error analysis in college algebra in the higher education institutions in la union
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This doctoral study looked into the error categories of the students in College ALgebra It provided an Instructional Intervention Plan as the output of the study It also provided a model framework on how specific error categories in students' solutions can be addressed, the Ragma's Error Interventions ModelTRANSCRIPT
ERROR ANALYSIS IN COLLEGE ALGEBRA IN THE HIGHER
EDUCATION INSTITUTIONS OF LA UNION
A Dissertation
Presented to
the Faculty of the Graduate School
Saint Louis College
City of San Fernando, La Union
In Partial Fulfillment
of the Requirements for the Degree
Doctor of Education
Major in Educational Management
by
FELJONE GALIMA RAGMA
January 11, 2014
ii
INDORSEMENT
This dissertation entitled, ―ERROR ANALYSIS IN COLLEGE
ALGEBRA IN THE HIGHER EDUCATION INSTITUTIONS OF
LA UNION,‖ prepared and submitted by FELJONE GALIMA RAGMA, in
partial fulfillment of the requirements for the degree DOCTOR OF
EDUCATION major in EDUCATIONAL MANAGEMENT, has been
examined and is recommended for acceptance and approval for ORAL
EXAMINATION.
NORA ARELLANO-OREDINA, Ed.D. Adviser
This is to certify that the dissertation entitled, ―ERROR ANALYSIS
IN COLLEGE ALGEBRA IN THE HIGHER EDUCATION INSTITUTIONS
OF LA UNION,” prepared and submitted by FELJONE GALIMA RAGMA,
is recommended for ORAL EXAMINATION.
MARIA LOURDES R. ALMOJUELA, Ed.D.
Chairperson
JOVENCIO T. BALINO, Ed.D. DANIEL B. PAGUIA, Ed.D. Member Member
AUGUSTINA C. DUMAGUIN, Ph.D. AURORA R. CARBONELL, Ed.D. Member Member
Noted by: ROSARIO C. GARCIA, DBA
Dean, Graduate School Saint Louis College
iii
APPROVAL SHEET
Approved by the Committee on Oral Examination as PASSED with
a grade of 96% on January 11, 2014.
MARIA LOURDES R. ALMOJUELA, Ed.D.
Chairperson
JOVENCIO T. BALINO, Ed.D. DANIEL B. PAGUIA, Ed.D.
Member Member
AUGUSTINA C. DUMAGUIN, Ph.D. AURORA R. CARBONELL, Ed.D. Member Member
Accepted and approved in partial fulfillment of the requirements
for the degree DOCTOR OF EDUCATION MAJOR IN EDUCATIONAL
MANAGEMENT.
ROSARIO C. GARCIA, DBA Dean, Graduate Studies
Saint Louis College
This is to certify that FELJONE GALIMA RAGMA has completed
all academic requirements and PASSED the Comprehensive Examination
with a grade of 96% on June 15, 2013 for the degree DOCTOR OF
EDUCATION major in EDUCATIONAL MANAGEMENT.
ROSARIO C. GARCIA, DBA
Dean, Graduate Studies Saint Louis College
iv
ACKNOWLEDGMENT
The researcher wishes to express his sincerest gratitude to the
following persons who contributed much in helping him structure the
research.
Dr. Nora A. Oredina, dissertation adviser, for always affirming and
supporting; and for giving necessary suggestions to better this study.
Dr. Maria Lourdes R. Almojuela, chairperson of the dissertation
panel, for her valuable critique, and most especially, for directing the
researcher to the correct structure of the research.
Dr. Aurora R. Carbonell, Dr. Augustina C. Dumaguin, Dr. Daniel
B. Paguia, Dr. Rosario C. Garcia and Dr. Jovencio T. Balino, the
panelists, for their brilliant thoughts.
The validators of the questionnaire and the research output for
giving suggestions that improved the study.
Presidents, registrars, academic deans, department
chairpersons, instructors and students of the Private Higher Education
Institutions in La Union, for lending some of their precious time in
dealing with the pre-survey and the questionnaires.
Mrs. Edwina M. Manalang and Mrs. Marilyn Torcedo, for sparing
some time for brainstorming for the built-in theory of the study.
v
Mesdames Grace, Lea, Melody, Graziel, Jay Ann, Abegail, Sister
Grace, Mafe, and Sir Roghene, the researcher’s friends, who gave him
inspiration.
Mr. & Mrs. Felipe and Norma Ragma, the researcher’s parents,
for always being there when the researcher needed some push.
