Available via license: CC BY 4.0
Content may be subject to copyright.
Teacher Education and Curriculum Studies
2024, Vol. 9, No. 2, pp. 40-51
https://doi.org/10.11648/j.tecs.20240902.12
*Corresponding author:
Received: 25 June 2024; Accepted: 11 July 2024; Published: 23 July 2024
Copyright: © The Author(s), 2024. Published by Science Publishing Group. This is an Open Access article, distributed
under the terms of the Creative Commons Attribution 4.0 License (http://creativecommons.org/licenses/by/4.0/), which
permits unrestricted use, distribution and reproduction in any medium, provided the original work is properly cited.
Research Article
Geometric Thinking of Prospective Mathematics Teachers:
Assessing the Foundation Built by University
Undergraduate Education in Ghana
Robert Benjamin Armah*
Department of Mathematics Education, University of Education, Winneba, Ghana
Abstract
This study investigates the geometric thinking levels of final year prospective mathematics teachers in Ghana, utilizing the van
Hiele model to evaluate their proficiency. The main purpose was to assess whether university undergraduate mathematics
education provides a sufficiently strong foundation for teaching senior high school geometry. A descriptive survey design was
employed, involving 1,255 prospective mathematics teachers from three universities: University of Education Winneba (UEW),
University of Cape Coast (UCC), and Akenten Appiah-Menka University of Skills Training and Entrepreneurial Development
(AAMUSTED). The van Hiele Geometry Test (VHGT) was administered to measure participants’ levels of geometric thinking.
The results revealed that 8.8% of participants attained van Hiele Level 1 (visualization), 30.0% reached Level 2 (analysis), and
32.4% achieved Level 3 (abstraction). However, only 15.9% and 12.9% of prospective teachers reached Levels 4 (deduction) and
5 (rigor), respectively. These findings indicate a significant gap between the current geometric thinking skills of prospective
teachers and the expectations of the Ghanaian mathematics curriculum, which anticipates higher-order thinking skills. The study
concludes that the current undergraduate mathematics education programs in Ghanaian universities may not be adequately
preparing future teachers to teach senior high school geometry effectively. It is recommended that these programs be revised to
include more focus on developing higher-order geometric thinking skills, with an emphasis on deductive reasoning, formal
proof-based learning and rigor in geometry thinking. Enhancing the curriculum and teaching methods could narrow this gap and
improve the overall quality of geometry education in Ghana.
Keywords
Geometric Thinking, van Hiele Levels, Prospective Mathematics Teachers, Undergraduate Education, Ghana
1. Introduction
Recent curriculum reform agendas consistently maintain
that learners’ geometric thinking is an essential requirement
which deserves significant attention in the teaching of
mathematics [19, 22]. Geometric thinking, which is more than
the ability to perform geometry tasks, refers to learners’ ap-
proaches to reasoning about shapes and other geometric ideas
[28]. Studies have shown that interest in students’ geometric
thinking abilities is topical and the significance of geometry in
school curriculum remains widely recognised in pedagogical
literature [11, 25, 30]. For example, Alex and Mammen con-
Teacher Education and Curriculum Studies http://www.sciencepg.com/journal/tecs
41
tended that it is the language of geometry conceptualized and
analysed in physical and spatial environments that helps
learners to develop the skills of critical thinking, deductive
reasoning and problem-solving [3].
Geometry plays a crucial role in modeling our surroundings.
For instance, the shapes of roofs often include triangles, tra-
peziums, and squares, while the design of dresses incorporates
symmetry. Additionally, tiling patterns on pavements, walls,
and floors commonly feature pentagons, triangles, and rec-
tangles. In school mathematics curriculum, shapes and space
are taught to foster the learning of higher mathematics such as
mechanics, vectors and mensuration [21]. Apart from the
significant role geometry plays in school mathematics cur-
riculum and the rich connection it has with other areas in
mathematics, geometry also plays a key role in advancing
engineering, computer technologies, physics, chemistry, ge-
ology, architecture and mathematics education [12, 17, 30].
Geometrical ideas are required in real life activities such as
building a house, designing an electronic circuit board, an
airport, a bookshelf, or even a newspaper page [29]. Given the
pervasive role geometry plays in stimulating students’
mathematics learning and highly skilled individuals, most
countries are concerned about how teachers teach and how
students learn various aspects of geometry [4, 6, 20].
In Ghana, geometry is taught at all levels of education with
its different proportions of knowledge ascribed in the math-
ematics curriculum from basic, secondary, through to college
of education and university level. Geometry forms a substan-
tial amount of the senior high school core mathematics cur-
riculum as it is treated in two of the seven content domains of
core mathematics curriculum as Plane Geometry and Men-
suration occupying approximately 29% of the core mathe-
matics teaching syllabus. Plane geometry covers angles of
polygons, Pythagoras’ theorem and its application and circle
theorems including tangents. Mensuration on the other hand
covers perimeters and areas of plane shapes, surface areas,
volumes of solid shapes and the earth as a sphere [18]. The
rationale for treating Plane geometry and Mensuration is to
assist students develop the skills of visualization, critical
thinking, deductive reasoning, logical argument and proof and
to give them the capacity to “organise and use spatial rela-
tionships in two or three dimensions, particularly in solving
problems [18]”. Though these skills present their own in-
structional challenges, they have life-long values beyond the
geometry classroom.
