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CHAPTER – III

METHODOLOGY

3.1. Introduction

This chapter deals with the methodology employed for this study. The

research design used in this study is experimental method. The study intends

to find out the ‘effectiveness of remediation on attainment of MLL in

Mathematics among V standard students from Shimoga district of Karnataka

State.’ This chapter presents a detailed description about the tools employed;

techniques used for data collection and method applied to analyze the data.

This chapter includes information on

• Design of the Study.

• Variables of the Study.

• Sample Selected for the Study

• Tools Used for the study.

• Data collection.

3.2. Design of the Study

This is an experimental study with pre and post-test design. In this study the

investigator has selected 18 sub competencies from V standard text book of

mathematics. These 18 sub competencies have drawn from seven main

competencies. These sub competencies were selected because in all the

selected schools these competencies were taught in first semester. Based on

these MLL competencies, investigator adapted a standard test developed by

Kashinath (2005). The adaptation was made in the light of competencies

taught. The adapted test was also tried out on 30 students from V standard.

The pre test was finalized by dropping the competencies which were very

easy and were also very difficult. The opinions of various experts were also

taken for finalizing the adapted test. The test was used as pre and post test

for assessing the effectiveness of the intervention for learning non mastered

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competencies. The investigator developed teaching strategies for teaching

each selected sub-competency. The detail of these strategies discussed at

the end of this chapter.

The investigator used these strategies for teaching all non-masters from rural

and urban schools taken for the study. The investigator took one session in

each school on alternative days for the experimental group. In this way the

investigator covered all the non-mastered competencies during two months of

intervention. After two months of intervention for the experimental groups, the

investigator conducted post test for both controlled and experimental groups.

The control group students were attending regular classes whereas students

from experimental group were attending the intervention class outside the

class room which was taken by the investigator himself. The performance of

the students from pre and post tests was analyzed to assess the effectiveness

of intervention on learning non mastered competencies.

3.3. Variables of the Study

The variable related for the study on “Diagnosis based Remediation on

Attainment of MLL in Mathematics among V Standard Students from Shimoga

District”, included remedial teaching which was considered as independent

variable, attainment of MLL competencies was dependent variable. The other

variables related to the gender, locales of the school (Rural/Urban) were

considered as demographic variables.

3.3.1. Dependent Variable

The dependent variable selected for the present study was level of attainment

on MLL competencies in Mathematics by the non-masters, selected from 5th

standard of rural and urban schools.

3.4. Sample and Sampling Procedure

The present study is conducted in three phases. In the First phase the

researcher visited schools to gather the data regarding the students’

backgrounds from selected schools. The investigator conducted pretest on all

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the students of 166 schools to diagnose non masters in mathematics. This

was done to select sample of the study representing the total population of

non-masters from these schools. Based on pre-test performance, the

investigator selected 10% of total population on random basis from these

selected schools of Shimoga district. Investigator covered the 10% 0f students

carefully from both rural, urban schools and also from boys and girls to have

equal representation. The investigator maintained the same 10%

representation in selection of gender. Total 166 schools selected from the all

seven blocks for meeting 10% representation of the sample selected.

Information regarding type of school, infrastructure etc., was obtained from

Sarva Shiksha Abhiyan (SSA) office, Shimoga district of Karnataka State.

After selection of 10 % of students from 166 Government primary schools of

seven blocks of Shimoga district, researcher visited all the Block Education

officers and Block Resource Persons to get necessary permission for

collecting data needed for the study.

In the Second Phase the researcher collected the data from these schools.

The researcher visited all the schools in person and administered the pre-test

himself with the help of BRC’s and teachers and assured the confidentiality of

the data. The sample of the study initially includes 1457 students from 166

schools in 7 Blocks. All the Government Primary schools of Shimoga district

formed the units of the study. 10% of students selected from each block to

achieve first objective of the study. This sample was stratified on the basis of

rural and urban locale. All the V standard students studying in these schools

initially formed sample of the study. At the end of first semester the students

were given pretest to assess attainment of the MLL competencies taught in

Mathematics.