Kuya Darwin, Ate Felinor and Ate Nailyn, the researcher’s
siblings, for always following up the researcher’s progress.
And lastly, to GOD Almighty, for giving the needed strengths in
the pursuit of this endeavor.
F. G. R.
vi
D E D I C A T O N
To my Parents,
Mr. & Mrs Felipe and
Norma Ragma
and
To my siblings,
Darwin, Felinor and
Nailyn
This humble work is
dedicated to all of you!
F.G.R.
vii
ABSTRACT
TITLE : ERROR ANALYSIS IN COLLEGE ALGEBRA IN
THE HIGHER EDUCATION INSTITUTIONS OF LA UNION
Total Number of Pages: 374
Text Number of Pages : 358
AUTHOR : FELJONE G. RAGMA
ADVISER : NORA ARELLANO-OREDINA, Ed.D.
TYPE OF DOCUMENT : DISSERTATION
TYPE OF PUBLICATION: Unpublished
ACCREDITING INSTITUTION: SAINT LOUIS COLLEGE
City of San Fernando, La Union CHED, Region I
KEY WORDS : Error Analysis, Math Performance, Error Categori- zation, Educational Management, Instructional
Intervention Plan, Mathematics Teaching Interven- tions, etc.
Synopsis
The descriptive study identified and analyzed the error categories
of students in College Algebra in the Higher Education Institutions of
La Union as basis for formulating a validated Instructional Intervention
Plan. Specifically, it determined the a) level of performance of the
students in College Algebra along elementary topics in sets and Venn
diagrams, real numbers, algebraic expressions, and polynomials; special
product patterns; factoring patterns; rational expression; linear
viii
equations in one unknown; systems of linear equations in two
unknowns; and exponents and radicals; b) the capabilities and
constraints of the students in College Algebra; and, c) the error
categories of the students along reading, comprehension, mathematising,
processing and encoding. Data were collected using a researcher-made,
all-word-problem test. The participants were 374 first year students
enrolled in College Algebra for first semester, school year 2013-2014. The
data gathered were treated statistically using frequency count, mean,
percentage and the Newmann’s tool for error analysis. It found out that
the students had fair performance in elementary topics, special products
and factoring while poor performance in rational expressions, linear
equations and systems of linear equations and very poor performance in
exponents and radicals; thus, the students, in general, had poor
performance. The performances of the student in the specified topics
were all considered as constraints. Mathematising and comprehension
were the major error categories of the students in elementary topics,
processing and reading errors in special products, reading and
Mathematising in factoring, reading and Mathematising in rational
expressions, reading and comprehension in linear equations; and reading
and Mathematising in systems of linear equations and exponents and
radicals. In general, their major error categories in College Algebra were
along reading and Mathematising. Moreover, the instructional plan is
ix
found to have very high validity. Based on the findings, it was concluded
that the students cannot competently deal with elementary topics,
special product and factoring patterns rational expressions, linear
equations, systems of linear equations and radicals and exponents.
Additionally, the instructional intervention plan is a very good material
that addresses problems on performance and errors. Based on the
conclusions, it is recommended that the schools should adopt the
Instructional Intervention Plan and let their mathematics instructors
attend the two-day seminar-workshop. The students should exert more
effort in understanding the different concepts in their College Algebra
course. They should spend more time dealing with drills and exercises.
The mathematics teachers should suit their instructional strategies to
the needs of the students. The English teachers must also intensify in
their classes the basic skill of reading with comprehension. A study
should be conducted to determine the effectiveness of the instructional
intervention plan. And, a similar study should be conducted in other
branches of Mathematics, applied sciences and English.
x
TABLE OF CONTENTS
Page
TITLE PAGE………………………………………………………………… i
INDORSEMENT…………………………………………………………… ii
APPROVAL SHEET…………....................................................... iii
ACKNOWLEDGMENT…………………………………………………… iv
DEDICATION……………………………………………………………… vi
ABSTRACT………………………………………………………………… vii
TABLE OF CONTENTS………………………………………………….. x
LIST OF TABLES…………………………………………………………. xiv
LIST OF FIGURES……………………………………………………….. xvi
CHAPTER
I INTRODUCTION……………………………………………… 1
Background of the Study.……......………….......... 1
Theoretical Framework……………………………..... 8
Conceptual Framework……………………………….. 15
Statement of the Problem…………........................ 19
Assumptions……………………………………........... 21
Importance of the Study……………...................... 21
Definition of Terms…………………………………..... 23
II METHOD AND PROCEDURES…………………………… 27
Research Design……………………………………… 27
xi
Page
Sources of Data………………………………………. 28
Locale and Population of the Study……………... 28
Instrumentation and Data Collection ..……….... 29
Validity and Reliability of the Questionnaire.