In the context of geometry instruction, van Hiele [28] posits
that the quality of teaching is one of the most significant
factors influencing students’ acquisition of geometric
knowledge in mathematics classes. He asserts that students’
progression from one geometric understanding level to the
next is more dependent on instructional quality than on other
factors like biological maturation or age. Additionally, vari-
ous factors such as teachers’ knowledge, gender, task diffi-
culty, learning environment, and curriculum also play crucial
roles in student achievement and motivation within the
mathematics classroom [6, 13]. Despite this, the quality of
geometry instruction stands out as a particularly influential
element.
Teachers’ mathematical and pedagogical content
knowledge are pivotal in positively impacting students’ mo-
tivation and learning in geometry. [1] further emphasize that
teachers’ content knowledge is crucial for students’ perfor-
mance. They suggest that the inadequate geometry knowledge
among prospective teachers is a significant factor contributing
to students’ poor performance in geometry. This observation
aligns with the arguments made by [5, 13], who both noted
that insufficient content knowledge in geometry among pro-
spective mathematics teachers may lead to subpar perfor-
mance in geometry examinations by students. Thus, while
multiple factors influence student outcomes in geometry, the
overarching theme is that the quality of instruction, under-
pinned by robust content and pedagogical knowledge among
teachers, is paramount in fostering better student performance
and motivation in geometry. Existing literature and prelimi-
nary observations indicate potential gaps in the geometric
understanding of prospective mathematics teachers in Ghana.
This study may help provide empirical evidence to confirm or
refute these observations, thereby informing policy and prac-
tice in undergraduate mathematics teacher education.
1.1. Problem Statement
In Ghana, the education system has undergone various re-
forms aimed at improving the quality of teaching and learning
mathematics in schools. Despite these efforts, concerns re-
main regarding the effectiveness of teacher training programs,
particularly in mathematics education. Several studies have
highlighted deficiencies in the geometrical competencies of
both in-service and pre-service teachers, raising questions
about the adequacy of teacher preparation programs [2, 5].
Geometry, being a fundamental component of the mathe-
matics curriculum, often poses significant challenges to stu-
dents and teachers alike. The abstract nature of geometric
concepts requires a solid foundation and a progressive de-
velopment of geometric thinking. However, anecdotal evi-
dence and preliminary studies suggest that many Ghanaian
mathematics teachers may not possess the necessary geomet-
ric understanding to facilitate effective learning among their
students [2, 5, 6].
Research indicates that students who have not reached van
Hiele Level 4 before enrolling in tertiary-level geometry
courses face a significantly lower likelihood of success [5, 7,
16, 23]. Consequently, it is crucial for students to achieve
level 4 by the end of their Senior High School education. In
this sense, it is anticipated that prospective mathematics
teachers should reach van Hiele level 5 of geometric thinking
before completing their undergraduate mathematics pro-
gramme. It follows logically that for senior high school stu-
dents to attain these necessary levels of geometric under-
standing, their teachers must possess a geometric thinking
Teacher Education and Curriculum Studies http://www.sciencepg.com/journal/tecs
42
level at or above these benchmarks. This competency allows
teachers to provide appropriate scaffolding and learning ex-
periences, essential for guiding students through the com-
plexities of geometric concepts. To effectively support stu-
dents in reaching these levels, it is imperative to assess the van
Hiele levels of prospective mathematics teachers. This ap-
proach is grounded in the van Hiele theory, which has sig-
nificantly contributed to the resurgence of interest in geome-
try education. Moreover, by ensuring that prospective
mathematics teachers attain a high level of geometric thinking,
we can better prepare them to facilitate student learning and
improve outcomes in geometry. This approach underscores
the importance of aligning teacher preparation with the cog-
nitive demands of the subject, ultimately fostering a more
robust understanding of geometry among students.
1.2. Purpose of Study and Research Question
The foundation of mathematical education, particularly in
geometry, is critical for the development of logical reasoning
and spatial understanding, skills essential for success in nu-
merous fields. Geometry is not only a cornerstone of mathe-
matics but also a subject that fosters critical thinking and
problem-solving abilities. In the context of Ghana, where
education is seen as a pivotal driver for socio-economic de-
velopment, ensuring that prospective mathematics teachers
possess a strong geometric foundation is of paramount im-
portance. This study aims to examine whether the current
undergraduate mathematics education in Ghanaian public
teacher training universities effectively prepares future
mathematics teachers in terms of their geometric thinking
capabilities. In pursuance of this purpose, the following re-
search question was formulated to guide the study: Which
stages of van Hiele Levels of geometric thinking do Prospec-
tive Ghanaian Mathematics teachers reach in their study of
geometry just before leaving the university undergraduate
level?
1.3. Theoretical Framework
The van Hiele framework, developed by Dutch educators
Dina van Hiele-Geldof and Pierre van Hiele, outlines five
levels of geometric thinking: Visualization, Analysis, Ab-
straction, Deduction, and Rigor. This model serves as a robust
framework for understanding how students learn geometry
and progress through different stages of geometric thought.
Each level represents a qualitative shift in thinking, empha-
sizing the need for education systems to facilitate transitions
between these stages effectively. For prospective teachers,
reaching higher levels of the van Hiele model is crucial as it
equips them with the necessary depth of understanding to
teach geometry effectively. The van Hiele Levels are de-
scribed as follows:
Level 1: Visualization
At this initial level, students recognize shapes and objects
based on their appearance, not their properties. They can
identify and name figures but do not understand the rela-
tionships between them. For example, a student might rec-
ognize a square because it looks like a “box” without con-
sidering its defining properties such as equal sides and right
angles [14, 26].
