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Paper—Problem-Based Geometry in Seventh Grade: Examining the Effect of Path-Based…
Problem-Based Geometry in Seventh Grade: Examining
the Effect of Path-Based Vs. Conventional Instruction on
Learning Outcomes
https://doi.org/10.3991/ijet.v16i12.21349
Andreja Klančar ()
University of Primorska, Koper, Slovenia
Hero Janez Hribar Elementary School, Stari trg pri Ložu, Slovenia
andreja.klancar@pef.upr.si
Andreja Istenič Starčič
University of Primorska, Koper, Slovenia
University of Ljubljana, Ljubljana, Slovenia
Federal University of Kazan, Kazan, Russia
Mara Cotič, Amalija Žakelj
University of Primorska, Koper, Slovenia
Abstract—This experimental study examined the impact of learning and
teaching geometry in seventh-grade geometry education comparing learning-
path and conventional instruction. According to constructivism, a learning path
with the use of learning objects should enhance learner autonomy and self-
directedness by providing differentiated instruction. We designed a model of
path-based geometry learning in a learning management system–based learning
environment with the use of dynamic geometry programs and applets, which
fosters visualisation and the exploration of geometric concepts through the ma-
nipulation of interactive virtual representations. The results show that the exper-
imental group (EG) achieved higher scores on all levels of knowledge and sta-
tistically significantly better results in taxonomy level-III tasks (problem-
solving knowledge) and overall score than the control group (CG). There was
initial equivalence between the EG and CG in prior knowledge. The authors
concluded that path-based geometry learning empirically develops knowledge
at higher cognitive levels.
Keywords—Mathematical education, geometry, path-based learning, learning-
management system, interactive learning environments, dynamic geometry pro-
grams, applets
1 Introduction
The fundamental role of teaching mathematics is to develop mathematical compe-
tence and thus the ability to use mathematics in everyday life. Various national and
international surveys systematically monitor and measure students’ knowledge at
different school levels and in different areas.
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Paper—Problem-Based Geometry in Seventh Grade: Examining the Effect of Path-Based…
The OECD’s Programme for International Student Assessment (PISA) survey
measures the abilities of 15-year-olds to use their knowledge and skills in reading,
mathematics and science to meet the challenges of real life. The basic skills in these
areas enable students to successfully continue their education and integration into
society and thus to be successful and efficient in everyday life. In 2012, the PISA
survey focused on mathematical literacy. The results showed that the performance in
mathematical literacy in Slovenia has been stable in recent years. In Slovenia, 80% of
students attained at least Level 2 proficiency in mathematical literacy (OECD aver-
age: 78%; EU average: 77%) [21].
Slovenian students are also involved in the National Assessment of Knowledge,
which is carried out by the National Examinations Centre and the National Education
Institute. In the nine-year primary education programme, education is divided into
three 3-year periods, at the end of which the students’ knowledge is assessed by the
National Assessment of Knowledge.
An analysis of the mathematical performance of students in the school years
2010/11 to 2014/15 [13], [14], [15], [16], [17] shows that the weaker academic areas
often included geometry (except the 2010/11 school year, in which ninth-graders
successfully solved geometric problem-solving tasks, which usually requires complex
procedures and problem-solving knowledge), measurement and, in 2013/14 and
2014/15, problem-solving, especially complex problems from everyday life.
In the field of geometry, sixth-graders were weaker in their understanding and
knowledge of basic geometric concepts, and ninth-graders were weaker in the field of
spatial representation, which manifests itself both in plane geometry and in the solu-
tion of complex problems in spatial geometry [13], [14], [15], [16], [17]. The ability
of spatial representation is crucial to understanding geometry. It plays an essential
role in mathematical thinking and influences success, not only in geometry, but in all
mathematical disciplines. Visualisation plays an important role in the development of
spatial representations of geometric concepts [28]. By improving the ability to visual-
ise, we can improve the ability to spatially represent shapes and think abstractly and
logically, which is crucial for solving everyday problems. With an appropriate teach-
ing approach and appropriate teaching and learning strategies, the teacher can help
students develop spatial representations [11].
2 Higher-Order Learning Outcomes and Path-Based Learning
Using LMS
The development of learning resources available on the Internet assists individual-
ised, differentiated instruction. For its effective, efficient integration in teaching and
learning, many researchers have explored the path-based approach [2], [5], [19], [22],
[25]. A learning path is a learning-environment functionality that integrates learning
objects in a road map for students [4] and is aimed at a student’s individual inquiry in
learning to select their learning activities [22]. The new concepts in learning design,
such as learning objects, appeared with the development of constructivism and the
Internet in the 1990s [24]. Learning objects are self-standing digital instructional
components available on the Internet that can be reused in a diverse learning envi-
ronment [24]. According to constructivism, a learning path with the use of learning
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Paper—Problem-Based Geometry in Seventh Grade: Examining the Effect of Path-Based…
objects enhances learner autonomy and self-directedness by providing differentiated
instruction. The learner must make decisions in their learning process about the in-
quiry and use of available learning resources.