Administration and Retrieval of the
Questionnaire ………………………………
30
31
Data Analysis ………………………………………….
Data Categorization……………………………….....
32
33
Parts of the Instructional Intervention Plan….……………………………………………….
36
Ethical Considerations…………………………...... 37
III RESULTS AND DISCUSSION…………………………….. 39
Level of Performance of Students in College Algebra…………………………………………….. 39
Elementary Topics………………………………
39
Special Product Patterns……………………… 41
Factoring Patterns ……………………………… 44
Rational Expressions…………………………… 46
Linear Equations in One Variable…………… 48
Systems of Linear Equations in Two Unknowns………………..………………….. 50
Exponents and Radicals……………………….
51
xii
Page
Summary on the Level of Performance of Students in College Algebra ………….
52
Capabilities and Constraints of Students in
College Algebra…………………………………..
54
Error Categories in College Algebra……………… 56
Elementary Topics……………………………… 56
Special Product Patterns……………………… 63
Factoring…………………………………………. 67
Rational Expressions…………………………… 74
Linear Equations in One Variable Systems 80 Systems of Linear Equations in Two
Unknowns……………………………………
85
Exponents and Radicals………………………. 91
Summary on the Error Categories in
College Algebra …………………………….
93
Validated Instructional Intervention Plan ………
96
Instructional Intervention Plan ……………………
Two-day Seminar-Workshop on the Utilization of the Instructional Intervention Plan………
Sample Flyer of the Two-Day Seminar/
Workshop ………………………………………..
Level of Validity of the Instructional Inter-
vention Plan ………………………………………
99
296
299
300
IV SUMMARY, CONCLUSIONS AND RECOMMEN-
DATIONS………………………………………………..
301
xiii
Page
Summary………………………………………………. 301
Findings………………………………………………… 302
Conclusions…………………………………………… 302
Recommendations…………………………………… 303
BIBLIOGRAPHY……………………………………………… 305
APPENDICES………………………………………………… 313
A Sample Computations on the:
Reliability of the College Algebra Test …
313
Validity of College Algebra Test ……….. List of Suggestions Made by the
Validators and the Correspond- ing Action/s by the Researcher …….
B Letter to Students-Respondents to Administer College Algebra Test ………..
The College Algebra Test ………………………
314
315
317
317
Math I – College Algebra Test (Table of Specifications) …………………..
C Letter to the Presidents/School Heads of
the HEIs understudy to Gather Data/Information ………………………….
324
326
D Sample of Corrected College Algebra Test…
336
CURRICULUM VITAE…………………………………….. 354
xiv
LIST OF TABLES
Table Page
1 Distribution of Respondents ………………………… 29
2 Level of Performance of Students in Elementary
Topics ………………………………………………..
40
3
4
Level of Performance of Students in Special
Product Patterns …………………………………..
Level of Performance of Students in Factoring Patterns ……………………………………………..
42
45
5
Level of Performance of Students in Rational Expressions ………………………………………..
47
6
7
Level of Performance of Students in Linear Equations in One Variable ……………………..
Level of Performance of Students in Systems of
49
Linear Equations ………………………………….
51
8 Level of Performance of Students in Exponents
and Radicals ………………………………………..
52
9 Summary Table on the Level of Performance of
Students in College Algebra …………………….
53
10 Capabilities and Constraints of Students in
College Algebra ……………………………………
55
11 Error Categories in Elementary Topics………..….. 57
12 Error Categories in Special Product Patterns……. 64
13 Error Categories in Factoring Patterns…………..... 68
14 Error Categories in Rational
Expressions ………………………………………..
75
xv
15
Error Categories in Linear Equations in One Variable…………………………………………….
Page
81
16
Error Categories in Systems of Linear Equations in Two Variables ........................
86
17 Error Categories in Exponents and Radicals…….. 92
18 Summary Table on the Error Categories in College Algebra…………………………………..
94
19 Level of Validity of the Instructional Intervention
Plan………………………………………………… 300
xvi
LIST OF FIGURES
Figure Page
1 Ragma’s Error Intervention Model…………………………… 13
2 The Research Paradigm ……………………………………….. 18
1
CHAPTER I
INTRODUCTION
Background of the Study
Education, in its general sense, is a form of learning in which
knowledge, skills, and values are imparted to a person or group of
persons through teaching, training, or research. Many countries adhere
to the principle that education is the key to a nation’s success. Some
experts even correlate the number of literate people to the nation’s
economic growth since national advancements are most commonly
achieved by people who have trainings and intellectual advancements
(www.educationworld.com).
