UNIVERSITY OF SCIENCE AND TECHNOLOGY OF SOUTHERN PHILIPPINES

TABLE OF CONTENTS

I. CHAPTER I …………………………………………………… i

Introduction ………………………………………………..ii

Theoritical Framework ……………………………………………………....ii

Research Question ……………………………………………………...iii

Hypothesis ……………………………………………………....iii

Conceptual Framework……………………………………………………....iv

Delimitation ……………………………………………………….iv

Significance of the Study ………………………………………………….....iv

II. Chapter II ………………………………………………………v

III. Chapter III ………………………………………………………vi

Methodology ………………………………………………………..vi

Research Design ………………………………………………………...vi

Sampling Procedure ………………………………………………………….vii

Locale Of the Study ………………………………………………………......vii

Respondents Of the Study …………………………………………………..viii

Data Gathering ………………………………………………………......ix

Statistical Tool …………………………………………………………….ix

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Chapter I

Introduction

Over the two decades, problems on how to teach mathematics emerged

including the difficulties on interpreting mathematical statements, conflict on the

mathematics lesson analysis and real life experiences, and precocious exposure to

the highly abstract mathematics lesson (Garfield & Ahlgren, 1988). Traditionally,

usual teaching in the middle school involves teacher-centered approach dominated

by lecturing, abstract concepts, theoretical lessons and chalks and talk technique

(Perin & Charron, 2006). However, these traditional approaches in teaching seem

not effective in the modern times.

In fact, school mathematics of today is viewed by the educators to be student

centred. Learner engaged in problem solving and reasoning. It should also promote

deep understanding and develop the learner’s critical and analytical thinking.

Strategy and instruction should not be limited to plain mastery of algorithms or the

development of certain mathematical skills. It should involve learners in investigation

through “exploring, conjecturing, examining and testing” (NCTM, 1990, p.95). And

should foster reflective thinking among students. Learners of today having hard

time to reflect knowledge in to authentic setting. Moreover, it was reported that

students have lacks of sense in the community and at work, does not reflect their

knowledge in the real world, and offers little room for the discussion (Artis, 2008;

Berns & Erickson, 2001). Studies indicated that traditional way of teaching

mathematics usually involve little active learning and causes students to become

unmotivated and disengaged (Caverly, Nicholson, & Radcliffe, 2004; Misulis, 2009;
 UNIVERSITY OF SCIENCE AND TECHNOLOGY OF SOUTHERN PHILIPPINES

Tilson, Castek, & Goss, 2010). To address this problem, teachers need to make a

paradigm shift of teaching pedagogies so that their students get involved in

teaching-learning process.

One of the strategy to aid this problem is contextualization, another way of

addressing the content of activities undertaken in the mathematics classroom

(Castek, & Goss, 2010). Also, teaching the lesson in the real life context increases

significantly the learning of students (Center for Occupational Research and

Development, 2012). Likewise, contextualization motivates the learners to know,

understand, and appreciate cultural heritage (Bringas, 2014).

The role of contexts in mathematics teaching and learning has gained much

attention. Lee (2012) presents examples of contextual problems dated over 1500

years ago in China so clearly the use of context is not a novelty. In Realistic

Mathematics Education theory, a context plays a significant role as a starting point of

learning for students to explore mathematical notions in a situation that is

‘experientially real’ for them (Gravemeijer & Doorman, 1999). Gravemeijer and

Doorman (1997) underlines that experientially real situation does not exclude pure

mathematical problem and “experiential reality grows with the mathematical

development of the student.” (p. 127). One of the key characteristics of good

contextual problems is its’ capacity to bring out a variety of mathematical

interpretations and solution strategies.

Moreover, Tomlinson et al. (2003) suggested to the teachers to conduct

contextualized instructions that will address students’ readiness, interest and

learning on a wide range classroom. This call for reform encourage maximum

participation and development of students’ learning. Teaching students through

concrete things before moving to abstraction lead them gradually from actual objects
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through symbols. This technique had shown to be particularly effective with students

who have difficulties in Mathematics (Jordan, Miller, & Mercer, 1998). Connecting

mathematical concepts through the use of objects create better retention and

integration of concepts in physical world.

