European Journal of Education Studies
ISSN: 2501 - 1111
ISSN-L: 2501 - 1111
Available on-line at: www.oapub.org/edu
Volume 3 │ Issue 1 │ 2017
doi: 10.5281/zenodo.223308
THE EFFECTS OF MATHEMATICAL MODELLING ACTIVITIES
ON THE DIFFICULTY PERCEPTION OF NUMBERS SENSE
AND ACHIEVEMENT AMONG 4TH GRADERSi
Isık N.1, Pilten P.2ii
National Education Ministry of Turkey, Konya, Turkey
1
N. Erbakan University, A.K. Faculty of Education
2
42090, Konya, Turkey
Abstract:
The purpose of the present research is investigating the effects of mathematical
modelling activities on the difficulty perception of numbers sense, which is perceived as
difficult by primary school 4th grade students, and achievement. The problem statement
of the present research was formed as Does mathematical modelling strategy has any effects
on th grade students levels of difficulty perceptions and their achievement related number
sense learning field?” The present research was conducted in accordance with
quantitative research methods in two steps. The first step was conducted in accordance
with survey model on 207 students, who studied in Selcuklu district of the province of
Konya in the spring semester of 2013-2014 School Year. The second step was also
conducted on 61 students from two equal classes of Esrefoglu Primary School in
accordance with experiment model with pre-test-post-test and control group. In order
to collect data for the present research,
Numbers Learning Domain Achievement and
Difficulty Perception Scale (NLDADPS) Form A and Form B were employed as pre-test
and post-test on experiment and control groups, Observation Form for the Evaluation of
the Experimental Procedure
experiment group, and
was employed to evaluate the implementation on the
Observation Form for the Evaluation of the Problem Solving
Activities Conducted in Control Group was employed to evaluate the teaching conducted
on control group. All of the scales were developed by the researcher. It was found that
mathematical modelling activities were more effective on procedural knowledge and
concept-procedure connections dimensions of the topics than traditional problem
solving activities, and enabled developing positive attitudes towards mathematics and
i
ii
This study is a summary of Doctoral (Phd) Thesis of Necip Isık.
Correspondence: email ppilten@hotmail.com
Copyright © The Author(s). All Rights Reserved.
© 2015 – 2017 Open Access Publishing Group
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THE EFFECTS OF MATHEMATICAL MODELLING ACTIVITIES ON THE DIFFICULTY PERCEPTION OF
NUMBERS SENSE AND ACHIEVEMENT AMONG 4TH GRADERS
contributed to the development of metacognitive skills required for establishing
concept-procedure connections.
Keywords: difficulty perception, mathematical modelling, mathematical modelling
activities
1. Introduction
Even mathematics subject as a whole is considered as a difficult subject by students, this
is not true for all topics and concepts; and not at the same level. Some topics are defined
as more difficult than others by the students. Studies on the topics that are easy for
students, and students have difficulty in learning are considered as important to guide
education, planners and teachers (Gürbüz et al., 2011). For this purpose, many studies
have been conducted to define the topics that students have difficulty in learning, and
the possible reasons for these Tall & Razali,
”aker,
“ydın,
Zachariades, Christou & Papageorgiou,
Durmuş,
a Dikici & İşleyen,
Yenilmez, 2007; Tatar, Okur & Tuna, 2008; Baki and Kutluca, 2009b; Gürbüz, Toprak,
Yapıcı & Doğan,
. These difficulties are mostly considered to have resulted from
deficiencies in basic concepts/pre-learning, inabilities in problem-solving and lack of algebraic,
geometrical and trigonometric skills Tall, 1993).
Durmuş
a , who stated that no studies on the topics that were more
problematic for students, students had problems in understanding and the reasons for
these problems at primary and secondary education levels were conducted in Turkey,
detected the difficulty indices of all topics in secondary school mathematics curriculum
with a Likert type questionnaire in his study carried to determine the topics that
students perceive as difficult in mathematics classes and the reasons for difficulties and
reported in accordance with the interviews conducted with students that lack of
motivation and the abstractness of the topics were the two important reasons. He also
conducted a similar study with primary school students in order to define the learning
difficulties in primary school mathematics and their reasons in which interviewed
students in order to question the reasons of difficulties and reported that students
defined the topics as complex, meaningless, and they didn t know where to use what
they learnt Durmuş,
b.
Lacking problem solving skills, as a commonly encountered problem in
mathematics learning and teaching, is considered as the basis of these difficulties. The
purpose of many activities intended for problem solving, either routine or not, is
overcoming problems encountered in daily life using alternative methods. At this point,
the vision of primary education mathematics curriculum was re-arranged as raising
individuals, who have such skills as using mathematics in their lives as necessary, establishing
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THE EFFECTS OF MATHEMATICAL MODELLING ACTIVITIES ON THE DIFFICULTY PERCEPTION OF
NUMBERS SENSE AND ACHIEVEMENT AMONG 4TH GRADERS
the relationship between real life situations and mathematics, producing alternative solutions to
the problems they encounter, thinking analytically, and reasoning and associating Ministry of
National Education, 2009). However, it is observed that mathematics course books
rarely include problematic situations that can be encountered in daily life. Yet,
providing students with experiences in which they can study with mathematical
situations that require different interpretations, and enabling them sharing these
experiences with their peers are of utmost importance. One way for students to acquire
these skills is making use model-establishing activities that involve mathematical
modelling (Lesh and Doerr, 2003; English and Watters, 2005).