In order to serve the second objective of the study, four schools out of total

selected schools were randomly selected from Shimoga district, these schools

were selected as the number of non masters were more and the basic

infrastructure, number of teachers, medium of instruction and socioeconomic

status of the families were found to be similar and matching. On the basis of

performance on competencies included in the pre- test of Mathematics, the

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masters and non-masters were identified. Those students who were found

achieving less than 80% of the competencies were non-masters and they

were considered for experimental group for this study.

Figure 3.4.1 Selection of the Sample is diagrammatically as shown below

Government Primary schools In Shimoga district (166)

MLL Competencies Based Pre-test (competencies, which were taught in I semester)

Non-Masters (100)

Experimental Group(50)

Total sample

MLL competencies based Post-test

Remedial teaching through appropriate method by investigator

GHPS - 1 GHPS - 2 GHPS - 3 GHPS - 4

Non Masters

Non Masters Non Masters Non Masters

Control Group(50)

Conventional teaching through regular teachers

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The sample of the study consisted 100 non-mastered students from class V.

These non-masters were randomly selected from the initial survey which was

conducted on 10% students from schools of seven blocks of Shimoga district

Karnataka state. Table 3.4.1(a) gives description of the sample.

Table 3.4.1 (a) : Selection of schools list(block,gender and locale wise) for the sample selection

Sl No Block Locale No. of Schools Boys Girls Total

urban 8 24 47 71 01 Bhadravathi

rural 21 105 81 186

urban 4 21 19 40 02 Sagar rural 23 91 102 193

urban 1 3 3 6 03 Hosanagar rural 19 75 75 150

urban 4 36 25 61 04 Shikaripura rural 22 80 107 187

urban 4 27 24 51 05 Shimoga rural 24 103 107 210

urban 1 9 9 18 06 Soraba rural 14 56 66 122

urban 1 4 8 12 07 Thirthahalli rural 20 70 80 150

Total 166 704 706 1457

Investigator conducted pre-test based on the preliminary survey conducted in

these schools immediately after the completion of first semester. The

investigator met the headmasters and authorities of the schools selected for

the initial survey and explained in detail the process of conducting test. The

investigator conducted pre-test after getting approval of the authorities to

know the MLL attainment levels in mathematics. After identifying the level of

attainments the non-masters were listed after matching basic infrastructure,

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number of teachers, medium of instruction and socioeconomic status of the

families and level of their attainments on the competencies covered in pretest

as masters and non-masters. Finally four schools were selected which were

matched on above mentioned dimensions. In the selected four schools, two

were urban schools and another two were rural schools. All these schools

were Kannada medium government primary schools and they followed the

same syllabus. The number of the schools and number of the students

selected are given in table 3.4.1b & 3.4.1(c).

Table: 3.4.1(b): List of the No of Non Master Students and Schools Selected for the Study

Sl. No.

Name of the school

No. of non master students

Rural/Urban Experimental/Control

group

1 GHPS, Hosamane

25 urban experimental

2 GHPS, Anandapuram

25 rural experimental

4 GHPS, Harakere

25 urban control

3 GHPS, Kashipura

25 rural control

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Table: 3.4.1(c ): Distribution of Sample Based on Gender and Locale

TOTAL SAMPLE

100 NON-MASTER STUDENTS

URBAN RURAL (50) (50)

CONTROLLED EXPERIMENTAL CONTROLLED EXPERIMENTAL GROUP GROUP GROUP GROUP (25) (25) (25) (25)

BOYS GIRLS BOYS GIRLS BOYS GIRLS BOYS GIRLS (12) (13) (12) (13) (13) (12) (13) (12)

The sample consisted of 100 non-master students of V standard from two

urban and two rural Government Primary schools. Among 50 students, 25

students from urban school and 25 students from rural school were selected

for experimental group and remaining 50 students, 25 students from urban

school and 25 students from rural school considered for control group for

assessing effectiveness of the experiment.

The investigator conducted intervention for teaching non mastered

competencies with the selected teaching strategies which had eight

components i.e. sequencing and segmenting, drill-repetition and practice-

review, directed questioning and responses, control difficulty or processing

demands of the task, use of technology, group instruction, supplements to

teacher and peer involvement, strategy cues to non mastered students of

experimental group from rural and urban areas for two months (Jan to Feb

2008). The controlled group selected was attending regular classes.

