Error analysis in college algebra in the higher education institutions in la union

ERROR ANALYSIS IN COLLEGE ALGEBRA IN THE HIGHER

EDUCATION INSTITUTIONS OF LA UNION

A Dissertation

Presented to

the Faculty of the Graduate School

Saint Louis College

City of San Fernando, La Union

In Partial Fulfillment

of the Requirements for the Degree

Doctor of Education

Major in Educational Management

by

FELJONE GALIMA RAGMA

January 11, 2014

ii

INDORSEMENT

This dissertation entitled, ―ERROR ANALYSIS IN COLLEGE

ALGEBRA IN THE HIGHER EDUCATION INSTITUTIONS OF

LA UNION,‖ prepared and submitted by FELJONE GALIMA RAGMA, in

partial fulfillment of the requirements for the degree DOCTOR OF

EDUCATION major in EDUCATIONAL MANAGEMENT, has been

examined and is recommended for acceptance and approval for ORAL

EXAMINATION.

NORA ARELLANO-OREDINA, Ed.D. Adviser

This is to certify that the dissertation entitled, ―ERROR ANALYSIS

IN COLLEGE ALGEBRA IN THE HIGHER EDUCATION INSTITUTIONS

OF LA UNION,” prepared and submitted by FELJONE GALIMA RAGMA,

is recommended for ORAL EXAMINATION.

MARIA LOURDES R. ALMOJUELA, Ed.D.

Chairperson

JOVENCIO T. BALINO, Ed.D. DANIEL B. PAGUIA, Ed.D. Member Member

AUGUSTINA C. DUMAGUIN, Ph.D. AURORA R. CARBONELL, Ed.D. Member Member

Noted by: ROSARIO C. GARCIA, DBA

Dean, Graduate School Saint Louis College

iii

APPROVAL SHEET

Approved by the Committee on Oral Examination as PASSED with

a grade of 96% on January 11, 2014.

MARIA LOURDES R. ALMOJUELA, Ed.D.

Chairperson

JOVENCIO T. BALINO, Ed.D. DANIEL B. PAGUIA, Ed.D.

Member Member

AUGUSTINA C. DUMAGUIN, Ph.D. AURORA R. CARBONELL, Ed.D. Member Member

Accepted and approved in partial fulfillment of the requirements

for the degree DOCTOR OF EDUCATION MAJOR IN EDUCATIONAL

MANAGEMENT.

ROSARIO C. GARCIA, DBA Dean, Graduate Studies

Saint Louis College

This is to certify that FELJONE GALIMA RAGMA has completed

all academic requirements and PASSED the Comprehensive Examination

with a grade of 96% on June 15, 2013 for the degree DOCTOR OF

EDUCATION major in EDUCATIONAL MANAGEMENT.

ROSARIO C. GARCIA, DBA

Dean, Graduate Studies Saint Louis College

iv

ACKNOWLEDGMENT

The researcher wishes to express his sincerest gratitude to the

following persons who contributed much in helping him structure the

research.

Dr. Nora A. Oredina, dissertation adviser, for always affirming and

supporting; and for giving necessary suggestions to better this study.

Dr. Maria Lourdes R. Almojuela, chairperson of the dissertation

panel, for her valuable critique, and most especially, for directing the

researcher to the correct structure of the research.

Dr. Aurora R. Carbonell, Dr. Augustina C. Dumaguin, Dr. Daniel

B. Paguia, Dr. Rosario C. Garcia and Dr. Jovencio T. Balino, the

panelists, for their brilliant thoughts.

The validators of the questionnaire and the research output for

giving suggestions that improved the study.

Presidents, registrars, academic deans, department

chairpersons, instructors and students of the Private Higher Education

Institutions in La Union, for lending some of their precious time in

dealing with the pre-survey and the questionnaires.

Mrs. Edwina M. Manalang and Mrs. Marilyn Torcedo, for sparing

some time for brainstorming for the built-in theory of the study.

v

Mesdames Grace, Lea, Melody, Graziel, Jay Ann, Abegail, Sister

Grace, Mafe, and Sir Roghene, the researcher’s friends, who gave him

inspiration.

Mr. & Mrs. Felipe and Norma Ragma, the researcher’s parents,

for always being there when the researcher needed some push.

Kuya Darwin, Ate Felinor and Ate Nailyn, the researcher’s

siblings, for always following up the researcher’s progress.

And lastly, to GOD Almighty, for giving the needed strengths in

the pursuit of this endeavor.

F. G. R.

vi

D E D I C A T O N

To my Parents,

Mr. & Mrs Felipe and

Norma Ragma

and

To my siblings,

Darwin, Felinor and

Nailyn

This humble work is

dedicated to all of you!

F.G.R.

vii

ABSTRACT

TITLE : ERROR ANALYSIS IN COLLEGE ALGEBRA IN

THE HIGHER EDUCATION INSTITUTIONS OF LA UNION

Total Number of Pages: 374

Text Number of Pages : 358

AUTHOR : FELJONE G. RAGMA

ADVISER : NORA ARELLANO-OREDINA, Ed.D.

TYPE OF DOCUMENT : DISSERTATION

TYPE OF PUBLICATION: Unpublished

ACCREDITING INSTITUTION: SAINT LOUIS COLLEGE

City of San Fernando, La Union CHED, Region I

KEY WORDS : Error Analysis, Math Performance, Error Categori- zation, Educational Management, Instructional

Intervention Plan, Mathematics Teaching Interven- tions, etc.

Synopsis

The descriptive study identified and analyzed the error categories

of students in College Algebra in the Higher Education Institutions of

La Union as basis for formulating a validated Instructional Intervention

Plan. Specifically, it determined the a) level of performance of the

students in College Algebra along elementary topics in sets and Venn

diagrams, real numbers, algebraic expressions, and polynomials; special

product patterns; factoring patterns; rational expression; linear

viii

equations in one unknown; systems of linear equations in two

unknowns; and exponents and radicals; b) the capabilities and

constraints of the students in College Algebra; and, c) the error

categories of the students along reading, comprehension, mathematising,

processing and encoding. Data were collected using a researcher-made,

all-word-problem test. The participants were 374 first year students

enrolled in College Algebra for first semester, school year 2013-2014. The

data gathered were treated statistically using frequency count, mean,

percentage and the Newmann’s tool for error analysis. It found out that

the students had fair performance in elementary topics, special products

and factoring while poor performance in rational expressions, linear

equations and systems of linear equations and very poor performance in

exponents and radicals; thus, the students, in general, had poor

performance. The performances of the student in the specified topics

were all considered as constraints. Mathematising and comprehension

were the major error categories of the students in elementary topics,

processing and reading errors in special products, reading and

Mathematising in factoring, reading and Mathematising in rational

expressions, reading and comprehension in linear equations; and reading

and Mathematising in systems of linear equations and exponents and

radicals. In general, their major error categories in College Algebra were

along reading and Mathematising. Moreover, the instructional plan is

ix

found to have very high validity. Based on the findings, it was concluded

that the students cannot competently deal with elementary topics,

special product and factoring patterns rational expressions, linear

equations, systems of linear equations and radicals and exponents.

Additionally, the instructional intervention plan is a very good material

that addresses problems on performance and errors. Based on the

conclusions, it is recommended that the schools should adopt the

Instructional Intervention Plan and let their mathematics instructors

attend the two-day seminar-workshop. The students should exert more

effort in understanding the different concepts in their College Algebra

course. They should spend more time dealing with drills and exercises.

The mathematics teachers should suit their instructional strategies to

the needs of the students. The English teachers must also intensify in

their classes the basic skill of reading with comprehension. A study

should be conducted to determine the effectiveness of the instructional

intervention plan. And, a similar study should be conducted in other

branches of Mathematics, applied sciences and English.

x

TABLE OF CONTENTS

Page

TITLE PAGE………………………………………………………………… i

INDORSEMENT…………………………………………………………… ii

APPROVAL SHEET…………....................................................... iii

ACKNOWLEDGMENT…………………………………………………… iv

DEDICATION……………………………………………………………… vi

ABSTRACT………………………………………………………………… vii

TABLE OF CONTENTS………………………………………………….. x

LIST OF TABLES…………………………………………………………. xiv

LIST OF FIGURES……………………………………………………….. xvi

CHAPTER

I INTRODUCTION……………………………………………… 1

Background of the Study.……......………….......... 1

Theoretical Framework……………………………..... 8

Conceptual Framework……………………………….. 15

Statement of the Problem…………........................ 19

Assumptions……………………………………........... 21

Importance of the Study……………...................... 21

Definition of Terms…………………………………..... 23

II METHOD AND PROCEDURES…………………………… 27

Research Design……………………………………… 27

xi

Page

Sources of Data………………………………………. 28

Locale and Population of the Study……………... 28

Instrumentation and Data Collection ..……….... 29

Validity and Reliability of the Questionnaire.

Administration and Retrieval of the

Questionnaire ………………………………

30

31

Data Analysis ………………………………………….

Data Categorization……………………………….....

32

33

Parts of the Instructional Intervention Plan….……………………………………………….

36

Ethical Considerations…………………………...... 37

III RESULTS AND DISCUSSION…………………………….. 39

Level of Performance of Students in College Algebra…………………………………………….. 39

Elementary Topics………………………………

39

Special Product Patterns……………………… 41

Factoring Patterns ……………………………… 44

Rational Expressions…………………………… 46

Linear Equations in One Variable…………… 48

Systems of Linear Equations in Two Unknowns………………..………………….. 50

Exponents and Radicals……………………….

51

xii

Page

Summary on the Level of Performance of Students in College Algebra ………….

52

Capabilities and Constraints of Students in

College Algebra…………………………………..

54

Error Categories in College Algebra……………… 56

Elementary Topics……………………………… 56

Special Product Patterns……………………… 63

Factoring…………………………………………. 67

Rational Expressions…………………………… 74

Linear Equations in One Variable Systems 80 Systems of Linear Equations in Two

Unknowns……………………………………

85

Exponents and Radicals………………………. 91

Summary on the Error Categories in

College Algebra …………………………….

93

Validated Instructional Intervention Plan ………

96

Instructional Intervention Plan ……………………

Two-day Seminar-Workshop on the Utilization of the Instructional Intervention Plan………

Sample Flyer of the Two-Day Seminar/

Workshop ………………………………………..

Level of Validity of the Instructional Inter-

vention Plan ………………………………………

99

296

299

300

IV SUMMARY, CONCLUSIONS AND RECOMMEN-

DATIONS………………………………………………..

301

xiii

Page

Summary………………………………………………. 301

Findings………………………………………………… 302

Conclusions…………………………………………… 302

Recommendations…………………………………… 303

BIBLIOGRAPHY……………………………………………… 305

APPENDICES………………………………………………… 313

A Sample Computations on the:

Reliability of the College Algebra Test …

313

Validity of College Algebra Test ……….. List of Suggestions Made by the

Validators and the Correspond- ing Action/s by the Researcher …….

B Letter to Students-Respondents to Administer College Algebra Test ………..

The College Algebra Test ………………………

314

315

317

317

Math I – College Algebra Test (Table of Specifications) …………………..

C Letter to the Presidents/School Heads of

the HEIs understudy to Gather Data/Information ………………………….

324

326

D Sample of Corrected College Algebra Test…

336

CURRICULUM VITAE…………………………………….. 354

xiv

LIST OF TABLES

Table Page

1 Distribution of Respondents ………………………… 29

2 Level of Performance of Students in Elementary

Topics ………………………………………………..

40

3

4

Level of Performance of Students in Special

Product Patterns …………………………………..

Level of Performance of Students in Factoring Patterns ……………………………………………..

42

45

5

Level of Performance of Students in Rational Expressions ………………………………………..

47

6

7

Level of Performance of Students in Linear Equations in One Variable ……………………..

Level of Performance of Students in Systems of

49

Linear Equations ………………………………….

51

8 Level of Performance of Students in Exponents

and Radicals ………………………………………..

52

9 Summary Table on the Level of Performance of

Students in College Algebra …………………….

53

10 Capabilities and Constraints of Students in

College Algebra ……………………………………

55

11 Error Categories in Elementary Topics………..….. 57

12 Error Categories in Special Product Patterns……. 64

13 Error Categories in Factoring Patterns…………..... 68

14 Error Categories in Rational

Expressions ………………………………………..

75

xv

15

Error Categories in Linear Equations in One Variable…………………………………………….

Page

81

16

Error Categories in Systems of Linear Equations in Two Variables ........................

86

17 Error Categories in Exponents and Radicals…….. 92

18 Summary Table on the Error Categories in College Algebra…………………………………..

94

19 Level of Validity of the Instructional Intervention

Plan………………………………………………… 300

xvi

LIST OF FIGURES

Figure Page

1 Ragma’s Error Intervention Model…………………………… 13

2 The Research Paradigm ……………………………………….. 18

1

CHAPTER I

INTRODUCTION

Background of the Study

Education, in its general sense, is a form of learning in which

knowledge, skills, and values are imparted to a person or group of

persons through teaching, training, or research. Many countries adhere

to the principle that education is the key to a nation’s success. Some

experts even correlate the number of literate people to the nation’s

economic growth since national advancements are most commonly

achieved by people who have trainings and intellectual advancements

(www.educationworld.com).

Furthermore, the central goal of education is to help a person

develop critical thinking, reasoning and problem-solving skills. Hence,

education prepares a person for life. One subject that helps people

prepare for life is Mathematics.

Mathematics is the science that deals with the logic of shape,

quantity, reasoning and arrangement. It is concerned chiefly on how

ideas, processes and analyses are applied to create useful and

meaningful knowledge that man can use throughout his life (Prakash,

2010). It has also become one of the powerful tools of man in cultural

adaptation and survival. Recorded history narrates that mathematical

2

discoveries have been at the forefront of every civilized society and in use

even in the most primitive of cultures. The needs of mathematics arose

based on the wants of society. The more complex a society is, the more

complex is the mathematical need. Primitive tribes needed little more

than the ability to count, but also relied on mathematics to calculate the

position of the sun and the physics of hunting (Hom, 2013).

Mathematics has played a very important role in building up

modern civilization by perfecting the sciences. In this modern age of

Science and Technology, emphasis is given on sciences such as Physics,

Chemistry, Biology, Medicine and Engineering. Mathematics, which is a

Science by any criterion, is also an efficient and necessary tool being

employed by all these Sciences. As a matter of fact, all these Sciences

progress only with the aid of Mathematics. So it is aptly remarked,

"Mathematics is the science of all sciences and the art of all arts." (Wells,

2006).

Furthermore, Mathematics is the language and the queen of the

Sciences. According to the famous Philosopher Kant, "A Science is exact

only in so far as it employs Mathematics." So, all scientific education

and studies which do not commence with Mathematics is said to be

defective at its foundation (Wells, 2006). Thus, neglect of mathematics

causes injury to all knowledge.

3

It is undeniable that Mathematics expresses itself everywhere, in

almost every facet of life - in nature and in the technologies in our hands.

It is the building block of everything in our daily lives, including mobile

devices, architecture, art, money, engineering, sports and many others.

Without mathematics, man can go astray (Petti, 2009).

Mathematical literacy is a must element in providing the students

with the basic skills to live their life. It is one of the basic pillars for the

student on which his life is, and would be standing. So the base of this

pillar needs to be really strong and clear. Mathematics helps the student

in developing conceptual, computational, logical-analytical, reasoning

and problem-solving skills. One Mathematics subject that trains such

skills is College Algebra. College Algebra is a pre-requisite subject in

higher education institutions. The National Center for Academic

Transformation (2009) labels it as the gateway course for freshmen in the

tertiary level. This means that a student who aspires to be a degree

holder must pass successfully through the course. This is the main

reason why most countries, through their ministry or department of

education, have mandated the inclusion of College Algebra in the course

curriculum.

No one can negate the importance of College Algebra. Cool (2011),

enumerates some of the uses of algebra in today’s world. Algebra is used

in companies to figure out their annual budget which involves their

4

income and expenditure. Various stores use algebra to predict the

demand of a particular product and subsequently place their orders. It

also has individual applications in the form of calculation of annual

taxable income and bank interest on loans. Algebraic expressions and

equations serve as models for interpreting and making inferences about

data (Okello, 2010). Further, algebraic reasoning and symbolic notations

also serve as the basis for the design and use of computer spreadsheet

models. Therefore, mathematical reasoning developed through algebra is

necessary through life, affecting decisions people make in many areas

such as personal finance, travel, cooking and real estate, to name a few.

Thus, it can be argued that a better understanding of algebra improves

decision-making capabilities in society (The Journal of Language,

Technology & Entrepreneurship in Africa, Vol. 2, No.1, 2010).

In addition, Algebra is one of the most abstract strands in

mathematics. This very nature of the subject makes it difficult for

students to appreciate and love Algebra. With this, Prakash (2010)

remarked that the place of mathematics in education is in grave danger.

The teaching and learning of College Algebra, with insufficient skills and

high anxiety levels, degenerated into the realm of rote memorization, the

outcome of which leads to satisfactory formal ability but does not lead to

real understanding or to greater intellectual independence. A testament

to this worsening scenario is the global move for educational reforms.

5

Countries around the world are alarmed by the lowering

performance of their students, especially in College Algebra. In America

alone, educational experts are tasked to improve performance in

Mathematics (Arithmetic, Algebra, Geometry and the like) so they can

bring back the glory days of the United States in topping Surveys of

Countries along students’ academic performance (Serna, 2011).

Bressoud (2012) added that even though there are interventions, College

Algebra failure rates are disappointing. Further, in a University in Africa

of Fall 2007, College Algebra examination results showed that only 23%

of the students performed well. This poor performance calls for the

establishment of the reason why College Algebra is challenging to many

students (Kuiyan, 2007). In addition, Shepherd (2005) revealed that most

students do not excel in their Algebra course. Most of them cannot

perform indicated operations, especially when fronted with word

problems. Students find it hard to solve problems in Algebra. Some just

do not answer at all. These situations reflect poor understanding of and

performance in the course (The Journal of Language, Technology &

Entrepreneurship in Africa, Vol. 2, No.1, 2010).

Although there are many causes of student difficulties in

mathematics, the lack of support from research fields for teaching and

learning is noticeable (The Journal of Science and Mathematics

Education, 2010). Egodawatte (2009) emphasized that getting the level

6

of performance among students would not help much in Mathematics

Education; researches need to dig deeper into the reasons by

characterizing students’ errors and misconceptions. With this situation,

error analysis is very essential. Egodawatte (2009) added that using error

analysis, it would be possible for teachers to design effective instruction

or instructional intervention to avoid this dismal performance. Thus, it

can be construed that research on student errors is a way to clearly plot

out a more valid action plan that could address issues on students’

mathematics performance.

Mathematical errors are a common phenomenon in students’

learning of mathematics. Students of any age irrespective of their

performance in mathematics have experienced getting mathematics

wrong. It is natural that analyzing students’ mathematical errors is a

fundamental aspect of teaching for mathematics teachers (Hall, 2007).

The Philippines is also not exempted from this global predicament

on the dismal performance in College Algebra. Garcia (2012) mentioned

that Filipino students enrolled in College Algebra regarded the subject

as challenging and a difficult subject which contributed to their low

performance. In addition, the national survey conducted by

Drs. Lambitco, Laz and Malab (2009) on the readiness of Filipino

students in College Algebra revealed that the students are not ready to

take up College Algebra course. Further, according to Professor Ramos

7

(2012), 40-50% of the students enrolled in College Algebra failed.

According to him, this performance is caused by poor instruction and

cognitive unpreparedness. This low performance was also highlighted

when Leongson (2003) revealed that Filipino students excelled in

knowledge acquisition but fared considerably low in lessons requiring

higher-order-thinking skills.

On the provincial scene, Picar (2009) strongly presented in his

study that students’ anxiety in College Algebra is high but their

performance is low. Pamani (2006) also mentioned that more than 60%

of the college freshmen in La Union have low to fair competence. Pamani

(2006) stressed that these results point out to a problematic situation in

education. These facts are also strengthened by Bucsit (2009) when she

revealed that out of 195 college freshmen in the Private Schools in

La Union, 113 or 58% of the students have fair performance. In addition,

Oredina (2011) revealed that the performance of SLC students in College

Algebra was at the moderate level only.

Furthermore, the researcher, being a College Algebra instructor,

observes that many students still have many misconceptions along

certain topics in College Algebra, even if most of the course contents are

just a recap of high school mathematics. To note, some students omitted

the signs when performing operations. Others did not know what to do

8

when presented with a word problem while many were not able to craft

their own procedures in solving the given problems

The aforementioned situationers on College Algebra performance

prompted the researcher to conduct an error analysis in College Algebra

in the Higher Education Institutions (HEIs) of La Union as basis for

formulating an instructional intervention plan.

Theoretical Framework

M. Anne Newman’s (1977) theory of errors and error categories

maintains that when a person attempts to answer a standard, written,

mathematics question, he has to be able to pass through a number of

successive hurdles, namely Reading (or Decoding), Comprehension,

Transformation or ―Mathematising,‖ Processing, and Encoding. From

these successive stages, students commit varied errors. According to the

theory, the reading errors are committed when someone could not read a

key word or symbol in the written problem to the extent that this

prevented him/her from writing anything on his/her solution sheet or

from proceeding further along an appropriate problem-solving path; the

comprehension errors are committed when someone had been able to

read all the words in the question, but had not grasped the overall

meaning of the words; thus, he can only indicate partially what are the

given and what are unknown in the problem; the transformation or

9

mathematising errors are committed when someone had understood

what the questions wanted him/her to find out but was unable to

identify the operation, or sequence of operations or the working equation

needed to solve the problem; the processing errors are committed when

someone identified an appropriate operation, or sequence of operations

or the working equation, but did not know the procedures necessary to

carry out these operations or equation accurately; and, the encoding

errors are committed when someone correctly worked out the solution to

a problem, but could not express this solution in an acceptable written

form. In some case, if the answer is not in its accepted simplified form

and does not indicate the unit.

Researchers which made use of the abovementioned theory were

Clement (2002), Ashlock (2006), Hall (2007) and Egodawatte (2011). All

of their studies were able to find out the specific error categories of their

student-respondents.

Furthermore, Vygotsky (1915) and Kolb’s (1939) constructivist

theory proposes that a person can construct and conditionalize

knowledge, especially after learning or experiencing something. As

applied to this study, the students are believed to be capable of showing

the desired competence after learning the contents of College Algebra

from their instructors.

10

Dewey (1899) and Roger’s (1967) active learning and experiential

learning theories propose that students are able to learn something and

apply what they have learned if they are engaged with their experiences.

As applied in the study, the problems in the researcher-made test were

anchored to the real-life encounters of the college students.

Also, Bruner’s (1968) intellectual development theory discusses

that intellect is innately sequential, moving from inactive through iconic

to symbolic representation. He felt that it is highly probable that this is

also the best sequence for any subject to take. The extent to which an

individual finds it difficult to master a given subject depends largely on

the sequence in which the material is presented. Further, Bruner also

asserted that learning needs reinforcement. He explained that in order

for an individual to achieve mastery of a problem, feedback must be

reviewed as to how they are doing. The results must be learned at the

very time an individual is evaluating his/her performance. This theory

supports the idea that solving written problems are successive in nature.

This also gave the idea to the researcher on how to check the all-word

problem test.

Further, Bandura’s (1963) social learning theory holds that

knowledge acquisition is a cognitive process that takes place in social

context and can purely occur through observation or direct instruction.

11

As applied in the study, the instructional interventions are student-

centered so that learning becomes more active.

In addition, when one attempts to address concerns on student’s

errors, instructional intervention can be a good scheme. Egodawatte

(2009) stresses that error analysis can pave away to clearly conceptualize

an action plan such as designing effective instruction or plotting out

instructional intervention. This idea by Egodawatte (2009) structures the

foundation of the output of the study.

Howell (2009) describes instructional intervention as a planned set

of procedures that are aimed at teaching specific set of academic skills to

a student or group of students. An instructional intervention must have

the following components: it is planned – planning implies a decision-

making process. Decisions require information (data); therefore, an

instructional intervention is data-based or research-based set of teaching

procedures; it is sustained – this means that an intervention is likely

implemented in a series of lessons over time; it is focused– this means

that an intervention is intended to meet specific set of needs for

students; it is goal-oriented – this means that the intervention is

intended to produce a change in knowledge from some beginning or

baseline state toward some more desirable goal state; and, it is typically

a set of procedures rather than a single instructional component/

12

strategy. Moreover, according to Manitoba Education Website (2010), an

instructional intervention plan contains the purpose or the background,

intervention objectives, specific topics, the error categories, the sample of

error, the proposed instructional strategy and or activities, and the

procedures of implementing the strategy. (http://www. edu. gov.

mb.ca/k12/specedu/bip/sample.html.)

The aforecited theories find their essence in the teaching and the

learning of mathematics and in the specific categories in the research’s

aim of identifying and analyzing errors. These also gave the researcher

the main reasons of formulating the research tool composed of all word

problems. Generally, they serve as the building blocks in structuring

this research. Further, the concept of instructional intervention plan

serves as the core idea in designing the output of this study.

Furthermore, these theories served as foundations in formulating

the proposed model of the researcher, the Ragma’s Error Intervention

Model. Figure 1 illustrates the model.

