The Geometric Thinking Levels of Pre-service Teachers in Ghana

Abstract

Teachers' geometrical competencies are very critical to the effective teaching of the subject. This study focused on the van Hiele Levels of geometric thinking reached by Ghanaian pre-service teachers before leaving for their Student Internship Programme (Teaching Practice) at the basic schools. In all, 300 second year pre-service teachers from 4 Colleges of Education were involved in this study. These pre-service teachers were given the van Hiele Geometry Test adapted from the 'Cognitive Development and Achievement in Secondary School Geometry Test' items during their second year, first semester. The results showed that 16.33% of pre-service teachers attained van Hiele Level 0 (i.e. the Pre-recognition Level or Level for those who have not yet attained any van Hiele Level), 27% of pre-service teachers attained Level 1, 32% attained Level 2 while 17.67% of pre-service teachers attained Level 3. However, only 6% and 1% of Pre-service Teachers attained Levels 4 and 5 respectively. These results show that majority (75.33%) of pre-service teachers' van Hiele Levels are lower than that expected of their future Junior High School 3 learners. This suggests that most of the pre-service teachers' geometry knowledge is not sufficient to teach at basic schools.

Full text

Higher
Education
Research
2017; 2(3): 98-106
http://www.sciencepublishinggroup.com/j/her
doi: 10.11648/j.her.20170203.14
The Geometric Thinking Levels of Pre-service Teachers in
Ghana
Robert Benjamin Armah1, Primrose Otokonor Cofie2, Christopher Adjei Okpoti2
1Center for Distance Education, Institute for Educational Development and Extension, University of Education, Winneba, Ghana
2Department of Mathematics Education, University of Education, Winneba, Ghana
Email address:
armahrb@gmail.com (R. B. Armah), pocofie@uew.edu.gh (P. O. Cofie), caokpoti@uew.edu.gh (C. A. Okpoti)
To cite this article:
Robert Benjamin Armah, Primrose Otokonor Cofie, Christopher Adjei Okpoti. The Geometric Thinking Levels of Pre-service Teachers in
Ghana. Higher Education Research. Vol. 2, No. 3, 2017, pp. 98-106. doi: 10.11648/j.her.20170203.14
Received: April 2, 2017; Accepted: April 13, 2017; Published: May 26, 2017
Abstract: Teachers’ geometrical competencies are very critical to the effective teaching of the subject. This study focused
on the van Hiele Levels of geometric thinking reached by Ghanaian pre-service teachers before leaving for their Student
Internship Programme (Teaching Practice) at the basic schools. In all, 300 second year pre-service teachers from 4 Colleges of
Education were involved in this study. These pre-service teachers were given the van Hiele Geometry Test adapted from the
‘Cognitive Development and Achievement in Secondary School Geometry Test’ items during their second year, first semester.
The results showed that 16.33% of pre-service teachers attained van Hiele Level 0 (i.e. the Pre-recognition Level or Level for
those who have not yet attained any van Hiele Level), 27% of pre-service teachers attained Level 1, 32% attained Level 2
while 17.67% of pre-service teachers attained Level 3. However, only 6% and 1% of Pre-service Teachers attained Levels 4
and 5 respectively. These results show that majority (75.33%) of pre-service teachers’ van Hiele Levels are lower than that
expected of their future Junior High School 3 learners. This suggests that most of the pre-service teachers’ geometry
knowledge is not sufficient to teach at basic schools.
Keywords: Van Hiele Levels, Geometric Thinking, College of Education Geometry, Pre-service Teachers, Ghana
1. Introduction
In Ghana, the key institutions that train pre-service
teachers for basic schools are the Colleges of Education. Pre-
service Teachers are students being trained to become first
time professional teachers. The subject of teacher knowledge
has been a key issue in mathematics education over the years.
Several researchers have indicated that teachers at all levels
need experiences studying geometry in order to attain the
content knowledge necessary to be effective instructors [12]
[18] [19]. Apart from the field of Mathematics, geometry is
important in other curriculum areas such as Science,
Geography, Art, Design and Technology [32]. It is therefore
not surprising that one of the aims of teaching Mathematics
in Ghana is to develop an understanding of geometric
concepts and relationships [22].
The College of Education geometry in Ghana is in two
different aspects; content aspects, which is studied in first
year and methodology aspects, studied in second year.