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3.5. Tool Used for the Study

The present study attempted on diagnosing non masters on the

selected competencies hence the investigator used MLL competency based

test in mathematics. Hypotheses 1 was matched on their MLL attainments,

age, gender and locale and other variables stated above regarding basic

infrastructure, number of teachers, medium of instruction and socioeconomic

status of the families, level of their attainments on the selected competencies.

3.5.1. MLL Competency Based Test in Mathematics

For assessing, MLL competencies selected for the study a test was

adapted after the test developed by Kashinath et:al.,( 2005).The investigator

adapted this test in the light of competencies taught in first semester to V

standard studying in the selected Government primary schools of Shimoga

district. This original test included more number of competencies and sub-

competencies. The investigator confined to the competencies taught only in

first semester to V standard students. Hence the test included the following

competencies listed below.

3.5.2. Competencies covered in Pre Test: i) Numbers

ii) Different numerals

iii) Fundamental operations

iv) Fractions

v) Decimals fundamental operations

vi) Decimals addition and subtraction with mixed operations

vii) Angles

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Table 3.5.1. (a) Blue print of Pre-test

S.No Contents Knowledge Understanding Application Skill Total1 Numbers 1(1) 2(1) 1(1) 1(1) 5(1) 2 Different

numerals 1(1) 1(1) 2(1)

3 Fundamental Operation

2(1) 1(1) 1(1) 4(1)

4 Fraction , decimal and percentage

2(1) 1(1) 1(1) 4(1)

5 Decimal fundamental operation

1(1) 1(1) 1(1) 3(1)

6 Decimal addition, subtraction and mixed fraction

1(1) 1(1) 1(1) 3(1)

7 Geometry angles

2(1) 1(1) 3(1)

Total 5(1) 10(1) 4(1) 5(1) 24(1)

Note:Number outside the bracket indicates number of questions and number

in the bracket indicates marks

Table 3.5.1. (b). Distribution of marks to different instrumental objectives Sl.No. Instrumental Objectives Marks Percentage

1 Knowledge 5 22 2 Comprehension(Understanding) 10 38 3 Skill 3 12.5 4 Application 6 25

Table 3.5.1(c). Content weightage Sl.No. Content/Competencies Marks Percentage

1 Numbers 5 22.5 2 Different Numerals 2 8 3 Fundamental Operations 4 16 4 Fractions, Decimals and Percentage 4 16 5 Decimals Fundamental Operations 3 12.5 6 Decimals Addition Subtraction with mixed

operations 3 12.5

7 Angles 3 12.5

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Table 3.5.1(d). Question Type Weightage

Sl.No. Type of Questions Marks Percentage 1 Objective Multiple Choice Questions 24 100%

One of the objectives of the study was to explore the MLL mastery level in

mathematics for this; the review of related literature was done which revealed

that the mastery level expectations were different for different studies. Ideally

speaking, all children achieving all competencies were indicative of mastery

achievement. However, this was hardly possible owing to various factors. It

was found that the researcher had favoured mastery of at least 80 percent of

the prescribed competencies as the mastery level hence the investigator also

fixed the mastery level at 80 percent. The precise instructional objectives

were formulated for adapting MLL competencies to be considered for

finalizing pre test for this study.

3.5.3. Item construction

Items were generated using the most popular and widely used amplified

objectives approach advocated by Pop Ham and Backer (1973). According to

this approach, a set of items generated to measure an objective includes a)

Response description, b) Content limits, c) Item format specification and d)

Item directions.

3.5.4. The Competencies

All the competencies selected in the remedial teaching were included for

developing criterion referenced achievement test to measure the attainment of

competencies. Following were the competencies on which the criterion

referenced achievement test was developed. The competencies listed in table

3.5.3 included for developing this test.

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3.5.5. Number of sub Competencies Selected

Determining the mastery of the competency depends on the nature of the

competency being measured. For the competencies like writing the numbers

of 5, 6, 7 digits, mastery can be determined by making students write the

numbers 5, 6, 7 digits once. Competencies like identifying their place values

and carryout addition, subtraction, multiplication and division in decimals

addition and subtraction with mixed fractions, mastery cannot be determined

by one or two items. Hence, depending upon the nature of the competency

the number of sub-competencies per competency needs to be covered. While

deciding on the sub competencies required for measuring mastery, items

were selected.

The performance of the students on sub competencies and overall

performance of main competency was considered for assessing their

attainment levels. Students who obtained 80 percent or more in each

competency were designated as masters of the competency measured.