The model, a corollary of Newmann’s (1977), highlights that when

someone answers a written mathematical problem, he has to undergo

different but successive stages such as reading, comprehension,

mathematising, processing and encoding stages. In simple words,

someone has to read the problem, understand what the problem says,

http://www.edu.gov.mb.ca/k12/specedu/bip/sample.html




13

Figure 1. Ragma’s Error Intervention Model

INSTRUCTIONAL

INTERVENTION

(Game-based,

visual/spatial-based,

motivational instruction,

technology-based,

cooperative learning,

tutorials,

differentiated teaching,

understanding-centered,

processing-centered,

reading strategies,

experiments,

dyads,

observations, and

scaffolding)

CAUSES OF ERRORS

(low Interest, attitude, high anxiety, Insufficient

recall, misconception, deficient mastery,

carelessness)

Encoding

Processing

Stage

Comprehension

Stage

Reading

Mathematising

Stage

Error Categories Stages in Problem Solving

Encoding Errors

Processing Errors

Mathematising Errors

Comprehension Errors

Reading Errors

Better

Performance

in College

Algebra

Mathematics Word

Problems

14

14

structure the working equation, solve and then finalize the answer/s. In

each of these successive stages, errors can be committed. These errors

are caused by low interest, high anxiety, negative attitude, insufficient

recall, misconception, poor mastery, and carelessness. To exemplify,

when someone does not bother to answer the problem, he is not

interested in mathematics or has high anxiety towards math. If he fails to

completely analyze what the problem is all about, he cannot completely

recall the essential mathematical details. If he cannot create a working

equation, he has poor mastery and deficient mathematical skills. If he

cannot proceed to the starting point of the mathematical solution, he

cannot recall the formulas or is unable to formulate the working

equation. If he cannot correctly and completely solve the problem, he has

deficient mastery and is careless in handling mathematical algorithms.

And, if he is unable to write a valid or unaccepted final answer, he is

careless or lacks the necessary mathematical skills.

Moreover, the different error categories and their causes can be

addressed through the varied instructional interventions. To illustrate,

reading errors caused by high anxiety and disinterest can be addressed

by providing motivational instructional activities and games;

differentiated instruction can also be a good instructional scheme.

Comprehension errors caused by misconception can be addressed by

15

concept attainment and processing. Mathematising errors caused by

poor mastery and insufficient recall can be addressed by direct

instruction, memory-bank game and the think-pair-share activities, to

name a few. Processing errors caused by poor mastery and insufficient

recall can be addressed by error targeting and correcting, explicit

instruction, etc. And lastly, encoding errors caused by carelessness can

be solved by solve-and-compare, cooperative learning groups, etc. When

all the error categories in each problem-solving stage together with their

respective causes are addressed through the instructional interventions,

better performance of the students in College Algebra will be achieved.

Conceptual Framework

Answering a standard, written, mathematics question requires a

person to undergo a number of successive stages: reading,

comprehension, mathematising, processing, and encoding. From these

successive stages, students commit varied errors.

The reading errors are committed when someone could not read a

key word or symbol in the written problem to the extent that this

prevented him/her from writing anything on his/her solution sheet or

from proceeding further along an appropriate problem-solving path.

The comprehension errors are committed when someone had been

able to read all the words in the question, but had not grasped the

16

overall meaning of the words; thus, he can only indicate partially what

are the given and what are the unknown in the problem.

The transformation or mathematising errors are committed when

someone had understood what the questions wanted him/her to find out

but was unable to identify the operation, or sequence of operations or the

working equation needed to solve the problem.

The processing errors are committed when someone identified an

appropriate operation, or sequence of operations or the working

equation, but did not know the procedures necessary to carry out these

operations or equation accurately.

The encoding errors are committed when someone correctly

worked out the solution to a problem, but could not express this solution

in an acceptable written form. In some case, if the answer is not in its

accepted simplified form and does not indicate the unit. This makes

mathematics teaching challenging.

Thus, for learning to take place, all the stages and aspects of

problem analysis and problem solving must be well understood by the

students.

Moreover, when someone aspires to help students to improve on

their performance, one needs to dig deeper into the reasons behind the

dismal performance. According to Newmann (1977), the type of errors

17

committed by the students when solving word problems can give baseline

data to teachers to help them improve on their mathematical skills.

Egodawatte (2009) and Hall (2007) stressed that mathematical

errors are a common phenomenon in mathematics learning. Students of

any age have experienced getting mathematics wrong (Hall, 2007). It is

natural that analyzing students’ mathematical errors is a fundamental

aspect of teaching for mathematics teachers.

Error Analysis is then an effective assessment approach that

allows one, especially teachers, to determine whether students are

making consistent mistakes when performing computations. By

pinpointing the error category or pattern of an individual student’s

errors, one can then directly teach the correct procedure for solving the

problem or can even formulate an effectively designed instructional

intervention scheme (Egodawatte, 2009).

It is in this light that the study is thought of, formulated and set

up. This conceptualization is logically designed in the Research Paradigm

in Figure 2. The paradigm made use of the Input-Process-Output (IPO)

model. The input is composed of the performance of the students along

elementary topics, special product patterns, factoring, rational

expressions, linear equations, systems of linear equations in two

unknowns and exponents and radicals. It also incorporates the error

18

Patterns

PROCESS OUTPUT INPUT

Validated

Instructional

Intervention Plan

for College

Algebra in the

Higher

Education

Institutions of

La Union

1. Interpretation

and Analysis of the

Performance of the

students along the

specified topics

2. Identification and

Analysis of the

capabilities and

constraints based

on the level of

performance

3. Identification and

Analysis of error

categories of the

students

4. Preparation and

Validation of

Instructional

Intervention Plan

1. Performance of the students along:

a. Elementary topics a.1. sets and Venn diagrams a.2. Real numbers a.3. Algebraic expressions

a.4. Polynomials

b. Special Product

c. Factoring Patterns

d. Rational Expressions

e. Linear Equations in One Unknown

f. Systems of Linear Equations in Two Unknowns

g. Exponents and Radicals

2. Error Categories along the specified topics in College Algebra along a. reading b. comprehension c. transformation d. process e. encoding

Figure 2. The Research Paradigm

19

categories of the students along the specified topics in Math 1 or College

Algebra along reading, comprehension, mathematising, processing and

encoding. These variables are indeed necessary to determine the

performance and error categories of the students in College Algebra.

The process incorporated the interpretation and analysis of the

performance of the students in College Algebra, the identification and

analysis of the capabilities and constraints and the identification,

categorization and analysis of errors in College Algebra. It also holds the

process of conceptualizing and validating the output of the study.

The output of the study, therefore, is a validated instructional

intervention plan for the Higher Education Institutions of La Union.

Statement of the Problem

This study identified and analyzed the error categories of students

in College Algebra in the Higher Education Institutions of La Union as

basis for formulating a Validated Instructional Intervention Plan.

Specifically, it sought answers to the following questions:

1. What is the level of performance of the students in College

Algebra along:

a. Elementary Topics;

a.1. Sets and Venn Diagrams

a.2. Real Numbers

20

a.3. Algebraic Expressions

a.4. Polynomials

b. Special Products;

c. Factoring Patterns;

d. Rational Expressions;

e. Linear Equations in One Unknown;

f. Systems of Linear Equations in Two Uknowns; and

g. Exponents and Radicals?

2. What are the capabilities and constraints of the students in

College Algebra?

3. What are the error categories of the students along the topics

in College Algebra along:

a. Reading;

b. Comprehension;

c. Mathematising or Transformation;

d. Processing; and

e. Encoding?

4. Based on the findings, what validated instructional intervention

plan can be proposed?

a. What is the level of validity of the instructional intervention

plan along face and content?

21

Assumptions

The researcher was guided with the following assumptions:

1. The level of performance of the students in College Algebra is

satisfactory.

2. The capabilities are along elementary topics while the

constraints are along factoring, special products, and systems of linear

equations in two unknowns.

3. The major error categories of the students are mathematising

and processing errors.

4. A validated instructional intervention plan addresses the errors

of the students in College Algebra.

Importance of the Study

This piece of work will greatly benefit the CHED, administrators,

heads, teachers, students, the researcher and future researchers.

The Commission on Higher Education (CHED). This study will give

the commission an idea of the reasons or causes of low performance in

College Algebra, which will help in developing improvements along

curriculum and human resource.

The school administrators of the HEIs in La Union. This study

will provide them with data that can be used as input to the curricular

programs.

22

The Mathematics department heads. This study will give them

insights about the performance and errors in College Algebra, which will

help them in designing mathematics instruction that suits the identified

errors of the students.

The Mathematics instructors. This study will give them baseline

data of the performance and errors of their students in College Algebra.

The output of the study, on the other hand, will make them more

prepared in addressing the errors since instructional interventions are

proposed for their utilization.

The students of the HEIs in La Union. This study will lead them to

a thoughtful understanding of mathematics since their errors will be

known. They will also be helped in improving their performance since

the instructional interventions will address their identified errors.

The researcher, a Mathematics instructor of Saint Louis College

(SLC). This study will make him more knowledgeable of his students’

performance and errors. This will also give him the opportunity to

structure an error intervention model that addresses students’ errors

which contributes to the improvement of the fields of mathematics

teaching and learning.

The future researchers. This study will motivate them to pursue

their research since this study can be used as basis for their future

23

study. This can also give them an idea on how to structure their own

instructional plan based on their students’ needs and interests.

Definition of Terms

To better understand this research, the following items are

operationally defined:

Capabilities. These refer to a performance with a descriptive

equivalent of satisfactory performance and above.

College Algebra. This is a 3-unit requisite subject in college which

includes elementary topics, special product and factoring patterns,

rational expressions, linear equations in one unknown, systems of linear

equations in two unknowns and exponents and radicals.

Elementary topics. These topics include concepts on sets,

real number system and operations, and polynomials.

Algebraic expressions. These are expressions

containing constants, variables or combinations of constants and

variables.

Polynomials. These are algebraic expressions with

integer exponents.

Real numbers. These are the numbers composing of

rational and irrational numbers.

Sets. These are collection of distinct objects.

24

Venn diagrams. These are diagrams proposed by the

mathematician A. Venn, which are used to show relationships among

sets.

Factoring patterns. These include the topics in factoring

given a polynomial. These include common monomial factor, perfect

square trinomial, general trinomial, factoring by grouping and factoring

completely.

Linear equations in one unkown. This includes topics on

equations with one variable such as 2x- 4 = 10 and 5x - 2x = 36. The

main thrust of this topic is for an unkown variable to be solved in an

equation.

Rational expressions. These are expressions involving two

(2) algebraic expressions, whose denominator must not be equal to zero.

The topics included are simplifying and operating on rational

expressions.

Special product patterns. These topics include the patterns

in multiplying polynomials easily. These patterns include the sum and

difference of two identical terms, square of a binomial, product of two

binomials, cube of a binomial and square of a trinomial.

Systems of linear equations in two unknowns. This topic

discusses how the solution set of a given system is solved. The methods

25

that are used in this certain topics include graphical, substitution and

elimination methods.

Constraints. These refer to a performance with a descriptive

equivalent of fair performance and below.

Error analysis. It is a diagnostic procedure aimed at determining

specific inaccuracies of the students in College Algebra. The analysis is

made using the Newmann Error Analysis tool (1977).

Error categories. These are the classes of inaccuracies according

to Newmann (1977). These error categories are reading, comprehension,

transformation or ―mathematising‖, process and encoding.

Encoding errors. These are committed when someone

correctly worked out the solution to a problem, but could not express

this solution in an acceptable written form. In some case, if the answer is

not in its accepted simplified form and does not indicate the unit of

measurement.

Comprehension errors. These are committed when someone

had been able to read all the words in the question, but had not grasped

the overall meaning of the words; thus, can only indicate partially what

are the given, what are unknown in the problem

Processing errors. These are committed when someone

identified an appropriate operation, or sequence of operations or the

26

working equation, but did not know the procedures necessary to carry

out these operations or equation accurately

Transformation errors. These are committed when someone

had understood what the questions wanted him/her to find out but was

unable to identify the operation, or sequence of operations or the working

equation needed to solve the problem

Reading errors. These are committed when someone could

not read a key word or symbol in the written problem to the extent that

this prevented him/her from writing anything on his solution sheet or

from proceeding further along an appropriate problem- solving path.

Higher Education Institutions (HEIs). This refers to the twelve

(12) respondent academic colleges and universities, public or private, in

La Union offering College Algebra for the school year 2013-2014.

Instructional intervention plan. This plan contains the teaching

approaches that address dismal performance. It is composed of the

background, the general objectives, the specific topics, the error

categories and causes, the sample error, the intervention and the

assessment strategy. This serves as the output of the study.

27

CHAPTER II

METHOD AND PROCEDURES

This chapter presents the research design, sources of data, data

analysis, the parts of the instructional intervention plan and ethical

considerations.

Research Design

The descriptive method of investigation was used in the study. This

design aims at gathering data about the existing conditions. Leary (2010)

defines such design as one that includes all studies that purport to

present facts concerning the nature and status of anything. This design

is appropriate for the study since it is aimed at gathering pertinent data

to describe the performance and errors of students in College Algebra.

Further, the quantitative research approach was also used.

Hohmann (2006) defines quantitative research approach as a component

of descriptive design making use of numerical analysis. It is aimed at

analyzing input variables using quantitative techniques such as

averages, percentages, etc. This approach is apt for this study since it

makes use of quantitative techniques to show the performance and

errors of the students in College Algebra.

28

Sources of Data

Locale and Population of the Study. The population of this

study was composed of College Algebra students enrolled in the Higher

Education Institutions (HEIs) of La Union for the first semester, school

year 2013-2014.

The total population of 5,849 students was pre-surveyed in this

study; however, since the population reached 500, random sampling was

employed.

To generate the sample population, the Slovin’s formula (Leary

2010) was used.

n = 𝑁

1+𝑁(𝑒2)

where:

n = the sample population

N = the population

1 = constant

e = level of significance @ .05

Using the Slovin’s formula, a total of 374 students distributed

among the 12 respondent Higher Education Institutions of La Union

constituted the respondents of this study.

Table 1 reveals the distribution of the sample population.

29

Table 1. Distribution of Respondents

Respondent HEIs N n

Institution A 78 5 Institution B 482 31 Institution C 230 15

Institution D 900 58 Institution E 609 39 Institution F 1349 86

Institution G 65 4 Institution H 196 13

Institution I 51 3 Institution J 1536 98 Institution K 170 11

Institution L 183 12

Total 5849 374

Instrumentation and Data Collection

A pre-survey was conducted to gather the contents of the syllabus

in College Algebra in each of the HEIs. The researcher was able to meet

the math instructors, department heads/chairs and academic deans who

gave data pertinent to the scope of College Algebra. The conglomerated

topics indicated in all the syllabi served as basis in the topics specified in

the research tool. (Please see appended table of specifications)

To gather the data pertinent to the level of performance and the

error categories, a researcher-made test was made. The researcher-made

test is an all-word-problem 20-item test, 5 points per item, covering all

the topics in College Algebra. Most of the questions were based on the

word problems from College Algebra books. All problem questions were

30

aligned along the synthesis-evaluation/evaluating-creating level under

the Bloom’s Taxonomy. As such, the questions dug into the overall

conceptualization and utilization of algebraic concepts and principles to

be able to carry out such problem. Hence, an item combined several

related subtopics to ensure that the scope of the course was still covered.

The whole test was administered by the math instructors handling

the classes through the permission of the presidents or concerned

authority in the HEI. The test was good only for one hour and did not

allow the use of calculators.

Validity and Reliability of the Questionnaire. To ensure the

validity of the research tool, it was presented to the members of the panel

and to experts in the field of mathematics. The experts are professors of

mathematics. Further, the suggestions made by the validators were

incorporated in the test (see suggestions in the appendix). The computed

validity rating was 4.32, interpreted as high validity (please see

appended computation). This means that the research tool was able to

measure what it intended to measure.

Moreover, to establish its reliability, it was pilot-tested to thirty (30)

students of Saint Louis College. The thirty (30) students were not

included as respondents of the study. The internal consistency or

reliability was determined using the Kuder-Richardson 21 formula. The

formula is (Monzon-Ybanez 2002):

31

𝐾𝑅21 = 𝑘

𝑘−1 1 −

𝑥 𝑘−𝑥

𝑘𝜎2

where:

k = number of items

𝑥 = mean of the distribution

𝜎2= the variance of the distribution

Thus, the computed reliability coefficient was 0.72 (please see

appended computation). This means that the test was highly reliable,

which pinpoints that the test was internally consistent and stable.

Administration and Retrieval of the Questionnaire. With the

necessary endorsement from the Dean of the Graduate School

(Dr. Rosario C. Garcia) of Saint Louis College, City of San Fernando,

La Union, the researcher sought permission from the president or head

of the different twelve (12) respondents-institutions to float the

questionnaire. The copies of the questionnaire was handed to the

deans/program heads of the various college institutions who were also

requested to administer the said questionnaire to the respondents of

which the answered questionnaires were retrieved on a specified date as

it was scheduled by the deans/program heads of the various

respondents-institutions.

32

Tools for Data Analysis

The data gathered, collated and tabulated were subjected for

analysis and interpretation using the appropriate statistical tools. The

raw data were tallied and presented in tables for easier understanding.

For problem 1, frequency count, mean and rate were utilized to

determine the level of performance in College Algebra. The formula for

mean is as follows (Ybanez, 2002):

M = ∑x

N

Where: M – mean

x – sum of all the score of the students

N – number of students

For problem 2, the capabilities and constraints were deduced

based on the findings, particularly on the level of performance in College

Algebra. An area was considered a capability when it received a

descriptive rating of satisfactory and above; otherwise, the area was

considered a constraint.

For problem 3, the Newmann Error Analysis Tool (1977) was used

to identify the errors and error categories of the students. (Please see the

error categories in the definition of terms.) Moreover, frequency count,

average and rate were used to determine the error categories of the

students.

33

The MS Excel Worksheet and StaText were employed in treating

the data.

Data Categorization

For the scoring/checking of the test, the scheme below was used:

Point Assignment Error Category

0 Reading Error

1 Comprehension Error

2 Mathematising Error

3 Processing Error

4 Encoding Error

5 No Error

For the level of performance in each topic in College Algebra, the

following scale systems were utilized.

Elementary Topics/ Factoring

Score Range Level of Performance Descriptive Equiva-

lent Rating

16.00-20.00 Outstanding Performance (OP) Capability

12.00-15.99 Satisfactory Performance (SP) Capability

8.00 -11.99 Fair Performance (FP) Constraint

4.00-7.99 Poor Performance (PP) Constraint

0-3.99 Very Poor Performance (VPP) Constraint

34

Special Products and Patterns/Rational Expressions/Linear Equations in One Variable


lent Rating 12.00-15.00 Outstanding Performance (OP) Capability


6.00-8.99 Fair Performance (FP) Constraint


0.00-2.99 Very Poor Performance (VPP) Constraint

Systems of Linear Equations in Two Variables


lent Rating 8.00-10.00 Outstanding Performance (OP) Capability





Exponents and Radicals


lent Rating

4.00-5.00 Outstanding Performance (OP) Capability



35

Score Range Level of Performance Descriptive Equiva- lent Rating



For the general performance in College Algebra, the scales below

were used:

Score Range Level of Performance

80.00-100.00% Outstanding Performance (OP)

60.00-79.99% Satisfactory Performance (SP)

40.00-59.99% Fair Performance (FP)

20.00-39.99% Poor Performance (PP)

0-19.99% Very Poor Performance (VPP)

The scale for interpretation on the reliability of the College Algebra

test was:

1.00 - Perfect Reliability (PR)

0.91-0.99 - Very High Reliability (VHP)

0.71-0.90 - High Reliability (HR)

0.41-0.70 - Marked Reliability (MR)

0.21-0.40 - Low Reliability (LR)

0.01-0.21 - Negligible Reliability (NR)

0.00 - No Reliability (NoR)

36

For the validity of the College Algebra test and the Instructional

Intervention Plan, the scale below was used:

Points Ranges Descriptive Equiva- lent Rating

5 4.51-5.00 Very High Validity (VHV)

4 3.51-4.50 High Validity (HV)

3 2.51-3.50 Moderate Validity (MV)

2 1.51-2.50 Poor Validity (PV)

1 1.00-1.50 Very Poor Validity (VPV)

Parts of the Instructional Intervention Plan

The instructional intervention plan contains the purpose or the

background, intervention objectives, specific topics, the error categories,

the sample error, the proposed instructional strategy and or activities,

the procedures of implementing the strategy and the assessment

strategy.

The instructional intervention plan is based on the level of

performance of the students in College Algebra, the culled-out

capabilities and constraints and the different error categories in each

topic of College Algebra. The foremost constraints and the two primary

error categories in each topic are given more emphasis on the

instructional intervention plan as seen on the number of indicated

37

interventions. There are still interventions for those considered as

capabilities for sustainability.

Ethical Considerations

To establish and safeguard ethics in conducting this research, the

researcher strictly observed the following:

The students’ names were not mentioned in any part of this

research. The students were not emotionally or physically harmed just

to be a respondent of the study.

There were HEIs which decided not be included in the study due to

some concerns and other priorities. This decision of opting not to join in

the study was respected by the researcher.

Coding scheme was used in reflecting the respondent HEI in the

table for distribution of respondents.

Proper document sourcing or referencing of materials was done to

ensure and promote copyright laws.

A communication letter was presented to the Registrar’s Office or

President’s Office to ask authority to gather the needed data on the

contents of the syllabi and number of students enrolled in College

Algebra.

A communication letter was presented to the President’s Office

asking permission to float the questionnaire.

38

The research instrument was subjected to validity and reliability.

Their suggestions were incorporated in the instrument. A list of summary

and the corresponding actions of the researcher is appended.

The instructional intervention plan was subjected for acceptability.

All the suggestions were incorporated.

39

CHAPTER III

RESULTS AND DISCUSSION

This chapter presents the statistical analysis and interpretation of

gathered data on the level of performance in College Algebra and the

error categories in each specified topic.

Level of Performance of Students in College Algebra

The first problem considered in this study dealt on the level of

performance of students in College Algebra along elementary topics - sets

and Venn diagrams, real numbers, algebraic expressions, and

polynomials; special product patterns, factoring patterns; rational

expressions; linear equations in one unknown; systems of linear

equations in two unknowns; and, exponents and radicals.

Elementary Topics

Table 2 shows the performance of the students in College Algebra

along elementary topics. It shows that the students had a mean score of

8.69 or 43.45%, a fair performance in elementary topics. This implies

that the students had not achieved to the optimum the needed skills in

elementary topics. It also reflects that the students had poor

performance in sets and Venn diagrams. This means that the students

were not capable of representing data relationships and solving problems

40

Table 2. Level of Performance of Students in Elementary Topics

Subtopic Mean Score Rate Descriptive

Equivalent

Sets and Venn Diagrams (5)

1.78

35.60%

Poor

Real Number System (5) 2.87 57.40% Fair

Algebraic Expressions (5) 1.64 32.80% Poor

Polynomials (5) 2.4 48.00% Fair

Overall 8.69 43.45% Fair

involving sets and Venn diagrams. Moreover, they had fair performance

in real number system. This means that the students could visualize, to

a moderate extent, the number line and perform operations on real

numbers. Further, they had poor performance in algebraic expressions.

This implies that the students could not perform well translations and

operations involving algebraic expressions. On the other hand, they had

fair performance in polynomials. This suggests that the students could

moderately recognize quantities represented by polynomials and perform

mathematical processes involving polynomials.

The findings of the study corroborate with the study of Oredina

(2011) revealing that the students had moderate level of competence in

Elementary topics. She mentioned that the students needed to achieve to

41

the fullest the needed competence in elementary topics in College

Algebra.

Further, the findings of the study conform to the study of Elis

(2013) revealing that the students had moderate performance in

Algebraic expressions. He stressed that this was caused by negative

attitude towards Mathematics.

On the other hand, the study of Pamani (2006) does not run

parallel to the findings of the study stating that the students had high

competence in pre-algebra, which included sets, real numbers, algebraic

expressions, etc. She explained that such level of performance reflected

that the students were highly capable of determining concepts and

performing mathematical procedures along these specified topics.

The findings of the study do not also harmonize with the study of

Okello (2010) revealing that 73% of the students failed in almost all

topics in College Algebra such as prerequisites, factoring and systems of

equations.

Special Product Patterns


along special product patterns. It reveals that the students had a mean

score of 7.41 or 49.40%, a fair performance in special product patterns.

This means that the students could not correctly perform special

42

Table 3. Level of Performance of Students in Special Product Patterns


Equivalent

Product of Two binomials (5)

2.69

53.50%

Fair

Square of a trinomial (5) 2.13 42.60% Fair

Cube of a Binomial (5) 2.59 51.80% Fair


product patterns implying that the students failed to master the skills

along special products. Further, it reveals that the students had fair

performance along product of two binomials. This implies that the

students could not productively use the FOIL method in getting the

product of binomials, implying that they cannot multiply and simplify

two alike or different binomials. Also, they had fair performance along

the square of a trinomial. This entails that the students cannot use the

(F + M +L)2= (F2 + M2 + L2 + 2FM + 2FL + 2ML) pattern reasonably.

Moreover, they also had fair performance along the cube of a binomial.

This indicates that the students cannot use the (F ± L)3= (F3 ± 3F2L ±

3FL2 ± L3) pattern correctly. Since the performance was within the fair

level only, it can be construed that the students had not attained to the

fullest the skills along the utilization of such patterns.

43

The findings of the study adhere to the study of Wood (2003)

emphasizing that the students performed fairly in College Algebra,

especially in special product and factoring patterns. He mentioned that

the students’ level of performance dug into a level of 39% and below.

The findings of the study also corroborate with the study of Pamani

(2006) stressing that the students had moderate competence in special

products. She mentioned that the students failed to master to the fullest

the needed skills in all the special product patterns.

Further, the study jibes with Oredina (2011) stating that the

students had moderate competence in special products. This means that

the students can handle special product patterns but had not fully

mastered the desired competencies. The students had very low

competence in squaring a binomial, low competence in monomial

multiplier, low competence in sum and difference of 2 binomials, high

competence in product of 2 different binomials but very high competence

on cube of a binomial and square of a trinomial.