Geometry in the first year content course covers areas such as
lines and angles, polygons, congruent and similar triangles,
geometrical constructions including loci, circle theorems, two
and three-dimensional shapes, movement geometry and co-
ordinate geometry. In the method course, geometry covers
areas such as developing ideas about shape and space,
teaching measurements, teaching geometrical constructions
and teaching rigid motion [22]. Geometry has a separate
subject status and forms a considerable amount of the content
of College of Education mathematics curriculum in Ghana. It
is a branch of mathematics that deals with the study of shape
and space. Without spatial ability, students cannot fully
appreciate the natural world [32].
The study of geometry contributes to helping students
develop the skills of visualization, critical thinking, intuition,
perspective, problem-solving, conjecturing, deductive
reasoning, logical argument and proof. Geometric
representations can also be used to help students make sense
of other areas of Mathematics: fractions and multiplication in
arithmetic, the relationships between the graphs of functions
(of both two and three variables), and graphical
representations of data in statistics [17]. Thus, it is imperative

99 Robert Benjamin Armah et al.: The Geometric Thinking Levels of Pre-service Teachers in Ghana
for pre-service teachers to attain a sufficient geometric
thinking level so that their subject matter knowledge in other
areas of Mathematics is enhanced for positive impact on their
future basic school learners.
However, a lot of concerns have been raised about the
levels of students’ geometric thinking in Ghanaian schools,
especially at the basic school level [3] [4] [6]. At the College
of Education level, the Chief Examiner’s annual reports for
End-of-Second Semester Mathematics Examination in
geometry, in the years 2011 and 2012 revealed that the pre-
service teachers’ presentations of solutions to most of the 2
and 3-dimensional geometrical problems were poor and
majority of them had problems solving questions involving
the concepts of exterior and interior angles of polygons and
their properties, among other concepts [23] [24]. In 2013 and
2014, the examiner’s report once again revealed candidates’
lack of adequate knowledge in geometry and application of
geometric concepts [25] [26]. The Conference Board of the
Mathematical Sciences (CBMS) notes that learning of
geometry is usually confronted by conceptual difficulties [8].
Teaching and learning of geometry still remain as one of the
most disappointing experiences in many schools across
nations [17].
Throughout the history of modern educational system,
there have always been students who have had difficulties
and thus, fallen behind others in the field of Mathematics
especially in geometry. This has encouraged teachers to
experiment with new methods of teaching in an attempt to
understand and correct this imbalance. A range of models to
describe learners’ spatial sense and thinking have also been
proposed and researched and these include Piaget and
Inhelder’s Topological Primacy Thesis [20] van Hiele’s
Levels of Geometric Thinking [28] and Cognitive Science
Model [11]. However, the theoretical framework on
geometrical thinking proposed by the van Hieles tended to
have attracted more attention than many others in terms of
giving an accurate description of students’ geometric
thinking and also impacting on geometry classroom
instructional practices. Although the van Hiele theory was
primarily aimed at improving teachers’ as well as learners’
understanding of geometrical concepts, it also appealed as an
ideal model for use as a theoretical framework as well as a
frame of reference to link geometry to educational principles
[10].
[29] argues that the quality of instruction has one of the
greatest influences on the students’ acquisition of geometry
knowledge in mathematics classes and that the students’
progress from one level to the next in geometry depends on
the quality of instruction more than other factors, such as
biological maturation or students’ age. Moreover, there are
many other factors, such as knowledge of teachers, gender,
task difficulty, environment, curriculum etc. appearing to
play vital roles on student achievement and motivation in the
mathematics classroom [13] [12] [21]. However, in view of
the fact that students spend most of their time in the schools
it is logical to say that the teacher is one of the most
important factors in student learning and thus, teacher ’s
mathematical and pedagogical content knowledge play vital
roles in impacting positively on students motivation and
mathematics learning. Furthermore, according to Stipek cited
in [12] teachers’ content knowledge plays prominent roles in
students’ performance, and the pre- and in-service school
teachers’ inadequate geometry knowledge might be another
important factor behind students’ poor performance in
geometry. This statement is consistent with the argument
made by [18] and [19] who stated that content knowledge in
geometry among pre-service teachers is not sufficient.
