Problem Solving, Reasoning and Conceptual Understanding in ...joics.org/gallery/ics-2641.pdf · decision making, problem solving, perception, and attention.. Cognition is a term referring

Problem Solving, Reasoning and Conceptual Understanding in

Mathematics among Senior Secondary School Students in Relation

to Gender and Cognitive Styles

Dr.Radha Arora1 , Dr.Pooja Arora2 Bharti Chadha3

1Associate Professor, M.G.N. College of Education, Jalandhar, 144021, Punjab, India 2Assistant Professor, M.G.N. College of Education, Jalandhar, 144021, Punjab, India

3M.ED Student, M.G.N. College of Education, Jalandhar, 144021,Punjab, India [email protected] /9646711883

INTRODUCTION

Abstract There is general dissatisfaction with the result of Mathematical Instruction despite of

pedagogic progress, in spite of teachers. The syllabi and methods should be such as to

project at least a reasonably correct picture of Mathematics in the mind of students.

Hence, there must be some factors like psychological, social and biographical affecting

the learner in learning of Mathematics at large. The purpose of this descriptive study

was to identify the prevalent cognitive styles of learners on Problem Solving,

Reasoning and Conceptual Understanding in Mathematics The survey was conducted

by using a Cognitive Style Inventory Test by Dr. Praveen Kumar Jha) for identifying

the students’ cognitive styles.. The association between students’ cognitive styles and

their academic performance was also explored. Mathematics achievement Test was

divided in to different subsets to find out Problem Solving, Reasoning ability and

conceptual understanding in mathematics of students prepared by investigator. The

Mathematics Achievement Test has reliability coefficient of 0.90. The population for the

study was the senior secondary school students from Govt. and Private Schools from

Jalandhar District. For their selection, random sampling technique has been employed.

Further students were bifurcate in to boys and girls. The responses of the both groups

of boys and girls to the instruments were scored and analyzed using mean and two way

analysis of variance. It was found from statistical evidence that Integrated Cognitive

Style has High Mathematical Achievement & High Mathematical Reasoning. So

integrated Cognitive Style helps students to make use of all other mathematical skills

and also able to reflect on solutions to problems and determine whether or not they

make sense. They appreciate the pervasive use and power of reasoning as a part of

mathematics. Girls are better than boys in Mathematical Problem Solving &

Mathematical concepts. So girls have more ability to do mathematical tasks and

potential to provide intellectual challenges for the enhancement of mathematical

understanding and development. Boys with Undifferentiated Style are good in

mathematics. Girls using Intuitive cognitive style will have high Mathematical

Achievement and rely on experience patterns characterized by universalized areas or

hunches and explore and abandons alternatives quickly. Boys with Cognitive Style II

use an unpredictable ordering of analytical steps when solving a problem. Girls with

Cognitive Style II also tend to be withdrawn, passive and reflective and often look to

others for problem solving strategies. Boys with Cognitive Style are intuition,

counterfactual thinking, critical thinking, backwards induction and adductive

induction. Girls with Cognitive Style II are way to give math a purpose and to help

students understand where formulas come from. They have ability to made argument,

to justify one’s process, procedure, or conjecture, to create strong conceptual

foundations and connections, in order to process new information .Boys with Cognitive

Style V have more clarity in concepts in Mathematics. Girls with Cognitive Style II

have ability to understand Numeric or quantitative entities, descriptions, properties,

relationships, operations, and events.

Key words-

Mathematical Problem Solving, Mathematical Reasoning, Conceptual Understanding in

Mathematics, Senior Secondary students, Gender, Cognitive Styles

Journal of Information and Computational Science

Volume 10 Issue 3 - 2020

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mailto:[email protected]

INTRODUCTION Mathematics is a necessity for people of all ages to be successful in life. Despite the usefulness of

Mathematics in daily life, there are factors that adversely affect the students’ ability to understand

and apply Mathematics concepts. Evidence exists that individuals possess habitual ways of

approaching tasks and situations associated with particular patterns in cognitive processes including

decision making, problem solving, perception, and attention..

Cognition is a term referring to the mental processes involved in gaining knowledge and

comprehension. These processes include thinking, knowing, remembering, judging and problem-

solving. These are higher-level functions of the brain and encompass language, imagination,

perception, and planning Brown et al ., (2006) defined that Cognitive styles, as “a psychological

construct relating to how individuals process information & it has many classifications. It once was

classified into field-independent style and field-dependent style, analytic style and global style,

reflective style and impulsive style, and tolerance and intolerance of ambiguity [4]. Shuells (1981)

defined that cognitive styles refers to the preferred way that different individuals have for

processing and organization and for responding to the environmental stimuli so it reflects aspect of

personality as well as aspect of cognition.[22] Chinn & Ashcroft (1993) defined that cognitive style

is an individual’s characteristic and relatively consistent way of processing incoming information

of all types from the environment.[5] Riding & Rayner (1998) stated that Cognitive style is a

person’s preferred and habitual approach to organizing and representing information.[17]

The concept of cognitive styles was originated in two dimensions in educational and vocational

psychological research circles. Learners’ different characteristics were explored because different

individuals retain and organize information in different fashions. Some researchers applied

cognitive styles in educational settings for observing the differences in academic performance of

students whereas others focus on different other domains like teaching and learning processes, and

introduced theories of learning and cognitive styles. T

Sternberg & Zhang, (2001) described that in the field of education, researchers have argued that

cognitive styles have predictive power for academic achievement beyond general abilities.[23] Pitta-