This approach is similar to the work of Jerome Bruner (Bruner, 1960) that

teachers should start with the concrete components that includes manipulatives,

tools, or any other objects that students can be handle during the instruction and

moving to abstract components that includes symbolic representations such as

numbers or letters that students can be write or interpret to demonstrate their

understanding of a task. Through representations of abstract concepts by real

objects, students can easily see the relevance of mathematics in their lives.

Based on the previous study students will have great experiences through the

use of contextualization and able to demonstrate understanding in the task through

the representation of authentic object found in the community they lived in. The

contextualized problem in this study is used as a strategy in helping students to

enhance academic achievement in math.

Thus, the main aim of this research is to examine the effect of the use of

contextualized problems in enhancing students’ achievement in mathematics. This

study will only focus on grade 11 students of Lagonglong Senior High School s.y

2021-2022 only focus on grade 11 students of Consuelo National High School s.y

2020-2021.

THEORITICAL FRAMEWORK

The conceptual context refers to the personal understanding of the situation.

These three contexts play vital role in developing mathematical understanding.

However, this study focused on conceptual context only to answer the research
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question. The socio-cultural theory of Vygotsky (1978) has gained recognition in the

mathematics education community. This theory speaks that students’ intelligence is

a result of social interaction in the world (Sutherland, 1993). Over which the students

have conscious control to language to build up a cognitive tool.

The conceptual context refers to the personal understanding of the situation.

These three contexts play vital role in developing mathematical understanding.

However, this study focused on conceptual context only to answer the research

question. The socio-cultural theory of Vygotsky (1978) has gained recognition in the

mathematics education community. This theory speaks that students’ intelligence is

a result of social interaction in the world (Sutherland, 1993). Over which the students

have conscious control to language to build up a cognitive tool.

This framework descriptions’ results will come up into understanding concept

as students create mental constructions. This mental construction was further

described by Sfard (1991) into two ways namely; operationally (process) or

structurally (objects). Also, Thompson (1994) described the development of

concepts in the terms of objects and process. He distinguishes that concepts were

developed through figural knowledge. On the other hand, the way a student interacts

with their family and friends influence the way they think, behave and speak, which

is transferred to other context including school and work (Gauvain, 2001).

Classroom setting seems to be a complex context because it is a part of a larger

world where common experiences of the students are associated yet individually,

students have unique experiences that define them as person (Santoro, 2009).

. The teacher’s pedagogy should reflect lessons that integrate these

intelligences at some stage of the lesson in an effort to improve students’ confidence

and reduce their anxiety in mathematics ( Dedrian Barnaby,2015)

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This author’s insight has prompted me to research how the use of integrating

local literature in the teaching of mathematics, how this strategy affect the student’s

achievement score particularly in problem solving, and the student’s opinion of the

said strategy.

Research Questions

1. How do student’s achievement scores compared in terms of the use two

strategies ; traditional self-learning and contextualized problem?

2. Is there significant difference in students’ achievement scores influenced by

the two strategies in teaching math?

Hypothesis

There is no significant difference between the use of contextualized problem and the

academic performance of students in math.

CONCEPTUAL FRAMEWORK

INDEPENDENT VARIABLE DEPENDENT VARIABLE

Achievement Scores
Contextualized Problem into
-Anxiety
Math
-Convention

Extraneous Variable

Pre- Test
-Attitude
-Personality
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Delimitation

This study is limited on three sections of Grade 11 senior high school

students having 50 plus learners each classroom for a total of 175 senior high

students in Lagonglong Senior High School school year 2021-2022.

Significance of the study

This study will help mathematics teachers who integrate contextualized problem

into mathematics recognize that mathematical understanding involves reading and

writing. For contemporary literacy educators this will be a guide to integrate reading

and writing across various content areas. For mathematics educators and

professional organizations to realize the effectiveness of the integration of reading

and writing will promote but are often presented simply as “tools for learning and

understanding” mathematics (Draper & Siebert, 2004, p. 928). Researchers were

able to investigate the classroom environment and students’ attitudes toward

reading, writing, and mathematics. Students will be able to develop critical thinking

skills which helpful to become a productive and competitive globally in the future.

