triangulation matrix
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USING MANIPULATIVES TO IMPROVE PROBLEM SOLVING SKILLS. 1
CHAPTER FOUR
Data Presentation and Analysis
This chapter focused on the presentation and analysis of data gathered using four data
collection tools: observation checklist, tests, questionnaires and an interview schedule.
This paper critically explored the use of manipulatives to improve students’ problem
solving skills in mathematics. The purpose of this research was to help my struggling
students:
Reach their full potential especially in the learning of mathematics.
To understand the concepts and functions in mathematics that cause
frustration and result in their inability to comprehend.
I would also like to bring greater comprehension to my students, when solving
mathematical problems. This would bring a fresh perspective from what they are
accustomed to. With the understanding that students learn differently it was quite
appropriate to research the effectiveness of using manipulatives in order to transition
students from the concrete level to the abstract.
The intervention was geared towards improving grade four students’ performance
in mathematics at a particular school. It was carried out over a period of eight weeks.
This intervention involved 15 students who made up the sample. Four teachers
participated by responding to a questionnaire. The data collected are presented using
tables and figures. The sources used to support the data analysis were garnered from the
USING MANIPULATIVES TO IMPROVE PROBLEM SOLVING SKILLS. 2
literature review presented in chapter two. Table 1, the triangulation matrix of data
sources, shows the sources of data for the respective research questions.
USING MANIPULATIVES TO IMPROVE PROBLEM SOLVING SKILLS. 3
Table 1
Triangulation Matrix of Data Sources
Questions Pre-test
Post-test
Interview
Questionnaire
Observation checklist
1. How were my students √ √ √performing in mathematicsprior to the intervention?
2. What were the reasons for my students' poor perform- √ √ance in mathematics?
3. How did the use ofmanipulatives impact on √ √ √my students' problemsolving skills?
4. What is the level of mystudents' performance in √ √mathematics after theuse of manipulatives?
5. How do my students nowfeel about mathematics? √ √
USING MANIPULATIVES TO IMPROVE PROBLEM SOLVING SKILLS. 4
Question 1. How are my students currently performing in mathematics?
The results and findings for this question were obtained from observing the students
along with their performance on the pre-test.
Table 1
Pre-test Results
Students Pre-test Scores %
A 0
AB 45
AC 15
AD 25
AE 15
AF 25
AG 60
AH 50
AI 40
AJ 45
AK 85
AL 65
AM 20
AN 40
AO 30
USING MANIPULATIVES TO IMPROVE PROBLEM SOLVING SKILLS. 5
The pre-test results indicated that six of 15 students or 40% received non- mastery scores which
is in the 0-25 range for this particular mathematics test. Before the intervention period I
observed that these students displayed low self-confidence in class. They did not actively
participate in the lesson and, did not work independently. The results showed that these students
were performing below average; they had difficulty understanding worded problems. Also
another six of 15 or 40% of the students received an average of between 26 and 50%. These
students were performing at the near mastery level. They also had difficulty interpreting the
questions. They were unaware of which operation to use when solving the mathematical
problems presented.
However, three students or 20% from the total number of 15 students received mastery which is
50% and over. The results indicate that only three students are performing at the required level.
These students followed written and oral instructions correctly and found it easier to make
connects especially relating to abstract contents. Based on these results they obviously
understood the problems they received. .
Question 2. What are the reasons for my students’ poor performance in
mathematics?
The results for this question were garnered from the following instruments: the teachers’
questionnaire and students’ interview schedule.
USING MANIPULATIVES TO IMPROVE PROBLEM SOLVING SKILLS. 6
30%
50%
15%5%
Results from the Teachers' Ques-tionnaire
Reading DifficultyMaking ConnectionDistractionAbsenteeism
Figure 1 Result from the Teachers’ Questionnaire
The result from the teachers’ questionnaire showed that there were many factors that
contributed to students poor performances in mathematics. The results indicated that 50% of the
students have difficulty making connections between mathematics and real life situations. It also
indicated that 30% of students have reading difficulties which pose a number of challenges when
solving worded problems. Another factor is that 15% of students are distracted in mathematics
class. This could have been contributed to a number of factors such as strategies employed, as
well as physical or emotional problems. There is also the issue of absenteeism. There were 5% of
the students who did not attend school regularly for various reasons. This could be reason for
their poor performance in mathematics. Overall it can be concluded that the majority of the
students had difficulties making connections between mathematics and everyday life and also the
ability to read.