Level 2: Analysis
At the analysis level, students begin to identify properties
and characteristics of shapes. They can recognize that a square
has equal sides and four right angles. However, their under-
standing is still largely descriptive, and they do not yet grasp
the relationships between different properties or figures. For
instance, they might understand that all squares have four
equal sides but may not connect this to the definition of a
rectangle [4].
Level 3: Abstraction
This level marks the beginning of logical reasoning about
geometric properties and relationships. Students can make
informal arguments about the properties of shapes and un-
derstand the relationships between different figures. For ex-
ample, they recognize that all squares are rectangles because
they meet the criteria of having four right angles and opposite
sides equal, but not all rectangles are squares [9].
Level 4: Deduction
At the deduction level, students can understand and con-
struct formal proofs. They can follow and create logical se-
quences of statements to establish geometric truths. This level
involves an understanding of the axiomatic structure of ge-
ometry, where students can work with definitions, theorems,
and postulates systematically. For instance, they can prove
that the base angles of an isosceles triangle are congruent
using deductive reasoning [4].
Level 5: Rigor
The highest level, rigor, involves a deep understanding of
the formal aspects of geometric systems. Students can com-
pare different axiomatic systems and understand the role of
undefined terms, definitions, and theorems within these sys-
tems. At this stage, they can work abstractly with various
geometric concepts and appreciate the nuances of different
geometric frameworks. This level is typically achieved in
advanced mathematics courses at the university level [24].
The van Hiele model has significant implications for
teaching geometry. One of its key insights is that instruction
should be tailored to the student’s current level of under-
standing to facilitate progression to higher levels. This sug-
gests a developmental approach to teaching geometry, where
educators provide experiences and tasks appropriate to each
level [10].
Critique and Extensions of the van Hiele Framework
While the van Hiele model has been widely praised for its
insights into geometry learning, it has also faced some cri-
tiques. One critique is that the model may oversimplify the
complexity of geometric thinking by categorizing it into dis-
crete levels. Some researchers argue that students’ under-
standing of geometry is more fluid and context-dependent
Teacher Education and Curriculum Studies http://www.sciencepg.com/journal/tecs
43
than the model suggests [15]. Additionally, recent research
has explored how the van Hiele levels apply to other areas of
mathematics beyond geometry. Extensions of the model have
been proposed to understand how students develop algebraic
thinking and other mathematical concepts [8].
2. Method
2.1. Research Design
The study employed a descriptive survey design. This de-
sign was chosen to gain a comprehensive and accurate un-
derstanding of the levels of geometric thinking attained by the
prospective Ghanaian mathematics teachers as they approach
the completion of their undergraduate education. The de-
scriptive survey method is particularly well-suited for this
type of research as it allows for the systematic collection and
analysis of data from a large sample, providing valuable in-
sights into the current state of geometric understanding among
the target population. Descriptive surveys are particularly
effective in obtaining a precise depiction of existing condi-
tions, behaviors, or phenomena without manipulating the
study environment. This aligns perfectly with the goal of
understanding the van Hiele levels of geometric thinking in a
natural educational setting.
2.2. Participants and Setting
Final year students from the 2022/2023 academic year in
the Department of Mathematics Education at three distinct
public teacher training universities namely, University of
Education, Winneba (UEW), University of Cape Coast (UCC),
and Akenten Appiah-Menka University of Skills Training and
Entrepreneurial Development (AAMUSTED) were purpose-
fully chosen for this study. These universities were deemed
ideal for the research as they are officially mandated to pro-
vide teacher education across various subject areas, including
mathematics. Furthermore, prospective mathematics teachers
in these institutions are recruited from all 16 administrative
regions of Ghana, enriching the sample with diverse abilities,
cultural, and social backgrounds. Consequently, the sample
effectively represents Ghanaian pre-service teachers nation-
wide who have completed at least two academic semesters
studying undergraduate geometry. To select the participants,
stratified random sampling was employed, resulting in a
sample of 1,255 final-year students from the three universities.
The participants included 81.27% males, 15.94% females, and
2.79% who did not disclose their gender identity. The major-
ity of participants (65%) were in the early adulthood age range
of 23 to 30 years. About 9% were under 23 years old, and 26%
were over 30 years old. Additionally, approximately 53% of
the participants entered university with diploma certificates in
teaching, while close to 47% were admitted with senior high
school certificates. This diversity indicates that the partici-
pants have varied background characteristics, with some
having prior teaching experience at the basic school level in
Ghana.
2.3. Overview of the Undergraduate Geometry
Course
The undergraduate geometry content course at these
teacher training universities, taught in the first and second
semester of the first year, encompasses topics that require
students to engage in advanced levels of geometric thinking.
This approach is designed to help students construct
knowledge and understanding through a structured process of
exploration, analysis, and both inductive and deductive rea-
soning [13, 27]. The course includes topics that enable stu-
dents to graph algebraic equations in two variables as lines
and circles, calculate angles and distances between lines and
circles, and define, prove, and construct Euclidean (plane)
geometry, as well as perform measurements in geometrical
shapes and solids. Other specific topics covered in the un-
dergraduate geometry course include conic sections, the
equation of a circle, the equation of a parabola, the equation of
an ellipse, the equation of a hyperbola, asymptotes to a hy-
perbola, polar coordinates, relations between polar and carte-
sian coordinates, area of a sector, length of a curve, arc length,
parametric equations, and polar equations.
These topics are not only fundamental to the mastery of
geometry but also pivotal for fostering a deep understanding
of mathematical principles that prospective teachers will
eventually impart to their students. By engaging with these
complex topics early in their academic journey, prospective
mathematics teachers are better prepared to develop the ana-
lytical and reasoning skills necessary for teaching senior high
school geometry effectively. This rigorous grounding in
geometric concepts ensures that future teachers can approach
the subject with confidence and inspire the same level of
understanding and appreciation in their future classrooms.