This will enhance their creativity in dealing mathematic integrated with local

literature. This research also will help students to contextualize and localize. For the

country, this research contributes patriotism.

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Chapter II

Limjap stated that,as modern civilization requires relentless quantification

and critical evaluation of information in daily transactions, it becomes necessary to

develop newer ways of thinking and reasoning that can be used to learn and do

mathematical activities. Through problem solving for instance, we acquire a

functional understanding of mathematics needed to cope with the demands of

society.

School mathematics of today is viewed by the educators to be student centred.

Learner engaged in problem solving and reasoning. It should also promote deep

understanding and develop the learner’s critical and analytical thinking. Strategy and

instruction should not be limited to plain mastery of algorithms or the development of

certain mathematical skills. It should involve learners in investigation through

“exploring, conjecturing, examining and testing” (NCTM, 1990, p.95). It should foster

reflective thinking among students.

Rivera and Nebres (1998) note specifically “the numerous published research

studies of Fennema and Carpenter on Cognitively Guided Instruction (CGI) in the

last quarter of this century [which] point to the pernicious effects of status quo ways

of thinking about mathematics and problem solving (i.e. existing mathematics

culture)”(p.11). CGI recognizes the “acculturation of school children to an algorithmic

approach to learning basic arithmetical facts” which pervade the current school

mathematics culture and which have been proven to be “detrimental to children’s

own ways of thinking about problem solving and computations” (p.12).

Bishop (1999) adds that “research has shown the importance of the idea of

situated cognition which describes the fact that when you learn anything you learn it
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in a certain situation” (p.41). Thus for learning to become meaningful, the learner

has to actively participate in the formation of mathematical concepts. She should not

passively receive knowledge from an authority but should be involved in the

construction of knowledge.

The theory of active construction of knowledge influenced many learning

theories formulated by staunch contemporary mathematics educator like Von

Glasersfeld, Cobb, Bauersfeld, Vygotsky and numerous others (Rivera, 1999). In

fact, “problem solving and mathematical investigations based on a constructivist

theory of learning, have been the main innovations or revivals for the last decade”

according to Southwell (1999, p.331).

Willoughby (1990) believes that the abundant books, pamphlets and courses

on critical thinking and problem solving that have been propagated in the 1980s

cannot be of help unless certain pedagogical misconceptions are clarified. This

includes prescribed rules such as finding key words in a problem to decide the

appropriate operations on the values given in the problem, or applying arithmetic

algorithm to any word problem. Developing critical and analytical thinking through

problem solving takes time and a lot of teacher’s commitment and dedication.

(Willoughby, 1990; Barb and Quinn, 1997).

Limjap stated that developing critical and analytical thinking involves

pedagogical conceptions with a philosophical basis. This paper adheres to the

constructivist theory of learning and promotes the belief that problem solving

processes rest on basic thinking skills which are best developed within a

constructivist framework.

In the light of existing literature base on mathematics instruction and

flourishing research studies on mathematics teaching and learning, this paper

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explores issues and finds ways of fostering critical and analytical thinking through

problem solving. Then it draws implications regarding the design of a contextualized

problem for General math at the grade 11 level that establishes problem type

schema. This design is supported by a philosophical basis of the role of technology

in the acquisition of mathematical knowledge. The design is not instrument specific,

since it is intended to be adaptable to whatever technology is available to both

teachers and students be it in progressive countries or in the third world countries.

Recent research studies on mathematics education have placed its focus on

the learners and their processes of learning. They have posited theories on how

learners build tools that enable them to deal with problem situations in mathematics.

Blais reveals that the philosophical and theoretical view of knowledge and learning

embodied in constructivism offers hope that educational processes will be

discovered that enable students to acquire deep understanding rather than

superficial skills. (Blais, 1988, p.631)

Limjap notes that as learners experience their power to construct their own

knowledge, they achieve the satisfaction that mathematical expertise brings. They

acquire the ability to engage in critical and analytical context of reflective thinking.

They develop systematic and accurate thought in any mathematical process.