Pantazi, Demetra & Christou, Constantinos. (2009) studied as a relationship between cognitive

styles, dynamic geometry and measurement performance. [16] The results are discussed in the terms

of nature of the measurement tasks administered to be students. Shabu, Subermony and Gupta

(2006) found that individuals with different cognitive styles do not significantly differ in their

intelligence. But they found significant correlations between all factors of problem solving

index.[20]

Deshi P.C. (1989) has studied the possible relationship between achievements in mathematics and

cognitive preference style in the cast, while for majority of commerce students, the recall style in

the first. No significant relationships found between cognitive preference style and mathematics. It

is an open question worth investigation whether by changing teaching strategies we can lead to

significant learning of mathematics.[7]

Hooda, Madhuri & Devi, Rani. (2017) remarked that the ministry of education should cautioned

teachers about the importance of cognitive styles during teaching and learning process. The

mathematics teacher should take importance of cognitive styles during preparing their lesson plan

and teaching aids.[9]

Anderson, Casey, Thompson, Burrage, Pearis and Kosslyn (2008) studied the effects of student’s

cognitive styles on their mathematical achievement, utilizing a new approach to the visual

verbalizer cognitive style dimensions. These studies supported the view that there exist three

different cognitive style dimensions, a verbal style as well as two types of visual cognitive styles

and indicated that the visual- spatial imagery rather than visual object imagery is related to success

in mathematics.[1] James J. Roberge and Barbara K. Flexer examined the effects of field

dependence/independence and the level of operational development on the mathematics

achievement of 450 students in grades 6-8. Field-independent students scored significantly higher

on total mathematics, concepts, and problem-solving tests. High-operational students scored

significantly higher on all tests.[10]



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Presumably teachers who understand leaner’s cognitive styles and implement them through flexible

instruction and assessment will develop confident learners who possess the necessary mathematical

number and reasoning skills. Good numbers skills without good mathematical reasoning and

problem solving skills makes mechanical learners who lack the necessary mathematical

competencies to solve problems, in spite of the importance of problem solving as a fundamental

philosophical basis for the study of mathematics. So present study is structured to find out Problem

Solving, Reasoning and Conceptual Understanding in Mathematics among Senior Secondary

School Students in Relation to Gender and Cognitive Styles

METHODOLOGY Objective The present study was designed to attain the following objective:

To study the Problem Solving Ability of senior secondary school students in relation to Gender &

cognitive styles.

To study the Reasoning Ability of senior secondary school students in relation to Gender &

cognitive styles

To study the conceptual understanding in mathematics of senior secondary school students in

relation to Gender & cognitive styles

Hypotheses

The proposed hypotheses were:

H1: There exists no significant difference in Mathematical achievement and its subsets (Mathematical Problem Solving, Mathematical Reasoning, and Conceptual Understanding in

Mathematics) among senior secondary school students in relation to Gender (Boys & Girls)

H2: There exists no significant difference in Mathematics Achievement and its subsets (Mathematical Problem Solving, Mathematical Reasoning, Conceptual Understanding in

Mathematics) among senior secondary school students in relation to their different Cognitive

Styles. (Systematic style, Intuitive style, integrated style, undifferentiated style and split style)”.

H3: There exists no significant interaction effect between Gender & Cognitive Styles on the score

of Mathematics Achievement and its subsets (Mathematical Problem Solving, Mathematical

Reasoning, Conceptual Understanding in Mathematics) among senior secondary school students.

Research Design:

The investigator was used survey method for studying the problem. Quantitative approach is applied

in this study. Furthermore, quantitative research is about identifying relationships between variables

through the use of data collection and analysis.

Identification and Recruitment of Participants:

In order to conduct the present study, six Govt. and Private Schools from Jalandhar District was

selected. For their selection, random sampling technique was employed. Out of the selected Schools

investigation has been carried out on 300 students of Govt. and Private Schools.

Sample: In order to conduct the present study, six Govt. and Private Schools from Jalandhar District

have been selected. For their selection, random sampling technique has been employed. Out of the

selected schools investigation has been carried out on 300 students of Govt. and Private Schools

Design of the Study:

To test the proposed hypotheses the design of the study was as follows:

Two way analysis of variance is employed on the score of Mathematical Problem Solving, Mathematical Reasoning& Conceptual Understanding in Mathematics .Mathematical Problem Solving, Mathematical Reasoning & Conceptual Understanding in Mathematics were studied as

dependent variables. Gender is studied as an independent variable and used for the purpose of



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classification viz-a-viz Boys And Girls . Cognitive styles are studied as an independent variable

and used for the purpose of classification viz-a-viz Systematic style, Intuitive style, Integrated

style, undifferentiated style and Split style.

MEASURES

The two instruments were used to collect data from the respondents. They include

Tool 1: Mathematics Scale Constructed By the Investigator

In order to develop lesson Mathematics Scale prepared by the investigator following steps were

followed:

Planning

Mathematics test was prepared keeping in view the universality of the various segments –

Mathematical Problem Solving, Mathematical Reasoning& Conceptual Understanding in

Mathematics. Teachers teaching in school were consulted and all of them reported poor

understanding of concept, reasoning and problem solving by the students. Also the investigator

herself checked the problems, conceptual questions and reasoning questions solved by the students.