Researchers have examined the effects of path-based methods on the learning out-
comes of students in comparison to conventional instruction and studied learning-
management system (LMS)-based learning environments [5].
Integrating technology into path-based geometry lessons, especially with the use of
dynamic geometry programs and applets, fosters (1) visualisation and (2) the explora-
tion of mathematical concepts through the manipulation of interactive virtual repre-
sentations [10]. It helps students visualise and so facilitates the comprehension of
geometric concepts. Computer-based graphical representations of geometric concepts
are more representative than paper-based representations. The clarity of interactive
displays facilitates observing and grasping concepts and conjectures, thus enabling a
more fluent transition between different representations, especially between the visual
and abstract levels [1].
For the purpose of the pedagogical experiment, we designed a model of path-based
geometry learning in an LMS-based learning environment considering the guidelines
of national [6], [7], [14], [15], [16], [17] and foreign literature [2], [5], [19], [22] to
address the challenges of teaching geometry in Slovenian education [7], [20], [21].
In path-based learning, students transition between geometric reasoning with prob-
lem visualisation, problem analysis and forecasting solutions. Our model of path-
based geometry learning integrates the principles of individualisation and differentia-
tion, meaning that students select their own paths and applications of learning objects
according to their prior knowledge and preferences.
Recommendations concerning the use of information and communications tech-
nology (ICT) in mathematics education vary from country to country in Europe. In
Slovenia, the use of ICT in mathematics education is included in the pedagogical
recommendations of the curriculum. Some contents provide for the use of ICT, while
others recommend using ICT only and leave the decision, whether to include ICT into
math lessons or not, to the teacher.
All these facts prompted us to study the impact of path-based geometry learning in
an LMS-based learning environment with the use of dynamic geometry programs and
applets to facilitate learner-centred instruction.
3 Description of The Model
We designed and evaluated a model of path-based learning of seventh-grade geom-
etry in an LMS-based learning environment with the use of dynamic geometry pro-
grams and applets. The learning path was designed in the LMS by applying a range of
learning objects. Individualised learning in the model of path-based geometry learning
was guided by a teacher and supported by guidelines. Students developed knowledge
through research. The path-based approach facilitates self-directedness and the auton-
omous selection of learning objects, and the transition between the stages of geomet-
ric reasoning is contingent on individual students and their selection of learning ob-
jects.
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Paper—Problem-Based Geometry in Seventh Grade: Examining the Effect of Path-Based…
As an example, we present learning path for researching the area of trapezoids
(Figure 1) which is explained in Table 1. The learning path is based on educational
theories and teaching approaches [3] that are focused on developing knowledge at all
levels, especially developing problem-solving knowledge, including visualisation and
critical thinking.
Fig. 1. The learning path
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Paper—Problem-Based Geometry in Seventh Grade: Examining the Effect of Path-Based…
Table 1. The learning path: Area of a trapezoid
Objectives: The student has to find a strategy to calculate the area of a trapezoid using their knowledge
about the area of a rectangle and then generalise the findings about the area of trapezoids and write them in
symbolic form as a formula.
1. Accessing pre-knowledge and motivation
Possible choices and examples of learning
objects
Description
The advantage of using computer
technology in learning path–based
geometry lessons
Real-life geometric problem: paving yards
Students use the ap-
plet-enabled overlay
option to draw the
requested geometric
shape (trapezoid) and
determine its area
using a non-standard
unit.
The applet enables students to ma-
nipulate objects to learn by trying to
find an appropriate solution, whereas
in paper-based materials, only fixed
visual support is given, and learning
by trying is not possible.
Paving using an applet enables the
development of the conceptual
knowledge of trapezoid area.
2. Research about the area of trapezoids
Possible choices and examples of learning
objects
Description
The advantage of using computer
technology in learning path–based
geometry lessons
Introductory activity: concrete materials
using guidelines in the form of a Power-
Point (PP) presentation:
By cutting the paper
where the trapezoid is
drawn, students trans-
form it into a rectangle.
The PP presentation is
designed to systemati-
cally guide the student
through the transfor-
mation of the trapezoid
into a rectangle.
Main activity: using an applet to explore
the trapezoid area
Students choose between two possibilities
of reshaping the trapezoid:
o Reshaping the trapezoid into a
rectangle
o Reshaping the trapezoid into a
parallelogram
Students learn the
process of reshaping
the trapezoid into a
rectangle/parallelogram
observing an anima-
tion, through which
they generalise the
findings and write
them in symbolic form
as a formula.
The animation helps students develop
conceptual knowledge and form
strategies to solve the problem of the
algebra text for the area of a trapezoid.
3. Practising and developing different types of skills and knowledge and applying them
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Paper—Problem-Based Geometry in Seventh Grade: Examining the Effect of Path-Based…
Possible choices and examples of learning
objects
Description
The advantage of using computer
technology in learning path–based
geometry lessons
Development of conceptual knowledge and
understanding
The trapezoid is also
introduced in the
coordinate grid. The
applet enables students
to draw corners, with
the learner testing their
understanding and
developing conceptual
knowledge.