Further, the study also agrees with the study of Bucsit (2009)

stating that the students had poor performance in special products. She

stated that this very dismal performance pointed out to the fact the

students could not really perform multiplication using polynomials. She

further explained that the students had not very well understood the

concepts and processes involved in special products.

44

Factoring Patterns

Table 4 illustrates the performance of the students in College

Algebra along factoring patterns. It shows that the students had a mean

score of 8.03 or 40. 15%, interpreted as a fair performance. This means

that the students could perform, to a restrained extent, factoring

patterns, pinpointing that the students failed to master, to the fullest, all

the skills along factoring.

It also shows that the students had poor performance in difference

of two perfect squares. It can be inferred that the students could not

distinguish and factor correctly polynomials of the form (x2-y2). Further,

the students had fair performance in perfect square trinomial. This

stresses that the students could not optimally recognize and factor

patterns of the form (F2 ± √2FL + L2). It also reveals that the students had

fair performance in factoring general trinomials. This means that they

were deficient along the required skills. It also reveals that the students

had poor performance in factoring by grouping. This implies that the

students failed to distinguish expressions within a polynomial that can

be grouped together for the purposes of simplification through factoring.

The study harmonizes with Gordon (2008) emphasizing that the

students had dismal performance in concepts involving algebraic

expressions, factoring and special product patterns.

45

Table 4. Level of Performance of Students in Factoring Patterns

Subtopic Mean Score Rate Descriptive Equivalent

Difference of 2 Perfect Squares (5)

1.05

21.00%

Poor

Perfect Square Trinomial (5) 2.64 52.80% Fair

General Trinomial (5) 2.67 53.40% Fair

Factoring by Grouping (5) 1.67 33.40% Poor


These findings also agree with the study of Pamani (2006) revealing

that students had moderate performance in factoring. It was stressed

that the students could perform factoring but needed to do more in order

for the students to attain the desired level of competency.

The findings of the study are in contrast with the study of Oredina

(2011) stating that the students had high competence in factoring

patterns. This means that the students could do well and perform very

satisfactorily factoring exercises.

It also does not jibe with the finding of the study of Bucsit (2009)

stating that the students had poor performance in factoring. She stated

that the students could not very well recognize and perform factoring

patterns.

46

Rational Expressions


along rational expressions. It shows that the students had a mean score

of 4.73 or 31. 53%, interpreted as a poor performance in rational

expressions. This pinpoints that the students failed to correctly simplify

and perform operations involving rational expressions or expressions

involving fractions.

Further, it reflects that the students had fair performance in

simplification of RAEs. This means that the students could not simplify

competently rational expressions to their simplest form by performing

cancellation and reduction. It also mirrors that the students had poor

performance in operations of RAEs. The students could not proficiently

add, subtract, multiply and divide rational algebraic terms or

expressions.

It also shows that the students had very poor performance in

simplification of complex RAEs. This means that the students failed to

perform procedures and algorithms pertinent to the simplification of

complex fractions.

The findings of the study run parallel to the study of Laura (2005)

stressing that students’ performance in College Algebra was in crisis. He

explained that the cohort of students passing College Algebra was only

about 33.33%. He pinpointed that factoring and rational expressions

47

Table 5. Level of Performance of Students in Rational Expressions (RAEs)

Subtopic Mean

Score

Rate Descriptive

Equivalent

Simplification of RAEs (5)

2.43

48.6%

Fair

Operations of RAEs (5) 1.52 31.40% Poor

Simplification of Complex RAEs (5) 0.78 15.60% Very Poor

Overall 4.73 31.53% Poor

were the most difficult for the students.

The findings jibe with the study of Bucsit (2009) revealing that her

respondents had poor performance along rational or fractional

expressions. She stressed that the students had deficient skills as

regards performing operations and simplifying involving rational

expressions. The students were not able to deal with finding the correct

LCDs to simplify correctly the expressions.

Contrary, the findings do not relate to the study of Oredina (2011)

showing that the students had moderate competence in rational

expressions. This means that the students had not fully acquired the

needed competence along the indicated areas. It was stressed that the

students could not correctly manipulate rational expressions, simplify

such and operate using the fundamental operations.

48

Linear Equations in One Variable


along linear equations in one variable. It shows that the students had a

mean score of 3.29 or 21. 93%, interpreted as a poor performance in

linear equations. This implies that the students had not mastered the

mathematical ways of representing data and forming linear equations to

be able to interpret and solve worded problems.

It also unveils that the students had poor performance in distance,

mixture, and age problems. This pinpointed to the fact the students were

deficient in analyzing, representing, crafting working equations and

solving problems related to linear equations in one variable. They could

not see how variables were related to each other; they failed to see

meaning among the algebraic verbal and numerical expressions that

could serve as their basis for structuring the solution of certain

problems.

The study agrees with Bucsit’s (2009) since it revealed that the

students were poor along word problems in linear equations in one

variable. She underlined that the students lacked the necessary skills in

understanding and translating expressions into useful data relevant to

the solution of a certain problem.

It also corroborates with the study of Pamani (2006) revealing that

the students had fair competence along linear equations. She stressed

49

Table 6. Level of Performance of Students in Linear Equations in One Variable


Equivalent

Distance Problem (5)

1.06

21.20%

Poor

Mixture Problem (5) 1.09 21.80% Poor

Age Problem (5) 1.14 22.80% Poor


that this performance points to the failure of students to understand the

complexities of word problems.

The findings of the study do not relate to the study of Oredina

(2011) revealing that the students had moderate competence in linear

equations in one variable. It was emphasized that students’

performances were fair-to-good only along this area. They had moderate

competence in solution of linear equations in one variable including coin,

distance and age problems, low competence in problems on involving

work, mixture, geometric relations and solid mensuration but had high

competence in number relation. She remarked that the students could

deal correctly with formulating, manipulating and finalizing formulas and

the linear equations in one unknown that best fit the main thrusts of the

word problems

50



along systems of linear equations in two variables. It shows that the

students had a mean score of 3.55 or 35.50%, interpreted as a poor

performance in systems of linear equations in two variables. This implies

that the students failed to represent and solve problems using systems of

linear equations. It can also be understood that the students failed to

perform elimination, substitution and other pertinent methods used in

solving systems of linear equations.

The findings of the study relate to the study of Denly (2009) stating

that the students performed unsatisfactorily in number system,

equations and inequalities. He noted that students did not consider

correctly the properties needed in solving equations.

This finding also harmonizes with Pamani’s study (2006) revealing

that the students had fair performance in systems of linear equations.

She stressed that the students were not able to apply the correct

mathematical methods to be able to get the correct solution sets to the

systems.

This study does not run parallel to the study of Oredina (2011)

disclosing that the students had moderate competence in Systems of

Linear Equations in Two Variables. This means that the students did

51

Table 7. Level of Performance of Students in Systems of Linear Equations in Two Variables


Equivalent

Applied Problems on fare (5)

1.28

25.60%

Poor

Applied Problems on numbers (5) 2.27 45.40% Fair


not achieve to the maximum the needed competencies in College Algebra.

They had moderate competence in graphing systems of linear equations

and solving worded problems; they also had low competence in slope and

systems in two (2) variables.


Table 8 unveils the performance of the students in College Algebra

along exponents and radicals. It discloses that the students had a mean

score of 0.39 or 7.80%, a very poor performance. This means that the

students had not mastered the needed skills for them to deal with

exponential and radical expressions competently. They were deficient in

manipulating expressions and equations involving exponents and

radicals. They were not able to correctly treat data inside the radical

symbols and express correctly the square of certain expressions.

52

Table 8. Level of Performance of Students in Exponents and Radicals

Subtopic Mean

Score

Rate Descriptive

Equivalent

Exponential and Radicals (5)

0.39

7.80%

Very Poor

Overall 0.39 7.80% Very Poor

The findings corroborate with the study of Li (2007) stating that

students had difficulty in dealing with exponents and radicals. He

explained that the students did not master the mathematical

principles behind simplification of such concepts. This dismal

performance points out to the fact that mastery was not attained.

In addition, the findings also jibe with the study of Pamani (2009)

showing that the students had fair performance in exponential and

radical expressions and equations. It was stressed that students failed to

understand the rudiments of these algebraic concepts.

Summary on the Level of Performance of Students in College Algebra in the HEIs in La Union

Table 9 shows the summary of the level of performance of students

in College Algebra. It can be clearly gleaned from the table that generally,

the students had a mean score of 36.08 or 36.08%, interpreted as poor

performance. This implies that students did not really achieve to the

53

Table 9. Summary Table on the Level of Performance of Students in College Algebra

TOPIC Mean

Score

Rate Descriptive

Equivalent

Elementary Concepts (20) 8.69 43.45% Fair

Special Product Patterns (15) 7.41 49.40% Fair

Factoring (20) 8.03 40.15% Fair

Rational Expressions (15) 4.73 31.53% Poor

Linear Equation in One Variable (15) 3.28 21.93% Poor

Systems of Linear Equations (10) 3.55 35.50% Poor

Exponents and Radicals (5) 0.39 7.80% Very Poor


maximum the needed or the desired competencies of the subject,

especially that such score did not even reach the mean score of 50 or

50%. This can be attributed to the fact that all the items were word

problems that require higher-order thinking and mathematical skills.

Wood (2003) stressed that when students are prompted with knowledge

or computation questions, students’ success rate is 86% or even higher;

but, when students are prompted with word problems, their success rate

dips down to a low of 39%. This is easy to understand since word

problems synthesize all the necessary skills, from knowledge to

evaluation, to be able to carry out the solution to a given problem. It is in

54

word problems where students are able to apply all the necessary

competencies learned to a situation that requires higher-order-thinking

skills.

Further, the students scored highest along special product

patterns; but, still within the fair level. It can be understood that the

students’ foremost moderate skill is along this subject matter. On the

contrary, they scored lowest along exponents and radicals. This means

that they had not gained competence in this area. This can be attributed

to insufficient time.

Capabilities and Constraints of Students

in College Algebra

The second problem in this study covered the capabilities and

constraints of students in College Algebra. Table 10 discloses the

capabilities and constraints in College Algebra as culled out from the

level of students’ performance. It can be clearly read from the table that

all content areas were regarded as constraints since the performance was

within the fair-to-very-poor levels only. Their foremost constraint was

along exponents and radicals. This means that they were weak along

treating exponential and radical expressions. Although still treated as a

constraint, they performed a little better along special product patterns.

The findings of the study corroborate with the study of Bucsit (2009)

stating that the students performed moderately in number

55

Table 10. Capabilities and Constraints of Students in College Algebra

TOPIC Mean Score Rate Classification

Elementary Concepts

8.69

43.45%

Constraint

Special Product Patterns 7.41 49.40% Constraint

Factoring 8.03 40.15% Constraint

Rational Expressions 4.73 31.53% Constraint

Linear Equation in One Variable 3.28 21.93% Constraint


3.55

35.50%

Constraint


0.39

7.80%

Constraint

system, poor in special product and factors, poor in linear equations and

systems, and fair in rationals, radicals and exponents. It can be deduced

that the constraints of the students in this study were along all the

topics in College Algebra.

Also, the study agrees with Denly (2009) when he revealed that all

students had difficulty in all the content areas in College Algebra. She

mentioned that College Algebra is indeed in crisis since most of the

students could not hurdle the demands of algebraic manipulations,

logic, and analysis of the different variables, especially in written word

problems.

56

Error Categories in College Algebra

The third problem considered in this study is on the error

categories of the students along elementary topics in College Algebra.

Elementary Topics

Table 11 shows the error categories of students along elementary

topics. It reveals that 85 or 22.72% of the errors in elementary topics

were along mathematising, 69.50 or 18.58% were along comprehension,

68 or 18.18% were along reading, 64 or 17.11% were along encoding,

and 61 or 16.31% were along processing. It also shows that 26.50 or

7.09% were not considered errors. This means that most of the students

committed Mathematising errors along elementary topics, implying that

they were able to understand what the questions wanted them to find

out; but failed to identify the series of operations or formulate the

working equation needed to solve the problem.

Specifically, 149 errors in sets and Venn diagrams were along

Mathematising errors. This means that the students were not able to

draw the relationships of the given data using the correct Venn

diagrams. Some made use of tables instead of Venn Diagrams. Others

had not written any equation, solution or diagram after identifying the

given data of the problem. Others also wrote an incorrect working

equation such as ―250 - 160 - 150 - 180 = x‖, ―250-20 = 30‖ and

57

Table 11. Error Categories in Elementary Topics

Subtopic Error Categories

R C M P E N Sets and Venn Diagram

94 51 149 35 18 27

Real Number System 45 15 62 91 142 19

Algebraic Expressions

62 185 41 35 20 31

Polynomials

71

27

89

83

75

29

Average 68 69.50 85 61 64 26.50

Rate 18.18% 18.58% 22.72% 16.31% 17.11% 7.09%

Legend: R- Reading Error C- Comprehension Error M- Mathematising Error P- Processing Error E- Encoding Error N- No Error

―160+150+180+75+90+20=775‖. Others did not write any equation after

presenting the data. This was caused by poor recall and mastery of the

course content. It is also good to note that 94 errors were along reading.

This means that the students had poor understanding regarding the

problem given, which led them not write any data from the given. It also

implies that the students really did not know what to do, leaving the item

unanswered. This highlights deficient mastery of the subject matter.

Moreover, 51 errors were committed along comprehension errors. This

implies that the students were able to read the problem but had not

completely understood the problem. This means that they were unable to

58

completely write the needed data. They missed out writing data such as

―20 customers chose all the brands‖. This was caused by deficient

mastery and carelessness. Also, 35 errors were committed along

processing errors. They were able to write the correct working equation;

however, failed to correctly write the solution. Students wrote on their

diagrams incorrect difference such as ―10‖ instead of ―5‖ for the

remaining number of people who chose Samsung brands. This was

caused by carelessness and deficient mastery of operations on sets.

Lastly, 18 errors were committed along encoding errors. The students

were not able to write the final answer in an acceptable form. The

students just left the answer 5 inside the Venn Diagram. Others just

indicated ―5‖ instead of indicating ―5 people chose other brands or love

other brands‖ as the final answer. This was caused by carelessness and

lack of critical thinking.

It also shows that 142 errors in real number system were along

encoding errors. This implies that the students failed to write the final

answer in an acceptable form. Most students only indicated ―11‖ as their

final answer instead of writing ―11 units‖. This was due to lack of critical

thinking among the students. It is also good to note that 91 errors in this

course content were along processing. It means that they were unable to

correctly perform the needed operations to be able to solve the problem.

The students committed errors on getting the distance of 9 from -2 and

59

10 from 8. Instead of writing ―9- (-2) = 11‖ and ―10 -8 = 2‖, students

wrote ―9- (-2) = 7‖ and ―10 + 8 = 18‖. Others also performed counting but

failed to consider the principle of counting from a number line, implying

an incorrect distance of 10 and 3 units. Some also left the answers

―9 units‖ and ―2 units‖ unadded even if the question was asking them to

get the sum of the distances.

Also, 62 errors were along Mathematising errors. The students did

not write anything as a working equation. Others wrote an incorrect one

such as ―7 + (-2) =d1 and10 + 8 = d2‖. Such error was caused by poor

recall of concepts and deficient mastery. Moreover, 45 errors were

committed along reading. This means that the students left the item

unanswered. This means that the students did not know what to do.

Lastly, 15 errors were committed along comprehension. They were able

to indicate only 7 and -2, but not 10 and 8. Others indicated the distance

to be from -2 being the least coordinate and 10, being the highest

coordinate. This was caused by deficient skill in mathematical

understanding.

Further, it also reveals that 185 errors in algebraic expressions

were along comprehension. This means that the students were able to

read all the words in the question, but had not grasped the Overall

meaning of the words; they only indicated partially what were the given,

what were unknown in the problem. Most of the students had written an

60

incomplete representation of the phrase ―the height is (x+9) cm more

than the base‖. Instead of writing ―(x+9) + (2x-5)‖, most of them wrote

―(x+9) cm‖ only. This was due to insufficient understanding of

mathematical expressions or poor skills along mathematical translations.

It is revealing that 62 errors were along reading. Students left this item

unanswered. This means that the students did not know what to do. This

error was caused by poor mastery or deficient recall.

Moreover, 41 errors were committed along Mathematising.

Students were not able to correctly indicate the formula for the area of a

right triangle. Others wrote ―A = bh, c2= a2+ b2 and A= 3s‖ instead of ―A =

½ bh‖. Others did not write any formula after indentifying the given from

the problem. This was due to poor recall. Further, 35 errors fall along

processing errors. Students committed errors in multiplying (2x-5) and

(3x +4). Instead of writing ―2x2 -7x -20‖, they wrote ―2x2 -23x -20, 2x2 +7x

-20 and 2x2 -7x +20‖. Others also committed errors in adding (2x-5) and

(x+9). Instead of writing ―3x + 4‖, they wrote ―3x-4‖. Others overdid their

analysis by applying the concept of the relationship and the

measurement of the 3 sides; so they wrote 2x-5< x+9. This was due to

deficient mastery and carelessness. Lastly, 20 errors were along encoding

errors. Students failed to indicate the correct unit of measurement. The

students wrote the answer in ―cm‖ instead of ―cm2‖. They also forgot to

61

write the unit of measurement. This was due to lack of critical thinking

and carelessness.

Moreover, 89 errors along polynomials were along Mathematising

errors. Most of the students failed to write the working equation. Others

wrote an incorrect equation such as ―(x4-1)-(x+1)‖ instead of ―(x4-

1)/(x+1)‖. This was caused by poor mastery and deficient recall. It is also

seen that 83 errors were along processing errors. Students performed

incorrect synthetic division while others performed incorrect factoring for

―(x4-1)‖ such as ―(x3)(x-1)‖ and ―(x + 1)(x -1)(x+ 1)(x + 1)‖. Others

performed incorrect cancellation in (x4-1)/(x+1). They immediately

cancelled x4 and x and subtracted 1 and -1; thereby, generating answers

x3 and x3-1. Others had written the correct working equation but had not

proceeded to the correct solution path. This was due to carelessness and

deficient mastery.

In addition, 75 errors were along encoding. Students just wrote ―x3-

x2+x-1 or (x2+1)(x-1)‖ without the word ―ice cream‖. Others had correctly

performed division but had not copied the correct sign, so instead of

writing ―(x3-x2 + x-1) ice cream‖, they wrote ―x3-x2-x-1) ice cream‖. Lastly,

27 errors were along comprehension. Students failed to completely write

the data from the given problem. This was due to laziness and

carelessness.

62

These results agree with the study of White (2007) revealing that

most misconceptions of his respondents along College Algebra were along

reading/ comprehension, transformation and carelessness in writing the

final answers. He revealed that most problems involving situations were

misunderstood by the students. He explained that these errors appeared

because the students did not have the critical ability to deduce major

concepts from a given problem. He also explained that the students’

insufficient exposure to this kind of problem and poor mastery caused

the errors.

Further, the findings of the study corroborate with Peng (2007)

revealing that students left items on Venn Diagrams, Polynomials and

Algebraic Expression integrating other concepts on Geometry,

Measurement and Basic Numerical Analysis unanswered. The

unanswered items pointed out to insufficient or even no knowledge of the

concepts. He explained that the items were unanswered because

students were new to this type of problem presentation or may not had

exposed well to diagram analysis. This type of error, according to Peng

(2007), is termed as ―beginning error for interpretation and logic‖.

This also relates to the study of Hall (2007) stressing that one of

the foremost problems of his students was their inability to understand

the language of mathematics. For some students, mathematical disability

was as a result of problems with the language of mathematics. Such

63

students had difficulty with reading, writing and speaking mathematical

terminologies which normally were not used outside the mathematics

lesson. They were unable to understand written or verbal mathematical

explanations or questions and therefore cannot translate these to useful

data.

Special Product Patterns

Table 12 unveils the error categories of the students in special

product patterns. It can be seen from the table that 151.33 or 40.46%

errors were committed along processing, 78.33 or 20.94% were along

reading, 47.67 or 12.75% were along Mathematising, 36 or 9.63% were

along encoding and 16.67 or 4.46% were along comprehension. It is also

good to note that 44 or 11.76% were not considered as errors. This

means that majority of the students committed processing errors in

special product patterns. They were able to read, understand and set up

the working equation but failed in proceeding to the correct solution

path, leaving incorrect answers.

Specifically, the table shows that 201 errors in product of 2

binomials were committed along processing errors. Students incorrectly

multiplied (3x2-5) to (3y+4) and (2x2+45) to (5y+2). Others committed

errors in evaluating (3x2-5); instead of writing ―(3(10)2-5 = 295)‖, they

wrote ―900-5 = 895‖. They also failed to multiply the measure of the lot

64

Table 12. Error Categories in Special Product Patterns


R C M P E N

Product of Two Binomials

64 16 15 201 26 52

Square of a Trinomial

86 18 82 146 30 12

Cube of a Binomial 85 16 46 107 52 68

Average

78.33

16.67

47.67

151.33

36

44

Rate

20.94%

4.46%

12.75%

40.46%

9.63%

11.76% Legend: R- Reading Error C- Comprehension Error M- Mathematising Error P- Processing Error E- Encoding Error N- No Error

by its respective price, leaving the solution process incomplete. This was

due to lack of critical thinking and deficient skill. Moreover, 64 errors

were along reading. The students left the item unanswered. This implies

that the students did not know what to do. This was caused by poor

mastery of content. It can also be gleaned that 16 errors were along

comprehension and 15 errors were along Mathematising. The students

failed to get the gist of the problem. The students, due to their

misunderstanding of the focus of the problem, failed to craft the working

equation or remember the formula suited to the problem.

Further, 146 of the committed errors in square of a trinomial were

along processing errors. The students failed to correctly square a

65

trinomial. Most of them answered (2x-4y+6z)2 as (4x2+16y2+36z2), worse

(4x2-8xy2+12y2) instead of 4x2+16y2+36z2+16xy+24xz-48yz. Others also

wrote 4x2+16y2+36z2–8xy +12xz -24yz. Others performed correctly the

pattern but failed to employ the rules of signs. This was caused by

deficient mastery of the subject matter.

It is also noted that 86 errors were along reading. This means that

some students left the item unanswered. The students had not

understood fully the problem or did not really know how to deal with the

problem. This was caused by poor competence. Also, 82 errors were

along Mathematising. The students failed to write the correct formula.

Instead of writing A= ∏r2, most of them wrote A= 2∏r, and A= 2∏r2. This

was misalignment of formulas. Others also were not able to write any

formula or working equation. This was caused by deficient recall. 30

errors were also committed along encoding errors. Most of them failed to

write the unit of measurement of the final answer. Others also committed

parenthetical error, a kind of encoding error. Instead of writing

(4x2+16y2+36z2–16xy +24xz -48yz)∏ cm2, they wrote 4x2+16y2+36z2–16xy

+24xz -48yz∏ cm2 . This was due to carelessness and lack of critical

thinking. Lastly, 18 errors were along comprehension. The students

failed to completely identify all the given from the data. They just listed

(2x-4y + 6z). Others even wrote (2x+4y+6z). This was due to carelessness

among students.

66

The table also shows that 107 errors in cube of a binomial were

along processing errors. The students failed to correctly cube the

binomial (2x+4). Most of them just wrote (8x3+63) or worse (8x3+12) and

(6x3+12) and (8x+64). The students failed to apply the pattern of (F+L)3 =

(F3+3F2L+3FL2+L3). This was caused by poor competence. In addition, 85

errors were along reading. The students left the items unanswered. They

did not know what to do to be able to arrive at the correct answer. This

was caused by poor mastery.

It can also be noted that 46 errors were along Mathematising errors.

The students failed to write the correct formula, V = s3. The students

wrote s2 or (s)(s). Some also wrote V= 3s3 and V= 4s. This was due to

poor retention of formulas taught to them even in the elementary. Also,

52 errors were along encoding errors. Students failed to write the final

answer with the correct unit of measurement. Others wrote cm, cm2 or

none at all. This was due to lack of criticality and carelessness among

students. Lastly, the 16 errors were committed along comprehension.

The students failed to write completely the given data. Instead of writing

(2x +4), some wrote (2x-4), (2+4), (x+4). This was due to carelessness.

The findings of the study corroborate with the study of Egodawatte

(2011) divulging that most students committed transformation and

processing errors along word problems involving algebraic expressions,

factoring and special products. He explained that the students failed to

67

remember and apply perfectly the special product and factoring patterns.

He further stressed that the students committed these kinds of errors

because the students had difficulty in carrying out several steps involved

in the mathematical process. He specifically itemized that the students

were poor in simplification, performing operations, exponential laws as

applied in factoring and product patterns, incorrect distribution and

invalid cancellation.

Also, the study of Allen (2007) harmonizes with the finding of the

study revealing that most students committed processing errors when

dealing with special products and factoring. He stressed that students

did not apply the correct rules in simplification of polynomials, algebraic

expressions, special products and factoring. He showed that many

students expanded (x+3)2 as x2+9 or worse x+6. Many of the errors were

caused by poor mastery of the mathematical principles in the said topics.

Factoring Patterns

Table 13 exposes the error categories of students in factoring

patterns. It shows that the students committed 128.25 or 34.29%

reading errors, 78 or 20.85% Mathematising errors, 60 or 16.17%

encoding errors, 39 or 10.42% processing errors and 25.58 or 6.75%

comprehension errors. It also shows that 43 or 11. 50% were not

considered errors. This implies that majority of the students failed to

68

Table 13. Error Categories in Factoring Patterns


R C M P E N

Difference of two Perfect Squares

182 46 111 17 15 3

Perfect Square Trinomial

88 29 53 37 92 75

General Trinomial 95 12 57 34 100 76

Factoring by Grouping

148 14 91 68 35 18

Average 128.25 25.25 78 39 60.5 43

Rate

34.29%

6.75%

20.85%

10.42%

16.18%


understand the applied problems along factoring. Majority left the items

unanswered since they did not know what to do. This is caused by poor

competence. This is even attested by the fact that only 43 students got

the item correctly.

