from previous study (Baturo & Nason, 1996, p. 245). In Task 1.1, the
subjects were asked to select the shapes (12 shapes) that have a perimeter. The objective
of this task was to determine the subjects‟ conceptual knowledge about the notion of
perimeter. According to Beaumont, Curtis, & Smart (1986), “the study of the concept of
perimeter is unified by considering the simple closed curve” (p. 5). Thus, four simple
closed curves (A, C, H, K), each of them has one interior, were used to ascertain whether
the subjects understood the concept of perimeter. For all simple closed curves, the
perimeter is the number of linear units it takes to surround it. Two simple but not closed
curves (B, G) were included to investigate further the subjects‟ understanding of the
concept of perimeter. Two closed but not simple curves (D, I), each of which has more
than one interior, were included to ascertain whether the subjects understood the concept
of perimeter.
Two 1-dimensional shapes (E, L) were included to ascertain whether the subjects
understood the concept of perimeter. Finally, two 3-dimensional shapes (F, J) were
included because review of research literature had shown that some students and
preservice teachers encountered difficulty in distinguishing between the attributes of
perimeter, area, and volume (Baturo & Nason, 1996; Beaumont, Curtis, & Smart, 1986;
Ramakrishnan, 1998; Reinke, 1997). Task 1.1 was also used to determine the subjects‟
linguistic knowledge of perimeter based on the language of mathematics (such as
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mathematical terms and symbols) that the subjects used to justify the selection of shapes
that have a perimeter. There are some good behaviors that the subjects needed to follow
when dealing with perimeter and area. Knowledge and justification of knowledge is an
important aspect in any discipline. Thus, this task was also used to determine the
subjects‟ ethical knowledge of perimeter by ascertaining whether the subject justify the
selection of shapes that have a perimeter.
Data Collection
Data for this study was collected using clinical interview techniques. The interview was
conducted in the Mathematics Teaching Room at a public university in Peninsula
Malaysia. The physical setting for each interview consisted of a table with two chairs, a
tape recorder and a digital video camera. Each interview was recorded through the tape
recorder and digital video camera positioned in front of the table. The camera was
focused on the subject, the working area, and the researcher. Blank papers, grid papers,
pencil, ruler, thread, compasses, and calculator were accessible to the subject throughout
the interview. Materials collected for analysis consisted of audiotapes and videotapes of
clinical interview, subject's notes and drawings, and researcher's notes during the
interview.
Data Analysis
The data analysis process encompassed four levels. At level one, the audio and video
recording of the clinical interview were verbatim transcribed into written form. The
transcription included the interaction between the researcher and the subject during the
interviews as well as the subject's nonverbal behaviors. At level two, raw data in the
forms of transcription were coded, categorized, and analyzed according to specific
themes to produce protocol related to the description of the subjects‟ knowledge of
perimeter.
At level three, case study for each subject was constructed based on information from the
written protocol. At this level, analysis was carried out to describe each subject's
behaviors in solving every tasks or problems. At level four, cross-case analysis was
conducted. The analysis aimed to identify pattern of responses of knowledge of perimeter
held by the subjects. Based on this pattern of responses, preservice secondary school
mathematics teachers‟ knowledge of perimeter were summarized.
FINDINGS OF THE STUDY
Findings of the study were presented in terms of the preservice secondary school
mathematics teachers‟ conceptual knowledge, linguistic knowledge, and ethical
knowledge of perimeter.
Conceptual Knowledge
Findings of the study suggest that four preservice secondary school mathematics teachers
(PSSMTs), namely Liana, Roslina, Tan, and Usha, have successfully selected all the
shapes that have a perimeter, namely shapes "A", "C", "D",”F”, "H", "I", “J”, and "K".
They have successfully selected all simple closed curves (A, C, H, K) as well as all
closed but not simple curves (D, I) that have a perimeter. Liana, Roslina, Tan, and Usha
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also selected the two 3-dimensional shapes (F, J) that have a perimeter. It indicated that
their notion of perimeter was not only limited to simple closed curves, and closed but not
simple curves, but also inclusive of 3-dimensional shapes. Table 3 shows each PSSMT‟s
selection of shapes that have a perimeter and their notion of perimeter.