Furthermore, the central goal of education is to help a person
develop critical thinking, reasoning and problem-solving skills. Hence,
education prepares a person for life. One subject that helps people
prepare for life is Mathematics.
Mathematics is the science that deals with the logic of shape,
quantity, reasoning and arrangement. It is concerned chiefly on how
ideas, processes and analyses are applied to create useful and
meaningful knowledge that man can use throughout his life (Prakash,
2010). It has also become one of the powerful tools of man in cultural
adaptation and survival. Recorded history narrates that mathematical
2
discoveries have been at the forefront of every civilized society and in use
even in the most primitive of cultures. The needs of mathematics arose
based on the wants of society. The more complex a society is, the more
complex is the mathematical need. Primitive tribes needed little more
than the ability to count, but also relied on mathematics to calculate the
position of the sun and the physics of hunting (Hom, 2013).
Mathematics has played a very important role in building up
modern civilization by perfecting the sciences. In this modern age of
Science and Technology, emphasis is given on sciences such as Physics,
Chemistry, Biology, Medicine and Engineering. Mathematics, which is a
Science by any criterion, is also an efficient and necessary tool being
employed by all these Sciences. As a matter of fact, all these Sciences
progress only with the aid of Mathematics. So it is aptly remarked,
"Mathematics is the science of all sciences and the art of all arts." (Wells,
2006).
Furthermore, Mathematics is the language and the queen of the
Sciences. According to the famous Philosopher Kant, "A Science is exact
only in so far as it employs Mathematics." So, all scientific education
and studies which do not commence with Mathematics is said to be
defective at its foundation (Wells, 2006). Thus, neglect of mathematics
causes injury to all knowledge.
3
It is undeniable that Mathematics expresses itself everywhere, in
almost every facet of life - in nature and in the technologies in our hands.
It is the building block of everything in our daily lives, including mobile
devices, architecture, art, money, engineering, sports and many others.
Without mathematics, man can go astray (Petti, 2009).
Mathematical literacy is a must element in providing the students
with the basic skills to live their life. It is one of the basic pillars for the
student on which his life is, and would be standing. So the base of this
pillar needs to be really strong and clear. Mathematics helps the student
in developing conceptual, computational, logical-analytical, reasoning
and problem-solving skills. One Mathematics subject that trains such
skills is College Algebra. College Algebra is a pre-requisite subject in
higher education institutions. The National Center for Academic
Transformation (2009) labels it as the gateway course for freshmen in the
tertiary level. This means that a student who aspires to be a degree
holder must pass successfully through the course. This is the main
reason why most countries, through their ministry or department of
education, have mandated the inclusion of College Algebra in the course
curriculum.
No one can negate the importance of College Algebra. Cool (2011),
enumerates some of the uses of algebra in today’s world. Algebra is used
in companies to figure out their annual budget which involves their
4
income and expenditure. Various stores use algebra to predict the
demand of a particular product and subsequently place their orders. It
also has individual applications in the form of calculation of annual
taxable income and bank interest on loans. Algebraic expressions and
equations serve as models for interpreting and making inferences about
data (Okello, 2010). Further, algebraic reasoning and symbolic notations
also serve as the basis for the design and use of computer spreadsheet
models. Therefore, mathematical reasoning developed through algebra is
necessary through life, affecting decisions people make in many areas
such as personal finance, travel, cooking and real estate, to name a few.
Thus, it can be argued that a better understanding of algebra improves
decision-making capabilities in society (The Journal of Language,
Technology & Entrepreneurship in Africa, Vol. 2, No.1, 2010).
In addition, Algebra is one of the most abstract strands in
mathematics. This very nature of the subject makes it difficult for
students to appreciate and love Algebra. With this, Prakash (2010)
remarked that the place of mathematics in education is in grave danger.
The teaching and learning of College Algebra, with insufficient skills and
high anxiety levels, degenerated into the realm of rote memorization, the
outcome of which leads to satisfactory formal ability but does not lead to
real understanding or to greater intellectual independence. A testament
to this worsening scenario is the global move for educational reforms.
5
Countries around the world are alarmed by the lowering
performance of their students, especially in College Algebra. In America
alone, educational experts are tasked to improve performance in
Mathematics (Arithmetic, Algebra, Geometry and the like) so they can
bring back the glory days of the United States in topping Surveys of
Countries along students’ academic performance (Serna, 2011).