It can also be read from the table that 182 errors in factoring

difference of two perfect squares were along reading errors. This means

that the students left the items unanswered. They did not understand

what the problem wants them to do or they did not know what to do.

This is due to the lack of competence of students. Moreover, 111 errors

were along Mathematising errors. This means that the students failed to

69

correctly write the correct formula or working equation demanded by the

problem. They failed to write the formula for the area of the rhombus, A=

½ d1d2. Others wrote the formula for the area of the square, A = s2. This

is clear sign of misalignment of formulas. This was due to insufficient

recall. This was due to poor exposure to this kind of geometric figure.

Also, 46 errors are along comprehension. This means that the students

did not fully understand the focus of the problem. This is attested by the

incomplete data or incorrect data written on their answer sheets.

Someonly wrote (2x2-162), forgetting (x-9). Others wrote (2x2-162) and

(x+9). This is due to carelessness. Further, 17 errors were along

processing. Most of the students after substituting the values to the

formula, committed factoring errors. Instead of writing 2(x2-81), they

wrote 2 (x2-162). They were able to factor out 2 from the first expression

but not in the 2nd expression. Others also left the items as (2(x2-162))/(x-

9). This means that the students failed to recognize the common factors

in the numerator which later on leads to the cancellation of the

expressions both for the numerator and denominator. This was due to

insufficient mastery in factoring. Lastly, 15 errors were along encoding

errors. This means that the students were able to correctly carry out the

solution process but failed to write the final answer in an unacceptable

form. Students forgot to indicate the unit of measurement, units2. This

was due to carelessness and lack of criticality,

70

Moreover, it can also be gleaned from the table that majority of the

errors along perfect square trinomial were along encoding. The students

failed to indicate the correct unit of measurement of the answer. Instead

of writing (2x-5) m, the students wrote simply (2x-5). Others wrote the

incorrect unit such as ―m2‖ and ―cm‖. This was due to carelessness and

lack of critical thinking. In addition, 88 errors were along reading. This

means that the students left the items unanswered. The students had

not understood the meaning of the problem which led them to leave the

item unanswered. They did not know how to hurdle such applied

problem. This was due to poor performance.

Additionally, 53 errors were along Mathematising. The students did

not write the formula or the working equation of the problem. Some

incorrectly wrote the formula. Instead of writing A = s2, they wrote A = 4s.

Others had incorrect derivation of the formula for ―s‖. Instead of writing s

=√A, they wrote s = A/2. This was caused by poor recall and poor

competence. Also, 37 errors were along processing errors. They failed to

get the factored form of the PST (4x2-20x+25). They divided the

expression by 2 instead of performing factoring. Lastly, 29 errors were

along comprehension. Students failed to completely understand what the

problem is asking them. They also incorrectly copied the given data. So

instead of writing (4x2-20x+25), some wrote (4x2+20x+25) and (4x-

20x+25).

71

Likewise, it is also reflected in the table that 100 errors in factoring

general quadratic trinomial were along encoding. They were able to

correctly get the answer (x+5) but failed to write the correct unit of

measurement, cm. This was due to lack of reflection among the students.

In addition, 95 errors were along reading. Students never wrote

something that leads to the solution of the problem. This implies that the

students did not know how to deal with the problem.

Further, 57 errors were along Mathematising. Students failed to

correctly write the working equation. Some did not write any formula

while the others wrote an incorrect one. The students wrote (x2+3x-40) -

(x-8) instead of (x2+3x-40)/ (x-8). This was in spite of the presence of the

word ―divide‖ in the problem. This was due to poor competence and

analytical thinking. Also, 34 errors were along processing. Students

failed to correctly factor (x2+3x-40) leaving it unfactored and unsimplified

with the denominator. Students also incorrectly cancelled x2 with x and

40 with 8 in their equation, (x2+3x-40)/ (x-8). This was invalid

cancellation. This implies that the students really did not know how to

factor trinomials of this form. This was due to poor competence and

mastery. Lastly, 12 errors were along comprehension. Students failed to

correctly write the two given data correctly. Instead of writing (x2+3x-40)

and (x-8), students wrote (x2+3x-40) and (x+8) or (x2-3x-40) and (x-8).

This was due to carelessness. Others wrote the number ―2‖ as an

72

important detail in the problem solution besides the fact that it only

details the equal measurements of the string when divided into two;

writing a given as (x2+3x-40)/2.

Additionally, the table also shows that 148 errors in factoring by

grouping were along reading errors. The students did not understand

what the problem is asking them to do. Others really did not know the

answer. Students even ignored a problem when prompted with series of

algebraic expressions such as x2+2xy+y2+x+y. Pamani (2006) stressed

that students with high anxiety and poor mathematical performance

often ignore expressions which were lengthy and contain complex

expressions and exponents. The errors were caused by high anxiety and

poor exposure to such kind of problem.

Also, 91 errors were along Mathematising. Students were not able

to write any working equation to solve the problem. Others performed

subtraction instead of division despite the implication of ―2 equal shares‖

in the problem; the working equations used were ―x2+2xy+y2+x+y – x+ y‖

and‖ x2+2xy+y2+x+y-xy‖. This was caused by poor understanding and

mastery.

Likewise, 68 errors were along processing. Students failed to factor

correctly and completely x2+2xy+y2+x+y. Others invalidly cancelled ―x+y‖

in (x2+2xy+y2+x+y) with (x+y), resulting in an incorrect answer x2+2xy+y2.

This was a clear reflection of misuse of cancellation rules. Others also

73

wrote the correct common factor (x+y) but failed to correctly factor the

remaining expressions. This was caused by poor mastery of factoring by

grouping.

Further, 35 errors were along encoding. Students were able to

correctly factor the given expressions but failed to write the correct unit

of measurement. Lastly, 14 errors were along comprehension. Students

did not completely and accurately analyze what the problem wanted

them to do. Students incompletely wrote the given while the others wrote

additional unnecessary data such as ―2‖ resulting in a data

(x2+2xy+y2+x+y)/2.

The findings of the study corroborate with the study of Egodawatte

(2011) divulging that most students committed transformation and

processing errors along word problems involving algebraic expressions,

factoring and special products. He explained that the students failed to

remember and apply perfectly the special product and factoring patterns.

He mentioned that students generated incorrect factored forms of x2+x,

which were x(x+x) and worse, x(1). He stated that the students

―oversimplified‖ the answer. They lacked critical analysis as to when and

how to end the factoring process correctly. He also explained that

―overdoing‖ existed as he pointed out to incorrect cancellation of

expressions.

74

This also agrees with the study of McIntyre (2005) revealing that

his respondents had misconceptions in writing final answers in algebraic

expressions and factoring patterns. The answer ―x+y‖ was still reduced to

xy. He explained that in factoring patterns and algebraic expressions,

students never leave an answer with an addition symbol present; the two

variables must be physically conjoined. According to him, students felt

that x+y can still be combined through the indicated operation. This

error, according to him, was caused by misassociation of arithmetic

principles; ―7+3= 10‖ is misassociated to ―x+y = xy‖.


Table 14 shows the error categories of students in rational

expressions. It can be gleaned from the table that 165.33 or 44.21% of

the errors were along reading, 90 or 24.06% were along Mathematising,

41 or 10.96% were along processing, 22 or 5.88% were along encoding

and 19.67 or 5.26% were along comprehension. It is also worthy to note

that 36 or 9.63% were not considered errors. This means that majority of

the students committed reading errors in simplifying rational algebraic

expressions. This implies that the students left the item unanswered.

They had not understood clearly and comprehensively the problem that

hindered them to write even a single data from the problem. This was

caused by the lack of exposure to such kinds of problems. According to

75

Table 14. Error Categories in Rational Expressions


R C M P E N

Simplification of RAEs

105 18 51 84 46 70

Operations of RAEs 138 17 161 20 17 21

Simplification of

Complex RAEs 253 24 58 19 3 17

Average

165.33

19.67

90

41

22

36

Rate

44.21%

5.26%

24.06%

10.96%

5.88%


some reactions of professors after retrieving the questionnaires, the

students failed to recognize the operations or the mechanical procedures

when expressions were converted to word problems. Blakelock (2013)

agrees with this observation of the professors when she mentioned that

when students just learned direct operation, direct cancellation or

simplification in the class, students would be hard up dealing with such

kind of expressions when written in word problems.

It can also be seen from the table that 105 errors in simplification

of rational algebraic expressions (RAEs) were along reading. This means

that most of the students left the item unanswered. This implies that the

76

students failed to write any data given by the problem. This means that

they had not understood what the problem is all about. It also means

that the students were not interested to solve problems involving

fractions or fractional expressions. Hall (2007) emphasized that most

students had difficulty dealing with exponents, fractions and radicals.

Most students, who find difficulty with these, often abandon solving such

problems. Further, 84 errors were along processing errors. This implies

that the students failed to correctly solve the given problems. Most of

them performed incorrect cancellation in (12x4y6/7xy) and (21/6x3y5).

Others placed the incorrect exponents in the denominator instead of in

the numerator such as (2/3xy). Others incorrectly placed the cancelled

form of 21/7 as 1/3 instead of 3/1 or 3.

Additionally, 51 errors were along Mathematising errors. Students

failed to write down the correct working equation of the problem. Others

wrote the incorrect working equation such as (12x4y6/7xy) ÷ (21/6x3y5)

or (12x4y6/7xy) - (21/6x3y5) or (12x4y6/7xy) = (21/6x3y5). This is due to

poor analytical skills. Further, 46 errors were along encoding. This

means that the answers were not written in a correct form. Others did

not write the unit of measurement. Others did not simplify 6/1 pesos.

Lastly, 18 errors were along comprehension errors. Students failed to

fully understand the given in the problems. Others wrote only partial

given such as (12x4y6/7xy) alone or (21/6x3y5) alone. Others wrote

77

(12x4y6/7xy) and (21/6x3y5) but without their corresponding units. This

was due to carelessness.

The table also reflects that 161 errors in operation of RAEs were

along Mathematising. The students failed to correctly write the working

equation of the problem. It is surprising that even if the students came

from different schools with different instructors, the students commonly

wrote the equation (1/2x)(8x/2) instead of (5/2x)(80x/2). This means

that the students failed to transform verbal expressions to numerical

expressions correctly. This was due to poor mathematical skills.

Also, 138 errors were along reading. The students failed to write

any data from the given problem. This means that the students failed to

understand the given problem which impeded them to deal with the

problem. Further, the 20 errors were committed along processing. The

students failed to correctly perform the mechanical procedures in solving

the given problem. Others placed (5/2x) ÷ (80/2x) instead of (5/2x) x

(80x/2). Others evaluated the value of 5 in (1/2x) and 10 in (8x/2). This

was due to carelessness and poor performance. Additionally, 17 errors

were committed along comprehension and encoding errors. This means

that they incompletely wrote the data, excluding 5 and 10 pesos as vital

in the solution of the problem. This also implied that the students left the

final answer without the correct unit. This was due to carelessness.

78

The table also exposes that 235 errors in simplification of complex

RAEs were along reading. The students left the items unanswered. This

means that they were not interested in solving the problems especially so

that the problem involves fractional expressions. They also forgot how to

deal with interest problems involving fractional items. This is due to their

low performance. Also, 58 errors were along Mathematising. This means

that the students failed to write the formula for interest, I = PRT. Others

did not write the formula and just multiplied the given. Others wrote the

formula I = 1 + PRT and I = PR. This agrees with the number of errors

along reading. Additionally, 24 errors were along comprehension. The

students failed to correctly indicate all the data in the problem. They had

not written correctly (1- 1/3) and wrote only 1/3 instead. Most of them

did not indicate a representation for time, which should had been ―x

years‖. This is due to insufficient critical analysis. Also, 19 errors were

along processing. The students failed to correctly compute the answer to

the given problem. Others incorrectly substituted the given to the

formula such as P = I/RT as (1/6 x 12,000) =P/[(1- 1t/3)]. This was due

to deficient mastery. Lastly, only 3 errors were along encoding. The 3

errors failed to write the expression ―t‖ in the final answer. The students

felt that an answer with a variable was still not the accepted final

answer. This was due to an incorrect thinking of oversimplification.

79

The finding of the study corroborates with Egodawatte (2011)

divulging that most students committed transformation and processing

errors along word problems involving algebraic, polynomial and rational

expressions. He explained that these errors were committed since the

problems were too symbolic and the most challenging part for students

was to find the correct method of solution or algorithm and making use

of the algorithm to produce a correct answer. He further stressed that

students had to choose the correct method from a wide range of possible

strategies which include but were not limited to determining common

denominators, common factors for cancellation, expansions using the

patterns, building up expressions, simplifications and comparisons.

Many of the incomplete answers of his students that were observed bear

evidence that the students could not select and apply the correct

strategy. He also explained that most students committed ―exhaustion

errors‖ when dealing with rational expressions and simplifying answers

in algebraic equations. Exhaustion errors are errors which were not

made at the beginning of the problem where an opportunity for its

commission existed. This type of Mathematising error may had existed

due to the incomplete concept recall of the students. This error can also

be attributed to the misleading background of the students pertaining to

the subject at hand. This error can also be caused by misapplication of

the algorithm learned.

80

It also agrees with Hall (2007) when he said that deletion and

cancellation errors were prevalent among the respondents of the study as

regards working on arithmetical equations and expressions, algebraic

and rational equations and fractional expressions. He explained that

―overgeneralizing‖ was the main cause of this type of error. He also added

that when students solve equations, they commit transposing errors

such as forgetting the change in signs of quantities.

Linear Equations in One Variable

Table 15 reflects the error categories of the students on linear

equations in one variable. It shows that 172.33 or 46.07% errors in

linear equations in one variable were along reading, 99.33 or 26.56%

were along comprehension, 55 or 15.71% were along Mathematising,

14 or 3.74% were along encoding and 10.67 or 2.85% were along

processing. It can also be seen that 22.67 or 6.06% had completely

answered the items with correct final answer. This implies that majority

of the students committed reading errors. This means that the students

failed to write any given data from the table; they left the item

unanswered in linear equations. This was caused by poor mastery of the

subject matter.

It also shows that 186 errors in distance problem were along

reading errors. This means that the students left the item unanswered.

81

Table 15. Error Categories in Linear Equations in One Variable


R C M P E N

Distance Problem

186

89

53

6

17

23

Money Problem 175 100 46 22 4 27

Age Problem 156 109 66 4 21 18

Average 172.33 99.33 55 10.67 14 22.67

Rate 46.07% 26.56% 14.71% 2.85% 3.74% 6.06%


They failed to write even a single data from the problem. This was due to

their lack of interest towards the problem. Blakelock (2013) asserts that

students’ interest in math is high when they were still toddlers, but when

they get older, this interest lowers down due to their experiences. This is

the reason why most college students do not bother solving problems,

especially so when such do not relate to their future profession. Further,

89 errors were along comprehension. This means that the students failed

to fully understand the thrust of the problem. Most of them incompletely

wrote the given data. Most of them did not present the data in a more

comprehensible format, such as using a table. This was caused by poor

skills. In addition, 53 errors were along Mathematising. This means that

the students were able to present the data but failed to write the correct

82

formula, D = RT. Others wrote Vf2= Vo

2 + 2 fusing Physics and College

Algebra. Others wrote 440-220= 220 as their working equation. This was

due to poor recall and deficient mastery of the subject matter. Also,

17 errors were along encoding. The students failed to write the correct

unit of the final answer. They wrote 240 and 200 as their final answers.

This was due to insufficient criticality and carelessness. Lastly, 6 errors

were along processing. These were committed because of the

incompleteness of the answers. The students failed to substitute the

value of x, which was 2, to the data presentation for the covered

distance. This was due to lack of criticality among the students.

Moreover, 175 errors in money problem were committed along

reading. This implies that most students left the item unanswered. This

means that the students did not know what to do. This also means that

the students failed to fully understand what the problem is talking

about.

Also, 100 errors were along comprehension errors. They failed to

fully understand the problem; only partially indicating what the problem

is giving. They also failed to understand that the problem data need to be

presented in a more organized way, such as using a table or column. In

addition, 46 errors were along mathematising. This implies that the

students failed to correctly write the formula or working equation needed

to solve the problem correctly, D x N = A or denomination multiplied to

83

the number of bills is equal to the OVERALL amount. Others added 1

and 20 and 1 and 50; then applied guess and check method for the two

numbers. This means that the students cannot transfer their ideas into

mathematical expressions. This is due to deficient mastery.

Likewise, 22 errors were along processing errors. Students

committed errors in multiplying 50 (27-x). Instead of writing 1350 – 50x,

others wrote 1350 – x or 135 – 50x or worse, 135 –x. This was due to

poor mastery and carelessness. Lastly, four errors were along encoding

errors. These 4 errors were along writing the correct unit. Instead of

writing km, they wrote kph; others left the answer with no unit. This was

due to carelessness and lack of reflective ability to verify if the final

answer is in its accepted form.

It can also be seen that 156 errors in age problem were along

reading. This means that majority of the students left the item

unanswered. This implies that they did not know what to do to be able to

get the correct answer needed by the problem. This is saddening since

high school mathematics had taught them topics on applied problems in

linear equations which started in first year, reinforced in the second year,

enforced in their 4th year and repeated in their tertiary year. This was

due to deficient skills in algebraic expressions and applied problems.

Also, 109 errors were along comprehension. The students failed to

completely present the data into tables. Others wrote in tables but failed

84

to represent the two time zones involved in the problem, the present and

the future. This was due to lack of criticality.

Also, 66 errors were along Mathematising errors. The students were

unable to write the correct equation x+20+10 = 2(x+10). Others wrote 2x

= 30 +10 and 30 + x = 2x. Others stopped when the data were already

presented in correct tables. Others tried to solve using trial and error

method by trying 2 numbers that fit the given categories. This was

caused by poor mathematical skills. Lastly, 4 errors were committed

along processing errors. The errors were along multiplication of

constants and variables and transposition. Others wrote x+2x = 30+10

instead of x-2x=30-20. Others wrote x+20+10 = 2x +10 instead of x +

20+10 = 2x (20). This was incomplete distribution. This was due to

deficient skills in handling algebraic expressions.

The findings of the study run parallel to Clement (2002) divulging

that most students’ errors on linear equations fall along transformation.

He stressed that his respondents had difficulty in translating words to

algebraic equations. He also expressed that analytical thinking falls short

among his students which led them to an incorrect process.

The findings of the study also run parallel to the study of

Egodawatte (2011) revealing that in linear equations and systems of

linear equations, most of the students got the correct answer; however,

some committed transformation and processing errors. Students failed to

85

produce a correctly transformed equation. Students failed to form correct

equations. The others failed to use correctly the methods of substitution,

elimination and the working backward methods. He explained that

students were unable to carry out these methods due to insufficient

skills on the procedures. Students failed to use the standard

mathematical practices. He also added that the number one problem of

his students is on variables. The students misinterpreted the product of

two variables. The students were not able to apply the laws of exponents.

He explained that the students misjudged the magnitudes of the

variables; he pointed out that the students lack the understanding of

variables.

Also, the findings agree with the study of Allen (2007) stressing

that students had trouble solving such items. He stressed that students

need to be skilled on fundamental principles pertaining to equalities. It

can be deduced that insufficient background causes the predicament.


Table 16 presents the errors of the students along linear equations

in two variables. It shows that 119.5 or 31.95% errors in systems of

linear equations in two variables were along reading, 95 or 25. 40% were

along Mathematising, 72.5 or 19.39% were along comprehension, 14.5 or

3.88% were along encoding and 10 or 2.67% were along processing. It is

86

Table 16. Error Categories in Systems of Linear Equations in Two Variables


R C M P E N

Problems on

fare/price

160 89 71 5 13 36

Problems on number relation

79 56 119 15 16 89

Average

119.5

72.5

95

10

14.5

62.5

Rate

31.95%

19.39%

25.40%

2.67%

3.88%


also good to note that 62.5 or 16.71% were not considered errors.

Further, the errors imply that majority of the students committed

reading errors. The students failed to fully understand the problem

thereby leaving the item undealt. This further means that the students

do not had the know-how in dealing with the given problems. This was

due to poor mastery of the expected competencies. This is really

saddening since this is not their first time to encounter such systems of

linear equations. They were able to deal with these even during their

secondary school days.

The table also points out that 160 errors in applied fare/price

problems were along reading. This means that majority of the students

87

left the items unanswered. The students failed to write even a single data

deduced from the given problem. This is caused by insufficient exposure

to such problem. It is true that when teachers facilitate the topic on

systems of linear equations, majority of them focused on the methods of

solving the value of x. Since this is one of the last topics offered in the

course syllabus, most teachers fail to teach how such systems were

transformed to applied problems due to lack of time.

Also, 89 errors were along comprehension. This means that the

students failed to completely and correctly understand the problem. They

failed to completely present the data into a more fathomable way, using a

tabular format. Also, they failed to correctly represent values for x and y.

This is due to lack of organization and criticality. Further, 71 errors were

along Mathematising. The students failed to correctly write the needed

working equations 8x + 10y = 200 and 3x + 10y = 150. Others simply

guessed and checked for 2 numbers that can satisfy the given

conditions. This clearly pointed out to the fact that the students cannot

transfer their ideas into mathematical expressions. Others did not write

any working equation. This is due to poor mastery of the subject matter.

Moreover, 13 errors were along encoding errors. The students

forgot to write the answers in an unacceptable written form. Most of

them failed to indicate the unit. The others were able to get the values for

x and y, but failed to pinpoint which among the two values answer the

88

question of the problem. Lastly, 5 errors were along processing. Students

failed to correctly apply substitution in the solution of the problem.

Instead of writing (3(200-10y)/8) + 10y = 150, they wrote ((200-10y)/8) +

10y = 150. Others did not proceed with their solutions when they

finished writing their working equation. This means that the students did

not know how to deal with the formulated system of linear equations.

This was due to carelessness and poor mathematical abilities.

It is also reflected in the table that 119 errors in number relation

were along Mathematising errors. The students were unable to correctly

write the formula or the working equation. Others wrote ―x +x = 100 and

x-x = 20‖ as their working equations. Most of the students applied trial

and error in solving the correct 2 numbers. This means that the students

were unable to correctly transform their ideas into mathematical

expressions. They were able to guess and check their answers to the

problem but find it hard to create a working solution to be able to get

their ―theorized‖ answers. This is not surprising since Ashlock (2006)

revealed the same finding in his study that students can jump into the

answers without any working solution. They had their solution in their

head but cannot write their solutions. Most instructors, even the

researcher, often meet students who can give the answers right away but

when asked of their solutions, fail to present any.

89

Further, 79 errors were along reading. This means that the

students left the item unanswered. This implies that the students did not

know what to do. Also, 56 errors were along comprehension. The

students failed to fully understand the given problem. They failed to

indicate representation of the given problem. This was due to lack of

critical and analytical ability. Further, 16 errors were along encoding.

The students were able to get the values for x and y but failed to indicate

a final sentence to be able to correctly answer the thrust of the

problem.

Lastly, 15 errors were along processing errors. Students failed to

solve the problem using a correct solution path. The students failed to

substitute correctly the derived equation to the other equation such as

y=100-x to x-y = 20. Others committed transposition errors in

transposing y in x +y = 100. This is due to carelessness and low mastery

of the subject matter. It is also good to note that 89 answered correctly

the given problem. This means that some students correctly and

completely answered the given problem. This contributed much on their

level of performance, a satisfactory performance.

The findings of the study agree with the study of Clement (2002)

stressing that most of his students committed transformation and

processing errors on systems of equations. He explained that these errors

were caused by insufficiency of skill or knowledge pertaining to how

90

certain variables were handled or how certain equation algorithms were

processed. He showed, too, that generally, students got the correct

answers but failed to simplify the answers in problems that need

simplification of answers. Others forgot to correctly indicate the unit for

the answers to be accepted. These errors were due to carelessness. He

explained that students forget to analyze their final answers. They did

not verify their answers by some accepted means.

Also, it agrees with Ashlock’s study (2006) divulging that students

can even produce the correct answer even if the solution is incorrect.

This situation abounds in problems involving numbers and number

relations. With this situation at hand, teachers do not only need to

correct the final answer but the process on how the answer is derived. He

stated further that students commit what he calls as ―overgeneralizing‖.

Students ―overgeneralize‖ data by jumping into the conclusions without

adequate data at hand. This overgeneralizing error leads them to

incorrect approach and answer. This error abounds in vast areas of

mathematics especially on number problems, arithmetic and

simplification problems. With this, he remarked that the students lacked

the needed computational fluency.

91


Table 17 presents the error categories in College Algebra,

specifically along exponential and radical expressions. It shows that 297

errors were along reading. This means that the students did not

understand the thrust of the problem. They really did not know what to

do to be able to answer the problem. They left the item unanswered. This

is so since the students had not touched this last topic of the course

syllabus. Many schools had festivities on their foundation, intramurals,

founder’s day and the like which limited the number of contact days for

discussion. This means that the students failed to write the working

equation. Others incorrectly wrote the working equation such as

√(2x+7) + 3x = 90 despite the fact that the problem indicated the word

―angle bisector‖ and ―equal parts‖. Others combined trigonometric

functions in the formula, including Sin x and Tan x. So, their working

equation is A = ½ (√(2x+7) – Sin 3x)r2. This clearly pointed out that the

students mixed up their concepts on College Algebra and Plane and

Spherical Trigonometry. This is misassociation of concepts of two

branches of College Mathematics. Further, they also wrote

90 = (√(2x+7)+3x, which pinpoints that the students jumped into the

incorrect conclusion that the angle is a right angle even if there is no

indication in the problem that the angle measures 90 degrees. Moreover,

29 errors were along comprehension errors. The students failed to

92

Table 17. Error Categories in Exponents and Radicals

Subtopic Error Categories R C M P E N

Exponential and

Radical expressions or equations

297 29 35 8 1 4

Average 297 29 35 8 1 4

Rate 79.41%

7.75%

9.36%

2.14%

0.27%

1.07%


indicate all the given data in the problem. Others wrote 2x- 7 only.