USING MANIPULATIVES TO IMPROVE PROBLEM SOLVING SKILLS. 7
Figure 2 Students’ Interview Results
Students were interviewed at the start of the intervention and at the end. The responses here
were obtained based on items one to six. Based on the students’ responses another reason for my
students’ poor performance in mathematics is their attitude towards the subject.
The result from the students’ interview schedule showed that the attitude displayed by students
when it pertained to the subject of mathematics varied. The responses in figure 3 were given by
the students. These students were allowed to select more than one option from the responses
given based on their feelings toward the subject area in question. Eleven or 73% stated that
mathematics was boring and lacked fun and games. Some of these students are also distracted
and they do not participate in class or work cohesively in groups. Ten or 67% of these students
claimed that there were too many rules to learn and they sometimes forgot which one to use. In
most cases these are the students who did not follow instructions appropriately. However, other
students expressed negative connotation about mathematics. Seven or 47% of the students
boring, lack fun games
too many rules
poor reading different operations
too much work
incorrect answers
0
2
4
6
8
10
12
14
16
1110
78
15
5
Results from the Students' Interview
Students' Attitude Toward Mathematics
No.
of s
tude
nts
USING MANIPULATIVES TO IMPROVE PROBLEM SOLVING SKILLS. 8
proclaimed that their reading was poor and they had difficulty processing the information. These
students were sometimes sad and showed little or no interest in mathematics class.
Subsequently, there were eight or 53% of the students who had difficulty using the different
operations at the correct time. These students along with others also attribute this to the many
rules to remember. On the other hand, the entire sample declared that mathematics was too
much work especially for young children. While five or 33% were disappointed and frustrated
that they always get the answers incorrect even if they worked very hard. These students’ self-
esteem was also very low.
Question 3. How did the use of manipulatives impact on my students’ problem
solving skills?
The results and findings for this question were obtained from a summary of the
observation carried out during the intervention.
Table 2
Summary of students’ observation checklist
Categories
Items observed Always Sometimes Rarely Never
-# % # % # % # %
Students have sense of 10 67 5 33 0 0 0 0self- confidence.
Students are able to use 11 73 1 7 3 20 0 0manipulatives to demonstraterules /concepts to associateaddition and subtraction.
Students require teachers' 6 40 3 20 5 33 1 7prompting and questioning to help remember the
USING MANIPULATIVES TO IMPROVE PROBLEM SOLVING SKILLS. 9
operations.
Students are able to use 12 80 3 20 0 0 0 0manipulatines to solve problems and completetask accurately.
Students can demonstrate using 9 60 4 27 2 13 0 0manipulatives in regrouping and showing representationin problem solving.
Students are able to make 10 67 5 33 0 0 0 0connections between mathematicalconcepts and everyday life.
Students can model concepts using manipulatives. 13 87 2 13 0 0 0 0
Students are able to demonstrate concepts learnt after using 11 73 3 20 1 7 0 0manipulatives.
Students can make the connection between concrete and abstract 10 67 4 27 1 6 0 0representation.
Students worked effectively in 13 87 2 13 0 0 0 0groups.
Students showed greater level 9 60 5 33 1 7 0 0of interest
Students participate more when 13 87 2 20 0 0 0 0using manipulatives
Students were curious to find 13 87 2 20 0 0 0 0
solutions to mathematics concepts when using manipulatives.