Studies have shown that learners who have not attained a
van Hiele Level 3 before taking a secondary school (Senior
High School) geometry course have a low chance of success
[19] [27]. Therefore, attainment of Level 3 upon completion
of elementary and middle school (i.e. Junior High School) is
desirable [27 [9]. Several researchers have also affirmed the
validity of the existence of the first four van Hiele Levels in
high school geometry courses [9] [19] [27]. Therefore, it is
expected that pre-service teachers attain at least, van Hiele
Level 4 of geometric thinking prior to the completion of their
Senior High School programme as well as their first year
geometry course in College of Education. It is also logical to
assume that for basic school learners to attain these
respective levels of geometric thinking, their prospective
teachers need to have attained a level of geometric thinking
at or above these levels in order to assist them by providing
appropriate scaffolding and learning experiences. These
arguments might be clearly explained by finding the van
Hiele Levels of pre-service teachers in geometry since the
van Hiele theory has been a facilitator for much of the
renewed interest in geometry.
1.1. Theoretical Framework
In the field of geometry, the best and most well-defined
theory for students’ levels of thinking is based on the van
Hiele theory [1] [31] [32]. The theory emerged from the
separate doctoral works of a husband-and-wife team of Dutch
Mathematics educators, Dina van Hiele-Geldof and Pierre
van Hiele, which were completed simultaneously at the
University of Utrecht, Netherlands in 1957 [27]. The couple
did research in the late 1950s on thought and concept
development in geometry among school children. Since Dina
died shortly after finishing her dissertation, it was her
husband, Pierre who clarified, amended, and advanced the
theory. The theory enables insight into why many students
encounter difficulties in their geometry courses, particularly
with formal proofs. The van Hiele theory comprises three
main aspects, namely: Levels of geometric thinking,
properties of the Levels and phases of learning which offers a
model of teaching that teachers could apply in order to
promote their learners’ levels of understanding in geometry
[9] [15] [29].
The van Hiele theory originally consists of five sequential
and hierarchical discrete Levels of geometric thought
namely: Recognition, Analysis, Order (Informal Deduction),
Deduction, and Rigor [27]. There are two different
numbering schemes that are commonly used to describe the

Higher Education Research 2017; 2(3): 98-106 100
van Hiele Levels: Level 0 through to 4, and Level 1 through
to 5. Originally the van Hieles numbering scheme used Level
0 through to 4, however, Americans [16] [27] and van Hiele’s
[29] [30] more recent writings make use of the Level 1
through to 5 numbering scheme instead. This according to
[16] allows for a sixth Level, Pre-recognition Level (i.e.
Level for learners who have not yet achieved even the basic
Level 1) to be called Level 0. This study used the Level 1 to
5 numbering scheme to allow utilization of Level 0.
The van Hiele Levels can be described as follows:
Level 1: Recognition (or visual level)
At this Level learners use visual perception and nonverbal
thinking. They recognize figures by appearance alone “and
compare the figures with their prototypes or everyday things
(“it looks like a door”), categorize them (“it is / it is not
a…”). They use simple language [31]”. Learners at this Level
do not identify the properties of geometric figures [30].
Level 2: Analysis (or descriptive level)
At this Level, “figures are the bearers of their properties. A
figure is no longer judged because it looks like one but rather
because it has certain properties [30]”. Learners start
analyzing and naming properties of geometric figures but
they do not understand the interrelationship between different
types of figures, and they also cannot fully understand or
appreciate the uses of definitions at this level [16].
Level 3: Order (or informal deduction level)
Learners at this Level are able to see the interrelationship
between different types of figures. They can create
meaningful definitions and give informal arguments to justify
their reasoning at this Level. Logical implications and class
inclusions, such as squares being a type of rectangle, are
understood [13] [16].
Level 4: Deduction
At this Level learners can give deductive geometric proofs.
They understand the role of definitions, theorems, axioms
and proofs. Learners at this Level can supply reasons for
statements in formal proofs [13] [31].
Level 5: Rigor
Learners at this Level “understand the formal aspects of
deduction, such as establishing and comparing mathematical
systems [16]”. Here, learners learn that geometry needs to be
understood in the abstract; see the “construction” of
geometric systems. Learners at this level should understand
that other geometries exist and that what is important is the
structure of axioms, postulates, and theorems [9].
1.2. Problem Statement
A teacher is viewed as someone that should possess
specific and adequate content knowledge. [5] indicated that
in various countries the need to improve the experience of
classroom mathematical learning through the development of
teachers’ knowledge of mathematics and knowledge of
pedagogy is still relevant. This is because the level of
understanding that learners achieve for any concept is limited
by the level of understanding of their teacher [2] [13]. [21]
work focused on elements of teacher knowledge (content and
pedagogical content knowledge) with teacher knowledge
being one of the factors that influence teacher behaviour.