Others wrote √(2x+7)/2and 3x/2 infusing the number 2 in the problem,

a clear sign of misinterpretation. Also, 8 errors were along processing.

The students just deleted the radical symbol in √(2x+7) = 3x without

performing the squaring process. Others squared 3 but not x, resulting

in the expression 9x instead of 9x2. Others chose the value -7/9˚ over 1˚

as the correct answer. This was due to carelessness and no criticality.

Lastly, only one error is committed along encoding error. One did not

indicate the unit for degrees ( ˚ ) in the final answer. This was due to

carelessness.

The findings agree with the study of Boon (2003) stressing that the

high occurrence of errors in exponents and radicals is due to over-

generalization. This over-generalization was due to carelessness and

insufficient practice. It also appeared that such error existed due to

93

misconceptions that students had actively construed when they use their

existing schema to interpret new ideas. He also explained that this error

may be brought by deficient mastery of concepts, rules and pre-requisite

skills which can be overcome by practice. He specifically stressed that

most students misconnect the rule on √4 = 2 to be true to √16 =8 or

worse, √6 =3. This was well explained by Allen (2007) when he

enumerated some of the errors of the respondents of the study on error

analysis in radical expressions and equations. He pointed out that

students had incorrect interpretation and representation of radicals,

especially on square roots. Students tend to divide the numbers when

getting the square of 16. So, instead of 4, the students wrote 8. This was

due to misalignment of rules. They applied the rule in √4 = 2 as true to

all numbers being extracted.

Summary on the Error Categories in College Algebra

Table 18 shows the summary on the error categories in College

Algebra. It reveals that 146.96 or 39.29% of errors in College Algebra

were along reading, 69.38 or 18.55% were along Mathematising, 47.42 or

12.68% were along comprehension, 45.86 or 12.26% were along

processing, and 30.29 or 8.10% were along encoding. This means that

majority of the students committed reading errors. This means that most

94

Table 18. Summary Table of Error Categories in College Algebra


R C M P E N

Elementary Concepts 68 69.50 85 61 64 26.50

Special Product

Patterns

78.33 16.67 47.67 151.33 36 44

Factoring

128.25 25.25 78 39 60.5 43


165.33 19.67 90 41 22 36

Linear Equation in One Variable

172.33 99.33 55 10.67 14 22.67

Systems of Linear Equation

119.5 72.5 95 10 14.5 62.5

Exponents and

Radicals

297 29 35 8 1 4

Average

146.96

47.42

69.38

45.86

30.29

34.10

Rate 39.29% 12.68% 18.55% 12.26% 8.10% 9.12%

Rank 1 3 2 4 6 5


of the students failed to critically understand the problem which led

them to leave the items unanswered. They never attempted to answer the

given items. This is due to their fair performance. It can also be deduced

and construed that students hardly can formulate the working equation

or remember the formula to solve the given problem.

95

It is also reflected that only 9.12% completed and correctly solved

the given problems in College Algebra. This means that majority of the

students really committed errors on the different categories.

The findings run parallel to Hall (2007) divulging the following

errors of his respondents in College Algebra: Computational Constraint.

Many students while they understand mathematical concepts are

inconsistent at computing mainly because they misread signs or carry

out numbers incorrectly or may not write numerals in the correct

column; Difficulty in transferring knowledge. Many students experience

difficulty in mathematics because of their inability to connect abstract or

conceptual aspects of mathematics with reality. Understanding what

mathematical symbols represent in the physical world proves to be

difficult to most students and this makes it common to find that some

students cannot visualize an equilateral triangle; Making Connections.

Some students cannot comprehend the relationship between numbers

and the quantities they represent and this makes mathematical skills not

to be anchored in any meaningful manner, making it harder for them to

recall and apply mathematical knowledge in new situations; incomplete

understanding of the language of mathematics.

Further, for some students, mathematical disability is as a result

of problems with the language of mathematics. Such students had

difficulty with reading, writing and speaking mathematical terminologies

96

which normally were not used outside the mathematics lesson. They

were unable to understand written or verbal mathematical explanations

or questions and cannot relate mathematical knowledge to physical

world; Difficulty in comprehending the visual and spatial aspects and

perceptual difficulties. Many students had the inability to visualize the

mathematical concepts. This makes students to memorize mathematical

formulae and facts - the difficulty in applying such knowledge in solving

unfamiliar mathematical problem.

Validated Instructional Intervention Plan in College Algebra

Rationale

Mathematics has always been regarded as a very essential element

in education for it does not only provide higher training for the human

mind but it is life, itself. Everyone, whether consciously or not, uses

mathematics in his daily life.

College Algebra, one branch of Mathematics, deals with elementary

topics, special products and factoring, rational expressions, linear

equation in one unknown, systems of equations in two unknowns and

exponents and radicals. It provides avenues for students to recall

important concepts learned in the secondary school. It also provides a

good foundation of readiness for students to hurdle the demands of

97

higher mathematics such as Trigonometery, Advanced Algebra, Geometry

and the like.

The noted dismal performance in this subject is caused by different

factors such as negative attitude, misconceptions, misapplications,

misalignmnet of rules, lack of criticality among others. With these

presents, it is apt to look into the reasons behind these. One good

mechanism that can address such dismal performance is an

instructional intervention plan.

The validated instructional intervention plan is based upon the

identified students’ level of performance, their capabilities and

constraints and the different error categories in College Algebra. All the

error categories are addressed in the plan since all of them were

considered constraints; but, more emphasis is given to a course content

with a very poor to poor performance level - these were the areas on

rational expressions, linear equations, systems of linear equations and

radicals and exponents. The instructional plan also gives emphasis on

addressing two (2) foremost error categories, reading and

mathematising. Further, there are some instructional interventions that

address two error categories.

Further, the instructional intervention plan details the specific

objectives; topics; level of performance, error categories and theoized

causes (arranged according to degree of error); error samples;

98

interventions, process, activities; and assessment strategy. The contents

serve as a comprehensive guide for teachers to improve performance and

check on errors.

General Objectives

The instructional intervention plan is formulated to:

1. Improve performance in all the topics in College Algebra; and

2. Address the different errors of students in solving problems in

College Algebra.

Matrix of the Instructional Intervention Plan

The Instructional Intervention Plan in College Algebra is detailed in

matrix form in the succeeding pages (see pages 99 – 295).

99

Instructional Intervention Plan in College Algebra

Specific Objectives

Topics Level of Perform-

ance

Error Cate-gories and

Theorized Causes

(arranged

according to degree of

error)

Sample Error Interventions, Process, Activities

Assessment Strategy

To use

Venn Diagrams in representing sets and set relationships

To utilize

Venn Diagrams in solving

applied problems.

To present complete

and accurate solutions

involving

A. Elemen-

tary Topics

Sets and the Venn Diagrams

Poor Mathema-

tising (dismal performance, insufficient recall)

Incorrect

working diagram- using tables

as solution diagram.

―250-160-150-180 = x‖

as the working equation.

Visual-Spatial

Processing This is a skill-based intervention that

emphasizes the skill on visualizing the given problem. It makes use

of diagrams and illustrations to show to

students how a certain problem is translated into an illustration or

diagram for easier understanding. It uses

direct instruction and the instructor models how problems are

illustrated. After, the students are given handouts and sample

The instructor

can check students’ learning during

the solving process of the students. The

learning is further

assessed when the students explain their

answers on the board.

100

Specific

Objectives


ance

Error Cate-

gories and Theorized

Causes

(arranged according to

degree of

error)

Sample Error Interventions,

Process, Activities

Assessment

Strategy

sets and

Venn Diagrams

problems for them to

demonstrate how such problems are

illustrated and solved. Procedures: 1. The instructor

presents a given problem and uses the Venn Diagram to

illustrate. He has to emphasize why the

Venn Diagram is the correct strategy to be used.

2. After the instructor models, he gives each

student a handout that contains an empty Venn Diagram where

students can write their answers.

101

Specific

Objectives


ance

Error Catego-

ries and Theorized

Causes


degree of

error)


Process, Activities

Assessment

Strategy

3. The students are

still guided by the instructor as he roams

around the classroom. The instructor checks students’ answers.

4. Students who are done with their answers and have

presented their correct answers can be

assigned to students who need assistance. 5. Presentation of

problems will be done after. Priority of

presentation is given to those who are assisted to check

102

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

Think-Pair-Share-

Explain Activity This is an interesting

cooperative-learning-based activity for students who struggle

to come up with correct answers. Procedures:

1. After the instructor finishes discussing the

lesson, the instructor pairs the students. The pairing is done

strategically pairing the fast and the

struggling ones. 2. The students will be given a problem to

solve. They will be given a problem to solve. They will help

Students’

answers

103

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

each other in reading,

analyzing and solving the problem. They are

required to discuss the problem until the two are convinced of the

solution. 3. The instructor monitors the students’

activity and checks for their answers. The

struggling ones will be required to explain the solution to the

instructor.

Model-Matching Activity Sheets This is an interactive

instructional activity that will lead students to match problems to

The students explain their answers to the

instructor and to the class.

104

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

their accomplished

Venn Diagrams or to match Venn Diagrams

to their corresponding working equations.

Procedures: 1. Instructors provide activity sheets

containing a matching type assessment.

2. The students match the items in column A

to Column B.

3. After, they will be asked to craft their own solution without

the models.

105

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

Reading (insufficient recall, deficient mastery, poor exposure)

The students left the items

unanswered

*Note: If instructors do

not want activity sheets, he can write

the items to be match on the board and let a matching exist among

students. Using this approach, the instructor can even

ask the students to explain how the

matching of concepts is done.

Direct Instruction with Paired reading

This is a type of instruction that focuses on the

essentials or the specific skill that needs to be targeted.

Students’ scores in the

activity sheets.

106

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

Procedures:

1. The instructor focuses on how applied

problems on sets and Venn Diagrams are understood or solved.

2. He can use technology in presenting the problem

or hand-outs. 3. The instructor pairs

the students for reading of the item assigned to them.

4. During the discussion of the item

assigned to the students, the instructor asks

questions on how students understood the problem. The

107

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

Comprehen-sion (poor exposure, lack of skills)

Students incompletely indicated all

data necessary to

the solution of the problem

students can switch to

the vernacular when not comfortable in

using English when explaining. Other students are asked to

give comments regarding the understanding of the

presenters.

Conceptual Processing This standards-based

mathematical inter-vention emphasizes the

need to build a deeper understanding of concepts. This involves

making ideas, facts and skills reflecting upon and refining

Students explain their understanding of

the problem.

Students are asked to give comments.

108

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

one’s own under-

standing. It utilizes concept-builder

materials such as diagrams and other manipulative.

Procedures: 1. After an interactive

discussion, the instructor asks

students to indicate all data from the problem. 2. After that, the

students are asked to explain the meaning of

each data, how the data must be sorted or how such data must

be treated.

109

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

Processing

(lack of practice, poor mastery, carelessness

Incorrect

difference of 180- 175 =10

Incorrect placement of

data and difference in the Venn

Diagrams.

3. The instructor gives

redirection or gives clarifying questions if

students are mislead. Trio Timed Drill

This is a variant of group learning that creates groups of 3

students.

Procedures: 1. The instructor assigns a student

leader in a group of 2 students, making them

a trio. 2. The instructor gives math worksheets that

the students will solve. 3. The leader facilitates the solution process of

Students

explain their understanding of the problem.

110

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

Encoding (carelessness, lack of criticality)

Students just left 5 inside

the Venn diagram; thus, there is

the given problem. The

leader is given a copy of the correct answer

for him to verify if his answer is correct and to check whether his

group mates get the correct answer. * Note, the instructor

gives only the copy of the answer if the

leader gets the correct answer. 3. The student leader

directs and redirects the students under

him. Self-Check This is a strategy that

directs and redirects students to check their personal work.

Students’ answers on the

drill exercises. Students’ answers on the

111

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

To

visualize

correctly real numbers

Real Number

System

Fair


no indication

of final answers.

Students just indicated 5,

instead of 5 people

Students just wrote 11

instead of 11 units as the final answer.

Procedures:

1. The students are given worksheets with

directing questions which include: Is your working solution

correct? Is your final answer in its simplified form? Does your final

answer address the question of the

problem? Does it have a unit? 2. The instructor

checks on the students’ answer.

Answer-switch-verify This interactive activity

asks for students to compare, contrast and give comments to the

self-check

exercises.

Students’ answer to the

given problem Seat works

112

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

using the

number line. To

perform operations involving

real numbers.

To solve

applied problems on

real numbers.

Incorrect

distance.

Incorrect Counting

answers of their fellow

students.

Procedures: 1. The instructor asks the students to answer

a given problem. 2. The instructor sets the time for all the

students to answer the given problem.

3. After the given time, the students exchange solution sheets with

each other. The students give or write

comments as to the completeness of the final answer, etc.

4. After the comment period, the students address the comments.

113

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

Processing


Students

failed to indicate the

correct formula or the working

equation. They wrote ―9 + (-2) = 7‖

and ―10 + 8 = 18‖; thus the

formula they used was D = P1 +P2.

Others did not indicate

any formula. Students left

the item unanswered.

Gallery Walk

This is a post-teaching instructional strategy

that assesses how students solve a given problem.

Procedures: 1. The instructor

divides the class into smaller learning

groups. 2. Each group is assigned an item to

solve. They are also given manila paper

and markers to present their solutions. 3. The students are

required to solve the items individually. They are only allowed

Student

solution presentations

114

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

Mathemati-sing (dismal performance deficient recall)

Students incompletely

wrote all the given in the problem

to write their answer

on the manila paper once everyone has

solved the problem at hand. 4. The students are

asked to present their answers to the class. The other groups can

give reactions to their answers.

Formula Match This is a strategy that

involves the formula used in solving items.

Students’ answers to the

activity Recitation

115

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy


Number line

pinpointing to the distance

of 10 and -2, as the largest and smallest

coordinates from among the 4

coordinates

Procedures:

1. The instructor presents the formulas

or the working equation and the different problems.

2. The students are asked to match the needed formula to the

respective problems. 3. The students will be

asked to explain their choices. 4. The class is free to

give comments.

Round Robin Reading This is a reading improvement strategy

that successively calls on students to read aloud a given problem.

Student reads

the given problem.

Answered hand-outs

116

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy


Procedures:

1. The teacher gives handouts on different

word problems or mathematical expressions.

2. The students will be asked to read on a round-robin basis.

3. After the reading sessions, the students

are asked to explain what they read. The students are free to

use the vernacular.

Comprehension Checker This is self-check

strategy that focuses on students’ understanding.

Understanding checker sheets

117

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

Procedures:

1. The instructor gives a work sheet that

contains word problems and comprehension

question item checklist. 2. The students read

the problem and check the item that

corresponds to their understanding. 3. The instructor

checks the items. If he sees that the students

have low scores, he gives direct instruction or assigns him to

someone with a perfect score. The instructor again gives another set

118

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

To translate verbal

expression to numerical

expression and vice versa.

To perform

operations on algebraic expression

To simplify all answers

Algebraic Expres-sions

Poor


Students did not understand

well the term ―the height is

(x+9) cm more than the base.

Students just wrote (x+9)

instead of (x+9) + (2x-5).

of comprehension

checker work sheets to the students to check

on improved understanding.

Explicit Instruction It is a dynamic, structured and

systematic methodology for

teaching academic skills. It is characterized by

learning guides or scaffolds, whereby

students are guided throughout the learning process.

Procedures: 1. Focus instruction on critical content, com-

Seatwork Assignments Student

Board work

119

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

prehension, analysis,

problem-solving strategies.

2. Break down the content on specific targets.

3. Tell students of what they need to learn before starting

instruction. 4. Review prior

knowledge and provide learning supports or guides for students to

learn the rudiments of the lesson.

5. Break the class into smaller learning groups to check on the

extent of attainment of the instructional objectives.

120

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy


Students left the items

unanswered. They do not know what to

do.

6. The rote classroom

activities can be done to assess learning

Systematic Instruction It

means breaking down complex skills into smaller, manageable

―chunks‖ of learning and carefully

considering how to best teach these discrete pieces to

achieve the overall learning goal.

Procedures: 1. Sequence learning

chunks from easier to more difficult and providing scaffolding,

Seatwork Assignments

Student Boardwork

121

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

or temporary supports,

to control the level of difficulty throughout

the learning process. 2. Teachers break

down a complex task, like analyzing and solving a math

problem, into multiple steps or processes with

manageable learning chunks and teach each chunk to mastery

before bringing together the entire

process. 3. In turn, the

students do the same process independently or by pair.

122

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

Reading

(insufficient recall, deficient mastery, poor exposure)

Students

failed to write the formula

for the problem.

Others wrote A = bh instead of A =

½ bh.

Sustained Silent

Reading (SSR) SSR is reading

instructional strategy that gives students instructional time to

read and analyze the problem.

Procedures: 1. Students are given

problem sets to be read silently. 2. The instructors give

the instructional item for them to analyze

and read the problem 3. After the SSR period, the teacher

asks questions that students will answer.

Seatwork

Assignments Student Board

work

123

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

Mathemati-sing (dismal performancedeficient recall)

Wrong addition of (2x+5) and (x-

9). Instead of writing (3x-4); others wrote

The questions focus on

how the students understood the

problem, how they can deal with the problem, the strategy and the

like. 4. The teacher again gives another problem

using the SSR method, but the difference is

the students will explain their understanding and

method on the board.

Quick Write It introduces a concept and connects this concept with prior

knowledge or experiences and allows students to

Students’ sharing of prior knowledge and

responses.

124

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

3x +4.

discuss and learn from

each other

Procedures: 1. Introduce a single word, phrase formula

to the class. 2. Students copy the concept on index

cards. 3. Students are given

two minutes to write whatever comes to their minds relative to

the concept. They may write freely using

single words, phrases, sentences, etc. 4. After time is called,

students may volunteer to share their thoughts on the

125

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

Processing (lack of practice, poor mastery, carelessness

Others overdid their solutions.

They wrote 2x+5 <0 and

3x<4 Others failed to write the

unit of measurement

.

subject.

5. The teacher gives direction, clarify or

affirm the student’s answers

Solve and React This allows students to solve whether

independently or independently. The

students will be asked to comment on the solutions to be

presented as regards the procedures of the

solution. Procedures:

1. The students will be asked to solve different items.

Students’ answers and reactions.

126

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

Encoding

(carelessness, lack of criticality)

Others performed

incorrect simplification of final

answers.

2. A student will be

asked to present solution on the board.

3. The students who are seated will be asked to comment on

the solution procedures as regards their correctness.

4. The students take note of this for future

use. Say Something

This is a variant of solve and react that

asks students to comment on the answers of the

students.

Students’

group work sheets

127

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

Procedures:

1. The students will be grouped into several

small learning groups (LG). 2. They will be solving

specific problem. 3. They will be exchanging and

commenting on the answers of the

students. 4. The teacher guides the students in the

correct examination and scrutiny of the

solution. Reflection Sheets

Teachers provide reflection sheets that ask the following:

Answered

Reflection sheets

128

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

To perform

operations on polynomials

To simplify polynomials accurately

To solve problems involving

polynomials

Polyno-mials

Fair

Mathemati-sing (dismal performance, insufficient recall)

Incorrect working

equation such as ―(x4-1)-(x+1)

No written

working equation

1. Is my answer in its

acceptable form? 2. Is my final answer

simplified correctly? 3. Does my answer contain unit? 4. Does my answer have the correct unit of measurement?

Five Word/

Formula Prediction Its purpose is to

encourage students to make predictions about text, working

equation or solution, to activate prior

knowledge, to set purposes for reading, and to introduce new

vocabulary

Quizzes Seat works Assignment

Recitation

Group work

129

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

Processing (lack of practice, poor mastery, careless-ness)

Incorrect factoring ―(x4-

1)‖ such as ―(x3)(x-1)‖

Incorrect cancellation (x4-1)/(x+1)

Procedures:

1. Select five key math words/ working equa-

tion from a set of problems that students are about to read.

2. List the words in order on the chalk-board.

3. Using Socratic Method, Clarify the

meaning of any unfamiliar words.

Carousel Brainstorm Purposes: This

strategy can fit almost any purpose intended, especially when

students find difficulty in understanding

Class presenta-tions and

reactions to solutions

130

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

problems or presenting solutions to problems.

Procedures:

1. Teacher determines what problems will be placed on chart paper.

2. Chart paper is placed on walls around the room.

3. Teacher places students into groups of

four. 4. Students begin at a designated chart.

5. They read the prompt, discuss with

group, and respond directly on the chart. 6. After an allotted

amount of time, students rotate to next

131

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

chart.

7. Students read next prompt and previous

recordings, and then record any new discoveries or

discussion points. 8. Continue until each group has responded

to each prompt. 9. Teacher shares

information from charts and conversations heard

while responding. 10. Students will be

asked to clarify points in the solution of the problem.

** This strategy can be modified by having the chart ―carousel‖ to

132

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy


Incorrect copying of

signs in the final answer; but correct

solution

No unit.

groups, rather than

groups moving to chart.

Say Something This encourages

students to react on one’s work and then eventually to react on

other’s work.

Procedures: 1. Instructor asks the students to solve

different problems. 2. The instructor gives

direction and time frame for students to solve.

3. After the specific time, the instructor reminds the students

Solution sheets

133

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

Reading


Students left

the items unanswered.

to finalize their

answer. 4. After 2 problems,

the students can exchange solution sheets and say

something about the solution and final answer.

GIST (Generating

Interactions between Schemata and Text) It directs students’

reflection on the content of the lesson

and leads them to summarize the problem and strategies

to differentiate between essential and non-essential information.

Students’ GIST

sheets

134

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

Procedures:

The task is to write a summary of the

keywords, the problem-solving strategy in groups. The

words, the notes and strategies capture the ―gist‖ of the text.

1. The instructor

models how to solve a certain problem. 2. Instructor models

the procedures by drawing blanks or

columns on the board. 3. Instructor thinks aloud as (s)he begins

to facilitate the intervention activity.

135

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

Comprehen-sion

(poor exposure, lack of skills)

Incorrect copying of

given data as to the signs

Incomplete representa-

tion of data

4. Students work with

a group or partner to complete a GIST for

the next chunk of problem. Students will eventually be asked to

create independent GISTs.

Copy-Solve-Cover It arouses students’

keen observation and comprehension about a certain text or

problem.

Procedures: 1. Instructor sets the objectives of the class.

2. The instructor demonstrates how certain data are

Solution sheets

136

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

organized and how

certain mathematical expressions are

properly understood. The students just copy.

3. On the succeeding items, the instructor covers the other half of

the item, then students will continue.

They will also be asked to explain their answers.

4. Gradually, students will do the same task

independently.

137

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

To get

the product of two

polynomials To solve

problems

involving product of two (2)

polyno-mials.

B. Special

Product Patterns

Product of two (2)

polyno-mials

Fair Processing


Incorrect

multiplication of (3x2-5) to

(3y+4) and (2x2+45) to (5y+2)

Incorrect evaluation in

―(3(10)2-5 = 295)‖, they

wrote ―900-5 = 895

Strategic Teaching

It is a teaching focused on specific lesson

contents. It is done after a diagnostic assessment is done.

Procedures: 1. Administer a

diagnostic test. For this study, the

research tool served as the diagnostic assessment.

2. From the results of the assessment, plan

or strategize the teaching based on the results. For this study,

the focus is on product of two (2) polynomials.

Quizzes

Worksheets Recitation

Group Presentation

138

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy


Item unanswered

3. The teacher teachers

using different approaches; then

assesses after the instructional time.

The Directed Reading-Thinking Activity (DRTA )

The DRTA is a discussion format that

focuses on making problems more understandable. It

requires students to use their background

knowledge, make connections to what they know, make

predictions about the text, set their own purpose for reading,

Activity Sheets

139

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

use the information in

the text and then make evaluative judgments.

It can be used with nonfiction and fiction texts.

FOCUS: Comprehension

Strategies: Prediction, Inference and Setting

Reading Purpose Procedures (begin by

explaining and modeling):

1. The teacher divides the reading assign-ments into meaningful

segments and plans the lesson around these segments.

140

Specific

Objectives


ance

Error Cate-


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degree of

error)


Process, Activities

Assessment

Strategy

2. In the class

introduction, the teacher leads the

students in thinking about what they already know about

the topic. (―What do you know about ...? What connections can

you make?) 3. The teacher then

has the students preview the reading segment examining the

illustrations, headings and other clues to the

content. 4. The teacher asks students to make

predictions about what they will learn.

141

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

5. Students may write

individual predictions, write with a partner or

contribute to an oral discussion creating a list of class

predictions. 6. Students then read the selection and

evaluate their predictions. Were their

predictions verified? Were they on the wrong track? What

evidence supported the predictions?

Contradicted the predictions? 7. Students discuss

their predictions and the content of the reading.

142

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy


Comprehen- sion


Incomplete

solution.

Correct solution but

incorrect generalization

8. The teacher and

students discuss how they can use this

strategy on their own and how it facilitates understanding and

critical thinking. 9. The teacher and students repeat the

process with the next reading segment that

the teacher has identified.

Self-Verification Procedures:

1. The teacher guides the students in the reflection of final

answers.