Mathematical modelling activities are defined as problem solving activities in
which teachers and students reason on real life situations, define, explain and estimate
about these situations, discover, expand and correct their own mathematical structures,
and meanwhile develop models by means of explain, test and review their
mathematical thinking Kaiser & Sriraman,
Eric,
Doerr and O Neill,
.
Previous studies have reported that students, who work with modelling
activities, can successfully overcome multi-component complex problems that reveal
thoughts and develop their existing understanding (English, 2006). Modelling activities
help students use various interpretations and methods in authentic content and develop
internal motivation (Mousoulides et al., 2007). Additionally, as students mathematize
patterns, relations or rules, they engage in important upper-level mathematical thinking
processes, such as explaining, analysing, building and reasoning (Lesh and Doerr,
2003). Therefore, unlike the traditional approach used in mathematics teaching,
modelling activities provide students with rich learning opportunities by encouraging
them to understand their previous learning by thinking more deeply on them and to
produce more generalizable solutions as they re-build them (English, 2003, 2006).
Mathematical modelling activities are believed to have positive contributions to
students difficulty perceptions of mathematics and the mathematics topics, and their
achievement levels. Accordingly, the present research is planned to be studied in two
dimensions. The first step is detecting the topics in numbers learning domain that
students perceive as difficult and define their achievement levels; and the second step is
investigating the effects of mathematical modelling activities on the difficulty
perception and achievement levels in the topics that are perceived as difficult.
The problem statement of the present research was formed as Does mathematical
modelling strategy has any effects on th grade students levels of difficulty perceptions and their
achievement related number sense learning field?
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THE EFFECTS OF MATHEMATICAL MODELLING ACTIVITIES ON THE DIFFICULTY PERCEPTION OF
NUMBERS SENSE AND ACHIEVEMENT AMONG 4TH GRADERS
2. Method
2.1.
Research Design
The present research is conducted in two steps in accordance with quantitative research
methods. The first step was conducted in accordance with survey model in order to
detect the topics in the numbers learning domain in the primary education mathematics
curriculum (Ministry of National Education, 2009) that are perceived as difficult by
students. The purpose of survey model is describing and defining a case that existed in
the past or still exists as it is. There are things that are wanted to be known, but what is
important is observing that appropriately and define it (Karasar, 2006). The second step
employed experiment model with pre-test-post-test and control group. Table 1, presents
the experimental design adopted in the present research through symbols.
Table 1: Experimental design of the research
Groups
Pre-test
Independent Variable
Post-test
GE
NLDADPS
Process projected in the curriculum + Mathematical Modelling
Activities (9 Weeks)
NLDADPS
GC
NLDADPS
Problem-solving activities projected in the curriculum
(9 Weeks)
NLDADPS
2.2. Participant Characteristics and Sampling Procedures
The first step of the present research was conducted on 100 female and 107 male, the
total of 207 students, who studied 4th grade in two state schools in Selcuklu district of
Konya province. These two schools were selected randomly among schools that had 4th
grade education and classes.
The work group of the second step of the present research consists of 61
students, who studied in 4/A and 4/C classes of Esrefoglu Primary Schools in Selcuklu
district of Konya province, which serves under the Ministry of National Education. Of
these students, 30 formed the experiment group and 31 formed the control group. The
students in experiment and control groups were taken in terms of mathematics
achievement as equivalent based on their school reports, and the remarks of their
teachers and the managers of their schools.
2.3.
Data Collection Tools
In order to collect data for the present research, Numbers Learning Domain Achievement
and Difficulty Perception Scale (NLDADPS) Form A and Form ” were employed as pretest and post-test on experiment and control groups, Observation Form for the Evaluation
of the Experimental Procedure was employed to evaluate the implementation on the
experiment group, and Observation Form for the Evaluation of the Problem Solving
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THE EFFECTS OF MATHEMATICAL MODELLING ACTIVITIES ON THE DIFFICULTY PERCEPTION OF
NUMBERS SENSE AND ACHIEVEMENT AMONG 4TH GRADERS
Activities Conducted in Control Group was employed to evaluate the teaching conducted
on control group. All of the scales were developed by the researcher.
NLD“DPS, Form “ and Form ” consists of two parts. Form “ consists of
questions prepared for achievement test, and Form ” consists of the questions in Form
“ besides a standard question with tick boxes below those questions, intended to
receive students remarks on the easiness or difficulty on the question. Students, who
answered questions in Form A, are not asked to answer the same questions in Form B,
but just to present their remarks on the easiness of difficulty of the questions through a
Likert type scale.