So by thorough checking of the errors committed by students in the selected fields and discussion

with the teachers, the investigator was able to collect relevant information about the types of errors

committed by the students in the selected field. After identification of students deficiencies few

topic from the ―Central board of secondary school syllabus of class 11th mathematics subject were

analyzed to develop the test on Mathematical problem solving, Mathematical reasoning and

Mathematical concept were selected. The investigator consulted the syllabus of mathematics subject

prescribed for class 11th and selected few topics. The items described were Fill in the blanks,

True/False, Short answer type questions, Objective type questions. Topics were

Mathematical Problem Solving

Relations and Functions

Conic sections

Coordinate geometry

Permutations and Combinations

Limits and Derivatives

Sequence and Series

Complex numbers and quadratic equations

Binomial theorem

Probability

Mathematical Reasoning

Blood relation type

Word problem

Conditional problems

Cause and effect problems

Mathematical Concept


Probability

Binomial theorem

Permutations and Combinations


Relations and Functions

Sequence and Series

Principal of mathematical induction

Designing and Construction



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The analysis of the content was done. Then the test items were written according to specific

objectives. In total 120 questions were selected. The questions were carefully written. The

Mathematics test thus, constructed was checked by the supervisor, with little modification in the

language of test items.

Preparation of Preliminary Draft of Test

Originally a comprehensive test was prepared including the different types of questions as indicated

by the subject teachers to be problematic. This test consisted of 120 items. The test comprised of

objective type items, short answer type, extended response questions, fill in the blanks and true/false.

The preliminary draft of the test was given to randomly select 300 students of XI class. The purpose

of the preliminary draft of the test was to find out functioning very easy and very difficult items and

also to examine the functioning of the item and distracters of multiple choice items.

TABLE I

THE DISTRIBUTIONS OF THE ITEMS IN THE PRELIMINARY TESTS

SR. No. NAME OF FIELD No. OF ITEMS

1. Mathematical reasoning 40

2. Mathematical problem solving 40

3. Conceptual understanding 40

Preparation of Final Draft

A careful scrutiny was made for the functioning of various distracters; dead distracters were

modified and replaced with more appealing and new ones. The final test comprised of 90 items.

While eliminating any item, care was taken that no basic concept is eliminated from the final draft of

the test is appended with the thesis. The items of the final draft were distributed in the same manner

as the preliminary draft:

Table II

THE DISTRIBUTIONS OF THE ITEMS IN THE FINAL TESTS

SR. No. PRILIMINARY DRAFT FINAL DRAFT

1. 40 30

2. 40 30

3. 40 30

Reliability

For determining the reliability, the test was administered to 50students of 7th class and reliability of

the achievement test was completed with the help of the following formula.

Reliability r = 1- M(K-M)/ K(S) 2

Where K = Number of items in the test, M = Mean of test score, S = Standard deviation of the score

Thus, the reliability of the test was found to be 0.90. Hence the constructed achievement test may be

considered as student’s achievement. The present test has content validity and a test presented fairly

well defined universe of content. In total 90 items were selected. 150 marks were allotted. Out of

which 1.5 marks for blanks, 1.5 marks for true/false, 1.5 marks for tick the right option, 2 marks for

short answer type questions.

Tool 2: TOOL3: Cognitive Style Inventory Test by Dr. Praveen Kumar Jha The cognitive style inventory is designed on the basis of the rationale as conceived by Martin (1983)

.it was planned to develop a comprehensive inventory to measure the dimension of cognitive style;

quite suitable for Indian sample. A pool of 92 statements including the suitable items for Indian

sample as suggested by Martin (1983) was prepared to make a prediction of cognitive style of

respondentsOngoing observational studies, along with effect to develop measurement devices for

assessing cognitive behavior, have resulted in an expanded version of the original model, which led

to the development of five following styles:

Systematic style: An individual who typically operates with a systematic style uses a well-defined

step to step approach when solving a problem; looks for an overall method or pragmatic approach;

then makes an overall plan for solving the problem. Intuitive style: The individual, whose style is

intuitive, uses an unpredictable ordering of analytical steps when solving a problem, relies on

experience patterns characterized by universalized areas or hunches and explores and abandons

alternatives quickly. Integrated style: A person with an integrated style is able to change styles



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quickly and easily. Such style changes seem to be unconscious and take place in matter of seconds.

Undifferentiated style: A person with such a style appears not to distinguish or differentiate between

the two styles extremes; i.e. systematic and intuitive, and therefore appears not to display a style.

Split style: An individual with split style shows fairly equal degrees of systematic and intuitive

specialization. However, people with split style do not possess an integrated behavioral response;

instead, they exhibit each separate dimension n completely different settings; using only one style at

a time based on nature of their tasks. To obtain reliability and validity of the final Hindi version of

CSI; 100 male students of Post graduate class were given to fill in the questionnaire and the data

obtained from them were taken into account. Reliability of test was determined by two methods Split

half method and Test retest method Split half method: The product moment co-efficient of

correlation between two halves; i.e., Split half was calculated for the whole scale and for each of the

five sub scales. The judges’ validity is considered to be the simplest method of examining validity of

a test. Here this method has been used to examine the validity of CSI. This method implies expert

evaluation whether the test items adequately reflect the objectives and content area. A test has

concurrent validity when it gives an estimate of certain performance. Product Moment correlation

was calculated between the obtained scores of Martin’s CSI and Hindi version of CSI as developed

by the author. A correlation coefficient of .262 was obtained which was satisfactorily significant

beyond .01 level of confidence. In this way CSI bears concurrent validity. Scoring of cognitive style

of an individual is in a five point Likert format. Five response categories are: Strongly Disagree,

Disagree, Undecided, Agree, Strongly Agree.