Students’

comments and reactions

Students’ work solutions

143

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

2. The students rectify

their solutions based on the reflection

directions. 3. The teacher gives another item to check

on understanding Question-Answer

Relationship (QAR) FOCUS:

Comprehension Strategies: Determining

Importance, Questioning and

Synthesizing QAR is a strategy that

targets the question ―Where is the answer?‖ by having the

144

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

classroom teacher and

eventually the students create

questions that fit into a four-level thinking guide. The level of

questions requires students to use explicit and implicit

information in the text: • First level: ―Right

There!‖ answers. Answers that are directly answered in

the text. • Second level: ―Think

and Search.‖ This requires putting together information

from the text and making an inference.

145

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

• Third level: ―You and

the Author.‖ The answer might be found

in the student’s background know-ledge, but would not

make sense unless the student had read the text.

• Fourth level: ―On Your Own.‖ Poses a

question for which the answer must come from the student’s own

background knowledge

Procedure (begin by explaining and modeling):

1. The teacher makes up a series of QAR questions related to

QAR Chart

146

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

the materials to known

to the students and a series of QAR

questions related to the next reading assignment.

2. The teacher introduces QAR and explains that there are

two kinds of information in a book

explicit and implicit. 3. The teacher explains the levels of questions

and where the answers are found and gives

examples that are appropriate for the age level and the content.

A story like Cinderella that is known by most students usually works

147

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

well as an example,

even in high school classes.

4. The teacher then assigns a reading and the QAR questions

he/she has developed for the reading. Students read, answer

the QAR questions and discuss their answers.

5. The teacher and students discuss how they can use this

strategy on their own and how it facilitates

understanding and critical thinking. 6. After using the QAR

strategy several times, the students can begin to make up their own

148

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

QAR questions and in

small groups share with their classmates.

7. The teacher closes this activity with a discussion of how

students can use this strategy in their own reading and learning.

The ultimate goal of this activity (and most

of the activities presented here) is for students to become

very proficient in using the activity and

eventually use the activity automatically to help themselves

comprehend text.

QAR Chart

149

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

Incomplete

encoding of data from the

problem No formula or

working equation written

KWL Chart and

Demonstration The know/want-to-

know/learned (KWL) chart guides students’ thinking as they begin

reading and involves them in each step of the reading process.

Students begin by identifying what they

already know about the subject of the assigned reading topic,

what they want to know about the topic

and finally, after they have read the material, what they have learned

as a result of reading. The strategy requires students to build on

KWL Chart

150

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

past knowledge and is

useful in making connections, setting a

purpose for reading, and evaluating one’s own learning.


Strategies: Activating Background

Knowledge, Questioning, Determining

Importance

Procedure (begin by explaining and modeling):

1. The teacher shows a blank KWL chart and explains what each

151

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy


Incomplete

encoding of data from the problem

No formula or

working equation written

column requires.

2. The teacher, using a

current reading assignment, demonstrates how to

complete the columns and creates a class KWL chart.

K W L

• For the know column: As students brainstorm

background knowledge, they

should be encouraged to group or categorize the information they

know about the topic. This step helps them get prepared to link

KWL CHART

152

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

what they know with

what they read. • For the want-to-

know column: Students form questions about the

topic in terms of what they want to know. The teacher decides

whether students should preview the

reading material before they begin to create questions; it depends

on the reading materials and

students’ background knowledge. Since the questions prepare the

students to find information and set their purpose for

153

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

reading, previewing the

material at this point often results in more

relevant questions. Students should generate more

questions as they read. • For the learned

column: This step provides students with

opportunities to make direct links among their purpose for

reading, the questions they had as they read

and the information they found. Here they identify what they have

learned. It is a crucial step in helping students identify the

154

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

important information

and summarize the important aspects of

the text. During this step, students can be reflective about their

process and make plans. 3. The teacher on the

next reading assignments can ask

students individually or in pairs to identify what they already

know and then share with the class, create

questions for the want to-know column either individually or in pairs

and share with class, and finally after reading, complete the

155

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

To square a

trinomial correctly

To

solve applied problems using

Square of a

trinomial

Fair

Processing (lack of practice, poor mastery, carelessness

Incorrect squaring of

(2x-4y+6z)2 as (4x2+16y2+36z2)

learned column.

4. The teacher closes

this activity with a discussion of how students can use KWL

charts in their own reading and learning. 5. The teacher

demonstrates the process in formulating

working equations and deriving formulas.

Error Bull’s-eye It directs students to

target specific errors in presented solutions.

Procedures: 1. The instructor presents different

List of errors culled out from

the solution. Students’

presentation of correct answers.

156

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

squaring a

trinomial

Reading


No answer

solutions with errors.

2. The students will be given instructional

time to study the solutions. 3. The students will be

asked to identify the errors. They will be asked to explain why

that certain part of the solution is wrong.

4. They will present the correct solution afterwards.

Group Reading with

Guide Sheets It directs reading comprehension by

giving questions that cull out student understanding.

Answers in

Reading Guide sheets

157

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy


Incorrect formulas as

A= 2∏r, and A= 2∏r2

Instead of writing

Procedures:

1. The instructor groups the students

and gives guide sheets in interpreting the applied problems.

2. The instructor checks the answers in the guide sheets.

3. The teacher gives comments and

redirections if necessary.

Comparison Matrix FOCUS:

Comprehension Strategies: Recognizing Similarities and

Differences

Answers in the Comparison

Matrix

158

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

(4x2+16y2+36

z2–16xy +24xz -48yz)∏ cm2,

they wrote 4x2+16y2+36z2–16xy +24xz

-48yz∏ cm2

(parenthetical error)

Procedures (begin by


1. The teacher writes the subjects/categories/to

pics/etc. across the top row of boxes. 2. The teacher writes

the attributes/characterist

ics/details/etc. down the left column of boxes.

3. Use as few or many of rows and columns

as necessary; there should be a specific reason students need

to recognize the similarities and differences between the

159

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

provided topics and

details. 4. Explain to and

model for students what each

column/row of the matrix requires.

Expressions

Given Mathemati-

cal expresions

Ans-wers

Operations

Related

vocabulary

Patterns

160

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

Encoding


Instead of

writing (2x-4y + 6z), others

even wrote (2x+4y+6z).

Response Notes


Strategies: Questioning, Inferring, Activating Background

Knowledge Procedures (begin by


1. The teacher introduces the response notes and

models how to respond to open-ended

questions, share understanding, make connections to

background knowledge, share feelings, justify

Students’

answers on their response

notes

161

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

To

correctly

perform cubing of a binomial.

Cube of a binomial

Fair

Comprehen-sion


Processing (lack of practice, poor mastery, carelessness)

Incorrect cubing of

(2x+4). Their answers were (8x3+63) or

pinions, etc.

2. Students then read and create their own

responses in their notebooks or journals. 3. The teacher then

asks students to share with the class and/or collects the notes.

4. The teacher and students discuss how

they can use this strategy on their own and how it facilitates

understanding and critical thinking.

Solution Theater This will present

different answers of students and will let them select the correct

Presented Solution

162

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

To

apply the correct

process of cubing a polynomial

in word problems.

worse

(8x3+12)

solution.

Procedures:

1. The students will be presented with a problem. They will be

asked to present solutions on the board. 2. After, the students,

by group, shall be watching or observing

(like in theater) all the solutions. 3. After, they will select

which is a wrong solution and which is

correct. 4. The students will explain the error of the

solution and to correct the error they found out,

163

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

Encoding



No unit

No answer

Self-Verification

(Please look at the details of the strategy in the earlier cells)

Listening Teams – prior to the lesson, the class is divided into 4

groups/sectors of the class:

FOCUS: CULLING OUT UNDERSTANDING OF

WORD PROBLEMS Procedures:

1. The teacher classifies students into:

* Readers – responsible for reading the applied

problem

Students’ work

sheets

Student

responses

164

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

* Strategists –

responsible for coming up with a solution

strategy. * Questioners – responsible for coming

up with 2 questions they have about the topic

*Agreers – responsible for coming up with 2

points they agree about on the topic *Nay Sayers – 2 points

about the lecture that they disagree with

*Example Givers – 2 examples that are applicable to the topic.

*Listeners – responsible to listen and list down key

165

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy


Incorrect formulas:

V= 3s3 and V= 4s

ideas.

2. The instructor facilitates the

presentation. 3. After some time, students do it alone.

Think-Alouds/ Metacognitive

Process

STRATEGY FOCUS: Comprehension Strategies: Monitoring

for Meaning, Predicting, Making

Connections Procedures (begin by

explaining and modeling): 1. The teacher chooses

Students’ answers in the instructional activity.

166

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

applied problems.

2. The teacher reads the text aloud and

thinks aloud as he/she reads. 3. Read the text slowly

and stop frequently to ―think-aloud‖ — reporting on the use of

the targeted strategies — ―Hmmm….‖ can be

used to signal the shift to a ―think-aloud‖ from reading.

4. Students underline the words and phrases

that helped the teacher use a strategy. 5. The teacher and

students list the strategies used.

167

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

Comprehen-

sion (poor exposure, lack of skills)

Incorrect

copying of the given (2x +4);

some wrote (2x-4), (2+4), (x+4)

6. The teacher asks

students to identify other situations in

which they could use these strategies. 7. The teacher

reinforces the process with additional demonstrations and

follow-up lessons.

COMPREHENSION CHECKER It helps teachers to

check whether students have correct

comprehension or not. It is a variety of anticipation-reaction

guide.

Student

responses Recitation

168

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

Procedures:

1.The instructor provides checklist on

the key words of the applied problems.

The students check the expressions that correspond to their

understanding.

The instructor directs the checking and redirects students who

are misled.

Daily re-looping of previously learned material

It is a process of always bringing in previously learned

169

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

material to build on

each day so that students have a base

knowledge to start with and so that learned structures are

constantly reinforced. This is for a topic that uses the same content area: linear equations, systems or rational expressions. Procedures:

1. Before beginning discussion, the teacher

elicits prior knowledge on the previous but related topics.

2. The students will be directed to relate the lesson at hand to the

170

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

To

perform factoring involving

difference of 2 perfect

squares, To

answer

applied problems

involving factoring of difference of

2 perfect squares.

C. Factor-

ing Patterns

Difference of 2

Perfect Squares

Pooor

Reading


Item left

unanswered

previous knowledge.

3. Discussion begins after the above-cited

processes. Structured Language

Experiences It is a well-structured learning activity where

students have abundant

opportunities to use language to describe their mathematical

understanding. It directs Students can

verbally explain/describe their math understanding,

they can write out their understanding, or

Student

presentations

171

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

they can draw pictures

and then explain.

Procedures: 1. Select a math concept/skill for which

students have received prior instruction and for which they have

demonstrated at least initial acquisition.

2. Develop a structured activity in which students can

describe their math understanding. The

activity should clearly relate the math concept/skill to the

language activity (e.g. students should clearly "see" the relationship

172

Specific

Objectives

Topics Level of Perform-mance

Error Cate-

gories and Theorized Causes

(arranged according to degree of

error)


Process, Activities

Assessment

Strategy

student elaborations in

a systematic way, ensuring that every

student receives feedback regarding their explanations (e.g.

for smaller groups, the instructor does this individually; for larger

groups, peer tutors evaluate each other’s

explanations while instructor monitors tutor pairs).

6. Instructor has opportunity to evaluate

at least one explanation/description for every student.

7. Review activity by modeling an accurate description of the math

173

Specific

Objectives

Topics Level of Perform-mance

Error Cate-

gories and Theorized Causes

(arranged according to degree of

error)


Process, Activities

Assessment

Strategy

concept/skill,

providing appropriate cueing (e.g. "think

alouds," visual, auditory, kinesthetic, tactile modalities).

Metacognitive strategy

It is a memorable "plan of action" that

provides students an easy to follow procedure for solving a

particular math problem. It is taught

using explicit teaching methods. It includes the student's thinking

as well as their physical actions.

174

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

1. Provide ample

opportunities for students to practice

using the strategy. 2. Provide timely

corrective feedback and remodel use of strategy as needed.

3. Provide students

with strategy cue sheets (or post the strategy in the

classroom) as students begin independently

using the strategy. Fade the use of cues as students demonstrate

they have memorized the strategy and how (as well as when) to

175

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

use it. (*Some students

will benefit from a "strategy notebook" in

which they keep both the strategies they have learned and the

corresponding math skill they can use each strategy for.)

4. Make a point of

reinforcing students for using the strategy appropriately.

5. Implicitly model using the strategy

when performing the corresponding math skill in class.

176

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

Mathemati-

sing (dismal performance, insufficient recall)

Incorrect

formula for area of a

square: A = s2 (A=2s)

Paired Think tank

It is variant of partner learning that uses

recalling of formulas encountered or taught.

Procedures: 1. The instructor asks the students to pair,

pairing must be according to degree of

mastery. 2. The teacher directs the recall of the

formulas in math. 3. The students, in

pair, will list down all the recalled formulas. 4. Checking of answers

will be done afterwards.

Student

responses

177

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

Comprehen-


Incorrect

copying of data from the

given

Reading for Meaning

Students become curious about printed

symbols or mathema-tical expressions once they recognize that

print, like talk, conveys meaningful messages that direct, inform or

entertain people. One goal of this curriculum

is to develop fluent and proficient readers who are knowledgeable

about the reading process. Effective read-

ing instruction should enable students to eventually become self-

directed readers who can:

construct meaning

Students

answers and responses

178

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy


Incorrect and incomplete

factoring of (2x2-162). They factored

from various types of

print material; recognize that there

are different kinds of reading materials and different purposes for

reading; select strategies appropriate for

different reading activities; and,

develop a life-long interest and enjoy-ment in reading a

variety of material for different purposes.

Independent Study (Using Learning

Activity Packages) This is a form of a seat work, using learning

Students answers and

responses

179

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

activity packages as

learning materials.

Procedures: 1. Students will be given some work to do,

based on prepared learning activity packages or skill book.

2. The students will be asked to check on their

answers by comparing with the answer key. 3. The students will be

asked to continue solving. The target is

for them to solve at least 5 items.

180

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

Problem solving

instruction: explicit instruction in the steps

to solving a mathematical problem including

understanding the question, identifying relevant and irrelevant

information, choosing a plan to solve the

problem, solving it, and checking answers.

Procedures: 1. The teacher

presents certain problems and how these items are solved

with different solution strategies.

Answers Sheets

181

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

To factor

PST.

To differentiate

a PST from other trinomials

To solve applied problems

Perfect Square

Trinomial

Fair


No unit in the answer.

Wrong unit of measurement

indicated: cm2, m instead of cm

2. The students chose

which among the strategies they should

use. 3. They solve individually but can

compare answers with their seatmates. They discuss their answers,

especially when the items

Self-help and self-correcting materials

Students practice a math concept/skill

using materials that provide them both math concept/skill

prompts (e.g. questions, math equations, word

Students’ answers

Quizzes

182

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

involving

PSTs.

problems, etc.) and the

solutions to each prompt.

Procedures: 1. Identify appropriate

math skill for student practice. 2. Incorporate

materials that include the features listed in

Critical Components. 3. Model how to perform the math skill

using each self-correcting material.

4. Ensure that students clearly understand how to use

the self-correcting material. Be especially sensitive to individual

183

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

students who have

difficulty with particular verbal or

nonverbal response modes that are required when using

each self-correcting material. Be especially sensitive to individual

students who have difficulty with

particular verbal or nonverbal response modes that are

required when using each self-correcting

material (e.g. for students who have significant writing

problems, then materials that require writing responses may

184

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

produce student

frustration and therefore would not be

appropriate). 5. Periodically monitor students who

are using self-correcting materials, providing them

feedback about appropriate or

inappropriate use of self-correcting materials.

6. Provide students with a way to record

their responses (e.g. a sheet of paper on which they record their

responses; have students record responses with dry-

185

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

erase-marker on

laminated response cards/sheets that

contain each math skill prompt). 7. Evaluate student

responses by examining student response sheets.

Provide students with corrective feedback

regarding their performance as soon as possible.

8. Do not grade student performance

using self-correcting materials! Grading performance will

detract from the motivation self-correcting materials

186

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

can elicit from

students and grading will inhibit student

willingness to "take risks," a crucial behavior for learning.

Scaffolding Instruction

It provides students who have learning

problems the crucial learning support they need to move from

initial acquisition of a math concept/skill

toward independent performance of the math concept/skill.

Also referred to as "guided practice."

187

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

Procedures:

1. Begin scaffolding after you have first

directly described and modeled the skill at least three times.

2. Perform the skill or learning task while asking questions aloud

and answering them aloud (questions

should pertain to specific essential features for specific

problem solving steps). Choose one or two

places during the problem solving process to question

your students.

188

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

3. Provide immediate

and specific feedback as well as positive

reinforcement with each student response. 4. When students

answer incorrectly, praise the student for his/her risk-taking

and effort while also describing and

modeling the correct response. When students answer

correctly, always provide positive

reinforcement by specifically stating what it is they did

correctly.

189

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

5. As your students

demonstrate success in responding to one or

two questions, then ask for an increased number of student

responses with the next example. (Corrective and positive

feedback continues as indicated by student

responses). 6. When your students demonstrate increased

competence, continue to fade your direction,

prompting students to complete more and more of the problem

solving process. Eventually, you only ask questions and

190

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

your students provide

all the answers. 7. As your students

demonstrate success in responding to one or two questions, then

ask for an increased number of student responses with the

next example. (Corrective and positive

feedback continues as indicated by student responses).

8. When your students demonstrate increased

competence, continue to fade your direction, prompting students to

complete more and more of the problem solving

191

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

process. Eventually,

you only ask questions and your students

provide all the answers. 9. When you are

confident that your students understand the problem-solving

process, invite them to actively problem-solve

with you (students direct problem-solving students ask question,

then both students and you respond).

10. Let student accuracy of responses and student nonverbal

behavior guide your decisions about when to continue fading your

192

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

Reading


No indication

of solution

direction.

Brain Storming

(Formulas) This is done by using learning circles.

Procedures: 1. Students will be

given different applied problems.

2. The task of the students is to give the corresponding working

equations or formulas that are needed for the

problems to be solved. 3. Presentation and critiquing of students’

answers will follow.

Student


193

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

Mathemati-


Incorrect

Formulas: A = 4s

S = A/2

Curriculum-Based

Probe It directs students to

solve 2-3 sheets of problems in a set amount of time

assessing the same skill. Instructor counts the number of

correctly written digits, finds the median

correct digits per minute and then determines whether

the student is at frustration,

instructional, or mastery level.

Students’


194

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

Procedures:

1. The instructor assesses the students’

mastery level through the quiz or seat work. 2. Based on the

results, the instructor gives students differentiated student

exercises based on their mastery level.

3. The instructor can focus on teaching the students under the

frustration level. The students in the

mastery level can facilitate drill for the instructional level.

195

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

Processing

(lack of practice, poor mastery, carelessness)

Incorrect

factoring of PST. They

simply divided by two.

Assigned Questions

(as Assignments) Focus: Reading,

Comprehension, Content

Procedures: 1. Students give assignments to

students to read at home.

2. When they enter the class, the instructor asks them to present

their work. 3. Other students will

be asked to give reactions. 4. Discussion on

critical concerns follow.

Student

Responses

196

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

To factor General Quadratic

Trinomials To solve

applied problems using

general quadratic trinomials.

General Quadratic Trinomial

Fair

Comprehen-



Reading (insufficient recall, deficient

mastery, poor exposure)

Wrong

indication of sign of the

copied data

No unit of measurement

No solution

Catching Signs

This is a strategy patterned sign

mnemonics.

Apply Self-Correcting Materials (See procedures above)

Adjusted speech: instructor changes speech patterns to

increase student comprehension. It includes facing the

Student

responses Solution Sheets

Recitation Students’

Answers Recitation Students’

Answers

Student responses Recitation

Recitation

197

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

students, paraphrasing

often, clearly indicating most important ideas,

limiting asides, etc Procedures: (This is

simply a variant of language switching) 1. Instructor can ask

the students to read. 2. When the instructor

directs students to understand the problem, he can switch

to the students’ mother tongue to stress the

essentials and to clarify vague thoughts, especially on

mathematical expressions.

198

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

Mathemati-


Encoding

Incorrect

working equation:

(x2+3x-40) - (x-8) instead of (x2+3x-40)

/ (x-8)

No unit

Puzzle Game

This is a variant of instructional game, or

another form of interactive worksheets.

Procedures: 1. Instructor gives an activity sheet that has

formulas and empty cells.

2. They match the formula and the letters to guess the magic

word.

Structured Peer Tutelage It is a well planned/

structured practice activities where students problem

Solution sheets

Activity sheets

Students’ answers on the activity sheets

199

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

solve in pairs

Procedures:

1. Determine goals for each peer tutelage activity.

2. Target specific math skills to be practiced.

3. Select appropriate materials that match

learning objectives and that can be implemented within a

peer tutoring format (i.e. provide both a

prompt sheet that contains problems to be solved and an

answer key that can be easily used by your students).

200

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

4. Design and teach

procedures/behaviors for tutoring.

5. Review classroom rules and teach new rules when

appropriate. 6. Pair students of varying achievement

levels. 7. Practice peer-

tutoring procedures before implementing them with academic

tasks. 8. Divide peer-

tutoring time into halves so each player has equal time as

coach and as player. 9. Signal students when it is time to

201

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

Processing


Incorrect

cancellation in (x2+3x-40)/

(x-8;x2 an x were immediately

cancelled.

switch roles.

10. Set goals for tutoring pairs and

provide positive reinforcement for tutoring pairs that

meet goals. 11. Provide response record sheets so you

can evaluate the performance of

individual students. Authentic Contexts

The purpose of Teaching Math

Concepts/Skills within Authentic Contexts is to explicitly connect

the target math

Students’

answers on the activity sheets

202

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy


Adding of superfluous data in the

given (x2+3x-

40)/2 instead of

concept/skill to a

relevant and meaningful context,

promoting a deeper level of understanding for students

Procedures: 1. Instructor chooses

appropriate context within which to teach

target math concept/skill. Refer to the assessment

strategy

Dynamic Assessment, for information about how to collect

information about students' interests and to use this information

Students’ answers

203

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

simply

(x2+3x-40)

to create authentic

contexts for assessment and

teaching Mathematics Student Interest Inventory

2. Instructor activates student prior knowledge of

authentic context, identifies the math

concept/skill students will learn, and explicitly relates the

target math concept/skill to the

meaningful context. 3. Instructor explicitly models math

concept/skill within authentic context.

204

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

4. Instructor involves

students by prompting student thinking

about how the math concept/skill is relevant to the

authentic context. 5. Instructor checks for student

understanding. 6. Students receive

opportunities to apply math concept or perform math skill

within authentic context. Instructor

monitors, provides specific corrective feedback, remodels

math concept/skill as needed, and provides positive reinforcement.

205

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

7. Instructor provides

review and closure, explicitly re-stating

how the target math skill relates to the authentic context and

remodeling the skill. 8. Students receive multiple opportunities

to apply math concept or practice math skill

after initial instruc-tional activity. Incor-porating the instructor

instruction strategies, Building Meaningful

Student Connections, Explicit Instructor Modeling, & Scaffold-

ing Instruction when teaching within

206

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

To apply the correct procedure in

factoring by grouping To use the

correct procedure in

answering word problems.

Factoring by grouping

Poor


Item left unanswered

authentic contexts can

be very effective.

Focusing on "Big Ideas" or the essentials It facilitates student understanding by concentrating student attention on key concepts and procedures. The linkages and connections between math concepts are made explicit by linking previously learned big ideas to new concepts and problem solving situations. By emphasizing the big ideas in each lesson,

instructors can build students' acquisition

Student responses

207

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

and use of key

conceptual knowledge across lesson content.

Procedures: 1. Choose math big

ideas that are foundational to the lesson and that

represent understandings that

can be applied across lessons (e.g. formula, mathematical

expressions). 2. Explicitly teach the

math big idea, linking it to previously learned information.

3. Explicitly teach the target math skill within the context of

208

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

4. The math big idea.

5. Provide multiple practice opportunities

for students using the Big Idea with the new math skill you taught.

6. Apply the math big idea to the target math skill using a variety of

problem solving situations.

7. Pair a visual cue with each math big idea (e.g. a picture of

an array for the Big Idea of "area").

8. Post the visual cue along with one sentence describing

why the big idea is important.

209

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

Mathemati-


Incorrect

working equation

(x2+2xy+y2+x+y) - (x+y) instead of

(x2+2xy+y2+x+y) / (x+y);

Structured

Cooperative Learning Groups

Students practice math concepts/skills they have previously

required with peers in teams or small groups.

Procedures: 1. Determine goals for

each cooperative learning activity. 2. Target specific

academic skills to be learned/practiced.

3. Select appropriate materials that match learning objectives.

4. Design and teach procedures/behaviors for team members to

Recitation

210

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

help each other.

5. Review classroom rules and teach new

rules when appropriate. 6. Assign students of

varying achievement levels to the same team.

7. Practice cooperative group procedures

before implementing them with academic tasks.

8. Set team goals and provide positive

reinforcement for teams that meet goals. 9. Evaluate success of

cooperative learning activity.

211

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

Processing


Invalid

cancellation in

(x2+2xy+y2+x+y) / (x+y); the expression

―x+y‖ was immediately cancelled.

Think-Pair-Share

Think-Pair-Share is a strategy designed to

provide students with "food for thought" on a given topics enabling

them to formulate individual ideas and share these ideas with

another student. It is a learning strategy

developed by Lyman and associates to encourage student

classroom participation. Rather

than using a basic recitation method in which a instructor

poses a question and one student offers a response, Think-Pair-

Students’

answers

212

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

Share encourages a

high degree of pupil response and can help

keep students on task. Procedures:

With students seated in teams of 4, have them number them

from 1 to 4. Announce a

discussion topic or problem to solve. (Example: Which

room in our school is larger, the cafeteria

or the gymnasium? How could we find out the answer?)

Give students at least 10 seconds of think time to THINK

213

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

of their own answer.