Those who mastered 80 percent or more of the selected competencies, on

the pre-test were considered as the masters of overall competency. Details of

number of sub competencies considered for in each competency along with

the cut of point fixed for mastery level by each competency is given in table

3.5.3.

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Table 3.5.3: Number of Competencies and Sub Competencies Selected for Pre test along with Mastery Level

Sl No Competency Sub competency

Total marks for sub

competencies

Cut of point fixed for

mastery level 80%

• Read and write big numbers according to their place values

• Find out the use of numbers having Ten thousand and Lakh place values

• Expand and arrange the numbers according to the place values, arrange them in ascending and descending orders

1 Numbers

• Write the numbers in order from unit to higher places by increasing the value ten times

5

4

• Know that the numbers indicating quantities in regional and international languages are different

2 Different numerals

• Write the international numerals and numerals of other languages and convert them mutually

2

1.6

• To carry out fundamental operations of five and six digits numbers and solve problems you come across in day to day situations

3 Fundamental operations

• To develop skills in writing numbers according to their place values and carry out addition, subtraction, multiplication and division

4

3.2

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• Ability to convert the given fractions into equivalent fractions

• Ability to analyze and solve problems of fractional operations(addition and subtraction) in daily life with proper understanding

• Ability to arrange the given three or four decimal places in ascending and descending order

4 Fractions, decimals and percentage

• Ability to find the relationship between fractions, decimals and percentage

4

3.2

• Skill in using fundamental operations of addition and subtraction with decimal numbers in your daily life

5 Decimals fundamental operations

• To carry out fundamental operations of 5 and 6 digit numbers and solve problems you come across in day-to-day situations.

3

2.4

• To carry out fundamental operations of decimal numbers and solve problems you come across in day to day situations.

6 Decimals addition and subtraction with mixed operations • To develop skill in writing

numbers according to their place values and carryout addition, subtraction with mixed operations.

3 2.4

• Constructing angles of different measurement using protractor

7 Geometry-angles

• Skill in using geometrical instrument accurately

3 2.4

3.6. Preliminary Try Out

3.6.1. Standardization of the Test

The quality of the MLL competency based test was assessed in terms of two

characteristics, Reliability and validity for this purpose. This test was

administered on 30 students from Government Lower Primary School,

Venkateshnagara and Government Lower Primary School, Lakkinakoppa,

Shimoga district to establish the reliability and validity.

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3.6.2. Administration

The investigator used the adapted test which was based on MLL

competencies. This pre-test was used for diagnosing non-mastery of the

students. The investigator with the help of teachers administered the test on

the 1457 students in 166 Government primary schools of seven blocks of

Shimoga district. The norms were followed for administration of the test as

given in the original test developed by H.M. Kashinath.

The pre-test adapted was subjected to a pilot study to establish the

discrimination power of each of the items in the test. Instructions to the

students were clearly given on the top sheet. The test was administered to a

sample of 30 students from Government Lower Primary School,

Venkateshnagara and Government Lower Primary School, Lakkinakoppa,

Shimoga district. These two schools were selected through random sampling

technique and selection of students for this was done in consultation with the

concerned teachers. The investigator selected five high achieving, five

average achieving and five low achieving students from each school as

sample.

Instructions to students for taking the test were adapted from the original test

keeping in mind their age level and mental ability. The general guideline for

giving the test was prepared for the use of the teacher. The teacher read out

the instructions in the beginning of the test, as the target group was V

standard students. Specific guidelines were included along with the questions.

All questions were answered in the space provided in the question paper as

all the questions selected were objective type. The cover page also included

items in the form of blanks to enable the investigator identify the respondent.

The identification data consisted of name, sex and name of the school.

3.6.3. Scoring

The test used was in Kannada hence the investigator explained to the

students how to write their responses in given blocks. The scoring of the

answer scripts were done according to the key prepared. Right answer carried

one mark and zero to the wrong answer.

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3.6.4. Item Analysis

An item analysis was done for adapting test to find out the effectiveness to

each competency and its capacity of discriminate between the high and low

group. The final selection of the competency was done based on the expert

opinion for each statement. The expert opinion were sought to know to what

extent to the given statement was appropriate. The pre-test was modified and

finalized according to expert’s opinion.

