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Journal of Mathematics Education p-ISSN 2089-6867 

Volume 11, No. 1, February 2022 e–ISSN 2460-9285 
 

  https://doi.org/10.22460/infinity.v11i1.p77-86  

  

77 

CLUSTERS OF PREVALENT PATTERNS OF GEOMETRIC 

THINKING LEVELS AMONG MATHEMATICS STUDENTS 
 

Joshua Edson Gorme Ordiz*, Ghanine Rhea Mecate 
Southern Leyte State University, Eastern Visayas, Philippines 

 

 

Article Info  ABSTRACT 

Article history: 

Received Oct 18, 2021 

Revised Dec 28, 2021 

Accepted Jan 10, 2022 

 

 Geometric thinking skills are the perceived abilities of an individual to think 
and reason in geometric contexts. These skills acquired by students in 

geometry remain poor and unsettling because of the misconceptions that 

hinder the students in learning the components of geometry. The study 

described the common unplaceable patterns in geometric thinking of 153 

mathematics education students in a state university in Eastern Visayas, 

Philippines. Frequency Analysis was employed in the study to determine the 

number of occurrences of the patterns stressing the cause for students placed 
under level 0 or unplaceable. Van Hiele Achievement Test was used to gather 

the students’ performance in geometry at all levels, namely: visualization, 

analysis, informal deduction, deduction, and rigor. The findings attested that 

only 13.1% of the students managed the third level of the Van Hiele Levels 
while 43.1% of them were unplaceable. Common patterns were drawn and 

describe to understand the consequences in geometric thinking ability at level 

0. These observable patterns were grouped into core-remedial, topical-

corrective, and close-corrective groups. The clusters will enable educational 
institutions to address the individual gaps in geometry. 

Keywords: 

Geometry, 

Mathematics Performance, 

Proving 

 

This is an open access article under the CC BY-SA license.  

 

Corresponding Author: 

Joshua Edson Gorme Ordiz,  

Faculty, College of Teacher Education, 

Southern Leyte State University 

Concepcion St, Sogod, Southern Leyte, Filipina 

Email: sirjosh.ordiz@gmail.com 

How to Cite: 

Ordiz, J. E. G., & Mecate, G. R. (2022). Clusters of prevalent patterns of geometric thinking levels among 

mathematics students. Infinity, 11(1), 77-86. 

 

1. INTRODUCTION 

Improving the students’ performance in mathematics, specifically in geometry, is a 

challenge by educators because students find it complicated. Despite the efforts made by the 

academic community to address the students’ performance in geometry in the discipline, it 

remained low and degrading. Concurrently,  number of students do not attain any of the 

geometric levels denoting poor achievement in geometry (Mullis et al., 2016). The content 

of the students’ performance calls the understanding of geometric thinking ability to 

determine the common patterns exist in level 0 or unplaceable and describe its’ 

consequences. Learning geometry measures on the Van Hiele Levels of geometric thinking 

https://doi.org/10.22460/infinity.v11i1.p77-86
https://creativecommons.org/licenses/by-sa/4.0/


 Ordiz & Mecate, Clusters of prevalent patterns of geometric thinking levels … 78 

ability of the students which is helpful in analyzing the learners’ performance (Alex & 

Mammen, 2012). 

Result of the study of Atebe (2008) which indicated 41% of the learners is at level 0. 

This study showed the difficulty that the learners have in recognizing figures in nonstandard 

positions. This finding is supported by Alex and Mammen (2012), presented that majority 

of the learners were unplaceable which means none of the students acquired any levels. The 

assignment of learners into levels showed the percentage in level 0 was 48%. This seeks for 

the need to deliver instruction at a level appropriate to learners’ level of thinking on the one 

hand and improving the quality of education starting from lower levels on the other hand. 

Besides, Marchis (2012) stressed out that students have confusion in geometry due 

to idea definition. Proper idea definition produces a self-idea of the image. This concept 

image may not develop in a few understudies, and in others, it may not identify with the 

formal definition. There is the need to address these misinterpretations when educating to 

enable the learners to consider where the misconception between the verbal meaning and 

mental image originates from. Characterizing and distinguishing shapes inclination is given 

to a visual model than a formal description (Özerem, 2012). Students prefer to rely on a 

visual prototype rather than a verbal definition when classifying and identifying shapes. To 

obtain the mathematical knowledge required in everyday living, educating the techniques on 

how to solve problems, and inhibiting reasoning strategies are the objectives of Mathematics. 

Troubles in learning geometry clarify cognitive improvement (Idris, 2009). 

Individual mental capacity is not just about visual discernment, breaking down components, 

knowing the connections between properties, building and appreciating proofs but also 

decision making, which is vital to accomplishing higher-level thinking in learning geometry. 