Development of procedural knowledge
To develop procedural
knowledge, students do
a quiz where they can
select different tasks
from within a set of
differentiated tasks or
generate new tasks
using applets.
Enables students to choose the com-
plexity of tasks and the individual
characteristics of trapezoids (acute,
right, obtuse, isosceles, etc.)
Development of problem-solving
knowledge
Students do a problem-
solving task with visual
support and a grid on
which squares are
drawn. The task is not
basic. The student must
persist in the reasoning
process to find a solu-
tion or make a conclu-
sion. Feedback is part
of the LMS learning
environment.
This enables students to use hints
which help them select an appropriate
problem-solving strategy. To check
the relevance of their solution, stu-
dents can use the solutions provided in
the unit.
4 Purpose of The Study
Due to the advantages of using technology in teaching and learning [27] and the in-
sufficient use of ICT in mathematics education and pedagogical approaches that de-
velop students’ spatial-visualisation abilities [7], [11] we developed a model of path-
based geometry learning in an LMS-based learning environment.
Based on Gagné's intellectual skills involved in problem solving [8], we considered
three levels of math knowledge: conceptual knowledge, procedural knowledge and
problem-solving knowledge [23].
The purpose of the study was to develop a model of path-based geometry learning
in seventh grade and examining whether path-based learning environment can be
more effective than conventional instruction for different levels of knowledge
(adapted version of Gagné’s taxonomy):
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Paper—Problem-Based Geometry in Seventh Grade: Examining the Effect of Path-Based…
• Taxonomy level I – conceptual knowledge – knowing and understanding basic
geometric concepts and facts
• Taxonomy level II – procedural knowledge
• Taxonomy level III – problem-solving knowledge – solving simple and complex
(geometric) problems
4.1 Research question and research hypotheses
The research was conducted among seventh-graders using a model of path-based
geometry learning to enable higher-order learning outcomes. By teaching seventh-
grade geometry in an LMS learning environment, we examined how students’ con-
ceptual, procedural and problem-solving knowledge was supported to answer the
following research question:
Can be path-based geometry learning (experimental group) more effective than
conventional instruction (control group) for different levels of knowledge?
General hypotheses: Seventh-grade students who have a model of path-based ge-
ometry learning will be more successful in solving geometric tasks than students who
receive traditional, discrete maths lessons.
H1. The experimental group (EG) will be more successful than the control group
(CG) on the level of basic, conceptual knowledge (first level).
H2. The EG will be more successful than the CG in solving tasks on the level of
procedural knowledge (second level).
H3. The EG will be more successful than the CG in solving tasks on the level of
problem-solving knowledge (third level).
H4. The EG will be more successful than the CG in overall score.
When testing hypotheses, we used a significance level of 0.05.
5 Methodology
A causal, descriptive experimental method of pedagogical research was used to
validate the hypotheses. The group of students who received the experimental factor
formed the EG, and the group of students who did not receive the experimental factor
formed the CG.
Data were collected quantitatively by pre- and post-participation tests.
The effectiveness of the path-based geometry learning model was tested using a
one-factor pedagogical experiment with classes as comparison groups. The experi-
mental factor had two modalities: geometry lessons based on the path-based geometry
learning model (EG) and geometry lessons based on the transmission approach with
traditional aids (CG).
5.1 Description of the lesson model in the control group
The CG followed the same learning objectives and content as the EG. The geome-
try lesson in the CG was based on transmission, a teacher-centred approach in which
the teacher is the dispenser of knowledge and the final evaluator of learning. Teachers
first explained the content in lectures, which was followed by students solving tasks
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Paper—Problem-Based Geometry in Seventh Grade: Examining the Effect of Path-Based…
on all three levels of knowledge. Students also received homework, where similar
tasks had to be solved. In geometry lessons, students in the CG used geometric tools,
textbooks and notebooks. In the textbooks, the instructions for most tasks which ap-
plied to calculating the perimeters and areas of triangles and quadrilaterals did not
include visual support. The tasks more often focused on the development of procedur-
al knowledge.
5.2 Description of the lesson model in the experimental group
We designed and evaluated a model of path-based learning of seventh-grade geom-
etry in an LMS-based learning environment (the model is described in Section 3). For
the EG, the learning path was designed in the LMS by applying a range of learning
objects including use of dynamic geometry programs and applets. Individualised
learning in the model of path-based geometry learning was supported by a teacher and
by guidelines. The EG developed knowledge on all three levels through research. The
path-based approach facilitates self-directedness and the autonomous selection of
learning objects.
5.3 Experiment sample
The study included 125 seventh-grade students randomly selected from three Slo-
venian schools. Sixty-three students from all three schools were included in the EG,
and 62 from all three schools were included in the CG. The EG consisted of 31 boys
and 32 girls, while the CG had 38 boys and 24 girls. A total of 69 boys and 56 girls
participated in the study. Six maths teachers participated, all with the same level of
education – university degrees and at least 10 years of work experience. All the
schools were provided with adequate ICT.