3.6.5. Reliability of the Achievement test

“The reliability of a test or any measuring instrument depends upon the

consistency with which it gauges the ability to which it is applied” (H. E.

Garret,1966). There are number of approaches to assess the reliability of a

test. The choice of approach depends on the type of information one is

seeking. As a test score is only interpretable when the test possesses

substantial internal consistency, that is, all the items in the test measure the

same construct, the coefficient of internal consisting would be interest

(Cronbach, 1951). In this study the coefficient of internal consistency has

been obtained by using the split-half method. Reliability refers to the

consistency with which a test measures whatever it measures. There are a

number of approaches to assess the reliability of test. The choice of approach

depends on the type of information one is seeking. The test scores can be

interpreted when it possesses substantial internal consistency. In this study

the coefficient of internal consistency was obtained by the investigator also

found out internal consistency through using the split half method. A test is

said to be internally consistent if all its items measures the same thing. To

estimate the internal consistency of the test the split half method has been

used. The coefficient of internal consistency has been computed for the test

by finding the product moment correlation between the score on odd and even

numbered sub competencies and then self correlation coefficient of the whole

test has been estimated by using spearman-brown prophecy formula.

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Nrt 2t/2 rtt = ——————————— (n-1)rt 1 + 2t/2

The reliability was found as 0.77. Thus the test has a high reliability.

3.6.6. Validation of the Test

The test is valid if the scores it assigns to examinees are free from constant

and systematic errors and hence the interference based on these scores is

justified. The content validity of the test refers to the extent to which the test

contains a representative sample of items, which define the content domain of

interest.

Copies of the this test were distributed, along with the copies of the list of

competencies selected to six content specialists in mathematics education

and mathematics teachers of regional Institute of Education and Subject

Inspector and experts in mathematics in Shimoga district. The experts are

requested to judge the relevance of each item in the test and to critically

examine them to ascertain the adequacy and clarity of the items. Based on

the opinion of the content experts, suitable modifications were made in the

test.

After the test was scrutinized by the experts, it was administered to a few

selected students as stated earlier. The choice of these schools was

purposive as these two schools were Kannada medium schools and they

followed state curriculum. As far as the selection of students was concerned,

the investigator in consultation with the concerned teachers selected five high

achieving, five average achieving and five low achieving students from each

school. The purpose of this testing was to determine the language and

content appropriateness of the test. The test was competency wise taking

three or four sub competencies at a time. The time taken to complete

answering to the questions on each competency by student was noted down.

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The total time taken to answer the written test by high achiever was

approximately 40 minutes and that of an average achiever was around 45

minutes and a low achiever was about 50 minutes. Based on this

administration of the test the investigator arrived at a decision that the time for

administering the test should be 45 minutes. This tryout also brought about

some modifications (review of difficulty level) in the tests for the test takers.

The final versions of the tests are given in appendix.

3.7. Experiment Conducted

The procedure adopted for conducting the experiment is as follows:

5) Administration of the pre-test

6) Experimental treatment

7) Administration of the post test

3.7.1. Administration of the Pre-Test

The researcher visited all the schools in person and administered the test

himself with the help of BRP’s( Block Resource Person) and teachers and

assured the confidentiality of the data. At the end of first semester the

students were given test to assess their attainments on the MLL

competencies taught in Mathematics.

Instructions to students for taking the test were adapted from the original test

keeping in mind their age level and mental ability. The general guideline for

giving the test was prepared for the use of the teacher. The teacher read out

the instruction in the beginning of the test, as the target group was V standard

students. Specific guidelines were included along with the questions. All

questions were answered in the space provided in the question paper as all

the questions selected were objective type. The time allotted for the test was

45 minutes. The cover page also included items in the form of blanks to

enable the investigator identify the respondent. The identification data

consisted of name, sex and name of the school.

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This test was used for diagnosing non-mastery of the students over the

covered competencies in the test. The investigator with the help of teachers

administered the test on the 1457 students from 166 Government primary

schools in seven blocks of Shimoga district. The norms were followed for

administration of the test as given in the original test. Since medium of

instruction of the selected Government primary schools was Kannada and the

test selected was also Kannada. The subjects were asked to answer all the

questions. The key was prepared simultaneously. For correct answers one

mark was assigned and for wrong answer zero mark.