An individual with better visual perception has an advantage in geometric reasoning (Walker 

et al., 2011). Learners need help to uncover these misconceptions and thus, build on correct 

perceptions. Learners need to develop and build up the proper schema about the previous 

knowledge before taking the new higher lessons in the upper educational level. Teachers 

must provide learning experiences that fit the level of thinking of the students. 

Concrete experiences of the learner in the primary level help to shorten the gap in 

abstract concept with the use of solid objects. In addition to that, visual assisted tools are 

being used to enhance the geometric thinking ability, and it functioned as a mental reference 

(Kamina & Iyer, 2009; Zanzali, 2000). Moreover, giving attention to the application of 

dialect is one of the pedagogical practices that support the development of the mathematical 

knowledge of the students (Schleppegrell, 2007). One of the main contributors to overall 

comprehension in many content areas, including mathematics, is vocabulary understanding.  

These observations were evident in this study to anchor the Van Hiele Theory (van 

Hiele, 1999) after supporting numerous knowledge that emphasizes the mediocre 

achievement in geometry. The Van Hiele Model considers significant imperative models in 

educating geometry and geometric thoughts and ideas. This model has five phases in which 

each level represents the development of the thinking process in geometry. The improvement 

of the geometric thinking of the students will lead to the summarization of the learning which 

is vital in using in a real-life situation (Pegg & Tall, 2010). It is one of the theories that are 

effective in teaching geometry to students through the school stages (Mistretta, 2000). As 

per Van Hiele, the five levels of geometric reasoning are Visualization, Analysis, 

Abstraction, Deduction, and Rigor (Groth, 2005).  

 

 

 

 

 



 Volume 11, No 1, February 2022, pp. 77-86

 

 

79 

 

 

Figure 1. The procedural pattern of learning geometry 

 

The illustration displays the levels of Van Hiele Model: the Visualization, Analysis, 

Informal Deduction, Deduction, and Rigor (see Figure 1). Furthermore, the patterns are 

formulated to identify the levels placed by the students, understand and describe the 

consequences of every unplaceable patterns in geometric thinking ability which is the main 

goal of the study. These consequences are necessary to address the concern that majority of 

the students are not able to reach even the first level of the model. These students are referred 

to be unplaceable into any of the levels of the Van Hiele Model.  

One of the problems of the teachers is that the ability of teachers to present problems 

related to geometry is weak. They cannot transfer their knowledge appropriately about 

geometric thinking levels and claimed that course contents to be designed for practice are 

thought to improve the subject matter knowledge related to geometry (Erdogan, 2020). 

Focusing on the geometric thinking skills of those students who were not able to be classified 

in any of the levels is indispensable in order to help the struggling educators to identify the 

possible interventions that is useful in improving the geometric levels by imparting the 

precise way of presenting knowledge to the learner. 

Levels are hierarchical, it is needed to fully acquire the previous levels in order to 

reach a higher level. Failing to do so leads to not acquiring any of the  Van Hiele Model. 

These levels are associated to geometric experiences (Van de Walle et al., 2014). Exploring 

the reasons why students failed to attain any of the Van Hiele Model. will result to providing 

teachers the possible solutions to the main concern by understanding what happened in the 

levelling of the said ability. Teaching geometry is the focus of the improvement of logical 

thinking and a vital factor of mathematical understanding (van Hiele, 1999). Educators have 

a critical role in teaching and learning geometry. 

 

 

 
 



 Ordiz & Mecate, Clusters of prevalent patterns of geometric thinking levels … 80 

2. METHOD 

This study utilized frequency analysis. It deals with the number of occurrences or 

frequency of the patterns stressing the cause for students placed under unplaceable. It 

describes the data set and provide a fair idea of what patterns the students are acquiring. The 

method used was Complete Enumeration in the conduct of the study.  

The locale of the study was all campuses of Southern Leyte State University. All 

mathematics students enrolled in Academic Year 2018-2019 were considered to be the 

participants of the study towards exploring the  placement of learners’ geometric ability (see 

Table 1). 

Table 1. Distribution of respondents 

Year level Number of Students Percentage 

Freshmen 46 30% 

Sophomore 36 24% 

Junior 33 21% 

Senior 38 25% 

Total 153 100% 

 

Distribution of the questionnaires to all students majoring Mathematics in the master 

list provided by the university followed. The Achievement test for the Van Hiele was 

composed of five questions in each level. In every query, students will choose the best 

answer from the options. Students must reach three points in each level to grasp the level 

need to attain. However, your previous score in the lower level is less than to 3 points, and 

you achieved a score that is <3 points in the next level, students can be classified as 

unplaceable. It is impossible for the students to obtain the higher levels without 

accomplishing the lower levels. Hence, students should have consistent score that is <3 

without failing the levels in between to grasp any of the levels. 