Respondents are classified according to following interpretation

A respondent who rates high on systematic scale and low on intuitive scale is identified as

having a systematic style.

Respondent who rates low on systematic scale and high on intuitive scale is designated as a

person having an intuitive style.

A testee with an integrated style rates high on both scales (systematic and intuitive) and is

able to change styles quickly.

An individual rating low on both systematic and intuitive scale is described as having

undifferentiated cognitive style.

The person rating in the middle range on both systematic and intuitive scale is considered to

have a split style.

PROCEDURE In order to conduct the study 300 students of 12th classes of senior secondary school of Jalandhar

district was selected as the sample. Students were segregated in to Boys and Girls. After that

Cognitive style inventory was administrator and students will be segregated in to different

Cognitive Styles. (Systematic style, Intuitive style, Integrated style, undifferentiated style and Split

style). There after the Mathematics scale was administered on segregated students and the score of

Mathematical Problem Solving, Mathematical Reasoning & Conceptual Understanding in

Mathematics was taken and data was given statistical treatment.

STATISTICAL TECHNIQUE: The data was analyzed using two ways analysis of variance

to find out the significant differences between groups. Mean and standard deviation of various

subgroups will be computed to understand the nature of data

The Data Obtained has been analyzed under the following headings:

RESULTS AND DISCUSSION This portion of the study presents the results of the data gathered by the researcher.

Mathematical Achievement and its subsets (Mathematical Problem Solving, Mathematical Reasoning, Conceptual Understanding in Mathematics) in Relation to Gender & their cognitive

styles In order to analyses the data means and standard deviation was computed on the obtained scores and

were further subjected to one way analysis of variance



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TABLE 1 Summary of Means and Standard Deviations of Mathematics Achievement and its subsets Mathematical

Problem Solving, Mathematical Reasoning, Conceptual Understanding in Mathematics in Relation to

Gender & their cognitive styles

Gender Mean Std.

Deviation

N

Mathematics

Problem solving

BOYS

Integrated 32.38 9.486 8

Intuitive 36.44 6.207 16

Split 34.56 6.069 63

Systematic 34.07 6.209 42

Undifferentiated Cognitive 35.33 5.955 6

Total 34.53 6.323 135

GIRLS


Intuitive 38.50 6.129 18

Split 34.58 6.896 90



Total 35.65 6.377 165

Total


Intuitive 37.53 6.161 34

Split 34.57 6.547 153



Total 35.15 6.367 300

Mathematical

Reasoning

Boys Integrated 39.63 6.906 8

Intuitive 36.25 6.904 16

Split 35.79 5.007 63



Total 36.14 5.409 135

Girls Integrated 37.17 7.360 6

Intuitive 37.28 4.787 18

Split 34.61 6.515 90



Total 35.01 7.025 165

Total Integrated 38.57 6.936 14

Intuitive 36.79 5.809 34

Split 35.10 5.951 153




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Total 35.52 6.364 300

Conceptual

understanding

in mathematics


Intuitive 35.00 6.851 16

Split 34.43 5.558 63



Total 35.30 6.234 135


Intuitive 34.50 5.864 18

Split 36.61 5.710 90



Total 35.84 6.195 165

Total


Intuitive 34.74 6.254 34

Split 35.71 5.732 153



Total 35.59 6.208 300

Total

Mathematics

Achievement


Intuitive 107.69 15.619 16

Split 104.78 10.518 63

Systematic 105.90 12.195 42


Total 105.97 12.245 135


Intuitive 110.28 9.833 18

Split 105.80 12.007 90

Systematic 105.93 12.695 43


Total 106.50 11.956 165

Total

Integrated 109.00 14.608 14

Intuitive 109.06 12.744 34

Split 105.38 11.393 153

Systematic 105.92 12.376 85


Total 106.26 12.069 300



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In Order To Analyze the Variable, the obtained scores were subjected to Anova. The Results have

been presented in Table 2

Table 2 Summary of Two Way Analysis of Variance on Score the of Mathematics Achievement and its subsets

Mathematical Problem Solving, Mathematical Reasoning, Conceptual Understanding in Mathematics in

Relation to their cognitive styles

Subsets

A

Gender

B

Cognitive

Styles

A×B ERROR TSS

(Mathematical

Problem

Solving)

Sum of

Squares

113.501 242.839 425.864 11628.849 38277.000

Mean

Square

113.501 60.710 106.864 40.099

F Ratio 2.83* 1.51 2.65*

(Mathematical

Reasoning)

Sum of

Squares

33.157 395.777 396.864 11753.337 390539.000

Mean

Square

33.157 98.944 99.216 40.529

F Ratio .818 2.44* 2.45*

(Mathematical

Concept)

Sum of

Squares

320.698 58.847 452.311 11199.180

Mean

Square

320.698 14.712 113.077 38.618 391590.000

F Ratio 2.83 .381 2.92*

Mathematics

Achievement

Sum of

Squares

.120 1101.238 1262.145 32795.081 3430912.000

Mean

Square

.120 275.309 315.536 113.08

F Ratio .001 2.43* 2.79*

Degree of freedom between (Gender) =1

Degree of freedom between (CS) =4

Degree of freedom within (CS) =290

* Significant at 0.05 Level of Confidence



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MAIN EFFECTS

Gender (A)

From the results inserted in the table 2 revealed that the variance ratio or

F is 2.83 & the degree of freedom between means is 1 and among groups is 290.