This has led to a growing recognition of the need for more
research studies on teacher knowledge. However, the
majority of the prior studies have focused on number
concepts [5] [7] [14] and studies regarding geometry are
limited.
The van Hiele theory has been applied to many curricula to
improve geometry classroom instruction in many developed
nations but in Ghana, the literature appears to suggest that
there has been little investigation involving the van Hiele
theory. Thus, very little studies have applied the van Hiele
theory to determine the level of geometric conceptualization
of Ghanaian pre-service teachers and also to improve
geometry instruction. Meanwhile, there is evidence that
many students in Ghana encounter severe difficulties with
school geometry [6]. Thus, this study was designed to fill this
void.
1.3. Purpose of Study and Research Question
The purpose of this study was to investigate the van Hiele
Levels of geometric thinking reached by Ghanaian second
year pre-service teachers before leaving for their Student
Internship Programme (Teaching Practice) at the basic
schools. The researchers therefore, seek to address the issue
of whether pre-service teachers possess enough
understanding of geometry to teach the subject well. In
pursuance of this purpose, the following research question
was formulated to guide the study: Which stages of van Hiele
Levels of geometric thinking do Ghanaian pre-service
teachers reach in their study of geometry before leaving
College of Education?
2. Method
2.1. Research Design
The researchers employed mainly the survey approach
using test. The survey in this study was used for descriptive
purposes. The researchers aimed at getting an accurate
description of the van Hiele Levels of geometric thinking
reached by pre-service teachers before leaving for their
Student Internship Programme (Teaching Practice) at the
basic schools.
2.2. Participants and Setting
Convenience sampling was used to select four Colleges of
Education in the Ashanti, Central and Greater-Accra Regions
of Ghana. The researchers are of the view that these Colleges
of Education were ideal for this study because the pre-service
teachers in these Colleges of Education are admitted from all
over the ten regions in Ghana. This has enriched the sample
used for the study in terms of pre-service teachers’ abilities,
cultural and social backgrounds. The sample used therefore
represents the characteristics of Ghanaian pre-service
teachers in any part of the country who had spent at least a
year studying geometry in the College. Stratified random
sampling was then used to select 300 second year pre-service

101 Robert Benjamin Armah et al.: The Geometric Thinking Levels of Pre-service Teachers in Ghana
teachers from these four Colleges.
2.3. Instrument
In order to address the research question in this study, the
participating pre-service teachers were given van Hiele
Geometry Test (VHGT) to identify their geometric thinking
levels. The VHGT was taken from the Cognitive
Development and Achievement in Secondary School
Geometry (CDASSG) project, developed by [27] which
found the van Hiele theory as a good predictor of students’
success in geometry courses. The test consists of 25-item
multiple choice test and is organized sequentially in blocks of
five Items (Part A). Items 1-5 deal with identification,
naming and comparing of geometric shapes such as squares,
rectangles and rhombi and measure students understanding at
Level 1. Items 6-10 deal with recognizing and naming
properties of geometric figures and measure students
understanding at Level 2. Items 11-15 deal with logical order
of the properties of figures previously identified, and the
relationships between these properties, these measure
students understanding at Level 3. Items 16-20 deal with
questions that require students to understand the significance
of deduction and the role of postulates, axioms, theorems and
proof, these also measure students understanding at Level 4.
While items 21-25 deal with the formal aspects of deduction
and measure student understanding at Level 5 [19]. The
researchers included a second part (Part B of the VHGT)
consisting of 3 items where participants were expected to
provide written responses. This was designed to further
explore the problem-solving abilities of the pre-service
teachers. These items included some commonly found in
texts and examination papers set for these learners. Item 1
required the pre-service teachers to calculate a missing value
in a given geometrical shape; item 2 also required the pre-
service teachers to find the surface area of a geometric figure;
and item 3 required the pre-service teachers to write a
complete proof of a theorem in geometry giving reasons.
Administration and Grading of the VHGT
The VHGT was administered in the second year, first
semester and written by all second year pre-service teachers
who were participating in the study. All participants’ answer
sheets from VHGT were read and scored by the researchers.