5.4 Variables
The experimental factor is the independent variable. The data, collected with pre-
and post-participation tests, represent the dependent variables in the statistical context.
For all students who took part in the study, we obtained data on gender (male and
female), the class they attended, their final grades in mathematics and Slovenian, and
their average final grades, which represent in the statistical context the control varia-
bles.
5.5 Data collection
We collected data with pre- and post-participation tests, which classified tasks ac-
cording to a three-level taxonomy of conceptual, procedural and problem-solving
knowledge. We used a pre-participation test to establish the initial equivalence be-
tween EG in CG students in their knowledge of seventh-grade maths (Table 3). We
conducted pre-participation testing before starting the experiment (first empirical
recording) before the model of path-based geometry learning was integrated into the
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Paper—Problem-Based Geometry in Seventh Grade: Examining the Effect of Path-Based…
EG. The post-participation test (the second empirical recording) was conducted one
week after the experiment under the same conditions and with the same tester.
The data, collected with pre- and post-participation tests, represent the dependent
variables in the statistical context.
The data presented in the statistical context by dependent variables were provided
to us by the class teacher and the mathematics teacher of the students involved in the
survey. Consent from the parents for the students’ participation in the research was
obtained beforehand.
Validity, reliability and objectivity: We designed the pre-participation test to ex-
amine the equivalence between the EG and CG and post-participation tests for the
purpose of this experiment after discussing the content of both tests and the applica-
bility of the three-level taxonomy to learning outcomes. We used Cronbach’s alpha
coefficient to calculate the reliability of both tests, both of which were higher than
0.8, indicating acceptable levels of reliability. We verified the validity of the tests by a
factor analysis, which demonstrated that the first factor in the pre-participation test
accounted for 25.3% of the variance, whereas the first factor in the post-participation
test accounted for 25.7% of the variance, meaning that the tests had construct validity.
We ensured the objectivity of the pre- and post-participation tests during both testing
and evaluation by conducting the evaluation according to previously established crite-
ria uniformly for all the tested students in both groups. The data collection was per-
formed according to the same procedure for both groups as well.
Difficulty and discrimination: Difficulty index is defined as the percentage of
those students who gave the correct response for a particular task.
The difficulty index (IT) was calculated according to the formula:
(1)
where represents the average number of points for task j, and represents the
maximum possible number of points for task j.
The difficulty index of most tasks was within an acceptable range from 31.6–
84.4%.
The index of discrimination (ID) showed that most tasks had acceptable values (ID
> 20%). Twenty-two subtasks had IDs above 40%, which shows a good discrimina-
tion of tasks; five subtasks had IDs between 20% and 40%, which indicates medium
discrimination; and one subtask had an ID under 20%, which indicates poor discrimi-
nation. The average ID value was 54%.
5.6 Statistical processing
We used the Statistical Package for the Social Sciences (SPSS) 22 to process the
quantitative data.
The pre-participation test established the initial equivalence between EG and CG
students. We used a t-test to determine the significance of the difference between the
EG and CG in the pre-participation tests for individual taxonomy levels (according to
the adapted version of three-level taxonomy of conceptual, procedural and problem-
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Paper—Problem-Based Geometry in Seventh Grade: Examining the Effect of Path-Based…
solving knowledge), and overall score. The EG and the CG were equal in the pre-
participation test on different levels of knowledge and in overall score.
We also used a t-test to determine the significance of the difference between the
EG and the CG in the post-participation test on all levels of knowledge.
5.7 Results and discussion
The results were interpreted in accordance with the proven hypotheses. In doing so,
we considered the maximum allowable risk of rejection of a hypothesis to be 5% (a
significance level of 0.05). If the level of statistical significance in the t-test was lower
than 0.05, this meant that the EG and CG were statistically significantly different
based on the tested hypothesis.
Results of pre-test: In pre-test we analyzed the differences in the knowledge of
geometry between the students in the experimental group and the ones in the control
group in the initial phase based on taxonomy levels and overall score on the pre-test.
The purpose of the pre-test was to test pre-knowledge in geometry acquired by the
students in the first and second three-year education periods.
In the initial phase, we tested the students’ knowledge of geometry at different tax-
onomy levels: understanding basic geometric concepts and facts (taxonomy level I),
using procedural knowledge (taxonomy level II), problem-solving knowledge (taxon-
omy level III).
Table 2 shows the basic statistical parameters of the pre-test: the number of stu-
dents in the EG and CG, the arithmetic mean, the standard deviation, the standard
mean error and the lowest and highest scores for different taxonomy levels and over-
all score.