While developing the scale, previous studies on students achievement and
difficulty perceptions of the numbers learning domain were studied and the
dimensions of the scale were defined accordingly. The dimensions were defined based
on Van de Wella
s idea that teaching that suits the structure of mathematics
should be intended for three purposes. Accordingly, teaching that suits the structure of
mathematics should help students with;
1. Understanding the conceptual knowledge,
2. Understanding the procedural knowledge,
3. Establishing connections between concepts and procedures.
Based on this basic purpose, the dimensions of the scale were defined as
conceptual knowledge, procedural knowledge and concept-procedure connection.
According to the related literature, conceptual knowledge is classified as Obvious
Conceptual Knowledge and Latent Conceptual Knowledge . “ccording to this
classification, obvious conceptual knowledge refers to processes, such as producing
definitions, choosing the correct definition among the provided definitions, evaluating
judgements, defining concepts related to the content, and explaining the reasons of the
provided procedure; while latent conceptual knowledge refers to such processes as,
making naming and classifying targeted choices, deciding on the correctness of
procedure process, evaluating sample procedure, converting between different
presentation formats and comparing multiplicities (Rittle Johnson and Schneider, 2014).
Accordingly, conceptual knowledge dimension items of the scale were developed based
on obvious conceptual knowledge evaluation criteria, taken its suitability with
students level into and considering that conceptual knowledge can be discriminated
more distinctively thorough procedural knowledge. The scale was developed in
accordance with the three steps suggested by Tracy and Gibson (2005);
First step: Previous studies on the mathematical achievement and difficulty
perception were studied through literature review and the process of mathematical
difficulty perception and evaluation was defined.
Second step: Scale items were developed in this step. This procedure was
conducted in five steps: (1) Related literature was studied and it was decided that
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THE EFFECTS OF MATHEMATICAL MODELLING ACTIVITIES ON THE DIFFICULTY PERCEPTION OF
NUMBERS SENSE AND ACHIEVEMENT AMONG 4TH GRADERS
answer formats of the developed scale were varied as questions that require openended, multiple choice and short answers, (2) an item pool of first forms of the items
was created, and in this trial scale 12 questions for each topic, the total of 82 questions
were included, (3) each item was re-evaluated considering the scale dimensions for
content validity, expert opinions were taken in this step, and validity indices were
calculated with Lawshe (1975) technique, (4) each item was studied in order to detect
any uncertainty in the content of problem/question, and expert opinions were taken in
this step as well, (5) items were tried on a broad sample, with this purpose, trial scale
was conducted on the 207 students, who formed the work group of the present
research.
Third step: In this step, analyses were conducted in three steps: (1) item analysis
was conducted; (2) content analysis was conducted in order to present the content
validity of the scale items and the scale, (3) reliability studies were conducted for the
scale.
In the analysis of the first step of the research, descriptive statistics techniques
used by Durmuş
a , such as difficulty index and arithmetic average X were
utilized. In the analysis of the second step, the relations between pre-test and post-test
scores were investigated. Independent samples t-test and paired samples t-test were
utilized in this procedure. The significance of the differences in the analyses was tested
at (p) 0,05 level.
2.4.
Experimental Manipulations or Interventions
Before starting the implementations of the research, necessary permissions were
received from the officials, pilot implementations were conducted and the teachers,
who carried the implementations, were informed.
During the preparation classes in the implementation process, students were
informed about the mathematical modelling activities in general terms. In these classes,
students were provided with steps of mathematical Modelling Activities (Figure 1),
they were informed about this process and that they would act in accordance with the
steps presented here during mathematical modelling activities.
While planning mathematical modelling activities during experimental
procedure, the modelling steps below suggested by Blum and Niss (1989) and Lesh and
Doerr (2003) were utilized.
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NUMBERS SENSE AND ACHIEVEMENT AMONG 4TH GRADERS
Understanding
and interpreting
the problem
Understanding
tables, graphics and
verbal information
Developing a
mathematical
model
Interpreting the
shared solution
Defining the
connections,
developing
hypothesis,
developing model
Decision Making,
Analysing The
System, Suggesting
New Solutions
Verifying and
Presenting The
Solution
Generalizing and
sharing the solution,
.evaluation
Figure 1: Mathematical Modelling Steps
The researcher prepared real life problems in order to use in the teaching process with
Mathematical Modelling Activities, and the following steps were followed by the
teacher during the implementation of the activities:
1. The teacher taught the lessons reminding the students the implementation
process of mathematical modelling activities, which both the teacher and the
students had.
2. The teacher made the students read the problems in the work sheets in due
course, and the teacher made stories of the problems when necessary in order to
help students understand the problems.
3. Students told each problem first to their group mates, then the group
representative, a different student in each activity, loudly to the whole class.
“dditionally, students answered teacher s guiding questions related to the
problem.
4. In groups, students discussed which of the helping elements, such as tables,
graphics, images, numerical axes, and figures, could be used in models, and used
the elements they decided on.
5. Students were guided for model development to solve the problems, by making
them establish connections between important concepts in the problems and the
other associated concepts, and asking them questions about the procedures they
would use and their reasons for using those.