2.4. Instrument
The van Hiele Geometry test (VHGT) was adopted and
used to collect data in order to address the research question in
this study. The VHGT, adapted from [26]’s Cognitive De-
velopment and Achievement in Secondary School Geometry
(CDASSG) project, is a well-crafted 25-item multiple-choice
test designed to assess varying levels of geometric under-
standing. Each set of five items targets a specific level of
cognitive development in geometry. Here is how the VHGT is
organized:
1. Items 1-5 (Subtest 1): These items focus on the identi-
fication, naming, and comparison of geometric shapes
such as squares, rectangles, and rhombi. They measure
students’ understanding at Level 1, where basic recog-
nition and description of shapes are assessed.
2. Items 6-10 (Subtest 2): These items deal with recogniz-
ing and naming the properties of geometric figures. They
Teacher Education and Curriculum Studies http://www.sciencepg.com/journal/tecs
44
evaluate students’ understanding at Level 2, where stu-
dents identify specific characteristics and properties of
the shapes they recognize.
3. Items 11-15 (Subtest 3): This section addresses the log-
ical order of properties and the relationships between
these properties of previously identified figures. It
measures students’ understanding at Level 3, where the
focus is on comprehending the connections and hierar-
chies among geometric properties.
4. Items 16-20 (Subtest 4): These items require students to
understand the significance of deduction and the roles of
postulates, axioms, theorems, and proofs. They assess
students’ understanding at Level 4, where formal logical
reasoning and the construction of geometric arguments
are key.
5. Items 21-25 (Subtest 5): This final block deals with the
formal aspects of deduction, measuring students’ un-
derstanding at Level 5. At this level, students engage
with advanced deductive reasoning, comparing and
contrasting different geometric systems and their un-
derlying axioms and theorems.
The VHGT’s design ensures a comprehensive assessment
of students’ geometric thinking across different levels,
providing insights into their cognitive development in geom-
etry. By organizing the test sequentially according to the van
Hiele levels, educators can diagnose specific areas where
students may need further instruction or support. This struc-
tured approach not only helps in identifying students’ current
levels of understanding but also guides the development of
targeted interventions to enhance their geometric reasoning
skills.
The VHGT has been widely recognized and utilized in
numerous studies, consistently demonstrating strong validity
and reliability [11, 26]. This robust track record justifies its
adoption in this study. By employing the VHGT, the re-
searcher ensured that the assessment tool is both credible and
capable of accurately measuring the levels of geometric
thinking among prospective mathematics teachers. This
choice not only enhances the reliability of the findings but
also aligns the study with established methods in the field of
mathematics education research.
2.5. Analysis of Data
The data collected from participants were meticulously
coded and entered into SPSS for comprehensive processing
and analysis. Descriptive statistical methods, including
measures of central tendency, frequency counts, and per-
centages, were employed to analyze the data. The results were
presented using tables and bar charts, providing a clear visual
representation. These descriptive statistics were instrumental
in understanding, interpreting, and describing the participants’
experiences and their levels of geometric conceptualization.
3. Results
3.1. Prospective Mathematics Teachers’
Performance in the VHGT
The objective of the study was to find out the van Hiele
levels (VHLs) of geometric thinking reached by prospective
mathematics teachers just before completing their university
undergraduate programme. The geometric thinking skills of
prospective mathematics teachers were defined according to
the van Hiele theory, encompassing their abilities to visualize,
analyze, abstract, deduce, and ability to display rigor in ge-
ometry thinking.
3.1.1. Visualization Skills: Performance on Subtest 1
(van Hiele Level 1)
Visualization skills, corresponding to VHL 1, imply stu-
dents’ ability to recognize and identify geometric shapes
based on their overall appearance. In other words, it implies
students’ ability to name and categorize shapes such as
squares, triangles, rectangles and parallelograms based on
visual characteristics. Students’ understanding at this stage is
primarily visual and intuitive, and students do not yet com-
prehend the formal properties or relationships that connect
different geometric figures. Table 1 presents the distribution
of participants’ correct responses to the five VHGT items
designed to assess their visualization skills.
Table 1. Distribution of Participants’ Visualization Skills by Frequency count and Percentage (VHL 1).
Level
Item Number
Geometry aspect examined
Correct response (n) (%)
VHL1
Visualization
1
Recognition of squares
1205
96.02
2
Recognition of triangles
1198
95.46
3
Recognition of rectangles
1150
91.63
4
Characterizing the Orientation of Squares
990
78.88
5
Identifying Orientation and Class Inclusion of Parallelograms
886
70.60
Teacher Education and Curriculum Studies http://www.sciencepg.com/journal/tecs
45
Table 1 depicts that as prospective mathematics teachers
approached the completion of their undergraduate education,
more than 90% demonstrated proficient visual recognition
and differentiation of squares, triangles, and rectangles.
However, fewer than 80% of participants accurately identified
the orientation of various squares and parallelograms based on
class inclusivity principles. This indicates that while a sig-
nificant majority of participants developed strong visualiza-
tion skills during their four-year program, few challenges
persisted in recognizing specific geometric orientations using
inclusive properties.