Table 2. Pre-test scores – basic statistical parameters for achievement (in %) for individual
taxonomy levels and overall score by group (EG, CG)
Variable
Group
N
M
SD
SE
Min
Max
Taxonomy level I (conceptual
knowledge) (%)
EG
63
54.1
19.9
2.5
22.2
100.0
CG
62
47.8
22.4
2.8
11.1
100.0
Taxonomy level II (procedural
knowledge) (%)
EG
63
53.3
21.9
2.8
8.3
100.0
CG
62
50.0
21.9
2.8
8.3
100.0
Taxonomy level III (problem-solving
knowledge) (%)
EG
63
37.6
25.2
3.2
0.0
100.0
CG
62
35.7
27.4
3.5
0.0
100.0
Overall score (%)
EG
63
48.8
19.2
2.4
10.0
100.0
CG
62
45.1
20.7
2.6
10.0
93.3
EG – experimental group, CG – control group, N – number of students, M – mean, SD – standard devia-
tion, SE – standard error of a mean, Min – minimum, Max – maximum
The students in the EG achieved the highest results in tasks on level I (54.1%), but
they had the lowest results in tasks on taxonomy level III (37.6%).
The students in the CG achieved the highest results in tasks on taxonomy level II
(50%), but they had the lowest results in tasks on taxonomy level III (35.7%).
The results show that the students in the EG were more successful in the pre-test on
all three taxonomy levels than the students in the CG (both in the knowledge and
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Paper—Problem-Based Geometry in Seventh Grade: Examining the Effect of Path-Based…
understanding of basic geometric concepts, the use of procedural knowledge, and
problem-solving knowledge). Furthermore, also in overall score achieved on the pre-
test students in the EG were more successful (48.8%) than the students in the CG
(45.1%).
A t-test (Table 3) was used to check and confirm that there were no statistically
significant differences in the pre-test between the EG and CG in the achievements on
different taxonomy levels and in overall score (p > 0.05).
Table 3. Pre-test scores – t-test for independent samples to verify the differences between the
EG and CG in achievement on individual taxonomy levels and overall score
Variable
t
df
p
Mean
Difference
SE Difference
Taxonomy level I (conceptual knowledge) (%)
1.663
123
0.099
0.063
0.038
Taxonomy level II (procedural knowledge) (%)
0.844
123
0.400
0.033
0.039
Taxonomy level III (problem-solving knowledge) (%)
0.404
123
0.687
0.019
0.047
Overall score (%)
1.060
123
0.291
0.038
0.036
t – value of the t-test for independent samples, df – degrees of freedom, p – level of statistical significance
(2-tailed)
Results of post-test: In post-test we analyzed the differences in the knowledge of
geometry between the students in the EG and the ones in the CG in the final phase
based on taxonomy levels and the overall score in the post-test.
Table 4 shows the basic statistical parameters of the post-test: the number of stu-
dents in the EG and CG, the arithmetic mean, the standard deviation, the standard
mean error, and the lowest and highest scores for different taxonomy levels and over-
all score.
Table 4. Post-test scores – basic statistical parameters for achievement (in %) for individual
taxonomy levels and overall score by group (EG, CG)
Variable
Group
N
M
SD
SE
Min
Max
Taxonomy level I (conceptual
knowledge) (%)
EG
63
67.9
23.9
3.0
0.0
100.0
CG
62
59.5
26.9
3.4
0.0
100.0
Taxonomy level II (procedural
knowledge) (%)
EG
63
59.3
29.0
3.7
0.0
100.0
CG
62
52.3
26.7
3.4
8.3
100.0
Taxonomy level III (problem-solving
knowledge) (%)
EG
63
29.5
24.1
3.0
0.0
100.0
CG
62
20.3
22.7
2.9
0.0
100.0
Overall score (%)
EG
63
52.9
22.6
2.8
3.3
100.0
CG
62
44.8
22.2
2.8
3.3
100.0
EG – experimental group, CG – control group, N – number of students, M – mean, SD – standard devia-
tion, SE – standard error of a mean, Min – minimum, Max – maximum
In the post-test, students in the EG achieved the highest results in tasks on taxono-
my level I (67.9%). The lowest results they achieved in tasks on taxonomy level III
(29.5%).
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The students in the CG also achieved the highest results in tasks on taxonomy lev-
el I (59.5%), while also achieving the lowest results in tasks on taxonomy level III
(20.3%).
The results show that the students in the EG were more successful in the post-test
on all three taxonomy levels than the students in the CG (in the knowledge and under-
standing of basic geometric concepts, the use of procedural knowledge and problem-
solving knowledge) and in the overall score achieved. The students in the EG
achieved an average of 52.9% of all points, whereas the students in the CG scored an
average of 44.8% of all points.
We further examined whether these differences were statistically significant.
As in the initial phase – before the introduction of the experimental factor – to de-
termine whether the EG and CG were even in terms of individual taxonomy levels
and the overall score, we used a t-test (Table 5) to check the statistical significance of
the differences between group achievements.