6. Students were made to discuss the model or models they developed for the
solution, then group representative, again different students in each activity,
introduced the model they developed. These developed models were discussed
in terms of their similarities and differences with the guidance of the teacher, and
they tried to find out the best model for the solution. The selected model was
revised and re-arranged.
7. At the end of each activity, students were asked to write one letter each to the
quasi people who encountered the problem, and a report describing their
models.
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NUMBERS SENSE AND ACHIEVEMENT AMONG 4TH GRADERS
8. Students were asked to develop problems similar to the ones in the activities,
and to check whether the model they developed provided a solution to that
problem as well.
9. The students discussed in which other situations the model they developed
could be used, and were asked to evaluate the modelling process.
During the total of nine-week process (27 class hours) of activities, students
worked on 9 real life problems in groups. The groups were formed of 4 students based
on their mathematics subject school report score as one student with high, two students
with medium and one student with low scores.
Control group lessons were planned according to 4th Grade Mathematics
Teacher s ”ook. Teachers and students conducted problem posing and solving
activities, as they did before.
3. Findings and Interpretations
3.1 Findings on the First Step of the Research
Table 2: Findings on the difficulty perception of numbers learning domain topics
Learning
domain
Sub-learning
domains
Dimensions
Difficulty index
averages
NUMBERS
Natural Numbers
Conceptual Knowledge
3.26
Procedural Knowledge
5,43
Concept-Procedure Connection
6,31
Conceptual Knowledge
3,00
Procedural Knowledge
5,40
Concept-Procedure Connection
4,39
Conceptual Knowledge
1,62
Procedural Knowledge
3,67
Concept-Procedure Connection
6,65
Conceptual Knowledge
1,53
Procedural Knowledge
7,22
Concept-Procedure Connection
9,31
Conceptual Knowledge
1,53
Procedural Knowledge
9,04
Concept-Procedure Connection
10,39
Conceptual Knowledge
2,20
Procedural Knowledge
4,84
Concept-Procedure Connection
13,72
Conceptual Knowledge
3,10
Procedural Knowledge
4,74
Concept-Procedure Connection
9,42
Addition
Subtraction
Multiplication
Division
Fractions
Decimal Fractions
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5,00
4,26
3,98
6,02
6,99
6,92
5,75
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THE EFFECTS OF MATHEMATICAL MODELLING ACTIVITIES ON THE DIFFICULTY PERCEPTION OF
NUMBERS SENSE AND ACHIEVEMENT AMONG 4TH GRADERS
As presented in Table 2, according to the averages of all dimensions of the sub-learning
domains, difficulty perception indices are .
for natural numbers, .
for addition,
.
for subtraction, .
for multiplication, .
for division, .
for fractions, and
.
for decimal fractions. According to these findings, division is perceived as the
most difficult topic by 4th graders, and this is respectively followed by fractions,
multiplication, decimal fractions, natural numbers, addition and subtraction.
Table 3: Findings on the Achievement Levels of Numbers Learning Domain Topics
Learning domain
Sub-learning domains
Dimensions
Achievement
averages
NUMBERS
Natural Numbers
Conceptual Knowledge
3.60
Procedural Knowledge
3,58
Concept-Procedure Connection
3,60
Conceptual Knowledge
4,43
Procedural Knowledge
3,12
Concept-Procedure Connection
2,84
Conceptual Knowledge
4,20
Procedural Knowledge
3,52
Concept-Procedure Connection
2,91
Conceptual Knowledge
3,91
Procedural Knowledge
3,10
Concept-Procedure Connection
2,56
Conceptual Knowledge
4,06
Procedural Knowledge
2,90
Concept-Procedure Connection
2,34
Conceptual Knowledge
3,74
Procedural Knowledge
3,49
Concept-Procedure Connection
2,55
Conceptual Knowledge
3,52
Procedural Knowledge
3,72
Concept-Procedure Connection
2,80
Addition
Subtraction
Multiplication
Division
Fractions
Decimal Fractions
3,59
3,46
3,31
3,19
3,10
3,26
3.34
As presented in Table 3, according to the averages of all dimensions of the sub-learning
domains, achievement levels are .
for natural numbers, .
for addition, .
for
subtraction, .
for multiplication, .
for division, .
for fractions, and . . for
decimal fractions. According to these findings, students were least successful in
division, which is followed respectively by multiplication, fractions, subtraction,
decimal fractions, addition and natural numbers.
Another interesting finding of the present research was that, even two students
got full marks from the achievement scale (Form A), they stated that they perceive the
same questions as difficult in the difficulty perception scale (Form B). As for the reason
for this, one of the students said I don t like mathematics, so I think all questions are
difficult , while the other said mathematics questions are difficult, and I like achieving the
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THE EFFECTS OF MATHEMATICAL MODELLING ACTIVITIES ON THE DIFFICULTY PERCEPTION OF
NUMBERS SENSE AND ACHIEVEMENT AMONG 4TH GRADERS
difficult . The opposite case was experienced with one student, who got bad scores on
the achievement scale (Form A) and stated that he perceived the same questions as easy
in the difficulty perception scale (Form B). When he was asked about that, he said So be
it, easy for me . According to these findings, even students generally have difficulty in
topics they achieve less, the differences in beliefs, attitudes, and self-perceptions of
students can result in some different findings.