3.7.2. Experimental Treatment

Based on the pre test it was found that students were lacking in

competency (major and minor competency) in solving problems in the

selected competencies. The intervention for developing non-mastered

competencies was given through teaching strategies which had 8 components

i.e. sequencing and segmenting, drill-repetition and practice-review, directed

questioning and responses, control difficulty or processing demands of the task, use of

technology, group instruction, supplements to teacher and peer involvement, strategy

cues to the experimental groups for a period of two months (from January to

Feb,2008) in the two schools selected each from rural and urban. The

teaching strategies for mastering the non-mastered competencies were

prepared by the help of available library information and discussed with the

guide and experts in mathematics teaching. The investigator requested the

authorities to allow him to teach the experimental groups according to the

features of the teaching strategy. One mathematics class was taken in each

school on alternative days for both urban and rural schools which represent

experimental group and the control groups were taught as usual by the

regular teachers.

3.7.2.1. Diagnosis Based Remediation

The investigator used varied of strategies designed for the study consisting of

the content and varied activities inside and outside the classroom to develop

the identified competencies in mathematics among the students of V

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standard. Here Investigator diagnosed the difficulties of the problems in each

competency in which students been non mastered. The remediation was

developed as a program which involved explicit Instruction on the basis of

cognitive strategy, incorporates research-based practices and instructional

procedures such as cueing, modeling, verbal rehearsal, and feedback. The

intervention sessions were well organized and structured in which appropriate

cues and prompts were built in so that students learnt and practiced the

cognitive and meta cognitive processes involved in learning competencies in

mathematics. Each non-mastered student was provided with immediate,

corrective, and positive feedback on his/her performance. The investigator

focused on mastery learning and automaticity were goals of intervention.

Explicit instruction allowed students to be active participants as they learnt

and practiced math problem-solving processes and strategies used by the

investigator during intervention session. The approach followed emphasized

active interaction among non-mastered students and investigator.

The remediation program as a whole consisted of eight components of

effective strategy instruction which were discussed below.

1. Sequencing and Segmenting

Sequencing and segmenting means breaking the task into component

subparts, providing short activities, and synthesizing the parts into a whole.

For example, each cognitive process/self-regulation strategy routine was

taught consecutively, began with reading the problem as a necessary first

step for solving the problem. The non-mastered students were taught to read

the problem and then asked about themselves if they understood it. They

were then taught to go back and reread it until they decided they understood

it. At the beginning of instruction, the investigator modeled the process and

provided plenty of step-by-step cues and prompts to the non-mastered

students for practice. Eventually these cues and prompts were phased out

after non-mastered students knew what to be done. They learnt how to

paraphrase or retell problems. The non-mastered students learnt the

paraphrasing routine, which was then added to the reading routine.

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Subsequently, non-mastered students had mastered a sequence of two

important processes for solving mathematical problems.

2. Drill-Repetition and Practice-Review

This component included progress checks to measure mastery skill,

sequenced review, repeated practice, distributed review and practice using

the same or similar practice problems, and ongoing and positive feedback.

For example, the paraphrasing routine was taught and then students

practiced on their own and also with peers.

a. PARAPHRASE: In this strategy the investigator asked the non-mastered

students to underline the important information in given sheets and putting the

problem in their own words. Then asked themselves whether the important

information had been underlined, what the question was and what they were

looking for.

b. Progress Check: On this the investigator modelled the routine (situation)

and guided the non-mastered students to follow the routine, they were

provided with practice until the routine became automatic. Then they learnt

how to paraphrase math word problem, through this they also learnt to

evaluate themselves using a progress check as a strategy.

3. Directed Questioning and Responses

Intervention used for remediation followed guided discussion technique to

promote overcoming difficulties adequately. All the non-mastered students

were engaged from the very beginning through an initial discussion of the

importance and strategies used for mathematical problems to the non-

mastered students to confirm whether they used this properly. In this session

the students also learnt setting the performance goals and they appeared as a

better problem solver. Investigator asked both “process-related” and “content-

related” questions. Non-mastered students were helped by the investigator

directed to ask questions for getting right solutions. Non-mastered students

were also taught when and how to ask for help.