Bressoud (2012) added that even though there are interventions, College
Algebra failure rates are disappointing. Further, in a University in Africa
of Fall 2007, College Algebra examination results showed that only 23%
of the students performed well. This poor performance calls for the
establishment of the reason why College Algebra is challenging to many
students (Kuiyan, 2007). In addition, Shepherd (2005) revealed that most
students do not excel in their Algebra course. Most of them cannot
perform indicated operations, especially when fronted with word
problems. Students find it hard to solve problems in Algebra. Some just
do not answer at all. These situations reflect poor understanding of and
performance in the course (The Journal of Language, Technology &
Entrepreneurship in Africa, Vol. 2, No.1, 2010).
Although there are many causes of student difficulties in
mathematics, the lack of support from research fields for teaching and
learning is noticeable (The Journal of Science and Mathematics
Education, 2010). Egodawatte (2009) emphasized that getting the level
6
of performance among students would not help much in Mathematics
Education; researches need to dig deeper into the reasons by
characterizing students’ errors and misconceptions. With this situation,
error analysis is very essential. Egodawatte (2009) added that using error
analysis, it would be possible for teachers to design effective instruction
or instructional intervention to avoid this dismal performance. Thus, it
can be construed that research on student errors is a way to clearly plot
out a more valid action plan that could address issues on students’
mathematics performance.
Mathematical errors are a common phenomenon in students’
learning of mathematics. Students of any age irrespective of their
performance in mathematics have experienced getting mathematics
wrong. It is natural that analyzing students’ mathematical errors is a
fundamental aspect of teaching for mathematics teachers (Hall, 2007).
The Philippines is also not exempted from this global predicament
on the dismal performance in College Algebra. Garcia (2012) mentioned
that Filipino students enrolled in College Algebra regarded the subject
as challenging and a difficult subject which contributed to their low
performance. In addition, the national survey conducted by
Drs. Lambitco, Laz and Malab (2009) on the readiness of Filipino
students in College Algebra revealed that the students are not ready to
take up College Algebra course. Further, according to Professor Ramos
7
(2012), 40-50% of the students enrolled in College Algebra failed.
According to him, this performance is caused by poor instruction and
cognitive unpreparedness. This low performance was also highlighted
when Leongson (2003) revealed that Filipino students excelled in
knowledge acquisition but fared considerably low in lessons requiring
higher-order-thinking skills.
On the provincial scene, Picar (2009) strongly presented in his
study that students’ anxiety in College Algebra is high but their
performance is low. Pamani (2006) also mentioned that more than 60%
of the college freshmen in La Union have low to fair competence. Pamani
(2006) stressed that these results point out to a problematic situation in
education. These facts are also strengthened by Bucsit (2009) when she
revealed that out of 195 college freshmen in the Private Schools in
La Union, 113 or 58% of the students have fair performance. In addition,
Oredina (2011) revealed that the performance of SLC students in College
Algebra was at the moderate level only.
Furthermore, the researcher, being a College Algebra instructor,
observes that many students still have many misconceptions along
certain topics in College Algebra, even if most of the course contents are
just a recap of high school mathematics. To note, some students omitted
the signs when performing operations. Others did not know what to do
8
when presented with a word problem while many were not able to craft
their own procedures in solving the given problems
The aforementioned situationers on College Algebra performance
prompted the researcher to conduct an error analysis in College Algebra
in the Higher Education Institutions (HEIs) of La Union as basis for
formulating an instructional intervention plan.
Theoretical Framework
M. Anne Newman’s (1977) theory of errors and error categories
maintains that when a person attempts to answer a standard, written,
mathematics question, he has to be able to pass through a number of
successive hurdles, namely Reading (or Decoding), Comprehension,
Transformation or ―Mathematising,‖ Processing, and Encoding. From
these successive stages, students commit varied errors. According to the
theory, the reading errors are committed when someone could not read a
key word or symbol in the written problem to the extent that this
prevented him/her from writing anything on his/her solution sheet or
from proceeding further along an appropriate problem-solving path; the
comprehension errors are committed when someone had been able to
read all the words in the question, but had not grasped the overall
meaning of the words; thus, he can only indicate partially what are the
given and what are unknown in the problem; the transformation or
9
mathematising errors are committed when someone had understood
what the questions wanted him/her to find out but was unable to
identify the operation, or sequence of operations or the working equation
needed to solve the problem; the processing errors are committed when
someone identified an appropriate operation, or sequence of operations
or the working equation, but did not know the procedures necessary to
carry out these operations or equation accurately; and, the encoding
errors are committed when someone correctly worked out the solution to
a problem, but could not express this solution in an acceptable written
form. In some case, if the answer is not in its accepted simplified form
and does not indicate the unit.