Table 5. Table 5: Post-test scores – t-test for independent samples to verify the differences
between the EG and CG in the achievement on individual taxonomy levels and overall
score
Variable
t
df
p
Mean Difference
SE
Difference
Cohen
d
Taxonomy level I (conceptual knowledge) (%)
1.847
123
0.067
0.084
0.045
0.33
Taxonomy level II (procedural knowledge)
(%)
1.397
123
0.165
0.070
0.050
0.25
Taxonomy level III (problem-solving
knowledge) (%)
2.195
123
0.030
0.092
0.042
0.39
Overall score (%)
2.015
123
0.046
0.081
0.040
0.36
t – value of the t-test for independent samples, df – degrees of freedom, p – level of
statistical significance (2-tailed)
We analysed the differences in the knowledge and understanding of basic geomet-
ric concepts between students from the experimental and the control groups in the
final phase.
In the initial phase, students in the EG achieved slightly better average results
(54.1%) in tasks on taxonomy level I (knowledge and understanding of basic geomet-
ric concepts) than the students in the CG (47.8%), but these differences were not sta-
tistically significant.
In comparing the arithmetic means of the achievements of the EG and CG on tax-
onomy level I (knowledge and understanding of basic geometric concepts), the EG
was also more successful (67.9%) than the CG (59.5%) on the post-test. The differ-
ence between the arithmetic means increased from 6.3% to 8.4% (Table 2 and Table
4, respectively).
Despite a slightly higher achievement of the students in the EG than the CG in the
knowledge and understanding of basic geometric concepts, the t-test showed that the
differences were not statistically significant (t(123) = 1.847, p = 0.064) (Table 5).
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The calculated arithmetic means, the results of the t-test and the level of statistical
significance indicate that the EG and CG were equal in the post-test in the knowledge
and understanding of basic geometric concepts, as there were no statistically signifi-
cant differences between the achievements of the experimental and the control groups.
The first hypothesis is therefore rejected.
The tasks of taxonomy level I checked the students’ knowledge and understanding
of basic geometric concepts, geometric shapes and their properties, the names of
shapes, and reading data from images to be used when performing procedures and
drawing shapes.
We expected the students in the EG to have more knowledge of basic geometric
concepts than the students in the CG based on the assumption that the knowledge
obtained through an active learning process is higher-quality. In addition, we attribut-
ed great importance to (1) visualising geometric concepts while using dynamic geom-
etry programs and applets and (2) ongoing feedback and its impact on the construc-
tion of concepts or changing conceptual representations. The students in the EG
achieved a higher average result in the post-test than the students in the CG, but the
differences between the two groups were not statistically significant. The students in
the CG had lessons in the traditional model, where the teacher first explained the
geometric concepts to the students, which was followed by consolidation and repeti-
tion by solving different tasks, enabling the students to memorise the concepts. In the
EG, students independently explored the content on the perimeters and areas of trian-
gles and quadrilaterals in an LMS-based learning environment, using dynamic geome-
try programs, applets and other interactive material. The material was supported by
various visual elements that helped students construct and understand conceptual
representations. Most students required more time when exploring content inde-
pendently, which means they solved fewer tasks in the consolidation phase compared
to the students in the CG.
Based on the results, the learning approach in the EG better enables students to
successfully obtain knowledge and understand basic geometric concepts.
We analysed the differences in the use of procedural knowledge between students
from the experimental and the control groups in the final phase.
In the initial phase, students in the EG also achieved a slightly better average result
(53.3%) in tasks on taxonomy level II (use of procedural knowledge) than the students
in the CG (50%), but these differences were not statistically significant.
If comparing the arithmetic means of the achievements of the EG and CG on tax-
onomy level II (use of procedural knowledge), we can see that the EG was also more
successful (59.3%) than the CG (52.3%) in the post-test. The difference between the
arithmetic means increased from 3.3% to 7% (Table 2 and Table 4, respectively).
Despite the slightly higher achievement of the students in the EG than the CG in
the use of procedural knowledge, the t-test showed that the differences were not statis-
tically significant (t(123) = 1.397, p = 0.165) (Table 5). The calculated arithmetic
means, the results of the t-test and the level of statistical significance indicate that the
EG and CG were even in the post-test in the use of procedural knowledge, as there
were no statistically significant differences between the achievements of the EG and
CG. The second specific hypothesis was therefore rejected as well.
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Paper—Problem-Based Geometry in Seventh Grade: Examining the Effect of Path-Based…
In the post-test, tasks on taxonomy level II checked the use of simple and advanced
procedural knowledge by means of direct and indirect tasks. Students calculated the
perimeters and areas of triangles and quadrilaterals, calculated the lengths of sides
based on the given data, and used them to calculate the perimeter and area of the giv-
en shape and drew shapes (also in the coordinate grid).
The aim of the study was to check whether a suitable approach could help students
successfully solve tasks on taxonomy level II (use of procedural knowledge), as the
applied model of path-based geometry learning lacked an emphasis on systematic
solving and recording procedures, unlike the traditional lessons given to the CG. Tra-
ditional instruction attributes great importance to procedural knowledge, so plenty of
time is dedicated to practising routine, complex procedures, often without knowing or
understanding the basic geometric concepts, which is a precondition for complex
procedures. CG students thus solved many tasks that helped them practise routine,
complex procedures, while in the EG, students acquired procedural knowledge from
didactic games and quizzes, where they could master both simple and complex proce-
dures with adequate visual support (visual material and animations). Students in the
EG were more successful in using procedural knowledge than those students in the
CG, but these differences were not statistically significant.