3.1.2. Analysis Skills: Performance on Subtest 2
(van Hiele Level 2)
Analysis skills in geometry refer to the ability to examine
and identify the properties and relationships of geometric
figures systematically and logically. This includes under-
standing how these relationships contribute to a deeper un-
derstanding of geometric concepts. Table 2 illustrates the
percentage distribution of participants who effectively applied
their analytical skills to achieve correct responses on the
VHGT test.
Table 2. Distribution of Participants’ Analysis Skills by Frequency count and Percentage (VHL 2).
Level
Item Number
Geometry aspect examined
Correct response (n) (%)
VHL2
Analysis
6
Relational properties of a square
1112
88.61
7
Diagonal property of rectangle
1014
80.80
8
Properties of rhombus
805
64.14
9
Properties of isosceles triangle
1220
97.21
10
Properties of kite
890
70.92
The table reveals that a significant majority of participants,
over 80%, demonstrated accurate analysis in understanding
the relationship properties of squares and the diagonal prop-
erties of rectangles. Specifically, 64.14% and 70.92% effec-
tively analyzed the fundamental properties of rhombuses and
kites, respectively. Notably, more than 97% successfully
identified the basic properties of isosceles triangles. These
findings underscore the participants’ strong ability to analyze
geometric properties and establish meaningful relationships
among different shapes.
3.1.3. Abstraction Skills: Performance on Subtest 3
(van Hiele Level 3)
Abstraction and ordering skills encompass the capacity to
arrange shapes logically, create abstract definitions, and dis-
cern essential from incidental properties within geometric
contexts. Table 3 illustrates the distribution of participants
who demonstrated proficient abstraction skills in the VHGT
assessment.
Table 3. Distribution of Participants’ Abstraction Skills by Frequency count and Percentage (VHL 3).
Level
Item Number
Geometry aspect examined
Correct response (n) (%)
VHL3
Abstraction
11
Logical reasoning using verbal cues: rectangles and triangles
695
55.38
12
Analytical reasoning based on triangle properties
798
63.59
13
Abstraction through rectangular orientation
769
61.27
14
Constructing logical arguments using inclusive properties
985
78.49
15
Establishing logical connections among parallelograms
755
60.16
From Table 3, the percentages of participants (55.38%,
63.59%, 61.27%, 78.49%, and 60.16%) who effectively ap- plied verbal logical reasoning and logical analysis to shape and
space were somewhat satisfactory. However, despite nearly
Teacher Education and Curriculum Studies http://www.sciencepg.com/journal/tecs
46
four years of studying university undergraduate mathematics,
their demonstration of verbal and logical reasoning was not as
robust as expected. Moreover, the proportion of participants
who accurately abstracted concepts was also modest.
3.1.4. Deduction Skills: Performance on Subtest 4
(van Hiele Level 4)
Deduction skills refer to the ability to derive conclusions
logically based on established principles, postulates, axi-
oms and theorems within geometry. This involves applying
deductive reasoning to make valid assertions about geo-
metric shapes, properties, and relationships. Table 4 pre-
sents the distribution of the percentage of participants who
applied deductive reasoning effectively to answer the
VHGT test items.
Table 4. Distribution of Participants’ Deduction Skills by Frequency count and Percentage (VHL 4).
Level
Item Number
Geometry aspect examined
Correct response (n) (%)
VHL4
Deduction
16
Deriving figural structures through deductive reasoning
779
62.07
17
Inferencing figural characteristics through deduction
619
49.32
18
Proof
859
68.45
19
Generalization
622
49.56
20
Deduction
602
47.97
Figure 1. Sample Item in Subtest 4.
Table 4 reveals that 62.07% and 68.45% of participants
effectively utilized deductive reasoning to derive geometric
structures and conduct geometric proofs. However, fewer than
50% (49.32%, 49.56%, and 47.97%, respectively) correctly
responded to items 17, 19, and 20, which required making
deductions, generalizing from observations, and applying
deductive reasoning. Figure 1 is an item from Subtest 4. The
correct answer for this item is choice A. Table 4 shows that
only 602 (47.97%) of prospective mathematics teachers had
this question correct. These results indicate a notable portion
of participants struggled with tasks involving higher-level
deductive reasoning.
3.1.5. Rigor in Geometry: Performance on Subtest 5
(van Hiele Level 5)
Rigor in geometry thinking involves the capacity to com-
pare axiomatic systems through formal theoretical approaches,
independent of concrete models. It refers to the ability to
engage in thorough and precise reasoning, applying formal
mathematical methods such as proofs and logical arguments
to analyze geometric concepts and properties systematically.
The analysis of participants’ responses to items 21 to 25
aimed to ascertain the extent to which they demonstrated
rigorous geometric thinking in drawing conclusions about
theories, axioms, or implicative statements. The findings are
detailed in Table 5.
Table 5. Distribution of Participants’ Rigor in geometry thinking in shapes by Frequency count and Percentage (VHL 5).
Level
Item Number
Geometry aspect examined
Correct response (n) (%)
VHL5
Rigor
21
Applying deduction with rigor
563
44.86
22
Conclusive deduction with rigor
456
36.33
23
Applying deduction with rigor
496
39.52
Teacher Education and Curriculum Studies http://www.sciencepg.com/journal/tecs
47
Level
Item Number
Geometry aspect examined
Correct response (n) (%)
24
Applying deduction with rigor
416
33.15
25
Implications in Geometry
403
32.11
Figure 2. Sample Item in Subtest 5.
Table 5 reveals that close to the end of their 4-year under-
graduate mathematics education programme, participants’
correct responses to items 21 to 25 were notably low, with
each item yielding less than 50% correct response. Sample
item from Subtest 5 is shown in Figure 2. The correct answer
for this item is choice D. However, Table 5 shows that only
403 (32.11%) of prospective mathematics teachers had this
question correct. This indicates significant challenges among
participants in applying rigorous methods for geometric con-
structions and generalizing implicative statements.