Entering table F with these degree of freedoms it may be observed that the F of

magnitude 2.83 >2.41 at .05 level of confidence. So F-ratio for the difference

between the means of two groups’ boys and girls on the score of Mathematical

Problem Solving was found to be significant at 0.05 level of confidence. Next F

of magnitude 2.83 >2.41 (df 2/290) for the difference between the means of two

groups’ boys and girls was found to be significant at on the score of Mathematical

concepts was found to be significant at 0.5 level of confidence. Hence, the data

provides sufficient evidence to reject the hypothesis in case of Mathematical

Problem Solving & Mathematical concepts namely H1 viz., “There exists no

significant difference in Mathematical achievement and its subsets (Mathematical

Problem Solving, Mathematical Reasoning, Conceptual Understanding in

Mathematics) among senior secondary school students in relation to Gender

(Boys & Girls)

Whereas F of magnitude .818 <2.41 (df 2/290) for the difference between the

means of two groups’ boys and girls on the scoreof Mathematical Reasoning

was not found to be significant even at 0.5 level of confidence. Next F of

magnitude .001<2.41 for the difference between the means of two groups’ boys

and girls on the score of Mathematical Achievement was not found to be

significant even at 0.05 level of confidence Hence, the data does not provides

sufficient evidence to reject the hypothesis in case of Mathematical Reasoning &

Mathematical Achievement.

Further the mean table 1 reveals that mean value is of girls are higher than mean

value of boys in Mathematical Problem Solving & Mathematical concepts. So

girls have more ability to do mathematical tasks and potential to provide

intellectual challenges for the enhancement of mathematical understanding and

development. Girls know better than the workings behind the answer. They know

why got the answer. They don’t have to memorize answers or formulas to figure

them out.



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The same results have been depicted through fig1:

Figure 1 : Graphical Representation of Mathematical Achievement and Its Significant Subsets(MP &MR) in

Relation to Gender

The results are in tune with the findings of.

Sabahat Anjum*(2015) in his research Gender Difference in Mathematics Achievement and its

Relation with Reading Comprehension of Children at Upper Primary Stage Research findings

revealed that Significant difference was found between mathematics achievement of girls and boys

at upper primary school stage. Significant difference was found between reading comprehension of

girls and boys at upper primary school stage.Significant positive correlation was found between

mathematics achievement and reading comprehension of children at upper primary school stage.[19]

MUSA, Danjuma Christopher & Samuel wanger Ruth(2019) revealed that Basic Mathematics

students in the Field Independence (FI) group achieved significantly better than the those in the

Field Dependence (FD). The findings also revealed that male students in both the Field

Independence (FI) and Field Dependence (FD) groups achieved better than the female students

significantly. Based on the findings of this study, it was recommendation that seminars and

workshops should be organized to adequately equip teachers with the needed skills to create an

environment where students with different cognitive styles can experience meaningful learning of

Mathematics.[15]

Cognitive Style (B)

From the results inserted in the table 2 revealed that the variance ratio or F is 2.44, the df between

means is 4 and among groups is 290. Entering table F with these df’s it may be observed that the F

of magnitude 2.44 > 2.41 for the difference between the means of five types of cognitive styles i.e.

Systematic, Intuitive, Integrated, Undifferentiated & Split Style on the score of Mathematical

reasoning are found to be significant at .05 level of confidence. Next F of magnitude 2.43>2.41 for

the difference between the means of five types of cognitive styles i.e. Systematic, Intuitive,

Integrated, Undifferentiated & Split Style on the score of Mathematical Achievement are found to

be significant at .05 level of confidence. Hence, the data provide sufficient evidence to reject the

hypothesis in case of Mathematical reasoning & Mathematical Achievement namely H2 viz.,

“There exists no significant difference in Mathematics Achievement and its subsets (Mathematical

Problem Solving, Mathematical Reasoning, Conceptual Understanding in Mathematics) among

33.834

34.234.434.634.8

35

35.2

35.4

35.6

35.8

36

BOYS

GIRLS

34.53

35.65

35.3

35.84

MP

MC



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senior secondary school students in relation to their different Cognitive Styles. (Systematic style,

Intuitive style, integrated style, undifferentiated style and split style)”.

Whereas F of magnitude 1.51<2.41 for the difference between the means of five types of cognitive

styles i.e. Systematic, Intuitive, Integrated, Undifferentiated & Split Style on the score of

Mathematical Problem Solving was not found to be significant even at .05 level of confidence. F

of magnitude .381<2.41 for the difference between the means of five types of cognitive styles i.e.

Systematic, Intuitive, Integrated, Undifferentiated & Split Style on the score of Mathematical

Concepts was not found to be significant even at .05 level of confidence. Hence, the data does not

provides sufficient evidence to reject the hypothesis in case of Mathematical Problem Solving &

Mathematical Concepts.

Further the mean table 1 reveals that students having Cognitive style (I) i.e. Integrated Cognitive

Style has High Mathematical Achievement and Cognitive Style (III) i.e. Split Cognitive Style has

Low Mathematical Achievement. It means integrated Cognitive Style is essential to identify

potential problems as well as opportunities in order to find better way of doing things. Again

students having Cognitive style (I) i.e. Integrated Cognitive Style has High Mathematical

Reasoning and Cognitive Style (III) i.e. Split Cognitive Style has Low Mathematical Reasoning. So

integrated Cognitive Style helps students to make use of all other mathematical skills and also able

to reflect on solutions to problems and determine whether or not they make sense. They appreciate

the pervasive use and power of reasoning as a part of mathematics.