USING MANIPULATIVES TO IMPROVE PROBLEM SOLVING SKILLS. 10
The result from the observation checklist indicated that all the students had some level of self-
confidence 10 students or 67% of the samples observed over the eight weeks period were always
confident, with five being confident sometimes. The same number of students also demonstrated
making connections between mathematical concepts and everyday life as well as concrete and
abstract representation. It also showed that five students or 33% of the sample were also able to
make connections sometimes. However, of the 15 students observed six or 40% required
prompting from teacher in order to model what was taught, 3 or 20% needed assistance
sometimes, while only five or 33% rarely did.
The data also revealed that 12 students or 80% of the students always were able to use
manipulatives to solve problems and complete tasks accurately. They visualized what was
required of them along with the use of hands on tools which helped them to make connections
more readily. It also indicated that 87% of the students observed worked effectively in groups;
they participated more and, were curious to find solutions. These students were also able to
model concepts using manipulatives. Nevertheless, it showed that 27% or less of these students
sometimes had difficulty in carrying out their task effectively when using manipulatives. They
needed prompting and extra help in making the connections of using objects to solve real life
problems.
Results from the observation checklist also indicated that majority of students; nine or 60%
showed a greater level of interest and were able to demonstrate concepts learnt after the
intervention. The minority were however rarely or sometimes able to demonstrate what was
taught. Similarly, 60% of the students were able to demonstrate using manipulatives in
regrouping and showing representation in problem solving. These students were also able to
make connections between mathematical concepts and everyday life situations.
USING MANIPULATIVES TO IMPROVE PROBLEM SOLVING SKILLS. 11
Question 4. What is the level of my student’s performance in mathematics after the
use of manipulatives?
The result and findings for this question were obtained from observing the students and
the students’ post- test results.
Table 4
Result of Students’ Post-test Scores
Students Percentage
A 25
AB 60
AC 40
AD 25
AE 50
AF 50
AG 75
AH 50
AI 90
AJ 75
AK 90
AL 65
AM 40
AN 55
AO 40
RANGE= 0-25 26-50 51-100
USING MANIPULATIVES TO IMPROVE PROBLEM SOLVING SKILLS. 12
non-mastery near-mastery mastery
For the post test the scores were divided into three ranges. Non mastery represents scores
for 25% and below, near mastery 26% to 50%. This indicates that they are performing at an
average level but need improvement. Mastery shows that student’ are performing above average
and have a good grasp on the subject matter. The scores presented in table 3 are an indication of
how the students were performing after the intervention programme. It showed that 12 students
or 80% of the students’ scores were improved. These students’ self-confidence and interest has
also shown remarkable improvement. Three students or 20% of the students’ scores remained the
same after the intervention period. These students are the ones who sometimes had difficulty in
modeling concepts.
Table 5
Comparison of students’ Pre-test and Post-test Scores
Students Pre-test Post-test % Gain (+ ) or Loss (-)
A 0 25 +25
AB 45 60 +15
AC 15 40 +25
AD 25 25 0
AE 15 50 +35
AF 25 50 +25
AG 60 75 +15
AH 50 50 0
AI 40 50 +10
USING MANIPULATIVES TO IMPROVE PROBLEM SOLVING SKILLS. 13
AJ 45 75 +30
AK 85 90 +5
AL 65 65 0
AM 40 40 0
AN 40 55 +15
AO 30 40 +10
RANGE= 0-25 26-50 51-100
non-mastery near-mastery mastery
This data presented indicated the improvement in students’ performance after the use of
manipulatives. When the pre-test was administered six students received non-mastery, six near-
mastery and three achieved mastery. However, after the intervention period a post test was given
to the same sample. Two students scored 25% and are still at the non-mastery level. After
comparing the pre-test and post- test results, although these students are still at the non-mastery
level it was shown that there was improvement in the test scores of student A. This student had
actually scored zero on the pre-test. Student AD scored 25% in both test.
Secondly, there was also improvement in the near-mastery category. After the post- test there
were now seven students in that category, one more than after the pre-test. It showed that 40%
gained improvement in their problem solving scores. Furthermore, six students instead of three
on the pre-test gained mastery, these students are now performing above the average range.