3.2. Overall Scores of Prospective Mathematics
Teachers in the VHGT
Table 6 illustrates the overall performance scores of pro-
spective mathematics teachers on the VHGT. Scores ranged
from a minimum of 32% to a maximum of 80%. The average
score among these prospective teachers was 52.32%, with a
standard deviation of 7.21%. This data indicates a moderate to
high proficiency level, with the average score slightly above
the midpoint of the scoring range. The standard deviation of
7.21 suggests some variability in performance, implying that
while many scores were close to the mean, there was a notable
spread in the participants’ abilities.
Table 6. Means, Standard Deviations, Minimum and Maximum VHGT Scores of Prospective Mathematics Teachers.
N
Mean
Standard Deviation
Maximum
Minimum
1255
52.32
7.21
80
32
Figure 3. Prospective Mathematics Teachers’ van Hiele Levels of Geometric Thinking.
Teacher Education and Curriculum Studies http://www.sciencepg.com/journal/tecs
48
3.3. Levels Reached Prospective Mathematics
Teachers in the VHGT
This study aimed to determine the van Hiele Levels of
geometric thinking achieved by Ghanaian prospective
mathematics teachers nearing the completion of their 4-year
undergraduate program, preparing them to teach at the senior
high school level. The bar chart in Figure 3 visually repre-
sents the geometric thinking levels attained by these future
educators. This graphical depiction offers a clear and im-
mediate understanding of their proficiency, highlighting the
distribution across different van Hiele levels. Analyzing
these results can provide valuable insights into the effec-
tiveness of the current educational program and identify
areas needing improvement to better prepare teachers for
their future roles.
Figure 3 shows that 8.8% of participants reached VHL 1,
demonstrating strong visualization skills, while 30.0%
achieved VHL 2, operating at the analysis level. Additionally,
32.4% of participants attained VHL 3, functioning at the
abstraction level. However, only 15.9% and 12.9% reached
VHL 4 and VHL 5, operating at the deduction and rigor levels,
respectively.
4. Discussion
The study’s purpose was to examine whether the current
undergraduate mathematics education in Ghanaian public
teacher training universities effectively prepares future
mathematics teachers in terms of their geometric thinking
capabilities. This was done by investigating the van Hiele
levels of geometric thinking of prospective mathematics
teachers involved in this study. These prospective mathe-
matics teachers were in the final (fourth) year of their uni-
versity undergraduate programme preparing to graduate to
teach at senior high schools in Ghana. Structured around van
Hiele’s geometric thinking levels, this study examined par-
ticipants abilities across visualization, analysis, abstraction,
deduction, and rigor stages. The results highlight the varying
degrees of geometric reasoning skills among prospective
mathematics teachers, emphasizing the need for targeted
interventions to enhance higher-level thinking skills.
The VHGT results indicated that 8.8% of prospective
mathematics teachers reached van Hiele Level 1, 30.0% at-
tained Level 2, and 32.4% achieved Level 3. Furthermore,
15.9% and 12.9% of the participants reached Levels 4 and 5,
respectively. This indicates that the majority of prospective
mathematics teachers are operating within the first three
stages of geometric reasoning as defined by the van Hiele
model, demonstrating foundational visualization, analytical,
and abstract thinking skills. However, a smaller proportion of
participants exhibit the advanced deductive and rigorous
reasoning skills necessary for higher-level geometric thinking.
In particular, the largest group of participants (32.40%)
reached the abstraction level where they understand rela-
tionships between properties of shapes and can logically de-
duce theorems based on these properties. Overall, only 28.8%
of participants reached the highest levels (i.e. level 4 and 5). In
other words, only this small proportion of participants can
understand and form formal proofs and comprehend the
structure of axiomatic systems. Also, these were the few who
understood working within different systems and under-
standing the implications of altering axioms. These findings
suggest the need for enhanced focus on developing high-
er-order reasoning capabilities within the mathematics edu-
cation curriculum.
The findings highlighted above align with earlier research
conducted in Ghana [2, 7, 4] and underscore the ongoing
concern about the methods of teaching and learning geometry
in Ghanaian schools. The studies referenced indicate that both
prospective mathematics teachers and senior high school
students are performing at lower levels of geometric thinking
than anticipated by the national mathematics curriculum. This
discrepancy suggests potential gaps in instructional strategies
and educational resources, highlighting the need for a com-
prehensive review of the geometry curriculum and teaching
practices to ensure they effectively promote higher-order
thinking skills including rigor and deductive reasoning skills.
Addressing these gaps is crucial for aligning educational
outcomes with curriculum standards and enhancing overall
mathematical proficiency.
It is essential to emphasize that deductive reasoning un-
derpins the comprehension of definitions, properties, axioms,
postulates, and other geometric elements used in geometric
proofs. Consequently, the absence of this critical reasoning
skill indicates that prospective mathematics teachers will
likely face challenges in explaining geometric concepts and
applying their knowledge to related fields such as algebra,
trigonometry, vectors and mechanics which are key compo-
nents of both core and elective mathematics at the senior high
school level. This gap in deductive reasoning not only hinders
their understanding of geometry but also impacts their overall
mathematical proficiency, making it imperative to enhance
instructional approaches that foster strong deductive reason-
ing skills in students.