3.2 Findings on the Second Step of the Research
Table 4: NLDADPS (Form B) T-test Analysis Results for the Comparison of
Difficulty Perception
Pre-test Scores
Sub-Learning
Domain
Test
Dimension
Group
N
X
S
Pre- Test
Conceptual
Knowledge
Experiment
30
1,14
0,42
Control
31
1,16
0,45
Procedural
Knowledge
Experiment
30
1,32
0,45
Control
31
1,38
0,52
ConceptProcedure Con.
Experiment
30
1,30
0,58
Control
31
1,37
0,69
Conceptual
Knowledge
Experiment
30
1,14
0,40
Control
31
1,14
0,45
Procedural
Knowledge
Experiment
30
1,44
0,60
Control
31
1,43
0,56
ConceptProcedure Con.
Experiment
30
1,43
0,77
Control
31
1,46
0,63
Conceptual
Knowledge
Experiment
30
1,13
0,39
Control
31
1,20
0,41
Procedural
Knowledge
Experiment
30
1,41
0,72
Control
31
1,25
0,56
ConceptProcedure Con.
Experiment
30
1,60
0,83
Control
31
1,58
0,77
Multiplication
Division
Fractions
Pre- Test
Pre- Test
Sd
T
P
59
-0,187
,85
59
-0,496
,62
59
-0,432
,66
59
-0,016
,98
59
-0,046
,96
59
-0,191
,85
59
-0,702
,48
59
-0,984
,32
59
-0,094
,92
“s presented in Table , there aren t significant differences between pre-test scores of
experiment and control groups, in terms of the dimension of sub-learning domain
(p>0,05). Accordingly, experiment and control groups are equal in terms of their pretest difficulty perceptions of multiplication, division and fractions sub-learning
domains.
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THE EFFECTS OF MATHEMATICAL MODELLING ACTIVITIES ON THE DIFFICULTY PERCEPTION OF
NUMBERS SENSE AND ACHIEVEMENT AMONG 4TH GRADERS
Table 5: NLDADPS (Form A) T-test analysis results for the comparison of
Achievement Level
Pre-test scores
Sub-Learning
Domain
Multiplication
Division
Test
Dimension
Group
N
X
S
PreTest
Conceptual
Knowledge
Experiment
30
4,10
0,72
Control
31
4,10
0,67
Procedural
Knowledge
Experiment
30
3,21
1,15
Control
31
3,40
1,25
ConceptProcedure Con.
Experiment
30
3,11
1,08
Control
31
2,96
1,13
Conceptual
Knowledge
Experiment
30
4,32
0,56
Control
31
4,17
0,72
Procedural
Knowledge
Experiment
30
3,26
1,20
Control
31
3,16
1,23
ConceptProcedure Con.
Experiment
30
2,80
1,14
Control
31
2,27
1,01
Conceptual
Knowledge
Experiment
30
3,90
1,01
Control
31
3,96
0,91
Procedural
Knowledge
Experiment
30
3,70
0,86
Control
31
3,40
0,79
ConceptProcedure Con.
Experiment
30
2,76
1,30
Control
31
2,27
0,90
PreTest
Fractions
PreTest
Sd
T
P
59
0,042
,96
59
0,604
,54
59
-0,523
,60
59
-0,901
,37
59
-0,312
,75
59
-1,895
,07
59
0,275
,78
59
-1,398
,16
59
-1,719
,09
“s presented in Table , there aren t significant differences between pre-test scores of
experiment and control groups, in terms of the dimension of sub-learning domain
(p>0,05). Accordingly, experiment and control groups are equal in terms of their pretest achievement levels of multiplication, division and fractions sub-learning domains.
Table 6: NLDADPS (Form B) T-test analysis results for the comparison of
Difficulty Perception
Post-test scores
Sub-Learning
Domain
Test
Dimension
Group
N
X
S
Multiplication
Post- Test
Conceptual
Knowledge
Experiment
30
1,08
0,28
Control
31
1,14
0,27
Procedural
Knowledge
Experiment
30
1,10
0,18
Control
31
1,34
0,40
ConceptProcedure Con.
Experiment
30
1,11
0,21
Control
31
1,27
0,56
Conceptual
Knowledge
Experiment
30
1,08
0,27
Control
31
1,08
0,24
Procedural
Knowledge
Experiment
30
1,11
0,37
Control
31
1,36
0,43
Division
Post- Test
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Sd
T
P
59
-0,864
,39
59
-2,970
,00
59
-1,440
,15
59
-0,058
,95
59
-2,386
,02
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THE EFFECTS OF MATHEMATICAL MODELLING ACTIVITIES ON THE DIFFICULTY PERCEPTION OF
NUMBERS SENSE AND ACHIEVEMENT AMONG 4TH GRADERS
Fractions
Post- Test
ConceptProcedure Con.
