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.
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.
Dean, Graduate Studies
Saint Louis College

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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

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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,

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 CategoriesStages 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 OUTPUTINPUT
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
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
Exponents and Radicals
lent Rating

35
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
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
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
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
Systems of Linear Equations in Two Variables
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
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
Systems of Linear Equations in
Two Variables 3.55 35.50% Constraint
Exponents and Radicals 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.
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:
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% 11.50%
Legend:
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,

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

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