Mean, Frequency, and Weighted Average. This was used to determine the mean 

rating of the sample and the exact number of each pattern exist. It is necessary for analysis 

and interpretation of any data and it indicates how well the data is. 
 

 

3. RESULTS AND DISCUSSION 

3.1. Results 

3.1.1. Van Hiele Geometric Thinking Ability 

Van Hiele Geometric Thinking designates how an individual acquires learning in the 

field of geometry that proposes five levels of geometric thinking. Each level utilizes its 

symbols and language and students can pass through this level “step-by-step.”  

Table 2. Geometric thinking ability according to Van Hiele levels 

VAN HIELE 

 MODEL 

YEAR LEVEL 

Freshmen Level Sophomore Level 

f <cf % f <cf % 

Unplaceable 24 46 52.2 16 36 44.5 

1 6 22 13.0 7 20 19.4 

2 7 16 15.3 7 6 19.4 

3 6 9 13.0 6 3 16.7 

4 0 3 0 0 0 0 

5 3 3 6.5 0 0 0 



 Volume 11, No 1, February 2022, pp. 77-86

 

 

81 

 

VAN HIELE 

MODEL 

YEAR LEVEL 

Junior Level Senior Level 

f <cf % f <cf % 

Unplaceable 17 33 51.3 9 38 23.7 

1 9 16 27.4 9 29 23.7 

2 7 7 21.3 8 20 21.1 

3 0 0 0 8 12 21.1 

4 0 0 0 0 4 0 

5 0 0 0 4 4 10.4 

VAN HIELE 

MODEL 

OVER ALL 

f <cf % 

Unplaceable 66 153 43.1 

1 31 87 20.3 

2 29 56 19.0 

3 20 27 13.1 

4 0 7 0.00 

5 7 7 4.5 
Legend: 0 - Unplaceable, 1- Visualization, 2- Analysis, 3- Informal Deduction, 4- Deduction, 5- Rigor 

 

Table 2 shows the achievement of the students across year levels in terms of their 

geometric thinking ability. Amongst respondents of the study, 20.3% were only level 1; 

19.0% are level 2; and 13.1% are level 3. There is a decreasing number od students who 

were able to reach a higher level in the Van Hiele Model. Looking at the distribution of 

students per geometric level, only 56.9% managed to attain visualization level. In the 

analysis level, only 36.6% achieved this level. Meanwhile, in the informal deduction level, 

17.6% were left to reached the third level. Among the respondents, there is only 4.5% who 

were able to developed the two highest level of geometric thinking which is deduction and 

rigor.  

On the other hand, a staggering 43.1% of the students were classified to be level 0 or 

unplaceable. These students were not able to reach visualization level which is the basic 

foundation of the geometric skills. 

The huge percentage of students found to be unplaceable initiated the query on what 

were the achievement made in the long period of time being exposed to geometry and other 

related areas in mathematics. Patterns were observed in the study as presented below. 

 

3.1.2. Common Patterns of Respondents under Unplaceable 

Outcomes of the international tests revealed that the leaners' geometry knowledge 

and skills were not at the preferred level. Geometry is the lowest performance area among 

all fields (Mullis et al., 2016). The data below represent the pattern of the scores of the 

students under the undesirable level called unplaceable. Each digit of the pattern represents 

each of the levels of the Van Hiele Achievement Test in an orderly manner.  

Table 3. Patterns of geometric thinking skills of unplaceable 

Patterns 
Mathematics Students 

f % 

00000 40 26 

01000 48 30 



 Ordiz & Mecate, Clusters of prevalent patterns of geometric thinking levels … 82 

Patterns 
Mathematics Students 

f % 

00100 17 11 

01100 15 10 

11101 17 11 

11001 7 4 

10100 7 4 

10010 7 4 

*0- Level not attained, 1- Attained level 

 

Core-Corrective Group 

The core-corrective group include the patterns 00000, 01000, 00100 and 01100 (see 

Table 3). This group barely succeeded the foundation of the geometric skills and abilities. In 

00000, it means that none of the levels were partially attained. Different aspects of the 

mathematics achievement of students can cause failure in achieving any of the levels (Meng 

& Sam, 2013). This means that students were really having a hard time visualizing concept, 

reasoning, synthesizing three to two dimensional objects, formulating assumptions, creating 

logical inferences, and also proof skills in geometry. Also, this is affected by the influence 

of teaching-learning process. Researches derive conclusions that students’ level of geometric 

thinking depends on how the instruction is delivered (Abu & Abidin, 2013; Alex & 

Mammen, 2012).  

In the 01000 and 00100 patterns (see Table 3), respondents succeeded in the second 

phase but failed in the rest of the levels. It was because students considered the properties of 

the figures which is not needed in the visualization level and not enough in attaining other 

levels. Students are unable to visualize both solids and positions of solids that they cannot 

see, neither the pattern of movements. There is also a lack of coordination between geometric 

presentation and visualization. 