These six students who gained mastery understood the concept of relating real life situations to
problem solving. .
USING MANIPULATIVES TO IMPROVE PROBLEM SOLVING SKILLS. 14
0-25 26-50 51-1000
1
2
3
4
5
6
7
8
Comparison of Pre-test and Post-test Result
pre-testpost-test
SCORES %
students
Figure 3 Comparison of students’ pre-test and post test scores
The data presented illustrated what has been presented on the pre-test and post-test table. The
comparison between these tests has shown that the students who did the eight weeks intervention
using manipulatives have improved in their level of problem solving skills. It also showed that
from a sample of 15 students and the score from the pre-test result that there were 6 students or
40% who were scoring non-mastery, after the post –test this number decreased to two students or
13%. The post-test has shown a decrease of 27% of the students who are performing at that
level. There was also improvement in the post test score as none of the students scored zero; this
was encouraging.
There was also an increase in the number of students who scored near mastery. There were six
students or 40% from the pre-test. Comparison with the post- test now showed seven students or
USING MANIPULATIVES TO IMPROVE PROBLEM SOLVING SKILLS. 15
46%. The score indicated that six students or 40% have gained mastery compared to three
students or 20% that was recorded in the pre-test.
Question 5. How do my students now feel about mathematics?
The results and findings for this question were obtained from interviewing the students
during week eight and observing them.
Figure 4 Students’ Interview Result after Intervention
The students were interviewed after the intervention period to establish their attitude toward
mathematics after the use of manipulatives. These students responded to part two of the
interview. There were 13 students who admitted that they were now more comfortable to solve
worded problems when they used manipulatives. There were 15 students who proclaimed that
their self -confidence was improved based on the class attendance, the fun they were having
while solving problems and getting better grades.
All 15 students were happy to attend classes based on the camaraderie in the classroom.
They were curious to find out what would happen next. They were all anticipating what they
think the class would be like for that session. Six of them really admitted that they wanted to
more com-fortable
more confident happy to attend class
being held back in class
excited02468
10121416
1315 15
1
12
Results from Students' Interview
Students Attitude toward Mathematics
No.
of S
tude
nts
USING MANIPULATIVES TO IMPROVE PROBLEM SOLVING SKILLS. 16
play with the manipulatives before and after mathematics class. They all stated that the class was
no longer boring. However, there was one student who believes that he was sometimes
uninterested because he believed was being held back in class. Overall, they were all excited
except for three students.
I also observed that the use of manipulatives to teach lessons can be time consuming and
it does not allow for flexibility. The teacher has to monitor the students and how they interact
with the objects. Students had to be given duty rosters because when they are not monitored
properly when using the tools it can result in a messy and disorderly classroom. However, when
these tools are used for the desired purposes and ground rules are set, then the students benefit
tremendously from the use of manipulatives, especially the slower learners. I discovered that
anything in the environment can be used as manipulative to teach a lesson. It would be more
sophisticated if the teacher could own the manufactured tools, but using the make shift or refuse
provided the similar experiences.
USING MANIPULATIVES TO IMPROVE PROBLEM SOLVING SKILLS. 17
Questions Pre-test Post-test
Interview Questionnaire
ObservationChecklist
How are my students currently performing in mathematics?
√ √ √
What are the reasons for my students’ poor performance in mathematics?
√ √
How does the use of manipulatives impact on my students’ problem solving skills?
√ √ √
What is the level of my students’ performance in mathematics after the use of manipulatives?
√ √
How do my students now feelabout mathematics? √ √
USING MANIPULATIVES TO IMPROVE PROBLEM SOLVING SKILLS. 18
Sub-Research Questions
1. What is the current performance of my students in Science?
2. What are the students’ attitudes toward Science after being taught using cooperative
learning?
3. What is students ‘time on task’ during the use of cooperative learning?
4. Will my student’s grades improve after the implementation of cooperative learning
strategy?
These questions should be used to do the triangular matrix plus others from the tools that are
relevant to the topic.
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