Experiment
30
1,13
0,34
Control
31
1,40
0,65
Conceptual
Knowledge
Experiment
30
1,11
0,31
Control
31
1,13
0,28
Procedural
Knowledge
Experiment
30
1,10
0,25
Control
31
1,23
0,44
ConceptProcedure Con.
Experiment
30
1,20
0,50
Control
31
1,50
0,73
59
-2,013
,04
59
-0,285
,77
59
-1,403
,16
59
-1,864
,05
As presented in Table 6, there are statistically significant differences between
experiment and control groups, in terms of multiplication operational knowledge
dimension (t= -2,970 and p<0,05), division operational knowledge (t= -2,386 and p <
0,05) and conceptual knowledge (t= -2,013 and p < 0,05) and fractions concept-operation
connection dimensions (t= - ,
and p ≤ ,
difficulty perception averages, which are
lower in favour of experiment group.
Table 7: NLDADPS (Form A) T-test Analysis Results for the Comparison of
Achievement Level
Post-test Scores
Sub-Learning
Domain
Multiplication
Division
Fractions
Test
PostTest
PostTest
PostTest
Dimension
Group
N
X
S
Conceptual
Knowledge
Experiment
30
4,30
0,75
Control
31
4,11
0,68
Procedural
Knowledge
Experiment
30
4,00
0,98
Control
31
3,33
1,34
Concept-Procedure
Con.
Experiment
30
3,75
1,12
Control
31
3,17
1,40
Conceptual
Knowledge
Experiment
31
4,38
0,81
Control
30
4,09
0,83
Procedural
Knowledge
Experiment
31
3,96
0,95
Control
30
3,25
1,31
Concept-Procedure
Con.
Experiment
31
3,55
1,38
Control
30
2,77
1,43
Conceptual
Knowledge
Experiment
31
4,26
0,86
Control
30
3,93
1,19
Procedural
Knowledge
Experiment
31
4,04
0,66
Control
30
3,55
0,91
Concept-Procedure
Con.
Experiment
31
3,65
1,26
Control
30
2,66
0,46
Sd
T
P
59
0,991
,32
59
2,236
,02
59
1,751
,08
59
1,390
,17
59
2,428
,01
59
2,150
,03
59
1,240
,22
59
2,362
,02
59
2,824
,00
As presented in Table 7, there are statistically significant differences between
experiment and control groups in terms of multiplication procedural knowledge (t=
2,236 and p<0,05), division procedural knowledge (t= 2,428 and p<0,05) and conceptEuropean Journal of Education Studies - Volume 3 │ Issue 1 │ 2017
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Isık N., Pilten P.
THE EFFECTS OF MATHEMATICAL MODELLING ACTIVITIES ON THE DIFFICULTY PERCEPTION OF
NUMBERS SENSE AND ACHIEVEMENT AMONG 4TH GRADERS
procedure connection (t= 2,150 and p<0,05), fractions procedural knowledge (t= 2,362
and p<0,05) and concept-procedure connection dimensions (t= 2,824 and p<0,05)
achievement levels averages, which are higher in favour of experiment group.
Table 8: NLDADPS (Form B) T-test Analysis Results for the Comparison of
Difficulty Perception
Pre-Test and Post-test Scores
Sub-Learning
Domain
Test
Multiplication
Experiment
Control
Division
Experiment
Control
Fractions
Experiment
Control
Dimension
Test
N
X
S
Conceptual
Knowledge
Pre-Test
30
1,14
0,42
Post-Test
30
1,08
0,28
Procedural
Knowledge
Pre-Test
30
1,32
0,45
Post-Test
30
1,10
0,18
Concept-Procedure
Con.
Pre-Test
30
1,30
0,58
Post-Test
30
1,16
0,21
Conceptual
Knowledge
Pre-Test
31
1,16
0,45
Post-Test
31
1,14
0,27
Procedural
Knowledge
Pre-Test
31
1,38
0,52
Post-Test
31
1,34
0,40
Concept-Procedure
Con.
Pre-Test
31
1,37
0,69
Post-Test
31
1,27
0,56
Conceptual
Knowledge
Pre-Test
30
1,14
0,40
Post-Test
30
1,08
0,27
Procedural
Knowledge
Pre-Test
30
1,44
0,60
Post-Test
30
1,11
0,37
Concept-Procedure
Con.
Pre-Test
30
1,43
0,77
Post-Test
30
1,13
0,34
Conceptual
Knowledge
Pre-Test
31
1,14
0,45
Post-Test
31
1,08
0,24
Procedural
Knowledge
Pre-Test
31
1,43
0,56
Post-Test
31
1,36
0,43
Concept-Procedure
Con.
Pre-Test
31
1,46
0,63
Post-Test
31
1,40
0,65
Conceptual
Knowledge
Pre-Test
30
1,13
0,39
Post-Test
30
1,11
0,31
Procedural
Knowledge
Pre-Test
30
1,41
0,72
Post-Test
30
1,10
0,25
Concept-Procedure
Con.