Researchers which made use of the abovementioned theory were
Clement (2002), Ashlock (2006), Hall (2007) and Egodawatte (2011). All
of their studies were able to find out the specific error categories of their
student-respondents.
Furthermore, Vygotsky (1915) and Kolb’s (1939) constructivist
theory proposes that a person can construct and conditionalize
knowledge, especially after learning or experiencing something. As
applied to this study, the students are believed to be capable of showing
the desired competence after learning the contents of College Algebra
from their instructors.
10
Dewey (1899) and Roger’s (1967) active learning and experiential
learning theories propose that students are able to learn something and
apply what they have learned if they are engaged with their experiences.
As applied in the study, the problems in the researcher-made test were
anchored to the real-life encounters of the college students.
Also, Bruner’s (1968) intellectual development theory discusses
that intellect is innately sequential, moving from inactive through iconic
to symbolic representation. He felt that it is highly probable that this is
also the best sequence for any subject to take. The extent to which an
individual finds it difficult to master a given subject depends largely on
the sequence in which the material is presented. Further, Bruner also
asserted that learning needs reinforcement. He explained that in order
for an individual to achieve mastery of a problem, feedback must be
reviewed as to how they are doing. The results must be learned at the
very time an individual is evaluating his/her performance. This theory
supports the idea that solving written problems are successive in nature.
This also gave the idea to the researcher on how to check the all-word
problem test.
Further, Bandura’s (1963) social learning theory holds that
knowledge acquisition is a cognitive process that takes place in social
context and can purely occur through observation or direct instruction.
11
As applied in the study, the instructional interventions are student-
centered so that learning becomes more active.
In addition, when one attempts to address concerns on student’s
errors, instructional intervention can be a good scheme. Egodawatte
(2009) stresses that error analysis can pave away to clearly conceptualize
an action plan such as designing effective instruction or plotting out
instructional intervention. This idea by Egodawatte (2009) structures the
foundation of the output of the study.
Howell (2009) describes instructional intervention as a planned set
of procedures that are aimed at teaching specific set of academic skills to
a student or group of students. An instructional intervention must have
the following components: it is planned – planning implies a decision-
making process. Decisions require information (data); therefore, an
instructional intervention is data-based or research-based set of teaching
procedures; it is sustained – this means that an intervention is likely
implemented in a series of lessons over time; it is focused– this means
that an intervention is intended to meet specific set of needs for
students; it is goal-oriented – this means that the intervention is
intended to produce a change in knowledge from some beginning or
baseline state toward some more desirable goal state; and, it is typically
a set of procedures rather than a single instructional component/
12
strategy. Moreover, according to Manitoba Education Website (2010), an
instructional intervention plan contains the purpose or the background,
intervention objectives, specific topics, the error categories, the sample of
error, the proposed instructional strategy and or activities, and the
procedures of implementing the strategy. (http://www. edu. gov.
mb.ca/k12/specedu/bip/sample.html.)
The aforecited theories find their essence in the teaching and the
learning of mathematics and in the specific categories in the research’s
aim of identifying and analyzing errors. These also gave the researcher
the main reasons of formulating the research tool composed of all word
problems. Generally, they serve as the building blocks in structuring
this research. Further, the concept of instructional intervention plan
serves as the core idea in designing the output of this study.
Furthermore, these theories served as foundations in formulating
the proposed model of the researcher, the Ragma’s Error Intervention
Model. Figure 1 illustrates the model.
The model, a corollary of Newmann’s (1977), highlights that when
someone answers a written mathematical problem, he has to undergo
different but successive stages such as reading, comprehension,
mathematising, processing and encoding stages. In simple words,
someone has to read the problem, understand what the problem says,
13
Figure 1. Ragma’s Error Intervention Model
INSTRUCTIONAL
INTERVENTION
(Game-based,
visual/spatial-based,
motivational instruction,
technology-based,
cooperative learning,
tutorials,
differentiated teaching,
understanding-centered,
processing-centered,
reading strategies,
experiments,
dyads,
observations, and
scaffolding)
CAUSES OF ERRORS
(low Interest, attitude, high anxiety, Insufficient
recall, misconception, deficient mastery,
carelessness)
Encoding
Processing
Stage
Comprehension
Stage
Reading
Mathematising
Stage
Error Categories Stages in Problem Solving
Encoding Errors
Processing Errors
Mathematising Errors
Comprehension Errors
Reading Errors
Better
Performance
in College
Algebra
Mathematics Word
Problems
14
14
structure the working equation, solve and then finalize the answer/s. In
each of these successive stages, errors can be committed. These errors
are caused by low interest, high anxiety, negative attitude, insufficient
recall, misconception, poor mastery, and carelessness. To exemplify,
when someone does not bother to answer the problem, he is not
interested in mathematics or has high anxiety towards math. If he fails to
completely analyze what the problem is all about, he cannot completely
recall the essential mathematical details. If he cannot create a working
equation, he has poor mastery and deficient mathematical skills. If he
cannot proceed to the starting point of the mathematical solution, he
cannot recall the formulas or is unable to formulate the working
equation. If he cannot correctly and completely solve the problem, he has
deficient mastery and is careless in handling mathematical algorithms.