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4. Control Difficulty or Processing Demands of the Task

Arranged tasks from easy to difficult and the investigator demonstrated on the

use of appropriate cues and prompts, and participating in guided discussion.

Since intervention was planned for primary school students for providing

problem-solving instruction started with one-step change addition problems in

which the “ending” was unknown. When students had mastered the problem

solving routine with problems of this type, they progressed to one-step change

problems in which the change was unknown. They progressed to change

problems in which the beginning was unknown, and so forth.

5. Use of Technology

Technology extended beyond audio visual aids/concrete materials which

included structured text, diagrams, flow charts, structured curricula, scripted

lessons, and various manipulatives. Students who were learned to be better

math problem solvers were taught how to use manipulatives to facilitate

computation after their understanding of math facts for addition and

subtraction had been mastered.

6. Group Instruction

Non masters who had math problem solving difficulties were taught in small

groups (5-8 students) to maximize investigator and student interaction.

Interaction between investigator and students and among peers was the

cornerstone of remedial teaching. Remedial teaching was intensive and time-

limited.

7. Supplements to Teacher and Peer Involvement

Non-masters in fifth grade, reading skills were given, cue cards to study for

homework as they memorized and learnt the various problem-solving

processes and self-regulation strategies. e.g., used the Jitendra, et al.

strategy for change problems, students first learnt to “find the problem” type

by reading the problem, retelling the problem, and asking themselves if it was

a change problem. They checked the box on the self-monitoring checklist as

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they completed the task. When they had mastered how to “read” and retell a

math word problem, they advanced to the organization step. Each step was

added successively until they had learned and applied the entire routine. After

small group instruction or homework, students were expected to return to the

general education math class and use what they had learned about solving

math problems. General education teachers must be made aware of the

instruction that non-mastered students were receiving and supplemented and

supported this remediation program in the general education math classes. To

do this, it was essential that general education teachers and investigator

communicated regularly about the non-mastered students and the instruction

and coordinated what was taught in the general education class with what

was taught in the remedial teaching class and vice versa. Continuity across

general and remedial teaching was essential for non-mastered students

success. General education teachers must reinforce what students had

learned to ensure that they applied appropriately, and also maintained,

acquired skills and strategies.

8. Strategy Cues

Non masters were given reminders and prompts such as individual Student

Cue Cards to carry with them for home and class use, Master Class Charts on

the classroom walls, problem type diagrams, think-aloud protocols, and

discussion about the benefits of using strategies.

i. Verbal Rehearsal Before non-mastered students actually solved problems, they first learned the

steps and memorized them by using verbal rehearsal. This was a memory

strategy that enabled non-mastered students to recall automatically the math

problem-solving processes and strategies. Non-mastered students in primary

school were learnt a SAY, ASK, CHECK routine similar to Montague’s (2003)

or Jitendra’s (2005) as they were learning how to represent problems.

Frequently, acronyms were created to help non-mastered students remember

as they verbally rehearsed and internalized the labels and definitions for the

processes and strategies. For Jitendra’s math problem-solving routine, the

acronym FOPS was created (F = Find the problem type, O = Organize the

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information using the change diagram, P = Plan to solve the problem, and

S = Solve the problem. Cues and prompts were used to help non-mastered

students as they memorized the processes and their definitions. When

students had memorized the math problem-solving routine, they cued other

non-mastered students and the investigator during practice sessions.

ii. Process Modeling Process modeling was thinking aloud while demonstrating an activity. For

mathematical problem solving, this mean that the problem solver said

everything she or he was thinking and doing while solving a problem. When

non-mastered students were first learning how to apply the processes and

strategies, the teacher demonstrated and modeled what good problem solvers

did as they solved problems. Students had the opportunity to observe and

hear how to solve mathematical problems. Both correct and incorrect

problem-solving behaviors were modeled. Modeling of correct behaviors

helped non-mastered students understood how good problem solvers used

the processes and strategies appropriately. Modeling of incorrect behaviors

allowed non-mastered students to learn how to use self-regulation strategies

to monitor their performance and located and corrected errors. Self-regulation

strategies were learned and practiced in the actual context of problem solving.

When non-mastered students learned the modeling routine, then they were

exchanged places with the investigator and become models for their peers.