Scoring of the part A of the VHGT was done as indicated
below;
First grading method: Each correct response to the 25-item
multiple-choice test was assigned 1 point. Hence, each Pre-
service Teacher’s score ranges from 0–25 marks.
Second grading method: the second method of grading the
VHGT was based on “3 of 5 correct” success criterion
suggested by [27]. By this criterion, if a Pre-service Teacher
answered correctly at least 3 out of the 5 items in any of the 5
subtests within the VHGT, the Pre-service Teacher was
considered to have mastered that level. Using this grading
system developed by [27], the pre-service teachers were
assigned weighted sum scores in the following manner:
(1) 1 point for meeting criterion on items 1-5 (Level-I,
Recognition);
(2) 2 points for meeting criterion on items 6-10 (Level-II,
Analysis);
(3) 4 points for meeting criterion on items 11-15 (Level-
III, Ordering);
(4) 8 points for meeting criterion on items 16-20 (Level-
IV, Deduction);
(5) 16 points for meeting criterion on items 21-25 (Level-
V, Rigor).
Thus, the maximum point obtainable by any Pre-service
Teacher was 1 + 2 + 4 + 8 + 16 = 31 points. The method of
calculating the weighted sum makes it possible for a person
to determine upon which van Hiele Level the criterion has
been met from the weighted sum alone. For example, a score
of 7 indicates that the learner met the criterion at Levels I, II
and III (i.e.1+ 2 + 4 = 7). The second grading system served
the purpose of assigning the learners into various van Hiele
Levels based on their responses.
According to [27], there are two different cases that can be
used in assigning Levels to students namely, the Classical
Case and the Modified Case. The study employed the
modified case in assigning Levels to pre-service teachers.
[27] posited that;
The assigning of Levels in either the classical or modified
case requires that a student’s responses satisfy Property 1 of
the Levels, i.e., that the student at level n satisfy the criterion
not only at that level but also at all preceding Levels. Thus a
student who satisfies the criterion at Levels 1, 2 and 5, for
instance, would have a weighted sum of 1 + 2 + 16 or 19
points, would have no classical van Hiele Level, but would
be assigned the modified van Hiele Level 2. A student who
satisfies the criterion at Level 3 only would not be assigned
either a classical or modified van Hiele Level. Neither of
these students would be said to fit the classical van Hiele
model.
The first case, identified as the Classical case, is based on
there being five distinct Levels. The second case, identified
as the Modified case, is based on four distinct Levels. The
decision to use the Modified Case to identify the van Hiele
Level of the test subjects was based on the fact that the
“modified van Hiele Levels fit more students more
consistently than the classical van Hiele Levels [27]” also it
gives a higher percentage of subjects that could be analyzed.
For the Part B, each of the 3 items was assigned 10 points.
Thus, pre-service teachers’ scores ranged between 0 and 30
marks.
2.4. Analysis of Data
This study aims to determine the van Hiele geometric
thinking Levels of the participating pre-service teachers. To
analyze data, descriptive statistics were used in an attempt to
understand, interpret and describe the experiences of the
research participants in terms of their levels of geometric
conceptualization. In specific terms, various descriptive
statistics such as frequency distribution, percentages, chart
and measures of central tendency, were used to analyse,
describe and compare the quantitative data in this study.

Higher Education Research 2017; 2(3): 98-106 102
3. Results
3.1. Pre-service Teachers’ Performance in Part A of the
VHGT
Table 1 presents the overall pre-service teachers’
performance on each item of the part A in the VHGT. As can
be seen in Table 1, each van Hiele Level (VHL) had five
items with five multiple choice options. However, some pre-
service teachers did not choose any of the options for some
items. This made the researchers include an additional option
(a “blank” option) in this table. For each item, the number in
bold corresponds to the right option and also represents the
total number of pre-service teachers who answered that item
correctly.
Table 1. Van Hiele Geometry Test (Part A): Item Analysis for each Level.
Choice
items A B C D E Blank
Level 1
1 0 278 0 14 8 0
2 2 0 50 236 2 10
3 26 4 250 0 16 4
4 14 198 20 60 4 4
5 28 0 146 110 12 4
Level 2
6 36 92 126 28 2 16
7 46 4 30 8 192 20
8 80 22 86 34 50 28
9 14 8 214 12 48 4
10 36 20 72 116 20 36
Level 3
11 64 70 56 40 56 14
12 70 134 30 18 38 10
13 30 4 4 28 230 4
14 48 6 102 18 112 14
15 26 120 12 62 66 14
Level 4
16 76 60 48 24 36 56
17 132 26 52 52 14 24
18 60 52 56 72 28 32
19 144 56 40 12 20 28
20 72 44 50 74 30 30
Level 5
21 80 34 52 36 42 56
22 68 26 48 90 32 36
23 140 48 18 30 38 26
24 28 30 30 148 34 30
25 32 84 100 24 28 32
*The figures in bold correspond to the right options and also represent the
total number of pre-service teachers who answered that item correctly.
n= 300.