Pre-Test
30
1,60
0,83
Post-Test
30
1,20
0,50
Conceptual
Knowledge
Pre-Test
31
1,20
0,41
Post-Test
31
1,13
0,28
Procedural
Knowledge
Pre-Test
31
1,25
0,56
Post-Test
31
1,23
0,44
Concept-Procedure
Con.
Pre-Test
31
1,58
0,77
Post-Test
31
1,50
0,73
European Journal of Education Studies - Volume 3 │ Issue 1 │ 2017
Sd
T
P
29
0,990
,33
29
2,873
,00
29
2,626
,01
30
0,338
,73
30
0,425
,67
30
1,793
,08
29
1,439
,16
29
2,549
,01
29
1,964
,05
30
1,438
,16
30
0,605
,55
30
0,391
,69
29
0,245
,80
29
2,316
,02
29
2,283
,03
30
0,983
,33
30
0,205
,83
30
0,530
,60
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THE EFFECTS OF MATHEMATICAL MODELLING ACTIVITIES ON THE DIFFICULTY PERCEPTION OF
NUMBERS SENSE AND ACHIEVEMENT AMONG 4TH GRADERS
As presented in Table 8, difficulty perception averages are lower among experiment
group after the experimental procedure for all topics, which were perceived as difficult
before the experimental procedure. These decreases are statistically significant in
multiplication procedural knowledge and concept-procedure connection, division
procedural knowledge and concept-procedure connection, and fractions procedural
knowledge and concept-procedure connection dimensions (p<0,05). There were also
decreases in the difficulty perception averages of all topics after the procedure
conducted in control group, however none of the differences between the pre-test and
post-test averages of control group students were statistically significant (p>0,05).
Table 9: NLDADPS (Form A) T-test Analysis Results for the Comparison of
Achievement Level
Pre-test and Post-test Scores
Sub-Learning
Domain
Multiplication
Test
Dimension
Test
N
X
S
Sd
T
P
Experiment
Conceptual
Knowledge
Pre-Test
30
4,10
0,72
29
-1,383
,17
Post-Test
30
4,30
0,75
Procedural
Knowledge
Pre-Test
30
3,21
0,98
29
-3,819
,00
Post-Test
30
4,00
1,08
ConceptProcedure
Con.
Pre-Test
30
3,11
1,12
29
-2,850
,00
Post-Test
30
3,75
0,21
Conceptual
Knowledge
Pre-Test
31
4,10
0,68
30
-0,083
,93
Post-Test
31
4,11
0,68
Procedural
Knowledge
Pre-Test
31
3,40
1,25
30
0,300
,76
Post-Test
31
3,33
1,34
ConceptProcedure
Con.
Pre-Test
31
2,96
1,13
30
-0,848
,40
Post-Test
31
3,17
1,40
Conceptual
Knowledge
Pre-Test
30
4,32
0,56
29
-0,367
,71
Post-Test
30
4,38
0,81
Procedural
Knowledge
Pre-Test
30
3,26
1,20
29
-2,719
,01
Post-Test
30
3,96
0,95
ConceptProcedure
Con.
Pre-Test
30
2,80
1,14
29
-2,726
,01
Post-Test
30
3,55
1,38
Conceptual
Knowledge
Pre-Test
31
4,17
0,72
30
0,560
,58
Post-Test
31
4,09
0,83
Procedural
Knowledge
Pre-Test
31
3,16
1,23
30
-0,344
,73
Post-Test
31
3,25
1,31
ConceptProcedure
Con.
Pre-Test
31
2,27
1,01
30
-1,944
,06
Post-Test
31
2,77
1,43
Conceptual
Knowledge
Pre-Test
30
3,90
1,01
29
-1,649
,11
Post-Test
30
4,26
0,86
Control
Division
Experiment
Control
Fractions
Experiment
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THE EFFECTS OF MATHEMATICAL MODELLING ACTIVITIES ON THE DIFFICULTY PERCEPTION OF
NUMBERS SENSE AND ACHIEVEMENT AMONG 4TH GRADERS
Control
Procedural
Knowledge
Pre-Test
30
3,70
0,86
Post-Test
30
4,04
0,66
ConceptProcedure
Con.
Pre-Test
30
2,76
1,30
Post-Test
30
3,65
1,26
Conceptual
Knowledge
Pre-Test
31
3,96
0,91
Post-Test
31
3,93
1,19
Procedural
Knowledge
Pre-Test
31
3,40
0,79
Post-Test
31
3,55
0,91
ConceptProcedure
Con.
Pre-Test
31
2,27
0.90
Post-Test
31
2,66
1,46
29
-1,749
,09
29
-3,248
,00
30
0,144
,88
30
-0,812
,42
30
-1,545
,13
As presented in Table 9, achievement level averages are higher among experiment
group after the experimental procedure for all topics, which were perceived as difficult
before the experimental procedure. These increases are statistically significant in
multiplication procedural knowledge and concept-procedure connection, division
procedural knowledge and concept-procedure connection, and fractions conceptprocedure connection dimensions (p<0,05). There were also increases in the
achievement level averages of all topics after the procedure conducted in control group,
however none of the differences between the pre-test and post-test averages of control
group students were statistically significant (p>0,05).