Table 3
PSSMTs’ Selection of Shapes That Have a Perimeter and Their Notion of Perimeter
Selection of shapes that
Notion of perimeter
PSSMTs
have a perimeter
"A", "C", "D",”F”, "H", "I", Simple closed curves,
Liana, Roslina, Tan, Usha
“J”, and "K"
closed but not simple
curves, and 3-dimensional
shapes
“A”, “C”, “D”, “H”, “I”, Limited to simple closed
Beng, Patrick, Suhana
and “K”
curves, and closed but not
simple curves
“A”, “C”, and “H”
Limited to common simple Mazlan
closed curves (triangle,
circle, and trapezium)
Three PSSMTs, namely Beng, Patrick, and Suhana, selected shapes “A”, “C”, “D”, “H”,
“I”, and “K” that have a perimeter. They have selected all simple closed curves (A, C, H,
K) as well as all closed but not simple curves (D, I) that have a perimeter. Nevertheless,
Beng, Patrick, and Suhana did not select the two 3-dimensional shapes (F, J) that have a
perimeter. It indicated that their notion of perimeter was limited to simple closed curves,
and closed but not simple curves, exclusive of 3-dimensional shapes.
One PSSMT, namely Mazlan, selected shapes “A”, “C”, and “H” that have a perimeter.
He has selected three simple closed curves (A, C, H) that have a perimeter. Nevertheless,
Mazlan did not select another simple closed curve (K) and the two closed but not simple
curves (D, I) that have a perimeter. He also did not select the two 3-dimensional shapes
(F, J) that have a perimeter. It indicated that his notion of perimeter was limited to
common simple closed curves (triangle, circle, and trapezium). All the PSSMTs did not
select the two simple but not closed curves (B, G) as well as the two 1-dimensional
shapes (E, L) that do not have a perimeter. In other words, they did not select an open
shape (including the lines) as having a perimeter.
Linguistic Knowledge
When asked to justify their selection of shapes that have a perimeter, six PSSMTs,
namely Liana, Mazlan, Roslina, Suhana, Tan, and Usha, used appropriate mathematical
term „closed‟ to justify their selection of shape “A” that have a perimeter. Liana, Mazlan,
Roslina, Suhana, Tan, and Usha explained that they selected shape “A” because it was
closed. Beng used appropriate mathematical terms „2-dimensional shapes‟ and „enclosed‟
to justify her selection of shape “A” that have a perimeter. Beng explained that she
selected shape “A” because it was a 2-dimensional shape and enclosed. Patrick used
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appropriate mathematical term „covered‟ to justify his selection of shape “A” that have a
perimeter. Patrick explained that he selected shape “A” because it was covered. Table 4
shows PSSMTs‟ selection of shapes that have a perimeter and the appropriateness of their
justification.
Four PSSMTs, namely Mazlan, Roslina, Suhana, and Usha, used appropriate
mathematical term „closed‟ to justify their selection of shape “D” that have a perimeter.
Mazlan, Roslina, Suhana, and Usha explained that they selected shape “D” because it was
closed. Tan used appropriate mathematical term „measure‟ to justify his selection of
shape “D” that have a perimeter. Tan explained that he selected shape “D” because its
perimeter can be measured using thread. It indicated that Tan appeared to associate the
notion of perimeter with the measurement of perimeter (i.e., perimeter does not exist until
it is measured).
Beng used appropriate mathematical terms „2-dimensional shapes‟ and „enclosed‟ to
justify her selection of shape “D” that have a perimeter. Beng explained that she selected
shape “D” because it was a 2-dimensional shape and enclosed. Patrick used appropriate
mathematical term „covered‟ to justify his selection of shape “D” that have a perimeter.
Patrick explained that he selected shape “D” because it was covered. Liana used
inappropriate words „joined together‟ to justify her selection of shape “D” that have a
perimeter. Liana explained that she selected shape “D” because all the sides are joined
together.
Two PSSMTs, namely Roslina and Usha, used appropriate mathematical term „closed‟ to
justify their selection of shapes “F” and “J” that have a perimeter. Roslina and Usha
explained that they selected shapes “F” and “J” because they were closed. Tan used
appropriate mathematical term „calculate‟ to justify his selection of shapes “F” and “J”
that have a perimeter. Tan explained that he selected shapes “F” and “J” because their
perimeter can be calculated on each surface of the 3-dimensional objects. It indicated that
Tan appeared to associate the notion of perimeter with the measurement of perimeter
(i.e., perimeter does not exist until it is measured). Liana used inappropriate words
„joined together‟ to justify her selection of shapes “F” and “J” that have a perimeter.
Liana explained that she selected shapes “F” and “J” because all the lines are joined
together.
Five PSSMTs, namely Mazlan, Roslina, Suhana, Tan, and Usha, used appropriate
mathematical term „closed‟ to justify their selection of shape “H” that have a perimeter.
Mazlan, Roslina, Suhana, Tan, and Usha explained that they selected shape “H” because
it was closed. Beng used appropriate mathematical terms „2-dimensional shapes‟ and
„enclosed‟ to justify her selection of shape “H” that have a perimeter. Beng explained that
she selected shape “H” because it was a 2-dimensional shape and enclosed. Patrick used
appropriate mathematical term „covered‟ to justify his selection of shape “H” that have a
perimeter. Patrick explained that he selected shape “H” because it was covered. Liana
used inappropriate words „joined together‟ to justify her selection of shape “H” that have
a perimeter. Liana explained that she selected shape “H” because all the lines are joined
together.