And, if he is unable to write a valid or unaccepted final answer, he is
careless or lacks the necessary mathematical skills.
Moreover, the different error categories and their causes can be
addressed through the varied instructional interventions. To illustrate,
reading errors caused by high anxiety and disinterest can be addressed
by providing motivational instructional activities and games;
differentiated instruction can also be a good instructional scheme.
Comprehension errors caused by misconception can be addressed by
15
concept attainment and processing. Mathematising errors caused by
poor mastery and insufficient recall can be addressed by direct
instruction, memory-bank game and the think-pair-share activities, to
name a few. Processing errors caused by poor mastery and insufficient
recall can be addressed by error targeting and correcting, explicit
instruction, etc. And lastly, encoding errors caused by carelessness can
be solved by solve-and-compare, cooperative learning groups, etc. When
all the error categories in each problem-solving stage together with their
respective causes are addressed through the instructional interventions,
better performance of the students in College Algebra will be achieved.
Conceptual Framework
Answering a standard, written, mathematics question requires a
person to undergo a number of successive stages: reading,
comprehension, mathematising, processing, and encoding. From these
successive stages, students commit varied errors.
The reading errors are committed when someone could not read a
key word or symbol in the written problem to the extent that this
prevented him/her from writing anything on his/her solution sheet or
from proceeding further along an appropriate problem-solving path.
The comprehension errors are committed when someone had been
able to read all the words in the question, but had not grasped the
16
overall meaning of the words; thus, he can only indicate partially what
are the given and what are the unknown in the problem.
The transformation or mathematising errors are committed when
someone had understood what the questions wanted him/her to find out
but was unable to identify the operation, or sequence of operations or the
working equation needed to solve the problem.
The processing errors are committed when someone identified an
appropriate operation, or sequence of operations or the working
equation, but did not know the procedures necessary to carry out these
operations or equation accurately.
The encoding errors are committed when someone correctly
worked out the solution to a problem, but could not express this solution
in an acceptable written form. In some case, if the answer is not in its
accepted simplified form and does not indicate the unit. This makes
mathematics teaching challenging.
Thus, for learning to take place, all the stages and aspects of
problem analysis and problem solving must be well understood by the
students.
Moreover, when someone aspires to help students to improve on
their performance, one needs to dig deeper into the reasons behind the
dismal performance. According to Newmann (1977), the type of errors
17
committed by the students when solving word problems can give baseline
data to teachers to help them improve on their mathematical skills.
Egodawatte (2009) and Hall (2007) stressed that mathematical
errors are a common phenomenon in mathematics learning. Students of
any age have experienced getting mathematics wrong (Hall, 2007). It is
natural that analyzing students’ mathematical errors is a fundamental
aspect of teaching for mathematics teachers.
Error Analysis is then an effective assessment approach that
allows one, especially teachers, to determine whether students are
making consistent mistakes when performing computations. By
pinpointing the error category or pattern of an individual student’s
errors, one can then directly teach the correct procedure for solving the
problem or can even formulate an effectively designed instructional
intervention scheme (Egodawatte, 2009).