O’Daffer and Thorquist (1993) define critical thinking as “a process of

effectively using skills to help one make, evaluate and apply decisions about what to

believe or do”(p.40). They cited the observations of Facett(1938) on a student using

critical thinking as one who 1. Selects the significant words and phrases in any

statement that is important and asks that they be carefully defined; 2. Requires

evidence supporting conclusions she is pressed to accept; 3. Analyzes that evidence

and distinguishes fact from assumption; 4 4. Recognizes stated and unstated

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assumptions essential to the conclusion; 5. Evaluates these assumptions, accepting

some and rejecting others; 6. Evaluates the argument, accepting or rejecting the

conclusion; 7. Constantly reexamines the assumptions which are behind her beliefs

and actions. Critical thinking abilities can only be developed in a setting which the

learner has ample knowledge and experience. Thus, fostering critical thinking in a

certain domain entails developing deep and meaningful learning within the domain.

Learners can acquire critical thinking strategies by using what cognitive and

developmental psychologists call a cognitive schema. Smith, Knudsvig and Walter

(1998, p.50) describe a cognitive schema to be “a scheme, method, process by

which (one) can see, organize and structure information” for better comprehension

and recall. Through the schema learners interpret, analyze, organize and make

sense of every information given in a problem situation through a constructive

process called reflective abstraction.

Reflective abstraction is introduced by Piaget, where critical thinkers are able

to assimilate information into their mathematical network and build from their prior

knowledge. They can accommodate new ideas including those that conflict with

what they know or believe and negotiate these ideas. They are willing to adjust their

belief systems after re-examining information. They are also able to generate new

ideas based on novel ideas that are available to them. They are expert problem

solvers who can handle abstract problem information and make sense of different

problem situations.

On the flip side, novice problem solvers are not able to handle abstract

mathematical concepts. They have difficulty recognizing underlying abstract

structures and often need to make detailed comparisons between current and earlier

problems before they can recognize the abstract information in the solution of the
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current problem ( Reed ,1987; Reed, Dempster, Ettinger, 1985; Anderson, 1984;

Ross, 1987, as cited by Bernardo, 1994). They usually resort to algorithmic activity

and not to the perception of essence. Blais (1988) observed that “they resist learning

anything that is not part of the algorithms they depend on for success”(p.627). They

tend to be very shallow in dealing with problem situations because of the lack of

depth in their experiences while engaging in mathematical activities.

All problem solvers, whether experts or novices, develop a cognitive schema

which cognitive scientists call problem-type schemata when confronted with a

mathematical problem. According to Bernardo (1994), “[k]nowledge about the

problem categories include information about the relevant underlying principles,

concepts, relations, procedures, rules, operations and so on”(p.379). Further, he

adds, “problem-type schemata are acquired through some inductive or

generalization process involving comparisons among similar or analogous problems

of one type”(p.379). Learners represent, categorize and associate problems to be

able to determine the appropriate solution. The expert’s schematic processing leads

to an accurate analysis of the problem which the novice hardly achieves.

Bernardo (1994) claims that “the novices’ schemata (expectedly) include[s]

mainly typical surface-level information associated with a problem type, whereas

experts’ schemata include[s] mainly statements of abstract principles that [are]

relevant to the problem type”(p.380). One example of the difference in the

processing of experts and novices given by Blais (1988) is on their reading process

of a mathematical material. Blais (1988) observes that, [w]hen novices read, the

process almost always appears to be directed toward the acquisition of specific

information that will be needed for algorithmic activity, (whereas) the reading

process used by experts is directed toward the perception of essence. (p.624)

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Limjap added that experts seem to readily categorize the mathematical

information in the material being read, thus facilitating the processing of information

that lead to the correct solution. They are able to attain some sort of a visual form of

say an algebraic expression and are able to communicate this before they perform

the algorithmic activity. Besides, they can determine errors and attain a deep

understanding of the underlying structure of the mathematical concept.

Experts rely not only on concepts and procedures when confronted with a

mathematical problem. They also have access to metacognition which is the

knowledge used by experts in “planning, monitoring, controlling, selecting and

evaluating cognitive activities” (Wong, 1989, Herrington, 1990, English, 1992 as

cited by English-Halford, 1992; Bernardo, 1997). With this higher order thinking skill,

problem solvers are assured of the success of every mathematical strategy they

employ

It is therefore the goal of education to help novices gain expertise in

mathematical activities such as problem solving. In the next section, we deal with a

few different views of studies conducted on didactics of problem solving.