The same results have been depicted through fig2:

Figure 2: Graphical Representation of Mathematical Achievement and Its Significant Subsets (MR&MA) in Relation

to Cognitive Styles

The results are in tune with the findings of

Van Gardener, (2006); Kozhevnikov et al., (2002); Presmeg, (1986) investigated the relationship

between Cognitive Styles and Mathematical Achievement. Study had shown that visual-spatial

imagery is beneficial for mathematics and that spatial imagery is an important factor of high

mathematical achievement.[24]

Dr. Parkash Chandra Jena (2014) conducted research on Cognitive Styles and Problem Solving

Ability of Under Graduate Students and The findings of the study revealed that there exists a

significance difference and positive relationship between cognitive styles and problem solving

abilities.[8]

Muhammad Shahid Farooq (2015) found that in the overall scenario the academic performance

differs significantly in relation to only the Auditory/Visual cognitive style. There is no significant

difference in performance of students at all other levels in relation to the other three cognitive

styles. The results of the study lead to the fact that further exploration is needed on a large data to

get more insight in the phenomena.[14]

39.63

39.2535.79

35.88

36.67

MR

CS1

CS2

CS3

CS4

CS5

108.25

107.69

104.78105.9

111.33

MA

CS1

CS2

CS3

CS4

CS5



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Sharma, Hemant & Ranjan, Pooja. (2018) conduced research on “Relationship of Cognitive Styles

with Academic Achievement among Secondary School Students”. The findings of the study

revealed that there is a significant positive relationship between cognitive styles (Field Independent

& Field Dependent) and academic achievement. Keywords: Cognitive Style, Field Independence,

Field Dependence, Academic Achievement.[21]

Ardi Dwi Susandi , Cholis Sa’dijah , Abdur Rahman As’ari , and Susiswo (2019) conducted

research on Students’ critical ability of mathematics based on cognitive styles. The results showed

that students who had the cognitive style of field-dependent and students who had the cognitive

style FI had good critical thinking skills in each step of problem-solving according to Polya. [2]

Rr C C Anthycamurty Mardiyana1 , and D R S Saputro( 2018) conducted research on Analysis of

problem solving in terms of cognitive style .The result of this research is to determine the mastery of

each type in cognitive style, that is Field Independent type and Field Dependent type on problem

solving indicator. The impact of this research is the teacher can know the mastery of student problem

solving on each type of cognitive style so that teacher can determine the proper way of delivering to

student at next meeting.[18]

Lusweti Sellah, Kwena Jacinta & Mondoh Helen (2018) revealed in their study that The equation

significantly predicted 62.8% of variance in performance in KCSE Chemistry (y′) based on four

regressor variables: performance in Mock (X1 ), level of student–teacher cognitive styles match (X2

), level of learner on the sequential–global scale (X3 ) and age of respondent (X4 ).[13]

Two Order Interaction

Gender and Cognitive Styles (A×B)

From the results inserted in the table 2 revealed that the variance 2.65 > 2.41 at .05 level of

confidence. So the F- ratio for the interaction between Gender and Cognitive Style on the score of

Mathematical Problem Solving was found to be significant at 0.05 level of confidence. Next the

variance ratio 2.45 > 2.41 at .05 level of confidence. So the F- ratio for the interaction between

Gender and Cognitive Style on the score of Mathematical Reasoning was found to be significant at

0.05 level of confidence. Next F of magnitude 2.92 > 2.41 at .05 level of confidence. So the F-

ratio for the interaction between Gender and Cognitive Style on the score of Mathematical concepts

was found to be significant at 0.05 level of confidence. Next F of magnitude 2.79 > 2.41 at .05

level of confidence. So the F- ratio for the interaction between Gender and Cognitive Style on the

score of Mathematical Achievement was found to be significant at 0.05 level of confidences Thus

the data provides sufficient evidence to reject the hypothesis H3viz, “There exists no significant

interaction effect between Gender & Cognitive Styles on the score of Mathematics Achievement

and its subsets(Mathematical Problem Solving, Mathematical Reasoning, Conceptual

Understanding in Mathematics) among senior secondary school students.

Further the mean table 1 reveals

In case of Total Mathematical Achievement

The mean score of Boys with Cognitive Style IV (Undifferentiated Style) is higher than

other Cognitive Styles such that Cognitive Style I, II,III, &IV namely, Integrated Style,

Intuitive Style, Split Style, Systematic Style &Undifferentiated Style. So Boys with

Undifferentiated Style are good in mathematics .

The mean score of Girls with Cognitive Style II (Intuitive Style) is higher than the other

Cognitive Style I, III, IV& V namely Integrated Style, Split Style Systematic Style, &

Undifferentiated Style. This mean that Girls using Intuitive cognitive style will have high

Mathematics Achievement and rely on experience patterns characterized by universalized

areas or hunches and explores and abandons alternatives quickly.

The examination of corresponding group mean from the table 1 revealed that

In case of S I (Mathematical Problem Solving) of Mathematics Achievement suggested that:

The mean score of boys with Cognitive Style II (Intuitive Style) is higher than other

Cognitive Styles I, III, IV & V. so boys with Cognitive Style II uses an unpredictable

ordering of analytical steps when solving a problem.