As already mentioned, the pre-condition for a successful execution of complex
procedures is knowing and understanding of concepts, which was also shown by the
results. Compared to the CG, the EG achieved a better average result in both the
knowledge and understanding of basic geometric concepts as well as in the use of
procedural knowledge.
We analysed the differences in solving simple and complex problems (problem-
solving knowledge) between students from the experimental and control groups in the
final phase.
In the initial phase, students in the EG achieved a slightly better average result
(37.6%) in tasks on taxonomy level III (solving simple and complex problems) than
the students in the CG (35.7%), but these differences were not statistically significant.
If comparing the arithmetic means of the achievements of the EG and CG on tax-
onomy level III (solving simple and complex problems), we can see that the EG was
also more successful (29.5%) than the CG (20.3%) on the post-test. The difference
between the arithmetic means increased from 1.9% to 9.2% (Table 2 and Table 4,
respectively).
As the EG and CG were even in terms of student achievements on taxonomy level
III, we used a t-test to check whether the differences between the groups were statisti-
cally significant.
The calculated arithmetic means, the t-test results (t(123) = 2.195, p = 0.030) and
the level of statistical significance (Table 5) indicate that the differences in the
achievements of the students from the EG and CG in solving simple and complex
problems were statistically significant in favour of the EG. This confirms our third
hypothesis.
In tasks on taxonomy level III, students were required to identify the relevant
shapes and their sides on a sketch, use the given visual support to transform the text
(including algebraic text) of the problem into mathematical language, read the rele-
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vant data from the sketch and create an appropriate strategy to solve the problem
using the acquired data.
The goals of the curriculum, national and international strategies and national stu-
dent assessment and international research, such as Trends in International Mathemat-
ics and Science Study (TIMSS) and PISA, are also to build students’ knowledge to
successfully solve mathematical problems and real-life problems. Students build prob-
lem-solving knowledge through different problem situations that are new to them,
cannot be predicted and are therefore unexpected, which promotes the development of
mathematical thinking: creative, critical, analytical, systemic thinking. For this pur-
pose, we designed a model of path-based geometry learning in an LMS-based learn-
ing environment with the use of dynamic geometry programs and applets based on a
process/didactic approach.
We analysed the differences between students from the experimental and the con-
trol groups in the final phase based on the total number of points.
In the initial phase, students in the EG achieved a slightly better average result
(48.8%) based on the total number of points than the students in the CG (45.1%), but
these differences were not statistically significant (t(123) = 1.060, p = 0.291) (Table
3).
In comparing the arithmetic means of the achievements of the EG and CG based on
the total number of points, the EG was also more successful (52.9%) than the CG
(44.8%) in the post-test. The difference between the arithmetic means increased from
3.8% (Table 3) to 8.1% (Table 5), which means that the average result of the post-test
was 8.1% better in the EG.
As the EG and CG were even in terms of student achievements based on the over-
all score, a t-test was used to check whether the differences between the groups were
statistically significant.
The calculated arithmetic means, the t-test results (t (123) = 2.015, p = 0.046) and
the level of statistical significance (Table 5) indicated that the differences in the
achievements of the students from the experimental and control groups in the total
number of points were statistically significant in favour of the EG. This confirms our
fourth specific hypothesis.
The t-test showed that the statistically significant differences in the post-test were
in favour of the EG on taxonomy level III (t (123) = 2.195, p = 0.030) and the overall
score (t(123) = 2.015, p = 0.046). There was medium effect size for problem-solving
knowledge (taxonomy level III) (r = 0.39) and for the overall score (r = 0.36). On
average, the result of the EG in the post-test was 8.1% better (Table 5). This answers
on our general research hypothesis: Seventh-grade students who had the model of
path-based geometry learning were more successful in solving geometric tasks than
the students that receive traditional, discrete maths lessons.
6 Conclusion
The model of path-based geometry learning in the LMS-based learning environ-
ment with the use of dynamic geometry programs and applets, which was based on a
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Paper—Problem-Based Geometry in Seventh Grade: Examining the Effect of Path-Based…
cognitive/constructivist method of teaching, was implemented in teaching practice to
check its effects on learning achievements.
The analysis of the results showed that the students who had the model of path-
based geometry learning achieved higher scores at all levels of knowledge, and the
EG achieved statistically significantly better results with a medium-size effect on
solving simple and complex problems, and the overall score. The results show that
path-based geometry learning empirically develops knowledge at higher cognitive
levels.
This model gives students the option to choose, as the online learning environment
enables them to encounter various physical and virtual situations. Students thus (co-
)create their learning paths and have the opportunities to make less linear, hierar-
chical, systematic progress. The online learning environment with the use of dynamic
geometry programs and applets allowed students to observe different representations
and transitions between them, which has a significant impact on the development of
the ability to visualise. Students built geometric representations on a concrete level,
followed by a transition to a symbolic level supported by visual aids, and the use of
animations allowed students to initiate and observe the process and repeat it if needed.