According to educational standards, senior high school
students should reach van Hiele Level 4 in geometric thinking
by the end of their secondary education [23]. Consequently, it
is expected that prospective mathematics teachers achieve
Levels 4 and 5 to effectively and confidently teach high
school geometry upon completing their undergraduate studies.
However, this study revealed that only 15.9% of these future
teachers attained Level 4, and just 12.9% reached Level 5.
This significant shortfall raises critical concerns about the
effectiveness of the current undergraduate mathematics edu-
cation programs in Ghanaian universities, especially in ge-
ometry.
Teacher Education and Curriculum Studies http://www.sciencepg.com/journal/tecs
49
5. Conclusion and Recommendations
The study concludes that majority of prospective
mathematics teachers preparing to graduate to teach at the
senior high school level in Ghana operate at the initial three
levels of geometric thinking and only a small proportion of
them operate at the higher levels. These results highlight a
persistent gap between the current competencies of pro-
spective mathematics teachers and the expectations of the
Ghanaian mathematics curriculum. The deficiency in
higher-order geometric thinking skills raises significant
concerns about the adequacy of the current undergraduate
mathematics education programs in Ghanaian universities.
This gap suggests that many future teachers may struggle to
effectively teach senior high school geometry, which re-
quires a solid understanding of deductive reasoning and the
ability to work with complex geometric concepts. Based on
the findings of this study, the following recommendations
are made:
1. Teacher training universities in Ghana should update
their undergraduate geometry education curriculum to
place greater emphasis on developing higher-order
geometric thinking skills. They should include more
content on formal proofs, logical reasoning, and the
structure of mathematical systems.
2. Teaching strategies that focus on problem-solving and
proof-based learning should be implemented in teacher
training universities in Ghana, encouraging prospective
mathematics teachers to engage in activities that require
logical reasoning and the formulation of formal geo-
metric proofs.
3. Teacher training universities in Ghana should provide
ongoing professional development opportunities for
prospective mathematics teachers to strengthen their
geometric reasoning skills. Workshops, seminars, and
advanced courses can help future teachers stay updated
on best practices and new developments in geometry
education.
4. Regular assessments should be conducted by lecturers in
teacher training universities in Ghana to monitor the
geometric thinking levels of prospective mathematics
teachers. These assessments should be used to identify
areas needing improvement and to provide targeted
support.
6. Limitations
The study admits three main limitations that should be
considered when interpreting the results. First, the use of the
van Hiele Geometry Test (VHGT) as the sole assessment tool
may have inherent biases or limitations in accurately captur-
ing the full range of geometric thinking skills. Incorporating
multiple assessment methods, including interviews and ob-
servational studies, could provide a more comprehensive
evaluation. Second, the study’s cross-sectional design pro-
vides a snapshot of geometric thinking levels at a specific
point in time, without accounting for the potential develop-
ment and progression of these skills over the course of the
undergraduate program. Longitudinal studies tracking the
same cohort over time could offer deeper insights into the
development of geometric reasoning. Third, the study did not
account for variations in educational contexts, such as dif-
ferences in teaching quality, curriculum implementation, and
resources available at different institutions. These factors
could significantly influence the development of geometric
thinking skills.
Abbreviations
AAMUSTED
Akenten Appiah-Menka University of
Skills Training and Entrepreneurial
Development
CDASSG
Cognitive Development and Achievement
in Secondary School Geometry
UCC
University of Cape Coast
UEW
University of Education, Winneba
VHGT
Van Hiele Geometry Test
VHL
Van Hiele Level
Acknowledgments
I extend my heartfelt gratitude to the Deans of Faculty,
Heads, and prospective mathematics teachers of the Depart-
ment of Mathematics Education at the three main universities
involved in this study: University of Education, Winneba
(UEW), University of Cape Coast (UCC), and Akenten Ap-
piah-Menka University of Skills Training and Entrepreneurial
Development (AAMUSTED). Their invaluable support and
cooperation during the data collection process were instru-
mental to the success of this research.
Author Contributions
Robert Benjamin Armah is the sole author. The author
read and approved the final manuscript.
Conflicts of Interest
The author declares no conflicts of interest.
References
[1] Akayuure, P., Asiedu-Addo, S. K., & Alebna, V. (2016). In-
vestigating the effect of origami instruction on pre-service
teachers’ spatial ability and geometric knowledge for teaching.
International Journal of Education in Mathematics, Science
and Technology, 4(3), 198-209.
https://doi.org/10.18404/ijemst.78424
Teacher Education and Curriculum Studies http://www.sciencepg.com/journal/tecs
50
[2] Akayuure, P., Oppong, R. A., Addo, D. A., & Yeboah, D. O.
(2022). Geometric Thinking Behaviours of Undergraduates
on-Entry and at-Exit of Online Geometry Course. Science
Journal of Education, 10(5), 155-163.
http://doi.org/10.11648/j.sjedu.20221005.12
[3] Alex, J. K. & Mammen, K. J. (2012). A Survey of South Af-
rican Grade 10 Learners’ Geometric Thinking Levels in Terms
of the van Hiele Theory. Anthropologist, 14(2), 123-129.
[4] Alex, J. K., & Mammen, K. J. (2018). Students’ understanding
of geometry terminology through the lens of Van Hiele theory.