3.1.1. Performance on Subtest 1: Van Hiele Level 1
The pre-service teachers performed well only in the first
four items of subtest 1. Table 1 shows that 278 (92.67%), 236
(78.67%), 250 (83.33%), 198 (66%) of the pre-service
teachers managed to correctly answer items 1, 2, 3 and 4
respectively, compared to item 5, 110 (36.67%) which was
not very encouraging. Figure 1 is an item from Subtest 1. The
correct answer for this item is choice D. Table 1 shows that
only 110 (36.67%) of the pre-service teachers had this item
correct, that is, knew that all the given quadrilaterals can be
referred to as parallelograms. This clearly indicates that 190
(63.33%) of the pre-service teachers who participated in this
research study lack knowledge of “class inclusion”.
Figure 1. Sample Item in Subtest 1.
3.1.2. Performance on Subtest 2: Van Hiele Level 2
Pre-service teachers did quite well on items 7 and 9. Out of
the 300 pre-service teachers, 192 (64%) and 214 (71.33%)
respectively answered items 7 and 9 correctly. However, pre-
service teachers’ performance on items 6, 8 and 10 was not
satisfactory. From a total of 300 pre-service teachers only 92
(30.67%), 80 (26.67%) and 116 (38.67%) of the pre-service
teachers were able to answer questions on these items
respectively.
Figure 2. Sample Item in Subtest 2.
Figure 2 is an item from Subtest 2. The correct answer for
this item is choice A. Table 1 shows that only 80 (26.67%) of
pre-service teachers had this question correct. A total of
73.33% of the pre-service teachers answered this item
wrongly. This shows pre-service teachers’ lack of knowledge
about the properties of a rhombus.
3.1.3. Performance on Subtest 3: Van Hiele Level 3
Subtest 3 is about learners knowing the interrelationship
between different types of figures. The performance of the
pre-service teachers for Subtest 3 was generally not
encouraging. Table 1 shows that 56 (18.67%), 134 (44.67%),
30 (10%), 48 (16%) and 120 (40%) of the pre-service
teachers correctly answered items 11, 12, 13, 14 and 15
respectively, which was not encouraging. The performance of
pre-service teachers on item 13 was abysmally poor. This
item is presented in Figure 3.

103 Robert Benjamin Armah et al.: The Geometric Thinking Levels of Pre-service Teachers in Ghana
Figure 3. Sample Item in Subtest 3.
The correct choice for item 13 in Figure 3 is A. However,
only 30 (10%) of pre-service teachers had this item correct.
This clearly shows that majority (90%) of the pre-service
teachers did not know that rectangles have common
properties with squares, in order words all squares are
rectangles. This implies that pre-service teachers have
difficulties in understanding “class inclusion”.
3.1.4. Performance on Subtest 4: Van Hiele Level 4
Subtest 4 is about learners being able to give deductive
geometric proofs, understanding the role of definitions,
theorems, axioms and proofs. Learners at this Level should
be able to supply reasons for statements in formal proofs.
This is the Level of development that high school students
need to be prior to completion of high school. However, the
performance of the pre-service teachers for Subtest 4 was
generally very poor. Table 1 shows that 48 (16%), 52
(17.33%), 72 (24%), 12 (4%) and 72 (24%) of the pre-service
teachers managed to correctly answer items 16, 17, 18, 19
and 20 respectively which was abysmally poor. This
generally indicates that the pre-service teachers have
difficulties understanding simple deductive geometric proofs,
understanding the role of definitions, theorems, axioms and
proofs.
3.1.5. Performance on Subtest 5: Van Hiele Level 5
Subtest 5 is about learners being able to work in a variety
of axiomatic systems that is, being able to study non-
Euclidean geometries comparing different systems and also
seeing geometry in the abstract. Similarly, Table 1 indicates
that 16 (5.33%), 18 (6%), 16 (5.33%), 16 (5.33%), and 10
(3.33%) of the pre-service teachers in the control group
managed to correctly answer items 21, 22, 23, 24 and 25
respectively.