Impersonal, impartial, real-time feedback also played an important role. Activities
were directed towards solving and researching different geometric problems. Prob-
lem-solving develops divergent and convergent thinking, creativity, argumentation
and decision-making abilities, and the ability to make interdisciplinary connections.
A study on learning and teaching basic geometric concepts in elementary school
[11] also found that the use of several means in the formulation of geometric concepts
had a positive effect on student achievements in geometry. We found similar argu-
ments in the baseline of the TIMSS 2011 [12] and PISA 2012 studies [21], which
emphasised that the last education period should focus on visualising basic geometric
concepts, as this is the only way for students to use spatial representations to transi-
tion between three-dimensional and two-dimensional shapes and their visualisations.
Raphael and Wahlstrom [18] and Fuys et al. [8] also found that the use of didactic
aids was crucial for a successful visualisation of geometric concepts. The aids that
allow both visualisation and manipulation simultaneously facilitate the construction of
basic geometric concepts and enable suitable cognitive visualisation.
7 Acknowledgement
The article was written as part of project activities of national project NA-MA
POTI - Natural Science and Mathematical Literacy: Promoting Critical Thinking and
Problem Solving and efficient use of ICT. Investment is co-financed by the Republic
of Slovenia and the European Union under the European Social Fund. Under the Op-
eration programme for the Implementation of the European Cohesion Policy in the
2016-2022 period.
Research by Andreja Istenič Starčič was financially supported by the Slovenian
Research Agency (P2-0210).
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9 Authors
Andreja Klančar, PhD, is teacher assistant and senior instructor of Educational
technology at the Faculty of Education, University of Primorska in Koper, Slovenia.
She is also math teacher at elementary school. Her field of research is the develop-
ment and evaluation of different teaching approaches and the effective use of technol-
ogy in the teaching and learning process. Email: andreja.klancar@pef.upr.s
Andreja Istenič Starčič, PhD, is professor in didactics at University of Primorska
and University of Ljubljana, Slovenia. Her teaching and research interests include
educational technology, teacher education, research evaluation, and particularly, in-
terdisciplinary research. Andreja was editor of British Journal of Educational Tech-
nology and is member of editorial boards in Educational Technology Research and
Development. Andreja has served as a visiting professor at the University of North
Texas, Macquarie University Sydney, and Kazan Federal University. Andreja is au-
thor of the book Educational technology and construction of authentic learning envi-
ronment. Email: andreja.starcic@pef.upr.si
Mara Cotič, PhD, is an Associate Professor in Didactics of Mathematics and Ele-
mentary School Mathematics at the Faculty of Education Koper, University of Pri-
morska. As a researcher in the area of didactic of mathematics develops new modern
models of teaching and learning mathematics, above all in the area of data processing
and problem knowledge. The results of her research work are besides scientific and
professional articles and papers at the international meetings seen also in the didactic
packages (handbooks for teachers, textbooks, exercises for consolidation) for all nine
grades of primary schools as well as in two independent scientific monographies. In
her articles and in her monography Data processing in the instruction of mathematics
she demonstrated her model of data processing instruction in the first five grades of
primary schools, which was a complete novelty in Slovenia. For the establishment of
this model she took into consideration the results of her own empirical researches and
the results of the researches of the most prominent scientists who deal with those
questions, and she respected the specificity and characteristics of the Slovenian envi-
ronment. In the monography Mathematical problems in the primary schools and in her
scientific articles she demonstrated the model of teaching and learning, where the
problems are the issuing point of all activities required to understand different math-
ematical contents and concepts. Her works represent a significant contribution to the
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Paper—Problem-Based Geometry in Seventh Grade: Examining the Effect of Path-Based…
theoretical issues and practical implementation of the development of Slovenian pri-
mary school didactics for mathematics. Email: mara.cotic@pef.upr.si
Amalija Žakelj, PhD, is professor in Didactics of Mathematics and Elementary
School Mathematics at the Faculty of Education, University of Primorska. Her re-
search is focused on the development of models of teaching and learning mathemat-
ics, involving problem situations and realistic problems as well as the complex ques-
tions about the meaning of mathematics education in connection with the effective-
ness of teaching and learning. As leading expert of the area of didactics of mathemat-
ics she transfers her knowledge and experiences into the field of mathematics educa-
tion. Her achievements in scientific, professional and pedagogical work in the field of
mathematics didactics are useful both for students and teachers as well as for re-
searchers in the field of pedagogical research. She is author of numerous scientific,
professional and pedagogical articles, handbooks for teachers, textbooks and contribu-
tions at conferences in Slovenia and abroad. Email: amalija.zakelj@pef.upr.si
Article submitted 2021-01-20. Resubmitted 2021-02-25. Final acceptance 2021-02-26. Final version
published as submitted by the authors.
iJET ‒ Vol. 16, No. 12, 2021
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