Pythagoras, 39(1), 376-384.
https://doi.org/10.4102/pythagoras.v39i1.376
[5] Armah, R. B., Cofie, P. O., & Okpoti, C. A. (2018). Investi-
gating the effect of van Hiele Phase- based instruction on
pre-service teachers’ geometric thinking. International Journal
of Research in Education and Science, 4(1), 314-330.
https://doi.org/10.21890/ijres.383201
[6] Armah, R. B. & Kissi, P. S. (2019). Use of the van Hiele
Theory in Investigating Teaching Strategies used by College of
Education Geometry Tutors. EURASIA Journal of Mathemat-
ics, Science and Technology Education, 15(4), em1694.
https://doi.org/10.29333/ejmste/103562
[7] Asemani, E., Asiedu-Addo, S. K., & Oppong, R. A. (2017).
The Geometric Thinking Levels of Senior High School stu-
dents in Ghana. International Journal of Mathematics and
Statistics Studies, 5(3), 1-8.
[8] Battista, M. T. (2007). The development of geometric and
spatial thinking. In F. K. Lester (Ed.), Second handbook of
research on mathematics teaching and learning (pp. 843-908).
National Council of Teachers of Mathematics.
[9] Breyfogle, M. L. & Lynch, C. M. (2010). Van Hiele Revisited.
Mathematics Teaching in the Middle School, 16(4), 232-238.
[10] Burger, W., & Shaughnessy, J. M. (1986). Characterizing the
van Hiele levels of development in geometry. Journal for Re-
search in Mathematics Education, 17(1), 31–48.
[11] Erdogan, F. (2020). Prospective Middle School Mathematics
Teachers’ Problem Posing Abilities in Context of Van Hiele
Levels of Geometric Thinking. International Online Journal of
Educational Sciences, 12(2), 132-152.
https://doi.org/10.15345/iojes.2020.02.009
[12] Gunhan, B. C. (2014). A Case Study on the investigation of
reasoning skills in Geometry. South African Journal of Edu-
cation, 34(2), 1-19.
[13] Halat, E. (2008). In-Service Middle and High School Mathe-
matics Teachers: Geometric Reasoning Stages and Gender.
The Mathematics Educator, 18(1), 8–14.
[14] Howse, T. D. & Howse, M. E. (2015). Linking the Van Hiele
Theory to Instruction. Teaching children mathematics, 21(5),
305-313.
[15] Jones, K. (2002). Issues in the Teaching and Learning of Geome-
try. In: Linda Haggarty (Ed), Aspects of Teaching Secondary
Mathematics: perspectives on practice. Routledge Falmer.
[16] Knight, K. C. (2006). An investigation into the van Hiele level
of understanding geometry of pre-service elementary and
secondary mathematics teachers. [master’s thesis, University
of Maine].
https://digitalcommons.library.umaine.edu/etd/1361/
[17] Luneta, K. (2015). Understanding students’ misconceptions:
An analysis of final Grade 12 examination questions in geom-
etry. Pythagoras, 36(1), 1-11.
[18] Ministry of Education (MOE) (2010). Teaching Syllabus for
Core Mathematics (Senior High School 1-3). Ministry of Ed-
ucation.
[19] Ministry of Education (MOE) (2012). Teaching Syllabus for
Core Mathematics (Senior High School 1-3). Ministry of Ed-
ucation.
[20] Moru, E., Malebanye, M., Morobe, N., & George, M. (2021).
A Van Hiele Theory analysis for teaching volume of
three-dimensional geometric shapes. Journal of Research and
Advances in Mathematics Education, 6(1), 17-31.
https://doi.org/10.23917/jramathedu.v6i1.11744
[21] Mukuka, A. & Alex, J. K. (2024). Student teachers’ knowledge
of school-level geometry: Implications for teaching and
learning. European Journal of Educational Research, 13(3),
1375-1389. https://doi.org/10.12973/eu-jer.13.3.1375
[22] National Council for Curriculum and Assessment (NaCCA)
(2019). Mathematics Curriculum for Primary Schools (Basic 4
- 6). Ministry of Education.
[23] Schwartz, J. E. (2008). Elementary Mathematics Pedagogical
Content Knowledge: Powerful Ideas for Teachers.
http://www.education.com/reference/article/why-people-have-
difficulty-geometry/
[24] Senk, S. L. (1989). Van Hiele levels and achievement in writ-
ing geometry proofs. Journal for Research in Mathematics
Education, 20(3), 309–321.
[25] Trimurtini, D. Waluya, S. B., Sukestiyarno, Y. L., & Khari-
sudin, Q. (2023). Effect of Two-Dimensional Geometry
Learning on Geometric Thinking of Undergraduate Students
During the COVID-19 Pandemic. Journal of Higher Education
Theory and Practice, 23(3), 177-187.
https://doi.org/10.17051/ilkonline.2021.01.91
[26] Usiskin, Z. (1982). Van Hiele Levels and achievement in sec-
ondary school geometry: Cognitive development and
achievement in secondary school geometry project. University
of Chicago Press.
[27] van Hiele, P. M. (1986). Structure and insight: A theory of
mathematics education. Orlando: Academic Press.
[28] van Hiele, P. M. (1999). Developing Geometric Thinking
through Activities that Begin with Play. Teaching Children
Mathematics, 6, 310–316.
[29] Yegambaram, P. & Naidoo, R. (2009). Better learning of
geometry with computer.
http://atcm.mathandtech.org/EP2009/papers_full/2812009_17
080.pdf
Teacher Education and Curriculum Studies http://www.sciencepg.com/journal/tecs
51
[30] Yi, M., Flores, R. & Wang, J. (2020). Examining the influence
of van Hiele theory-based instructional activities on elemen-
tary preservice teachers’ geometry knowledge for teaching 2-D
shapes. Teaching and Teacher Education, 91, 1-12.
https://doi.org/10.1016/j.tate.2020.103038