Even though some pre-service teachers were able to
answer some items in subtests 4 and 5 correctly, the number
of Pre-service Teachers attaining Level 4 and Level 5 in the
VHGT was very small.
3.2. Performance on Part B of the VHGT
Table 2 summarizes the overall performance of pre-service
teachers in the section B part of the VHGT. There were 3 test
items; item 1 was on triangles, properties of parallel lines and
transversal, item 2 was on area of two-dimensional shapes
while item 3 was a short proof on congruent triangles. The
responses of pre-service teachers who demonstrated good
knowledge and provided the right responses for the items
were described as correct. Responses of pre-service teachers
who attempted items but did not get the total marks allotted
per test item were described as partially correct, while the
responses that exhibited lack of knowledge about the items
were described as completely wrong. However, few pre-
service teachers did not attempt some of the items at all;
these were described as “blank”.
Table 2. Van Hiele Geometry Test (Part B): Item Analysis.
Correct Partially
Correct
Completely
Wrong Blank
Item (%) (%) (%) (%)
1 88(29.33) 110(36.67) 88(29.33) 14(4.67)
2 30(10) 190(63.33) 70(23.33) 10(3.33)
3 26(8.67) 66(22) 198(66) 10(3.33)
* n= 300
The results in Table 2 show that the pre-service teachers
performed well only in the first item. Majority of the pre-
service teachers (110) representing 36.67% had item 1
partially correct while 88 pre-service teachers representing
29.33% had item 1 correct. The performance of pre-service
teachers in item 2 was not encouraging; out of a total of 300
pre-service teachers only 30 pre-service teachers representing
10% answered this item correctly. Again, pre-service
teachers’ performance in item 3 was extremely poor;
majority (198) pre-service teachers representing 66% had this
item completely wrong. This again revealed pre-service
teachers difficulties in understanding simple deductive
geometric proofs, understanding the role of simple
definitions, theorems, axioms and proofs.
3.3. Overall Scores of Pre-service Teachers in the VHGT
Table 3 presents the overall scores of pre-service teachers
in both Parts of the VHGT. The minimum score pre-service
teachers obtained in the VHGT was 28%, while the
maximum score was 69%. The mean score of pre-service
teachers was 49.25% while the standard deviation was 5.01.
Table 3. Means, Standard Deviations, Minimum and Maximum VHGT
Scores of Pre-service Teachers.
N Mean Stand Dev Maximum Minimum
150 49.25 5.01 69 28
3.4. Levels Reached by Pre-service Teachers in the VHGT
The purpose of this study was to find out the the van Hiele
Levels of geometric thinking reached by Ghanaian pre-
service teachers’ before leaving for their Student Internship
Programme (Teaching Practice) at the basic schools. The bar
chart in Figure 4 provides a visual confirmation of the van
Hiele Levels of geometric thinking attained by these pre-
service teachers.

Higher Education Research 2017; 2(3): 98-106 104
Figure 4. A Bar Chart showing Pre-service Teachers van Hiele Levels of
Geometric Thinking.
As shown in Figure 4, 16.33% of pre-service teachers
attained VHL 0 (i.e. the Pre-recognition Level or Level for
those who have not yet attained any van Hiele Level). For
VHL 1, 27% of pre-service teachers attained that Level. 32%
of the pre-service teachers attained VHL 2. In addition,
17.67% of pre-service teachers attained VHL 3. However,
only 6% and 1% of pre-service teachers attained VHL 4 and
5 respectively.
4. Discussion
The results of the VHGT revealed that 27% of the pre-
service teachers attained van Hiele Level 1, 27% reached level
2 and 17.67% reached Level 3 by the theory. Again, only 6%
and 1% of pre-service teachers attained Level 4 and Level 5 in
the VHGT respectively. In other words, majority of pre-service
teachers showed only the first three reasoning stages described
by the van Hiele Levels in different percentiles. However,
16.33% of the pre-service teachers did not attain any of the van
Hiele Levels suggesting that these pre-service teachers who are
about to leave for their Student Internship Programme
(Teaching Practice) at the basic schools are at the Pre-
recognition Level. These findings concur with those of
previous research studies [12] [18] [19]. The findings indicate
that majority of pre-service teachers were found to be
operating at the basic van Hiele Levels (i.e. Levels 1 and 2) as
well as the pre-recognition level, and that a very small number
of pre-service teachers operated at van Hiele Levels 3, 4 and 5.