4. Discussion and Conclusion
In the first step of the research, there were differences in rankings in terms of difficulty
perception and achievement, yet especially the first three topics (division,
multiplication and fractions) were common in both terms. Accordingly, division,
multiplication and fractions are the topics that students have most difficulty in numbers
learning domain. This finding is in agreement with some previous similar studies in the
related literature Toluk,
“rdahan and Ersoy,
Soylu,
Durmuş,
”irgin and G(rb(z,
Mısral,
Işık,
Kubanç,
.
Another finding of the first step of the present research was obtained by
studying the scale in terms of its dimensions. Accordingly, students have less difficulty
in conceptual knowledge dimension, while they have more difficulty in procedural
knowledge dimensions. This finding is in agreement with ”aykul
s finding
related to conceptual knowledge that there aren t any concepts that students in the first
five years of primary education will have difficulty in learning among the mathematical
concepts that are aimed to teach to these students.
Pre-learned conceptual knowledge is the basis of procedural knowledge.
Conceptual knowledge covers procedural knowledge, as procedural knowledge covers
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NUMBERS SENSE AND ACHIEVEMENT AMONG 4TH GRADERS
conceptual knowledge. Therefore, there isn t a distinct line separation procedural and
conceptual knowledge (Baki, 1998). Concepts are for procedures that advance step-bystep in mental presentations (Van de Walle, 2004). In other words, conceptual
knowledge covers procedural knowledge and is the pre-condition for it. According to
the findings on the conceptual knowledge and procedural knowledge dimensions in
this context, the difficulties students have is procedural knowledge dimension result
from that the relations between concepts cannot be established adequately, and
students have knowledge of concepts only at cognitive knowledge level that includes
rules and generalizations.
Students have most difficulty in concept-procedure connection dimension. The
failure in acquiring conceptual basis of procedural knowledge results in failure in
establishing the connection between procedural knowledge and concepts, establishing
models, and deciding in where to use the procedures, which presents itself as failure in
problem-solving (Baykul, 2006). This difficulty experienced in concept-procedure
connection dimension in this context, result in difficulty in problem solving and
developing processes in all topics of numbers learning domain. Because problemsolving is a scientific method as well, it requires critical thinking, creative and reflective
thinking and use of analysis and synthesis skills (Reusser and Stebler, 1997; Cited in:
Soylu and Soylu, 2006). From this aspect, the difficulty in concept-procedure connection
results from the deficiencies related to upper-level cognitive skills. Among the reasons
for failure, that we cannot provide students with help in relational understanding plays
an important role (Baykul, 2006).
According to the findings obtained in the second part of the present research,
mathematical modelling activities conducted on the experiment group were more
effective than the problem-solving activities conducted in control group in both
procedural knowledge and concept-procedure connection dimensions. As stated in the
findings of the first of the present research, procedural knowledge cannot be separated
from conceptual knowledge distinctly, and considering that these concepts and
connections between procedures underlie the procedural knowledge, we can claim that
mathematical modelling activities are pretty effective in providing concept-procedure
connection.
Another finding of the present research is that, in the class on which
mathematical modelling activities were conducted, students were more willing to
participate in the lesson, enjoyed implementing activities, and the teacher was more
willing to teach the lesson. Accordingly, we can claim that mathematical modelling
activities are also effective in developing positive attitudes towards mathematics.
Consequently, it can be stated that mathematical modelling activities enable
students be active in learning process more than traditional problem-solving activities
(Doruk and Umay (2011), result in developing positive attitudes towards mathematics
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NUMBERS SENSE AND ACHIEVEMENT AMONG 4TH GRADERS
(Boaler, 2001; Korkmaz, 2010; Mehraein and Gatabi, 2014a), is a very effective method in
establishing connections between concepts and procedures and acquiring metacognitive skills (English and Wattters, 2004; Blum and Borromeo Ferri, 2009; Olkun,
Şahin, Dikkartın and G(lbağcı,
Sağırlı,
Hıdıroğlu,
.
The findings of the present research are on the effects of mathematical modelling
activities on mostly cognitive processes. Further studies can be conducted on both
cognitive and affective processes. Teaching with mathematical modelling activities can
be compares with other methods, and the differences of modelling process can be
presented more clearly. Considering that mathematical modelling competencies are
more process oriented that being product oriented, therefore process-oriented
evaluations instead of studying the modelling products can be provided.
Through the use of mathematical modelling activities in daily mathematics
classes, students can develop modelling skills, and achieve in modelling a real-life
problem on their own (Maaß, 2005). Accordingly, for the students to be more successful
in problem-solving, mathematical modelling activities can be included in the
curriculum more in order to develop students modelling skills.
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THE EFFECTS OF MATHEMATICAL MODELLING ACTIVITIES ON THE DIFFICULTY PERCEPTION OF
NUMBERS SENSE AND ACHIEVEMENT AMONG 4TH GRADERS
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