(Research shows that the quality of

student responses goes up significantly when you allow

"think time.") Using student

numbers, announce

discussion partners. (Example: For this

discussion, Student #1 and #2 will be partners. At the

same time, Student #3 and #4 will talk

over their ideas.) Ask students to PAIR

with their partner to

discuss the topic or solution.

214

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

Finally, randomly

call on a few students to SHARE their ideas

with the class. Instructors may also

ask students to write or diagram their responses while doing

the Think-Pair-Share activity. Think, Pair, Share helps students

develop conceptual understanding of a

topic, develop the ability to filter information and draw

conclusions, and develop the ability to

consider other points of view.

215

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

Encoding


No indication

of unit.

Structured

Controversy Using structured controversy in the classroom can take many forms. In its most typical form, you select a specific problem. The closer the problem is to multiple issues central to the course the better. This strategy involves providing students with a limited amount of background information and asking them to construct an argument based on this information. This they do by working in groups.

Solution sheets

216

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

Procedures:

Choose a discussion topic that has at

least two well documented positions.

Prepare materials: o Clear expectations

for the group task.

o Define the positions to be

advocated with a summary of the key arguments

supporting the positions.

o Provide reference materials including a

bibliography that support and elaborate the

217

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

arguments for the

positions to be advocated.

Structure the controversy: o Assign students to

groups of four. o Divide each group

into dyads who are

assigned opposing positions on the

topic. o Require each

group to reach

consensus on the issue and turn in

a group report on which all members will be evaluated.

218

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy


Incorrect data. Adding unnecessary

data. Instead of

(x2+2xy+y2+x+y) only, they wrote

(x2+2xy+y2+x+y) / 2

Conduct the

controversy: o Plan positions.

o Present positions. o Argue the issue. o Reverse positions

and argue the issue from those perspectives.

o Reach a decision.

Explicit Instructor Modeling The purpose of explicit

instructor modeling is to provide students

with a clear, multi-sensory model of a skill or concept. The

instructor is the person best equipped

Students’ responses

219

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

to provide such a

model.

Procedures: 1. Ensure that your

students have the

prerequisite skills to perform the skill.

2. Break down the

skill into logical and learnable parts (Ask

yourself, "what do I do and what do I think as I perform

the skill?"). 3. Provide a

meaningful context for the skill (e.g. word or story

problem suited to the age & interests of your students).

220

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

4. Provide visual,

auditory, kinesthetic

(movement), and tactile means for illustrating

important aspects of the concept/skill (e.g. visually display

word problem and equation, orally cue

students by varying vocal intonations, point, circle,

highlight computation signs

or important information in story problems).

5. "Think aloud" as you perform each step of the skill

221

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

(i.e. say aloud what

you are thinking as you problem-solve).

6. Link each step of the problem solving process (e.g. restate

what you did in the previous step, what you are going to do

in the next step, and why the next

step is important to the previous step).

7. Periodically check

student understanding with

questions, remodeling steps when there is

confusion.

222

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

To

simplify correctly RAEs

To apply simplification of RAEs in

D. Ration-

al Expres-sions

Simplifi-cation of RAEs

Fair

Reading


Item

unsolved.

8. Maintain a lively

pace while being conscious of

student information processing difficulties (e.g. need

additional time to process questions).

9. Model a

concept/skill at least three times

before beginning to scaffold your instruction.

Assigned Questions

(as Seat work) Focus: Reading and Formulas

Students’

homework

223

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

applied

problems.

Procedures:

1. Students give assignments to

students to read. They can be grouped and mixed according to

degree of ability. 2. When they enter the class, the instructor

asks them to present their work.

3. Other students will be asked to give reactions.

4. Discussion on critical concerns

follow.

224

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

Processing


Incorrect

cancellation in

(12x4y6/7xy) and (21/6x3y5).

Incorrect

placing of simplified

form. Instead of 3 in the numerator, it

was placed in the

denominator.

Planned Discovery

Activities The purpose of

Planned Discovery Activities is to provide students who have

learning problems the opportunity to make meaningful

connections between two or more math

concepts for which they have previously received instruction

which they have previously mastered. It

is important to remember that this is a student practice

strategy and it is not intended for initial instruction.

Students’

solutions

225

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

Procedures:

1. Determine two or more concepts that

have a mathematical relationship which

students would benefit from understanding.

These concepts must have already

been taught and must have been already mastered by

the students. 2. Develop a well

organized/structured activity that provides students

the opportunity to understand the desired mathema-

226

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

tical relationship

between the selected math

concepts. 3. Provide explicit

directions for

completing the activity.

4. Develop and provide

to students a structured learning

sheet or other appropriate prompt that guides

students toward the learning objective.

5. Monitor students as they participate in the activity.

Circulate the classroom, provide specific corrective

227

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

feedback, model

appropriate skills as needed, prompt

student thinking, and provide positive reinforcement.

6. At the conclusion of the activity, provide students with the

solutions to the structured learning

sheet/prompt and explicitly state/ illustrate the

desired mathema-tical relationship(s).

7. How Does This Instructional Strategy Positively

Impact Students Who Have Learning Problems?

228

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

Mathemati-


Incorrect

working equation such

as (12x4y6/7xy) ÷ (21/6x3y5) or

(12x4y6/7xy) - (21/6x3y5) instead of

(12x4y6/7xy) x (21/6x3y5)

Experiential Learning

(focus: solving math problems)

Experiential learning is inductive, learner centered, and activity

oriented. Personalized reflection about an experience and the

formulation of plans to apply learning to other

contexts are critical factors in effective experiential learning.

The emphasis in experiential learning is

on the process of learning and not on the product.

Experiential learning can be viewed as a

Students’

answers Quizzes

Recitations

229

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

cycle consisting of five

phases, all of which are necessary:

experiencing (an activity occurs);

sharing or

publishing (reactions and observations are shared);

analyzing or processing (patterns

and dynamics are determined);

inferring or

generalizing (principles are

derived); and, applying (plans are

made to use learning

in new situations).

230

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

Encoding


No unit.

Answers were

not simplified:6/1

Procedures:

1. Instructor presents solved problems.

2. Instructor leads the students to analyze the solved problems, to

direct the students to analyze solution patterns.

3. Students are given items to solve. They

can clarify misconceptions if necessary.

Graphic organizers:

visual displays to organize information from the problem. They

help students to consolidate informa-

Organizer

sheets

231

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

Comprehen-


Incomplete

data

tion into meaningful

whole and they are used to improve

comprehension of stories, organization of writing, and

understanding of difficult concepts in word problems.

Procedures:

1. Instructor presents problems. 2. Instructor

Demonstrates how graphic organizers are

used. 3. Understanding of students will be

checked based on the teaching techniques of

232

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

To

perform the basic operations

on RAEs To apply

operations on RAEs to applied

problems

Operation

on RAEs

Poor

Mathemati-


Incorrect

working equation (1/2x)(8x/2)

instead of (5/2x)(80x/2)

the teachers. Follow-

up questions that will probe into the in-depth

understanding of the students must be structured.

4. Students use the organizers independently.

Think-Pair-Share

Think-Pair-Share is a strategy designed to provide students with

"food for thought" on a given topics enabling

them to formulate individual ideas and share these ideas with

another student. It is a learning strategy

Students’

responses

233

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

developed by Lyman

and associates to encourage student

classroom participation. Rather than using a basic

recitation method in which a instructor poses a question and

one student offers a response, Think-Pair-

Share encourages a high degree of pupil response and can help

keep students on task.

Procedures: With students seated in teams of 4,

have them number them from 1 to 4.

234

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

Announce a

discussion topic or problem to solve.

(Example: Which room in our school is larger, the cafeteria or the

gymnasium? How could we find out the answer?)

Give students at least 10 seconds of

think time to THINK of their own answer. (Research shows that

the quality of student responses goes up

significantly when you allow "think time.") Using student

numbers, announce discussion partners.

235

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

(Example: For this

discussion, Student #1 and #2 will be

partners. At the same time, Student #3 and #4 will talk over their

ideas.) Ask students to PAIR with their partner

to discuss the topic or solution.

Finally, randomly call on a few students to SHARE their ideas

with the class.

Instructors may also ask students to write or diagram their

responses while doing the Think-Pair-Share activity. Think, Pair,

236

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

Share helps students

develop conceptual understanding of a

topic, develop the ability to filter information and draw

conclusions, and develop the ability to consider other points

of view.

Learning Partners: discuss a document, interview each other

for reactions to a document or

presentation, critique or edit each others’ work, recap a lesson,

develop a test question together, compare

237

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

Reading


Items

unanswered

notes, stump your

partner.

Procedures: 1. Instructor provides problems after the

students are paired. 2. The students are asked to recall the

correct formulas. 3. Critiquing of

answers is done. Continuous

Performance Charting The goal of continuous

monitoring and charting of student performance is twofold.

First, it provides you, the instructor, information about

Student

responses

Recitation

238

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

student progress on

discrete, short-term objectives. It enables

you to adjust your instruction to review or re-teach concepts or

skills immediately, rather than waiting until you've covered

several topics to find out that one or more

students didn't learn a particular skill or concept. Second, it

provides your students with a visual represen-

tation of their learning. Students can become more engaged in their

learning by charting and graphing their own performance.

239

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

Procedures:

1. Determine a specific instructional

objective (classify objects according to color, size, shape,

pattern; add two digit numbers without regrouping, solve story

problems with + and - fractions, select the

relevant information in a story problem). 2. Design a

"curriculum slice" using the C-R-A

assessment strategy (see Additional Information for an

example of a curriculum slice below.) Choose

240

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

3. appropriate items

that accurately reflect the target math skill at

the appropriate level of understanding (concrete,

representational, abstract) and that can be administered in a

short time period (perhaps a 1-3 minute

timing). Include more items than you think the student can

complete within the designated time period

so that you get an accurate indication of their optimal

performance.

241

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

Procedures:

1. Determine a specific instructional

objective (classify objects according to color, size, shape,

pattern; add two digit numbers without regrouping, solve story

problems with + and - fractions, select the

relevant information in a story problem). 2. Design a

"curriculum slice" using the C-R-A

assessment strategy (see Additional Infor-mation for an example

of a curriculum slice below.) Choose appro-priate items that

242

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

3. accurately reflect

the target math skill at the appropriate level of

understanding (con-crete, representa-tional, abstract) and

that can be administered in a short time period (perhaps a

1-3 minute timing). Include more items

than you think the student can complete within the designated

time period so that you get an accurate

indication of their optimal performance. 4. Administer and

score the assessment. 5. Have students plot incorrect and correct

243

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

responses on a graph

(see Additional Information for an

example of a graph/chart). 6. Discuss and draw

goal lines on graph. 7. Repeat process. 8. Determine success

of your instruction based on the "learning

picture" depicted on the student's chart/ graph (see Additional

Information for examples of different

learning pictures and what they mean). 9. Make appropriate

instructional decisions based on the student's learning picture.

244

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

Processing


Others wrote

(5/2x) ÷ (80/2x)

instead of (5/2x) x (80x/2)

Drill & Practice

As an instructional strategy, drill &

practice is familiar to all educators. It "promotes the

acquisition of knowledge or skill through repetitive

practice." It refers to small tasks such as

the memorization of spelling or vocabulary words, or the

practicing of arithmetic facts and may also be

found in more sophisticated learning tasks or physical

education games and

Student

responses

Recitation Performance

Chart

245

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

sports. Drill-and-

practice, like memorization, involves

repetition of specific skills, such as addition and subtraction, or

spelling. To be meaningful to learners, the skills built through

drill-and-practice should become the

building blocks for more meaningful learning.

Procedures:

1. Students will be given exercises to solve independently, by pair

or by small groups. 2. The instructor roams around to check

246

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy


Incomplete data. They

missed out 5 and 10 for the denomination

.

No unit.

on students’ answers.

3. The instructor assists students who

are experiencing difficulty. 4. Students are asked

to present their solutions.

Drill & Practice As an instructional

strategy, drill & practice is familiar to all educators. It

"promotes the acquisition of

knowledge or skill through repetitive practice." It refers to

small tasks such as the memorization of spelling or vocabulary

Student responses

Recitation

Solution sheets on drills

247

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

words, or the

practicing of arithmetic facts and may also be

found in more sophisticated learning tasks or physical

education games and sports. Drill-and-practice, like

memorization, involves repetition of specific

skills, such as addition and subtraction, or spelling. To be

meaningful to learners, the skills built through

drill-and-practice should become the building blocks for

more meaningful learning.

248

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy


Procedures:

1. Students will be given exercises to solve

independently, by pair or by small groups. 2. The instructor

roams around to check on students’ answers. 3. The instructor

assists students who are experiencing

difficulty. 4. Students are asked to present their

solutions.

Didactic Questions This is variant of partner learning that

focuses on students’ understanding. It focuses on how

Student’s responses

Recitation

249

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

To

simplify complex

RAES. To use

simplificatio

n of RAEs in solving

applied problems.

Simplifi-

cation of Complex

RAEs/ fractions

Very

Poor

Reading


Items left

unanswered

students should

understand applied problems. They will

focus on how answers are solved and simplified for

acceptance. Apply SSR

(see mechanics above)

Response journal:

Students record in a journal what they

learned that day or strategies they learned or questions they have.

Students can share their ideas in the class,

Student

responses

Recitation Journals

250

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy


Incorrect formulas: I =

1 + PRT and I = PR

with partners, and

with the instructor.

Procedures: 1. Students are given assignments. They are

asked to write their questions pertinent their reading

assignment. 2. The questions shall

be shared and discussed in class. 3. The questions shall

serve as the starting point of the instructor.

Didactic Questions This is variant of

partner learning that focuses on students’ understanding. It

Student responses

251

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

Comprehen-sion


Time (t) was not

represented and indicated.

focuses on how

students should understand applied

problems. They will focus on how answers are solved and

simplified for acceptance.

Instructional Game The goal of each

student practice strategy in this program is to provide

students who have learning problems

multiple opportunities to respond to a particular learning

task. This is certainly true for Instructional Games as well.

Recitation

252

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

Instructional Games

can also make practicing math skills

fun due to its game format.

Procedures: 1. Determine math skill(s) for which target

students have received prior instruction and

which they can perform with at least moderate success.

2. Select a student age/interest appro-

priate game context in which the target math skill can be performed.

3. Develop game procedures that allow for many math-skill

253

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

practice opportuni-

ties (complexity of game procedures

should not detract from skill practice).

4. Provide students

with necessary materials to play the game.

5. Model the math skill(s) to be practiced

at least once in isolation and at least once within the game

context before the game is played.

6. Provide explicit directions for playing the game and model

game procedures.

254

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

7. State behavioral

expectations and model essential game-

playing behaviors (e.g. turn-taking, respond-ing appropriately when

I am not the winner, etc.) 8. Invite several

students to model playing the game

before game begins. Provide an opportunity for students to ask

questions and to clarify misconceptions.

9. Monitor students as they play the game, providing specific

corrective feedback, modeling skills when appropriate, and

255

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy


Incorrect substitution:

P = I/RT as (1/6 x

12,000) =P/[(1- 1t/3)].

providing positive

reinforcement. Demonstrate

enthusiasm for game as students play Provide a way for

students to show their work so that you can evaluate their

performance after the game is completed.

Procedures: Model-lead-test

strategy instruction (MLT): 3 stage process

for teaching students to independently use learning strategies: 1)

instructor models correct use of strategy; 2) instructor leads

256

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy


No indication of ―t‖ in the final answer.

No unit.

students to practice

correct use; 3) instructor tests’

students’ independent use of it. Once students attain a score

of 80% correct on two consecutive tests, instruction on the

strategy stops.

Apply self-help

257

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

To solve

applied problems on

distance using linear equations.

To solve the value of x

E. Linear

Equations in One

Unknown Applied

Problems on Distance

Poor Reading


No answer.

Explicit vocabulary

building through random recurrent

assessments: Using brief assessments to help students build

basic subject-specific vocabulary and also gauge student

retention of subject-specific vocabulary.

Procedures: 1. Instructor asks

students to read certain problems.

2. They will be asked to share their ideas and even problems

regarding the problem. 3. Discussions follow.

Student

responses

Recitation

258

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

Comprehen-


Incomplete

indication of data.

Native language

support: providing auditory or written

content input to students in their native language.

Procedures: 1. Using GLCs,

students will be redirected to

understand the problem. 2. They are allowed to

explain using the vernacular.

Student developed glossary: Students

keep track of key content and concept words and define them

Student

responses

Recitation

259

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

Mathemati-


They failed to

write the correct formula, D =

RT. Others wrote Vf

2= Vo2

+ 2 fusing Physics and College

Algebra. Others wrote

440-220= 220

in a log or series of

worksheets that they keep with their text to

refer to. Correcting formula

mismatch A variant of formula matching

Procedures:

1. Students will be presented with columns of problems

and formulas. The items are already

matched, but incorrectly done. 2. The instructor will

then give the task to correct the incorrect matching.

Student


Quizzes

260

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

Encoding


No or

incorrect unit.

Unit Matching

This will let students match the conditions

of the problem to its corresponding unit of measurement.

Procedures: 1. Instructor provides

activity sheets that contain a column for

problem conditions and a column for unit of measurement.

2. After answering, discussion of answers

follows. 3. Correction and redirection of

misconceptions shall be a follow-up teaching procedure.

Student

responses

Recitation

261

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

Processing


Incomplete

solution. After getting the

value of x, they did not go back to the

tabular repre-sentation.

Columnar Battle

This is an instructional game that integrates

group learning circles (GLCs)

Procedures: 1. Students will be grouped into 3-4.

2. Leaders and assistant leaders will

be assigned. 3. A representative per group shall be called to

solve given items. 4. Students who are

seated will also be asked. Monitoring by the instructor and

assistant leader must be done.

Student

responses

Recitation

262

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

To

represent data

correctly To

interpret

problems correctly

Applied

Problems on

Money/ Amount

Poor

Reading


Not answered.

5. This continues until

majority of students have gone to the

board. 6. Discussion of misconceptions will be

done. Timed Reading

This is a variant of structured reading

strategy. Procedures: 1. The instructor gives

students applied problems to solve.

2. After several minutes or the instructional time for

reading, the instructor gives guide questions for students to be led

Student

responses

Recitation

263

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

Comprehen-


Partial

indication of data.

towards the correct

understanding. 3. The students are

asked to present their answers highlighting keywords and

translations. Student-led

discussions This is a variant of

cooperative learning groups. Its focus is to hasten reading,

comprehension and mathematical skills.

Procedures: 1. Using clustered

groups, a leader will be assigned to facilitate the analysis and

Seatwork

264

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

Mathemati-sing

(dismal performance, insufficient recall)

Others added 1 and 20 and

1 and 50 as the working equation.

solution of certain

problems. 2. The leader will ask

his group members to read. The leader, in turn, will give

directions as to how the problem should be hurdled.

3. Checking of answers will be done.

4. Critiquing will also be done.

Table Completion This is a form of data

collection strategy.

Completed Table

Student responses

Recitation

265

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy


Instead of writing 1350 – 50x, others

wrote 1350 – x or 135 –

Procedures:

1. The instructor provided supplemental

materials on applied problems and tables where headings for all

required data are indicated. 2. After silent reading,

the students will be asked to fill in the

table. 3. The teacher checks and redirects students

when necessary.

Equation Generator This will ask students to present solutions on

the board.

Generated equations Quiz

266

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

50x or worse,

135 –x

Procedures:

1. After teaching, students will be called

to go in front to generate equations or to recall formulas.

2. Students will be asked to write answers on the board. Students

who are seated will be asked to verify written

formulas. 3. The students will explain whether a

generated equation is correct or not. It

should be clear when an equation is correct or not.

4. A seat work may follow for assessment.

267

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

Encoding


No unit.

Incorrect unit of KPH

instead of KM only.

Quiz bees

This is a form of instructional activity

that tests students’ mastery of the subject matter.


divides the class into three. The instructor

provides flaps for students to answer. 2. The flaps will be

raised once the bell is signaled.

3. Checking of answers follow. 4. If an item is not

solved by majority of the groups, the teacher stops temporarily the

Student

responses

Recitation Answers

268

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

game and discusses

with the class the correct mechanical

procedures. Fishbowl of Units

This is another form of Q&A technique.

Procedures. 1. The teacher provides

a bowl/ box where units of measures are indicated.

2. Students will be called to pick a strip

paper from the bowl/box. 3. The students are

asked to explain when they should use the unit of measure.

Students’

explanation

269

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

To

formulate working

equations on age problems

To find the value of x

To represent

data correctly

To solve

age problems

Applied

Problems on Age

Poor Reading


Item

unanswered.

Reciprocal peer

tutoring (RPT) to improve math

achievement This is a general strategy in improving

performance. Focus: How to deal with problems -

1. Have students pair, choose a team

2. Explain to the students the goal of the activity.

3. Let the fast learners tutor on math

problems, and then individually work a sheet of drill problems.

Students get points for correct problems and work toward a goal.

Student

responses

Recitation

270

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

Send me a problem

This is a reading game that leads students to

read and solve certain problems.

Procedures: 1. 10 students shall be selected to write

certain mathematical expressions or

problems. 2. The problems or mathematical

expressions shall be sent to certain

students. 3. The students, in turn, will solve the

problem and will give the answer to the students who gave

271

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

Comprehen-sion


No represen-tation for the

age of two persons

involved.

them the question. The

student-sender determines if the

answer is correct or not. If incorrect, the sender will give guides.

Question Generation This lets students to

write questions and give their

corresponding understanding as regards the given and

the requirement of the problem.

Procedures: 1. Students are asked

to create five types of questions from a reading assignment,

Generated Questions

272

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

with each question

moving to a higher level of thinking. Place

the questions on note cards to be passed and discussed or handed

in. 2. Students are then asked to write their

opinions regarding the thrusts of the problem:

the given and the required. 3. The instructor

checks and reinforces topics not understood

by majority of the class.

273

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

Mathemati-


No written

formulas or working

equations. Others wrote

30 +x = 2x as the working equation.

Concentric Circles –

small circle forms inside a larger one,

smaller circle discusses while the larger circle listens and

then roles are reversed.

Procedures: 1. The students shall

form 2 concentric circles. 2. On the first round,

the inner circle shall compose a problem.

The outer circle shall write the corresponding

equation or formula.

Student

responses

Recitation

274

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy


No unit.

3. On the 2nd round,

the tasks are reversed. 4. Discussion of

answers is a follow-up procedure.

Deck of Cards This is an interactive seatwork.

Procedures:

1. Students are asked to fold a sheet of paper, creating a card.

2. On the left part, a problem is written.

3. The students will exchange cards. 4. After, they will be

solving the given problems. The solutions will be placed

Student responses

Recitation

275

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

To

formulate expressions

To apply

concepts on linear equation in

Applied

Problems on Fare

Poor

Processing


Reading


Incorrect

transposition; Others wrote x+2x = 30+10

instead of x-2x=30-20

Items

unanswered.

on the right portion of

the card. 5. Then, the card shall

be given back to the owner for checking. 6. The students will

determine if the answers are correct or not.

7. Students who did not get the answer

correctly shall be helped by the person who gave the problem.

Re-teaching

This is a form of instruction where the instructor re-teaches

the topic but with a different strategy.

Student


Seatwork

276

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

word

problem

Procedures:

1. The teacher, after assessing students’

difficulties, plans for re-teaching.

2. The teacher has to alter the strategies like using math websites or

instructional games.

3. The target of the intervention is to refocus the skill and

target the difficulties.

277

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

Comprehen-


No tabular

representa-tions or

representation of the missing data.

Idea Spinner

This is an interesting way to elicit students’

knowhow and knowwhy of the subject matter

Procedures: 1. instructor creates a

spinner marked into 4 quadrants and labeled:

Predict, Explain, Summarize and Evaluate.

2. After new material is presented, the

instructor spins and asks a student to respond accordingly.

3. Redirection is done when necessary.

Student

responses

Recitation Seat work

278

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

Mathemati-


No working

equation. Majority

applied guess and check.

Comprehension

Builder This is an instructional

strategy that helps students to structure their understanding of

the applied problems. Procedures:

1. Table utilization shall be demonstrated

to the class – how it is filled up completed. 2. After, the students

will be asked to do it independently or by

pair. 3. Redirection is done when necessary.

Completed

tables

279

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

Encoding


No unit.

Find the Rule –

students are given sets of examples that

demonstrate a single rule and are asked to find and state the rule.

Procedures: 1. Students will be

presented with PowerPoint

presentations on algebraic expressions, formulas and working

equations per each problem.

2. The instructor models how each equation is derived for

the working equation. 3. After ample items, the students are asked

Student

responses

280

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

Processing


Incorrect

substitution; Instead of writing

(3(200-10y)/8) + 10y

= 150, they wrote ((200-10y)/8) + 10y

= 150.

to do it themselves.

4. Units of measure can also be included in

the presentations. Solution Inspector

This is an instructional activity that will lead students to be critical

about presented problems.


presents the objectives of the activity and to

elicit from the class the job of an inspector. 2. The instructor

presents to the class solutions to applied problems.

Inspectors’

reports

281

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

To

formulate equations involving

verbal expressions

To solve problems in number

relation

Applied

Problems on Number

Relation

Fair

Mathemati-


Incorrect

Working Equation. Others wrote

―x +x = 100 and x-x = 20

as the working equation.

3. As inspectors, they

will look into the errors of the solutions. They

will explain why that is an error and where did it start.

4. The students are asked to provide the necessary corrections.

Jumbled Equations

This activity elicits students’ prior knowledge on formulas

and working equation.

Procedures: 1. Instructor provides a randomly ordered set

of equations.

Student


282

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

Reading


No working

equation. Others

applied trial and error

No solution.

No

representa-tions for x

and y.

2. The students will be

asked if the equation is correct or not.

3. They will be asked to reorder the jumbled formulas and

equations based on the dictates of the word problems.