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Table 4
PSSMTs’ Selection of Shapes That Have a Perimeter and the Appropriateness of Their
Justification
Selection of
shapes that have
a perimeter
“A”
“C”
“D”
Justification
PSSMTs
Appropriate
Inappropriate
Closed shape
Liana, Mazlan,
Roslina,
Closed object
Tan
Closed surface
Usha
Closed boundary
Suhana
2-dimensional shape and enclosed
Beng
Covered
Closed shape
Patrick
Liana, Mazlan,
Roslina
Closed surface
Usha
Closed boundary
Suhana
Its perimeter can be measured (using
thread)
Tan
2-dimensional shape and enclosed
Beng
Covered
Closed shape
Patrick
Roslina
Closed surface
Usha
Closed boundary
Suhana
Its perimeter can be measured (using
thread)
Tan
2-dimensional shape and enclosed
Beng
Covered
Patrick
All the sides are joined together
“F” and “J”
Closed shape
Liana
Roslina
Closed surface
Usha
Its perimeter can be calculated on each
surface of the 3-dimensional objects.
Tan
All the lines are joined together
“H”
Closed shape
Liana
Mazlan, Roslina
Closed object
Tan
Closed surface
Usha
Closed boundary
Suhana
2-dimensional shape and enclosed
Beng
Covered
Patrick
All the lines are joined together
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Table 4 (continued)
Selection of
shapes that have
a perimeter
“I” and “K”
Justification
PSSMTs
Appropriate
Inappropriate
Closed shape
Roslina
Closed surface
Usha
Closed boundary
Suhana
Its perimeter can be measured (using
thread)
Tan
2-dimensional shape and enclosed
Beng
Covered
Patrick
All the lines are joined together
Liana
Three PSSMTs, namely Roslina, Suhana, and Usha, used appropriate mathematical term
„closed‟ to justify their selection of shapes “I” and “K” that have a perimeter. Roslina,
Suhana, and Usha explained that they selected shapes “I” and “K” because they were
closed. Tan used appropriate mathematical term „measure‟ to justify his selection of
shapes “I” and “K” that have a perimeter. Tan explained that he selected shapes “I” and
“K” because their perimeter can be measured using thread. It indicated that Tan appeared
to associate the notion of perimeter with the measurement of perimeter (i.e., perimeter
does not exist until it is measured). Beng used appropriate mathematical terms „2dimensional shapes‟ and „enclosed‟ to justify her selection of shapes “I” and “K” that
have a perimeter. Beng explained that she selected shapes “I” and “K” because they were
2-dimensional shapes and enclosed. Patrick used appropriate mathematical term
„covered‟ to justify his selection of shapes “I” and “K” that have a perimeter. Patrick
explained that he selected shapes “I” and “K” because they were covered. Liana used
inappropriate words „joined together‟ to justify her selection of shapes “I” and “K” that
have a perimeter. Liana explained that she selected shapes “I” and “K” because all the
lines are joined together.
Ethical Knowledge
Findings of the study suggest that all the PSSMTs had taken the effort to justify the
selection of shapes that have a perimeter. All the PSSMTs provided appropriate
justification for selecting shapes “A” and “C” that have a perimeter, as shown in Table 4.
All the PSSMTs who had selected shapes “D”, “F” and “J”, “H”, and “I” and “K” that
have a perimeter provided appropriate justification for their selection, except Liana. She
had provided inappropriate justification for selecting shapes “D”, “F” and “J”, “H”, and
“I” and “K” that have a perimeter.
CONCLUSION AND IMPLICATION
In conclusion, half of the eight PSSMTs in this study had the correct notion of perimeter
that simple closed curves, closed but not simple curves, and 3-dimensional shapes have a
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perimeter. The implication of this finding is that mathematics educators as well as
mathematics teacher educators need to organize teaching and learning activities that
provide opportunity for their students and preservice teachers to investigate examples and
nonexamples of shapes that have and do not have a perimeter. They included 1dimensional shapes, simple closed curves, simple but not closed curves, closed but not
simple curves, and 3-dimensional shapes because previous studies had shown that some
students and preservice teachers encountered difficulty in distinguishing between the
attributes of perimeter, area, and volume (Baturo & Nason, 1996; Beaumont, Curtis, &
Smart, 1986; Ramakrishnan, 1998; Reinke, 1997).
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APPENDIX A
Task 1.1: Notion of perimeter
(Puts a handout comprise 12 shapes in front of the subject).
Probes:
What do you mean by ____ ?
Could you tell me more about it?
Why did you select this shape?
Why didn't you select this shape?
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