Initially, non-mastered students needed plenty of prompting and reinforcement

as they became more comfortable with the problem-solving routine. However,

they soon became proficient and independent in demonstrating how good

problem solvers solved math problems. One of the instructional goals was to

gradually move non-mastered students from overt to covert verbalization. As

non-mastered students became more effective problem solvers, they began to

verbalize covertly and then internalized. In this way, they not only became

more effective problem solvers, but they also became more efficient problem

solvers.

iii. Visualization Visualization was critical to problem representation. It allowed non-mastered

students to construct an image of the problem on paper or mentally. Non-

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mastered Students showed how to select the important information in the

problem and develop a schematic representation. To do this, investigator

modeled how to use manipulatives to represent a problem, and then how to

draw a picture or make a diagram that showed the relationships among the

problem parts using both the linguistic and numerical information in the

problem. These three-dimensional and two-dimensional visual

representations took many forms and varied from one non-mastered student

to another. As non-mastered students became better problem solvers, they

used a variety of visual representations including manipulatives, pictures,

tables, graphs, or other types of displays. Initially, non-mastered students

showed how to use the manipulatives and also how to translate the results of

their manipulations with concrete objects to more symbolic representations

using paper and pencil, e.g., the problem type diagrams. Later, as non-

mastered students became more proficient, they progressed to mental

images. Interestingly, if the problem was novel or challenging, they frequently

returned to conscious application of processes and strategies, which was

typical of good problem solvers.

iv. Role Reversal Role reversal was an important remedial teaching activity that promoted

independent learners. As non-mastered students became familiar with the

math problem-solving routine, they took the role of investigator as model and

actually changed places with the investigator. They used a manipulative just

as the investigator did and engaged in process modeling to demonstrate that

they applied effectively the cognitive and meta cognitive processes and

strategies they had learned. Other non-mastered students prompted or asked

questions for clarification. In this way, non-mastered students learned to think

about, explanation, and justification of their visual representations and their

solution paths. Investigator also took the role of the non-mastered student,

guided the “non-mastered student as investigator” through the process.

v. Performance Feedback Performance feedback was critical to the success of the program. Progress

checks were given throughout instruction to determine mastery of the routine.

Investigator and parents assisted non-mastered students with graphing their

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progress and visually displaying their performance, which wais very

reinforcing for them. Investigator carefully analyzed performance during

practice sessions and provided each student with immediate, corrective

feedback. Appropriate use of processes and strategies was reinforced

continuously until non-mastered students became proficient. Non-mastered

Students needed to know the specific behaviors for which they were praised

so they repeated these behaviors. Non-mastered students were taught how to

give and receive reinforcement and to reinforce themselves, and had plenty of

opportunities to practice doing it. The goal was to teach non-mastered

students to monitor, evaluate, and reinforce themselves as problem solvers.

vi. Distributed Practice Distributed practice was the cornerstone for ensuring that non-mastered

students maintained what they had learned. To become good math problem

solvers, non-mastered students learnt to use the processes and strategies

that successful problem solvers used. As a result, their math problem-solving

skills and performance levels improved. However, to achieve high

performance, non-mastered students were given ample opportunity to

practice initially as they learned the math problem-solving routine and, then, to

maintained high performance, they continued to practice intermittently over

time. They practiced individually, in teams and in small groups. They involved

in solving a range of problems from textbook-type problems to problems

encountered in real life.

vii. Mastery Learning Prior to remedial teaching, a pretest was given to determine baseline

performance levels of individual students. During remedial teaching, mastery

checks were given to monitor non-mastered students’ progress over time and

to determine effectiveness of the program. If students were not making

sufficient progress, booster sessions provided to improve performance levels

to mastery. Booster sessions were brief lessons to review and refresh what

non-mastered students had previously learned and mastered.

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3.7.3. Administration of the post test

After the proposed treatment had been given to the subjects, the

second phase of the data collection was started. At the end of the treatment

MLL based achievement test was administered to the experimental and

control group students. The instructions were strictly adhered to as given in

the preliminary information and note in the test paper.

3.8. Statistical Techniques Employed in the study

Using SPSS for windows (version 16.0) following statistical methods were

employed for the data collected in the present investigation.

1. Contingency coefficient analysis

2. Independent samples ‘t’ test

3. One-way Analysis of Variance

4. Duncan’s Multiple Range test

5. Repeated measure ANOVA