This is problematic, since Level 4 skills are desirable prior to
the completion of Senior High School and thus, required to
successfully begin College of Education geometry.
From the item-by-item analysis, it was evident that the pre-
service teachers could identify plane shapes by mere
visualization and also identify the properties of these plane
shapes, which is a van Hiele Levels 1 and 2 geometric
competences respectively. However, the pre-service teachers’
responses to items in subtest 3 suggested that majority of them
could not identify the interrelationship between different types
of figures. Pre-service teachers could not create meaningful
definitions and give informal arguments to justify their
reasoning. In addition, logical implications and class
inclusions, such as squares being a type of rectangle were not
understood by most of these pre-service teachers. This finding
is also consistent with [6] observation that class inclusion
which theoretically according to van Hiele belongs to Level 3,
was frequently found most difficult among learners. Also, only
few (6% and 1% respectively) of the pre-service teachers
exhibited geometric reasoning at van Hiele Levels 4 and 5.
This suggests that teaching and learning in geometry is mainly
focused on the basic van Hiele Levels (i.e. van Hiele Levels 1
and 2), with a small amount of geometry work being done at
the advanced Levels (i.e. Levels 3, 4 and 5).
5. Limitations of the Study
From a total of 38 public Colleges of Education in Ghana,
only 4 Colleges of Education were selected for this study.
Though this was compensated for by the strategic location of
the Colleges to attract students from several regions of the
country, it is still difficult to generalize the findings for the
whole country. Also, the range of activities or tasks that the
pre-service teachers were tested on was limited due to time
constraints. So too was the range of instruments used. Further
investigations with larger groups and a wider range of
activities and instruments might yield different results.
6. Conclusion
This study was an attempt to measure the van Hiele Levels
of geometric thinking among pre-service teachers in Ghana.
It specifically sought to find out the stages of the van Hiele
Levels of understanding Ghanaian second year pre-service
teachers reach in their study of geometry before leaving for
their Student Internship Programme (Teaching Practice) at
the basic schools. In all, 300 second year pre-service teachers
from 4 Colleges of Education were involved in this study.
These pre-service teachers were given the van Hiele
Geometry Test (VHGT) from the Cognitive Development
and Achievement in Secondary School Geometry (CDASSG)
test items. The results show that majority (59%) of the pre-
service teachers attained the basic van Hiele Levels 1
(Recognition) and 2 (Analysis). In addition, 16.33% attained
Level 0 (Pre-recognition), a Level of thinking which is not
even expected from the Junior High School learner. It can be
counter argued, that much depends on whether, during their
Senior High School years and first year geometry course in
College of Education, the pre-service teachers were taught
such concepts as identifying the interrelationship between
different types of figures, creating meaningful definitions,
giving informal arguments to justify their reasoning, class
inclusions and simple deductive geometric proofs to enable
them operate at higher van Hiele Levels.
In order to teach geometry successfully at the basic school
level, the expected geometric reasoning stage for the basic
school teachers is Level 3 (Order) or above [9] [19] [27].
However, this study found that 75.33% of Pre-service basic
school teachers’ van Hiele Levels was below Level 3
(Order). This indicates that these pre-service teachers, about
to leave for their Student Internship Programme (Teaching

105 Robert Benjamin Armah et al.: The Geometric Thinking Levels of Pre-service Teachers in Ghana
Practice) in the basic schools, demonstrate a van Hiele Level
that is lower than that expected of their target audience.
These findings are alarming, and raise a concern about how
to break the cycle of limited geometric understanding.
Recommendations
From the findings of this study, it is recommended that;
(1) Pre-service Teachers geometry course should be
revised in terms of content and scope and these
courses may be reorganized according to the
geometrical thinking levels of van Hiele.
(2) Geometry instruction should be supportive and
appropriate to the van Hiele geometrical thinking
levels. This should involve more hands-on
investigations that will actively engage the learners.
Geometry instructors should ensure that learners
understand and know the properties of all geometric
shapes as well as their interrelationships to enable
them establish class inclusion, which according to
this study is sorely lacking. More work need to be
done on class inclusions and simple deductive
geometric proofs to enable pre-service teachers
operate at van Hiele Levels higher than that expected
of their future basic school learners.
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