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The mean score of Girls with Cognitive Style II (Intuitive Style) is higher than other

Cognitive Style I, III VI & V. So Girls with Cognitive Style II also tend to be withdrawn,

passive and reflective and often look to others for problem solving strategies.

In case of S II (Mathematical Reasoning) of Mathematics Achievement suggested that:

The mean score of boys with Cognitive Style I (Integrated Style) is higher than other

Cognitive Styles II, III, IV & V. so boys with Cognitive Style are intuition, counterfactual

thinking, critical thinking, backwards induction and adductive induction.

The mean score of Girls with Cognitive Style II (Intuitive Style) is higher than other

Cognitive Style I, III VI & V. So Girls with Cognitive Style II are way to give math a

purpose and to help students understand where formulas come from. They have ability to

made argument, to justify one’s process, procedure, or conjecture, to create strong

conceptual foundations and connections, in order to process new information

In case of S III (Mathematical Concepts) of Mathematics Achievement suggested that:

The mean score of boys with Cognitive Style V (Undifferentiated Style) is higher than other

Cognitive Styles I, II III,& IV. So boys with Cognitive Style V have more clarity in concepts

in Mathematics.

The mean score of Girls with Cognitive Style II (Split Style) is higher than other Cognitive

Style I, II, and VI & V. So Girls with Cognitive Style II have ability to understand Numeric

or quantitative entities, descriptions, properties, relationships, operations, and events.

The same results have been depicted through Fig 3:

Figure 3: Graphical Representation of Mathematical Achievement and Its Significant Subsets with Interaction of

Gender and Cognitive Styles

The results are in tune with the findings of

Arnup, Jessica & Murrihy, Cheree & Roodenburg, John & McLean, Louise.

(2013) found that A significant gender/cognitive style interaction was found. Boys

with an Analytic/Imagery style achieved significantly higher results than the girls

with an Analytic/Imagery style, supporting the contention that certain cognitive

styles affect boys and girls mathematics performance differently. Implications of

results and strategies for improving mathematics achievement among girls are

discussed.[3]

0

20

40

60

80

100

120

BOYS GIRLS BOYS GIRLS BOYS GIRLS BOYS GIRLS

MP MR MC MA

39.6336.44 38.5 37.28

110.28

36.6139.33

111.33

CS1

CS2

CS3

CS4

CS5



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Chrysostomou, Marilena & Tsingi, Chara & Cleanthous, Eleni & Pitta-Pantazi,

Demetra. (2011). In their research indicated that spatial imagery , in contrast to the

object imagery and verbal cognitive styles, is related to the achievement in algebraic

reasoning and number sense . The study also revealed that as prospective teachers’

spatial imagery style increases, the use of conceptual strategies in solving the tasks

also increases. Implications of these findings concerning the teaching process are

discussed. [6]

W Kusumaningsih, H A Saputra and A N A (2019) in conducted research on

Cognitive style and gender differences in a conceptual understanding of

mathematics students The results showed that cognitive style between male and

female students have a similarity understanding of mathematics concepts in

indicators that give examples or examples of the counter of the learned concepts

and have differences in understanding of mathematics concepts; redefine the

concepts in mathematics; connect the various concepts in mathematics as well as

outside mathematics; identify the characteristic of procedure or theory and present

the idea in various forms of mathematical representation. Based on data analysis, it

can be concluded that the understanding of mathematics concepts of cognitive

style on the male is better than on the female. [11]

Findings of the Study

It was found from statistical evidence that Integrated Cognitive Style has

High Mathematical Achievement & High Mathematical Reasoning. So

integrated Cognitive Style helps students to make use of all other

mathematical skills and also able to reflect on solutions to problems and

determine whether or not they make sense. They appreciate the pervasive

use and power of reasoning as a part of mathematics.

It was found from statistical evidence that Girls are better than boys in

Mathematical Problem Solving & Mathematical concepts. So girls have

more ability to do mathematical tasks and potential to provide intellectual

challenges for the enhancement of mathematical understanding and

development..

It was found from statistical evidence that So Boys with Undifferentiated

Style are good in mathematics. Girls using Intuitive cognitive style will

have high Mathematical Achievement and rely on experience patterns

characterized by universalized areas or hunches and explore and abandons

alternatives quickly.

It was found from statistical evidence that boys with Cognitive Style II

uses an unpredictable ordering of analytical steps when solving a problem

Girls with Cognitive Style II also tend to be withdrawn, passive and

reflective and often look to others for problem solving strategies..

It was found from statistical evidence that boys with Cognitive Style are

intuition, counterfactual thinking, critical thinking, backwards induction

and adductive induction. Girls with Cognitive Style II are way to give math

a purpose and to help students understand where formulas come from.

They have ability to made argument, to justify one’s process, procedure, or



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conjecture, to create strong conceptual foundations and connections, in

order to process new information

It was found from statistical evidence that boys with Cognitive Style V

have more clarity in concepts in Mathematics. Girls with Cognitive Style II

have ability to understand Numeric or quantitative entities, descriptions,

properties, relationships, operations, and events.

Conclusion and Educational Implications Academic qualifications, technical competencies, and professional skills are no

longer the only key skills for success now-a-days. Hooda, Madhuri & Devi, Rani.