In the 01100 patterns (see Table 3), students attained the mid-levels; however, 

previous and succeeding levels failed to reach and considered as unclassified. It was because 

students used the properties possessed by each of the shapes, thus failed in obtaining the first 

level. Considering characteristics acquired by the figures were not enough to reach into the 

higher levels. This stresses that learners have a hard time to produce a formally deduced 

definitions and properties in order to prove and provide a logical concept which is necessary 

to reach in its desired level. 

 

Topical-Corrective Group 

The patterns 11101 and 11001 (see Table 3) were classified to be topical-corrective 

group. This group managed to complete the foundation skills but not the higher levels of the 

model.  In these patterns, 11101 and 11001 (see Table 3), students failed between the levels, 

thus considered it as unplaceable. In the first pattern, failing in deduction level (Level 4) but 

reaching into the higher-level means failing to attain any of the Van Hiele levels. It was due 

to lack of knowledge in proving and understanding axioms, corollaries, and postulates. In 

the second pattern, students used their instincts rather than logic in creating conceptual 

statements and properties of the geometric figures and not considering the necessary and 

sufficient conditions in the Euclidean and non-Euclidean geometry. Both patterns failed in 

attaining the formal deduction level. In this level, students failed to provide logical reasons 

or proofs of the following mathematical structure of the elements. Students’ mathematical 



 Volume 11, No 1, February 2022, pp. 77-86

 

 

83 

understanding of proofs, axioms and postulates and corollaries is at risk. It implies that 

students are struggling in terms of proving. Thus, students need to build a strong foundation 

of the basic understanding of geometry to achieve the higher levels. Recently, a growing 

number of studies have provided evidence that students have difficulty with mathematical 

proofs (Ko & Knuth, 2009). 

 

Close-Corrective Group 

This group of patterns managed to achieved the foundation skills in geometry which 

is visualization, however, failed to achieved even partially and struggling to complete the 

levels of the Van Hiele Model. 

10100 and 10010 patterns were categorized as unplaceable. Students attained the first 

levels but failed in reaching the second and third level. It was unplaceable since students 

must achieve the second level to surpass to the next levels. Students cannot skip into t he 

higher levels unless attained the previous levels. It was because students only focused on 

their visual perception, and nonverbal thinking but neglecting the properties, relationships, 

proofs and manipulating axioms. In this level, they can measure, fold and cut paper, use 

geometric software, etc. It implies that undertakers only focused on their visual recognition, 

considered as difficult in applying the properties and unable to perceive relationship between 

figures and components of each shape used to justify and support for the students to answer. 

The pattern reveals that difficulty in understanding the properties and perceiving the 

relationships between properties and figures can affect the students’ performance in the 

higher level of mathematics.  

After identifying patterns, we classified all the existing patterns into three groups, 

topical- corrective, close-corrective, and core- remedial groups. 

 

3.2. Discussion 

As a result of the Van Hiele geometric achievement test, it is concluded that majority 

of the undertakers are under unplaceable. Thus, exploring the patterns existed in the said 

level give us the keys in observing and providing insights on remedial actions in geometry 

that assess the geometric thinking skills of the students in order to improve the levels of their 

geometric ability. 

Topical- corrective group, these are the group of students that have higher chances 

in learning the levels of geometric thinking however, teachers should take consideration in 

providing exercises or activities that can help improve in proving and logical reasoning. The 

group that already captured visualization level but finds it difficult for them attain the 

succeeding levels are placed in close-corrective group because they cannot break down 

information into concepts. Under core-remedial group, students need recalibration of the 

geometrical concepts from simple to complex. Students placed in this group failed to 

visualize geometric figures. Thus, students were having a hard time understanding geometry.  

All these remedial actions provide us analysis of errors observed in every unplaceable 

patterns. Each group introduced differentiated instruction based from the individual group 

needs to improve the geometric levels. These actions will contribute plenty intervention 

strategies and techniques to educators that can be integrated in their geometry lessons. 

 
 

4. CONCLUSION 

Individual gaps in the performance of geometry in the early levels of education is 

important to nurture because of the relevance of the topics in geometry in higher education. 



 Ordiz & Mecate, Clusters of prevalent patterns of geometric thinking levels … 84 

The progress of a student determined the ability of them to perform well in any mathematics 

related disciple. The levels of Van Hiele’s geometric thinking showed that many of the 

student were not able to attain any of the levels namely visualization, analysis, informal 

deduction, deduction, and rigor. 

Appropriate strategies were needed to ensure full understanding by the students are 

needed by all mathematics educators especially in the basic education to address the issue of 

geometric thinking performance. It will showcase the distinction of every student and the 

required remedial action based from their performance. 

 
 

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