4. They will be asked to explain their

answers. Apply Structured

Reading Variable Basket

This is a variant of fishbowl method for formula selection but

used in improving comprehension.

Student

responses

283

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

Comprehen-


They

committed transposition

errors in transposing y in x +y = 100

No unit. No indication of final

answer in the conclusion.

Procedures:

1. The instructor presents the objectives

of the instructional activity. 2. In the basket are

variables and the expressions where the variables are culled

out. 3. They will explain if

the representing variables are correctly written or not. They

will be asked to explain their answers and

correct the representations when necessary.

Recitation

284

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

Processing


Encoding


Problem solving

instruction: explicit instruction in the steps

to solving a mathematical problem including

understanding the question, identifying relevant and irrelevant

information, choosing a plan to solve the

problem, solving it, and checking answers.

Procedures: 1. The teacher

presents certain problems and how these items are solved

with different solution strategies.

Student

responses

Recitation Solution sheets

285

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

To

simplify exponents and radicals

correctly

G. Expo-

nents and Radicals

Very

Poor

Reading


Item not

answered.

2. The students chose

which among the strategies should they

use. 3. They solve individually but can

compare answers with their seatmates. They discuss their answers,

especially when their answers are different.

4. Board work can be a good assessment strategy.

Direct Instruction

with Paired reading This is a type of instruction that

focuses on the essentials or the specific skill that needs

Student


286

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

To solve

exponential and radical

equations To solve

problems

involving radical and exponential

expressions and equa-

tions.

to be targeted.

Procedures:

1. The instructor focuses on how applied problems on sets and

Venn Diagrams are understood or solved. 2. He can use

technology in presenting the problem

or hand-outs. 3. The instructor pairs the students for

reading of the item assigned to them.

4. During the discussion of the item assigned to the

students, the instructor asks

287

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy


Incorrect formulas/

working equation: √(2x+7) + 3x =

90 A = ½ (√(2x+7) – Sin

questions on how

students understood the problem. The

students can switch to the vernacular when not comfortable in

using English when explaining. Other students are asked to

give comments regarding the

understanding of the presenters.

Graffiti This is an instructional

activity that elicits students ideas on a certain problem.

Student responses

Recitation

Board work

288

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

3x)r2

Procedures:

1. An issue/question/ problem is indicated on flipchart paper and

there may be many in the room on tables. 2. As individuals or

groups (with different colored markers) the

students visit each station and write their opinions/answers/que

stions. 3. Sharing of ideas is

done. Redirecting and processing of answers will also be done.

289

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

Comprehen-


Incorrect data

representation

Others wrote √(2x+7)/2 and 3x/2 instead

of √(2x+7) and 3x only.

Reading Corners

This is a variant of reading stations or

centers. Procedures:

1. Students are presented with word problems and different

options of solving the problem. Theses sets

are placed on the corners of the room. 2. Groups or batches

of students go to the corners and read the

problem. They will also select the best strategy listed

in the options column.

Student

responses

Recitation

290

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy


Incorrect deletion of the radical

symbol in √(2x+7) = 3x

without squaring.

3. After, the students

will be asked to share answers.

4. Discussion of answers will be done

Formula Derivation and Analysis This is a form of direct

instruction.

Procedures: 1. The instructor directly teaches

students on how to generate working

equation out of the given applied problems.

2. Demonstration and chunking of data will be done.

Student work sheets

291

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy


No unit Wrong

selection of the value of x.

3. Instructor asks the

students the area of the process where they

find difficulty. 4. Essentials will be stressed on the area of

difficulty. 5. Analysis and derivation of formulas

will be done. 6. Pairing can be done

so the students can help each other. 7. Activity sheets are

good supplemental materials.

Using data diagram This is a form of

tabular data collection.

Completed diagram

292

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

Procedures:

1. Instead of tables, the teacher

demonstrates how to use diagrams in collecting data from

the given problem. 2. The teacher explain how the items are

translated into workable expressions.

3. The students are asked to do it.

Policy Recall

This is an interactive strategy that will lead students to critique

their own work.

Student


Student responses

293

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

Procedures:

1. The students will be presented with rules

when square root is done, deleting radical symbols, squaring, etc.

2. The students will also be presented with correct and incorrect

application of policies. 3. In the correction of

policies, students will be asked to give comments on what

policy is violated. 4. They can also be

paired for supplement or assistance. 5. The students will be

given task to perform

Recitation

Quiz

294

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

operations involving

radicals and exponents.

Direct Instruction focusing on the

essentials This is an integrative approach of teaching

students emphasizing on the essential

aspects, specifically their point of error.

Procedures: 1. Using the results of

the assessment, the instructor plans direct instruction focusing on

key elements in data

295

Specific

Objectives


ance

Error Cate-


Causes


degree of

error)


Process, Activities

Assessment

Strategy

selection, conclusion,

simplification and unit utilization.

2. The students will be given the task to solve

something. 3. The instructor

monitors and checks immediately on

students’ errors.

Two-day Seminar-Workshop on the Utilization of the Instructional Intervention Plan

I. Rationale

The challenges of teaching mathematics to students of the 21st

century are not easy. One challenge is the lowering performance of

students in Mathematics. But, the most pressing according to

Egodawatte (2010) are the causes why students really fail in their

performance in mathematics.

Educational experts contend that teachers, for them to be effective,

must not only know what to teach; they must also know how to teach.

Instructional strategies and principles of teaching are very necessary in

the life of the teacher. But, how can these instructional strategies or

interventions be known and mastered by the instructors, especially if

they are not sent to seminars and are attending graduate school, more so

with instructors who are not graduates of a teaching course? Indeed,

seminar-workshop is necessary.

Seminar-workshop is the process of acquiring specific skills to

perform a better job. It helps people to become familiar with essential

tools and elements necessary for them to be more effective. Through

seminar-workshop, people’s behavior towards a task becomes modified.

Such modified behavior contributes to the successful attainment of goals

and objectives.

296

297

The proposed two-day seminar-workshop is based upon the

foremost constraints and error categories of the students in College

Algebra. Their foremost error categories are along reading and

mathematising.

II. General Objectives

1. Improve instructional pedagogical competencies; and,

2. Apply and adopt the different instructional interventions.

III. Seminar-Workshop Course Contents

Instructional Interventions on the different error categories

IV. Methodologies

Participative Lectures and demonstration will be the main

methodologies of the seminar-workshop.

V. Facilitators

The facilitators for the proposed seminar-workshop were chosen

based on their extent of involvement in the research, qualifications,

trainings and seminars attended and organized.

Name Position/Extent of Involvement in the Research

Qualifications

Feljone G. Ragma Instructor Proponent

Ed.D

298

Nora A. Oredina Professor Proponent’s Adviser

Ed.D

Lea L. de Guzman Professor

Ed.D

Jovencio T. Balino Professor

External Evaluator

Ed.D.

VI. Participants

All mathematics instructors in the Higher Education Institutions

(HEIs) of La Union

VII. Duration

Two consecutive Saturdays: April 14 and 21, 2014 (refer to the

proposed program of activities)

VIII. Logistics

Registration fee (P500 per participant)

e.g. 50 participants P 25,000.00

Expenses

Honoraria for speakers (2,500/speaker) P 10,000.00

Meals/Snacks for the speakers (P250/speaker) P 1,000.00

Meals/Snacks for the speakers (P250/participant) P 12,500.00

Certificates and kits P 1,500.00

IX. Success Indicator

The mathematics instructors of the Higher Education Institutions

of La Union shall be able to utilize the instructional interventions.

299

SAMPLE FLYER OF THE TWO-DAY SEMINAR-WORKSHOP

300

Level of Validity of the Instructional Intervention Plan

Table 19 shows the level of validity of the instructional intervention

plan. It shows that the level of validity of the intervention plan is 4.51,

interpreted as very high validity. This means that the instructional

intervention plan is very highly functional, acceptable, appropriate,

timely, implementable and sustainable. It further implies that the plan is

a very good material that can address the dismal performance and the

different error categories.

Table 19. Level of Validity of the Instructional Intervention Plan

Criteria Validators Mean

A B C D E

I. Face 3 5 4 4 4 4.0

II. Content

a. Functionality

5

4

5

5

5

4.80

b. Acceptability 4 5 5 4 4 4.40

c. Appropriateness 5 5 5 4 5 4.80

d. Timeliness 3.5 5 5 4 5 4.50

e. Implementability 3.5 5 5 5 4 4.50

f. Sustainability 4 5 5 4 4 4.60

Average 4.0 4.86 4.86 4.29 4.43 4.51

301

CHAPTER IV

SUMMARY, CONCLUSIONS AND RECOMMENDATIONS

This chapter shows the summary, findings, conclusions and

recommendations of the study.

Summary

The study identified and analyzed the error categories of students

in College Algebra in the Higher Education Institutions of La Union as

basis for formulating a Validated Instructional Intervention Plan.

Specifically, it determined the level of performance of the students in

College Algebra along elementary topics, special products patterns,

factoring patterns, rational expression, linear equations in one unknown,

systems of linear equations in two unknowns and exponents and

radicals; the capabilities and constraints of the students in College

Algebra; the error categories of the students along reading,

comprehension, mathematising, processing and encoding errors; and the

validated instructional intervention plan.

The study is descriptive with a researcher-made College Algebra

test as the instrument of the study. The test was administered to 374

students of the HEIs in the province of La Union for 1st semester of the

school year 2013-2014. The data collected were treated using frequency

count, mean, rate and the Newmann’s error analysis tool.

302

Findings

The researcher found out the following:

1. The students had fair performance in elementary topics, special

products and factoring while poor performance in rational expressions,

linear equations and systems of linear equations and very poor

performance in exponents and radicals. The students had a general

performance of poor.

2. The performances of the student in the specified topics were all

considered as constraints.

3. Mathematising and comprehension were the major error

categories of the students in elementary topics, processing and reading

errors in special products, reading and Mathematising in factoring,

reading and Mathematising in rational expressions, reading and

comprehension in linear equations, and reading and Mathematising in

systems of linear equations and exponents and radicals. In general, their

major error categories in College Algebra were along reading and

Mathematising.

4. The instructional intervention plan is very highly valid.

Conclusions

Based on the findings of the study, the following are concluded:

303

1. The students cannot competently deal with elementary

topics, special product and factoring patterns, rational expressions,

linear equations, systems of linear equations and radicals and

exponents.

2. The students are deficient in terms of their skills of the

topics in College Algebra.

3. Majority of the students cannot start the problem-solving

process which leads them not to successfully finish all the stages of

problem solving.

4. The instructional intervention plan is a very good material

that can address the dismal performance and errors of the students.

Recommendations

Based on the conclusions of the study, the following are humbly

recommended:

1. The schools should adopt the Instructional Intervention Plan

and let their mathematics instructors attend the two-day seminar-

workshop.

2. The students should exert more effort in understanding the

different concepts in their College Algebra course. They should spend

more time in dealing with drills and exercises rather than dealing with

social media and entertainment.

304

3. The instructional interventions plan should be used not only

in the province of La Union but in all schools experiencing the same

student error patterns in College Algebra.

4. The mathematics teachers should suit their teaching

strategies based on their students’ needs and interests.

5. The English teachers should intensify the development of the

students’ skill of reading with comprehension in their classes.

6. A study should be conducted to determine the effectiveness

of the instructional intervention plan.

7. A similar study should be conducted in other branches of

Mathematics, applied sciences and English, especially in the subjects

where percentage of failures is high.

305

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APPENDIX A Sample Computations

Sample Computations on the:

Reliability of the College Algebra Test of

College Algebra Test

Validity of College Algebra Test

List of Suggestions Made by the

Validators and the Corresponding

Action/s by the Researcher

313

Sample Computation of Reliability of the College Algebra Test

For the College Algebra Test

Scores:{38,39,29,25,28,38,41,37,33,36,28,9,9,30,34,34,24,29,20,19,41,22,27,27,21,22,19,26,27,24}

Data from StaText:

k=100

k-1=99

𝑥 = 27.87

𝜎2=69.77

𝐾𝑅21 = 𝑘

𝑘 − 1 1 −

𝑥 𝑘 − 𝑥

𝑘𝜎2

𝐾𝑟21 = 100

99 1 −

27.87(100 − 27.87)

100 (69.77)

KR21= 0.71906

KR21 ≈ 0.72;high reliability

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Sample Computation on the Validity

of College Algebra Test

Criteria Validators Mean

1 2 3 4 5 6

1. The directions are clear

and specific and do not warrant misconceptions among

students.

3 5 5 5 3 4 4.17

2. The sentences are free

from grammatical errors and other construction lapses.

5 4 4 5 2 3 3.83

3. The test items are clearly and specifically

formulated based on student’s level of

understanding.

5 4 4 5 4 4 4.33

4. Mathematical expressions and

equations are encoded clearly to avoid student

misunderstanding.

4 4 4 5 5 3 4.17

5. There are provisions for

students to show their solutions.

4 5 5 4 5 4 4.50

6. The test items cover the

course content as indicated by the table

of specifications.

4 5 5 5 5 4 4.67

7. The test items are

written to cull out the specific errors of students in College

Algebra.

4 5 5 5 4 3 4.33

8. Generally, the test

items represent what they ought to measure.

4 5 5 5 4 4 4.50

Overall 4.13 4.63 4.63 4.88 4.00 3.63 4.32

315

List of Suggestions Made by the Validators and the Corresponding

Action/s by the Researcher

Suggestions Remarks

The 30-item test cannot be accomplished by the students in the specified time frame.

Incorporated The 30 items were reduced to 20

items only; however, the researcher saw to it that the scope of the

College Algebra test still covered the specified scope of the syllabi. For example, instead of separate

items for sets and Venn Diagrams, an item was constructed to deal

with these 2 related topics; instead of separate items for addition, subtraction, multiplication and

division of polynomials and rational expressions, an item that conglomerates the four basic

operations was formulated.

An item is solved by a student in at most 3 minutes.

Provide more space for the students to show their answers.

Incorporated More spaces were provided for the

students to clearly and completely show their solutions.

Delete the line for working equation and illustration since it will take much of the student’s time;

anyways, these will be reflected when they start writing their

preliminary steps for the solutions. This will also give the students the freedom of what specific strategy to

use in solving the given word problems.

Incorporated The provisions for working

equations and illustration were deleted.

316

Suggestions Remarks

Emphasize on the instructions that the students need to show their complete solutions.

Incorporated The instruction on showing the

complete solutions and the non-utilization of calculators was made

bold and of bigger font size.

Check on some lapses on grammar. Incorporated

Grammar lapses were checked.

Add more spaces between and

among numerical coefficients, variables and their exponents for

clarity.

Incorporated

Spaces were provided between and

among the numerals, variables and their exponents.

Some data need to be more realistic.

Incorporated Some data were changed to be

more realistic. Instead of a problem on a concert, a problem on fare in a jeep was written to replace the said

item.

The scoring scheme should be

revised, in consultation with the adviser, so that an item will not

just be 1 point. The points should be distributed along the different levels specified along the error

categories.

Incorporated.

The scoring scheme was revised.

Please see data categorization.

APPENDIX B Research Tool

Letter to Students-Respondents to Administer

the College Algebra Test

The College Algebra Test

Math I - College Algebra Test (Table of Specifications)

317

SAINT LOUIS COLLEGE


GRADUATE SCHOOL

September 2013

My dearest students,

The undersigned is a Doctor of Education Major in Educational

Management (Ed.D-EdM) student of Saint Louis College undertaking the

study entitled, ―Error Analysis in College Algebra in the HEIs in La

Union.‖ It is with this cause that your support is sincerely solicited so

that this study can be carried out and may greatly contribute to the

improvement of the teaching-learning process.

Please lend an hour to answer this word problems set. It may take

much of your precious time but your answers to these problems will

contribute much to the success of this study.

Rest assured that all information obtained herein will be held

strictly confidential. Your immediate attention to this request is highly

cherished.

Thank you so much!

Sincerely yours,

Mr. Feljone G. Ragma

Researcher/ Ed.D. student

318

COLLEGE ALGEBRA TEST

Name (optional)_______________________________School:____________________________________

INSTRUCTIONS: Please read, analyze and solve the problems that follow. Please indicate all

information being asked in the given problems on the test sheets.

PLEASE SHOW ALL SOLUTIONS. NO USING OF CALCULATORS!

1. 250 customers were asked in a survey as to what cell phone brands they like the most. The results

reveal that 160 chose Samsung, 150 chose Nokia and 180 chose iPhone, 75 chose Samsung and Nokia,

100 chose Samsung and iPhone, 90 chose Nokia and iPhone. 20 customers choose all the 3 brands. How

many love other brands?

Given data:

Solution:

2. What is the sum of the distance of 7 from -2 and 10 from 8 on the number line?

Given data:

Solution:

3. The base of a right triangle is expressed as (2x-5) cm and its height is (x+9) cm more than the base,

what is its area in cm2?

Given:

Solution:

319

4. Juan de la Cruz finds out that his money is expressed in (x4-1) pesos. If he wantsto buy (x+1)ice cream,

how many ice cream can he buy?

Given:

Solution:

5. Don Mario is choosing between lots A and B. Lot A is (3x2-5) square meters sold at P (3y+4) per square

meter while lot B is (2x2+45) square meters sold at P (5y+2). If x = 10 and y = 2, which is cheaper?

Given:

Solution:

6. The radius of a circular table is expressed as (2x-4y+6z) cm, what is its area in cm2?

Given:

Solution:

7. The side of a cube measures (2x+4) cm, what is its volume?

Given:

Solution:

320

8. The area of a rhombus is (2x2-162) square units. If one of the diagonals measures (x-9), what is the

measure of the other diagonal?

Given:

Solution:

9. The area of a square garden is expressed as (4x2-20x+25) meters2. What is the measure of its side?

Given:

Solution:

10. A string measuring (x2+3x-40) cm is divided into 2 parts. If one part measures (x+8)cm, what is the

measure of the other part?

Given:

Solution:

11. A truck has (x2+2xy+y2+x+y) loads of stone to be delivered to 2 customers. If the first customer shall be

delivered (x+y) loads only, what is the share of the second customer?

Given:

Solution:

321

12. Aling Maria wishes to buy 12𝑥4𝑦6

7𝑥𝑦 kilo of tomatoes for

21

6𝑥3𝑦5 pesos per kilo. How much will she pay?

Given:

Solution:

13. Agnes has 1

2𝑥 pieces of 5-peso coin and

8𝑥

2 of 10-peso coin. What is the product of the 5-peso and 10-

peso amounts?

Given:

Solution:

14. The interest of an amount invested in a bank at simple interest is 1/6 of 12,000. If the rate is at (1-1/3), how much is the principal investment?

Given:

Solution:

15. Two vehicles travel at the same time but in opposite directions. Vehicle A runs at 120 kph while vehicle B runs at 100kph. After some time, their distance from each other is calculated to be 440 km. What is the distance traveled by each of the two vehicles?

Given:

Solution:

322

16. Feljone has 27 bills consisting of 20-peso and 50-peso bills, If he has a total of 990 pesos, how many 20-peso bills does he have?

Given:

Solution:

17. Lorna is 20 years older than her daughter, Rudylyn. In ten years, she will be twice as old as her

daughter, how old is Rudylyn?

Given:

Solution:

18. The fare for a jeepney was P200 for 8students and 10regular passengers. The fare, on another day,

was P150 for 3students and 10regular passengers. How much was the fare for a regular passenger?

Given:

Solution:

323

19. The sum of 2 numbers is 100 while their difference is 20. What are the two numbers?

Given:

Solution:

20. An angle bisector divides an angle into 2 equal parts. If one of the equal angles measures ( 2𝑥 + 7)˚

while the other measures (3x)˚, what is the measure of one the smaller angles?

Given:

Solution:

MATH 1 - COLLEGE ALGEBRA TEST

TABLE OF SPECIFICATIONS

TOPICS TOTAL

HOURS KNOWLEDGE

COMPREHENSION

REMEMBERING

UNDERSTANDING

ANALYSIS APPLICATION

ANALYZING

APPLYING

SYNTHESIS EVALUATION

EVALUATING

CREATING

ITEM PLACEMENT

TOTAL ITEMS

PRELIMS 15 7 7

Elementary Topics - Sets and Venn Diagrams - Real Number System - Algebraic Expressions - Polynomials

8 4 1-4 4

Special Products and Patterns - Product of 2 polynomials - Square of a Trinomial - Cube of a Binomial

7 3 5-7 3

MIDTERMS 15 7 7

Factoring - Difference of 2 Perfect Squares - Perfect Square Trinomial - General Trinomial

8 4 8-11 4

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313

- Factoring by grouping

Rational Expressions - Simplification of RAEs - Operation on RAEs - Simplification of Complex RAEs/ fractions

7 3 12-14 3

FINALS 15 6 6

Linear Equations in One Variable Applied Problems on: - Distance - Mixture/Money/Coin - Age

6 3 15-17 3

Systems of Linear Equations Applied Problems on: - Fare/Price - Number Relation

6 2 18-19 2

Exponents and Radicals - Exponential and Radical expressions and equations

3 1 20 1

45 30 20

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APPENDIX C Communications

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331

332

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APPENDIX D Sample of Corrected College Algebra Test

336

Legend: Guide to Checking 5 pts. – No Error 4 pts. - Encoding Error (EE) 3 pts. – Processing Error (PE) 2 pts. – Mathematising (ME) 1 pt. – Comprehension Error (CE) 0 pt. – Reading Error (RE)

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350

351

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354

CURRICULUM VITAE

PERSONAL DATA

Name: Feljone Galima Ragma

Date of Birth: July 31, 1986

Place of Birth: San Isidro, Candon City, Ilocos Sur

Home Address: San Isidro, Candon City, Ilocos Sur

e-mail Address: [email protected]

Civil status: single

EDUCATIONAL ATTAINMENT Pre-Elementary: UCCP

Candon City, Ilocos Sur Graduated 1991 With honors

Elementary: Candon South Central School Candon City, Ilocos Sur Graduated 1997

With honors Secondary: Santa Lucia Academy

Santa Lucia, Ilocos Sur Graduated 2003 With honors

Tertiary: Saint Louis College San Fernando City, La Union Graduated 2007

Bachelor in Secondary Education Cum Laude

Major in Mathematics Recognition Award

Graduate Studies: Saint Louis College

San Fernando City, La Union Graduated 2011

mailto:[email protected]

355

Master of Arts in Education Cum Laude Major in Mathematics Best in Research

Post-Graduate Studies: Saint Louis College

San Fernando City La Union Graduated 2014

Doctor of Education Magna Cum Laude Major in Educational Best in Research

Management

BOARD EXAMINATION/ CIVIL SERVICE ELIGIBILITY

Licensure Examination for teachers (LET) 2007

P.D. 907 Civil Service Eligible

WORK EXPERIENCE, POSITIONS/SPECIAL ASSIGNMENTS

School/Institution Position Inclusive Dates

Saint Christopher Academy Classroom Teacher 2007-2008 Bangar, La Union

Christ the King College Classroom Teacher 2008-2013 San Fernando City, La Union Subject Area Coordi- 2010-2013

nator

Saint Louis College Instructor I 2013-2014 City of San Fernando, La Union

OTHER WORK-RELATED EXPERIENCES

Adviser and Panelist, Graduate School Researches Saint Louis College


External Evaluator, Undergraduate Researches

Saint Louis College City of San Fernando, La Union

356

Review Facilitator, Civil Service Exam

Dacanay Hall San Fernando City September-October, 2013

TRAININGS/SEMINAR-WORKSHOPS FACILITATED

January 2014

Giving Feedback to Improve Student’s Learning and Behavior Christ the King College City of San Fernando, La Union

2013

Seminar on How to Love and Like Mathematics Saint Louis College City of San Fernando, La Union

Back to Basics of Test Construction

Christ the King College City of San Fernando, La Union June 29, 2012

Understanding by Design and K-12 Christ the King College

May 20, 2012

Problem-Solving Techniques in Secondary Mathematics Association of Private Schools City of San Fernando, La Union

July, 2010

Seminar-Workshop on Creating Gradebooks through MS EXCEL Christ the King College City of San Fernando, La Union

2009 Seminar-Workshop on Creating Interactive Slides through MS

PowerPoint Christ the King College

City of San Fernando, La Union 2009

357

CONFERENCES/ SEMINARS PARTICIPATED

Engaging Learners in the Mathematics Classroom Saint Louis College September, 2013

Sustainability in the Classroom Saint Louis College

September, 2013

Colloquy in Thesis Advising Saint Louis College September, 2013

International Education Conference: How to be an Effective and

Successful Teacher SMX Conventional Hall, Pasay City August, 2012

Formative Assessment in the K-12 Curriculum Christ the King College

August 3-4, 2012

Mathematics Trainer’s Guild Seminar on Singaporean Math Association of Private Schools July 6-7, 2012

Moving Forward with Backward Design: A Deeper look at UBD Saint Louis University Laboratory Elementary School

January, 2011

Understanding and Planning for the 2010 SEC for Mathematics Phoenix Hall, Quezon City

November, 2010

Training Program for Mathematics Teachers University of the Cordilleras September, 2010

358

Seminar on Yoga and Relaxation Christ the King College

August, 2010

Critical Questions to Elicit Critical Thinking Christ the King College July, 2010

Seminar-Workshop on Homeroom Guidance and Counseling Techniques

Christ the King College June, 2010

Utilizing and Interpreting CEM Test Data University of Baguio

May, 2010

In-Service Training and Workshop on Curriculum Programs and Teaching Strategies Christ the King College

November, 2009 Seminar on Innovations in Teaching and Learning Approaches

Christ the King College July, 2009

Understanding and Planning for the SEC Phoenix Hall, Pangasinan

September, 2009

PROFESSIONAL ORGANIZATION

National Organization for Professional Teachers (NOPTI)

PAFTE

325