In the light of all the issues and conflicts on various aspects of problem

solving, particularly on developing cognitive strategies among students, and with the

assumption that teachers hold wholesome beliefs and attitudes towards

mathematics teaching, this paper attempts to offer suggestions on effective ways of

fostering critical and analytical thinking through problem solving at different school

levels

Contextualized math problem and open-ended and involves aspects of both

problem-solving and mathematical modeling. Lesh and Zawojewski (2007) define

problem solving as the process of interpreting a situation mathematically, which

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usually involves several iterative cycles of expressing, testing and revising

mathematical interpretations – and sorting out, integrating, modifying, revising, or

refining clusters of mathematical concepts from various topics within and beyond

mathematics. (p. 782)

With respect to mathematical modeling, when learners work on a problem

involving a real-world context, part of the problem solving process may involve the

construction of mathematical models, or systems of objects, relationships, and rules

that can explain or predict the behavior of other systems (Doerr & English, 2003).

Although we do not claim that the problem discussed in this paper is a modeling

problem per se, participants engage in aspects of the modeling process (e.g.,

developing a model and interpreting solutions) as they solve the problem. The

problem used in this study is contextualized and ill-structured, and requires that the

learner find and use information from the real world.

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Chapter III

Methodology

Research Design

This study examine the effects of the approaches/strategy in teaching

General math on student achievement toward mathematics.

Quasi-experimental design using one experimental group and one control

group was used in the study,

The independent variable of the study where the two strategies in teaching

General Math and the dependent variables are the achievement scores of the

students in modified-teacher test.

The Pre-test- Post-test Non-comparable Experimental Control Group Design was

used in the study. The design of the study is illustrated as follows

TREATMENT PRE- TREATMENT POS-

GROUP TEST TEST

EXPERIMENTA Q1 X1 Q2

L
CONTROL Q1 X2 Q2
In experimental group, students were exposed to contextualized problem

that were parallel to the topics covered by the researcher. Every student was

provided with contextualized problem through self-learning learning module

needed for the activity The procedures and time allocation for every activity is

clearly written in provided hand-outs .

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In the control group, the students were taught in traditional manner. The

teacher give the self-learning module, introduced the new lesson, applied the

concept by giving examples, conducted exercises to master the concepts which was

followed by an evaluation.

Students were not informed that they were the subjects of the study.

Both control and experimental group were given the same set of exercises every

meeting. Parallel quizzes on the topics covered which were prepared by the

researcher were also given to both groups.

Sampling Procedure

There were two strand in the Grade 11 level, namely : HUMSS and ABM , which

has General Mathematics subject in the first semester. The researcher purposively

choose the HUMSS class with heterogeneous students as the experimental group

and the remaining section is the control group. The total number of the students are

175 students of Senior High School-Senior High. The pilot section is the Nobility.

Locale of the Study

This study is conducted at Lagonglong Senior High School which is located at

Talahiran Poblacion, Misamis Oriental. The school has a population of 380 students

from Grade 11 to Grade 12. There are 175 senior high school student-respondents

which are all Grade 11 who took General Mathematics for the 1st semester of the

school year 2021-2022.

Respondents of the Study

The respondents of the study are 175 Senior High School Academic Strand

students of Lagonglong Senior High School in 11th grade in General Mathematics

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class. There are fifteen males and twenty females ranging from 16 to 25 years old

which are present by the time of distribution of the questionnaire.

Gathering Data Procedure

The researchers secured a permit in the school administration to conduct a

survey. After securing the said permit, the researchers administer the questionnaires

to the respondents. They were asked to bring the questionnaire since it is modular

learning. The researchers let the students be aware of the purpose of the activity.

The respondents are given 1 hour to complete their answers (pre-test and post-test)

and the mean is appropriate for scale option. Researchers assured the respondents

that their responses are to be kept with confidentiality.

Statistical Tools

 Frequency

 Pearson’s Correlation

 Percentage

 ANCOVA
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