(2018) suggested the mathematics teacher should take importance of cognitive

styles during preparing their lesson plan and teaching aids.[9] Mathematics

Teachers’ roles in school are complex. First of all, as is known to everyone,

teachers have their own cognitive styles as well as learners, which mean teachers’

own style will have influence on their performance in classrooms. Teachers prefer

to have classroom discussion, and they have the tendency of making students the

center of the math class and let students assume the responsibility of arranging

class. Because there are different types of students with different cognitive styles

teachers tend to preaching and discovering, and they but not students’ function as

the manager of the classroom. The teaching-learning situation with interplay

between teachers and students is the pursuing goal of teacher. Teachers intend to

have their teaching-learning situation in agreement with the cognitive

characteristics of teaching. So, under this circumstance, teachers take the role of

students’ learning facilitator. As is discussed in this article, integrated cognitive

style is an important cognitive concept in mathematics class. So teachers should

learn to make use of learners’ cognitive styles and optimize learners’ learning

effect by overcoming the drawbacks of learners’ cognitive styles. The relationship

between learners and teachers in schools should be complementary and the

teachers’ roles should be dynamic and should not be stereotyped. And with the

help of teachers, learners can form an optimized cognitive model, which will

enable learner to attain knowledge more efficiently.

Learners can be encouraged to think about their cognitive style and how it affects

their learning by trying a quiz to identify their preferences. If they understand how

they prefer to think then they can learn how to optimize their work in the

classroom, and also try alternative ways.

Understanding our learners, and identifying ways they learn, ‘cognitive styles can

be a helpful concept. However, any approaches to teaching and learning that sift

and sort learners on the basis of pre-determined judgments about how they

‘should’ learn are likely to be ineffective Consideration cognitive styles in math

class is most frequently manifested in attempts to cater to what are perceived to be

the differing needs of different learners. So teacher has to engage with learning or

interested in learning something, Learning is not automatic. And students can't

pretend to understand something unless they enjoy lying to them self. Learning

needs a goal. Our brains are designed to learn. But our brains require deliberate

control of the information input, if students will teach without understanding their



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styles then knowledge becomes vague and awareness is more subconscious than

conscious.

Studies to explore the effectiveness of these proposals with different cognitive

style students and its influence on academic achievement of these subjects are

clearly needed. In the light of the results of this research, the mediating role of

learning strategies in the influence of cognitive style on academic achievement

should be also analyzed in adolescence levels.

References

[1] Anderson, K. L., Casey, M. B., Thompson, W. L., Burrage, M. S., Pezaris,

E., & Kosslyn, S. M. (2008). Performance on Middle School Geometry

Problems With Geometry Clues Matched to Three Different Cognitive

Styles. Mind, Brain, and Education, 2(4), 188– 197.

https://doi.org/10.1111/j.1751-228X.2008.00053.x

[2] Ardi Dwi Susandi ,Cholis Sa’dijah1 , Abdur Rahman As’ari , and Susiswo

(2018);Students’ critical ability of mathematics based on cognitive styles

International Seminar on Applied Mathematics and Mathematics

Education 2019 IOP Conf. Series: Journal of Physics: Conf. Series 1315

(2019) 012018 IOP Publishing pg 1-10

[3] Arnup, Jessica & Murrihy, Cheree & Roodenburg, John & McLean,

Louise. (2013). Cognitive style and gender differences in children’s

mathematics achievement. Educational Studies. 39. 355-368.

[4] Brown, Elizabeth, et al, (2006), Reappraising Cognitive Styles in Adaptive

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[5] Chinn, C.J. and Ashcroft, J.R (1993). Mathematics for Dyslexics: A

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[6] Chrysostomou, Marilena & Tsingi, Chara & Cleanthous, Eleni & Pitta-

Pantazi, Demetra. (2011). Cognitive styles and their relation to number

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[8] Dr. Parkash Chandra Jena (2014) Cognitive Styles and Problem Solving

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[9] Hooda, Madhuri & Devi, Rani. (2017). Significance Of Cognitive Style For

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[10] James J. Roberge and Barbara K. Flexer(1983) ; Cognitive Style,

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[11] Kusumaningsih, Widya & Saputra, H & Aini, A. (2019). Cognitive

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[13] Lusweti Sellah, Kwena Jacinta & Mondoh Helen | (2018)

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[14] Muhammad Shahid Farooq(2015) Cognitive Styles and Quality of

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[15] Musa, Danjuma Christopher & 2 SAMUEL, Iwanger Ruth

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2019

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Web-Resources

https://www.verywellmind.com/what-is-cognition-2794982#reducing-

sensory-information

http://www.seidatacollection.com/upload/product/201107/2011jyhy101a21

.pdf

https://www.teachingenglish.org.uk/article/learning-styles-discussion-

forum

https://www.basicknowledge101.com/subjects/learningstyles.html

http://www.scielo.br/scielo.php?script=sci_arttext&pid=S0102-

79722012000100013



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https://www.verywellmind.com/what-is-cognition-2794982#reducing-sensory-information

https://www.verywellmind.com/what-is-cognition-2794982#reducing-sensory-information

http://www.seidatacollection.com/upload/product/201107/2011jyhy101a21.pdf

http://www.seidatacollection.com/upload/product/201107/2011jyhy101a21.pdf

https://www.teachingenglish.org.uk/article/learning-styles-discussion-forum

https://www.teachingenglish.org.uk/article/learning-styles-discussion-forum

https://www.basicknowledge101.com/subjects/learningstyles.html

http://www.scielo.br/scielo.php?script=sci_arttext&pid=S0102-79722012000100013

http://www.scielo.br/scielo.php?script=sci_arttext&pid=S0102-79722012000100013

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