Triangulation Matrix

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


literature review presented in chapter two. Table 1, the triangulation matrix of data

sources, shows the sources of data for the respective research questions.


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? √ √


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


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.


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.


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


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


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.


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.


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


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


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. .


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


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


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.


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? √ √


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.

Documents

Triangulation Matrix