It is in this light that the study is thought of, formulated and set
up. This conceptualization is logically designed in the Research Paradigm
in Figure 2. The paradigm made use of the Input-Process-Output (IPO)
model. The input is composed of the performance of the students along
elementary topics, special product patterns, factoring, rational
expressions, linear equations, systems of linear equations in two
unknowns and exponents and radicals. It also incorporates the error
18
Patterns
PROCESS OUTPUT INPUT
Validated
Instructional
Intervention Plan
for College
Algebra in the
Higher
Education
Institutions of
La Union
1. Interpretation
and Analysis of the
Performance of the
students along the
specified topics
2. Identification and
Analysis of the
capabilities and
constraints based
on the level of
performance
3. Identification and
Analysis of error
categories of the
students
4. Preparation and
Validation of
Instructional
Intervention Plan
1. Performance of the students along:
a. Elementary topics a.1. sets and Venn diagrams a.2. Real numbers a.3. Algebraic expressions
a.4. Polynomials
b. Special Product
c. Factoring Patterns
d. Rational Expressions
e. Linear Equations in One Unknown
f. Systems of Linear Equations in Two Unknowns
g. Exponents and Radicals
2. Error Categories along the specified topics in College Algebra along a. reading b. comprehension c. transformation d. process e. encoding
Figure 2. The Research Paradigm
19
categories of the students along the specified topics in Math 1 or College
Algebra along reading, comprehension, mathematising, processing and
encoding. These variables are indeed necessary to determine the
performance and error categories of the students in College Algebra.
The process incorporated the interpretation and analysis of the
performance of the students in College Algebra, the identification and
analysis of the capabilities and constraints and the identification,
categorization and analysis of errors in College Algebra. It also holds the
process of conceptualizing and validating the output of the study.
The output of the study, therefore, is a validated instructional
intervention plan for the Higher Education Institutions of La Union.
Statement of the Problem
This study identified and analyzed the error categories of students
in College Algebra in the Higher Education Institutions of La Union as
basis for formulating a Validated Instructional Intervention Plan.
Specifically, it sought answers to the following questions:
1. What is the level of performance of the students in College
Algebra along:
a. Elementary Topics;
a.1. Sets and Venn Diagrams
a.2. Real Numbers
20
a.3. Algebraic Expressions
a.4. Polynomials
b. Special Products;
c. Factoring Patterns;
d. Rational Expressions;
e. Linear Equations in One Unknown;
f. Systems of Linear Equations in Two Uknowns; and
g. Exponents and Radicals?
2. What are the capabilities and constraints of the students in
College Algebra?
3. What are the error categories of the students along the topics
in College Algebra along:
a. Reading;
b. Comprehension;
c. Mathematising or Transformation;
d. Processing; and
e. Encoding?
4. Based on the findings, what validated instructional intervention
plan can be proposed?
a. What is the level of validity of the instructional intervention
plan along face and content?
21
Assumptions
The researcher was guided with the following assumptions:
1. The level of performance of the students in College Algebra is
satisfactory.
2. The capabilities are along elementary topics while the
constraints are along factoring, special products, and systems of linear
equations in two unknowns.
3. The major error categories of the students are mathematising
and processing errors.
4. A validated instructional intervention plan addresses the errors
of the students in College Algebra.
Importance of the Study
This piece of work will greatly benefit the CHED, administrators,
heads, teachers, students, the researcher and future researchers.
The Commission on Higher Education (CHED). This study will give
the commission an idea of the reasons or causes of low performance in
College Algebra, which will help in developing improvements along
curriculum and human resource.
The school administrators of the HEIs in La Union. This study
will provide them with data that can be used as input to the curricular
programs.
22
The Mathematics department heads. This study will give them
insights about the performance and errors in College Algebra, which will
help them in designing mathematics instruction that suits the identified
errors of the students.
The Mathematics instructors. This study will give them baseline
data of the performance and errors of their students in College Algebra.
The output of the study, on the other hand, will make them more
prepared in addressing the errors since instructional interventions are
proposed for their utilization.
The students of the HEIs in La Union. This study will lead them to
a thoughtful understanding of mathematics since their errors will be
known. They will also be helped in improving their performance since
the instructional interventions will address their identified errors.
The researcher, a Mathematics instructor of Saint Louis College
(SLC). This study will make him more knowledgeable of his students’
performance and errors. This will also give him the opportunity to
structure an error intervention model that addresses students’ errors
which contributes to the improvement of the fields of mathematics
teaching and learning.
The future researchers. This study will motivate them to pursue
their research since this study can be used as basis for their future
23
study. This can also give them an idea on how to structure their own
instructional plan based on their students’ needs and interests.
Definition of Terms
To better understand this research, the following items are
operationally defined:
Capabilities. These refer to a performance with a descriptive
equivalent of satisfactory performance and above.
College Algebra. This is a 3-unit requisite subject in college which
includes elementary topics, special product and factoring patterns,
rational expressions, linear equations in one unknown, systems of linear
equations in two unknowns and exponents and radicals.
Elementary topics. These topics include concepts on sets,
real number system and operations, and polynomials.
Algebraic expressions. These are expressions
containing constants, variables or combinations of constants and
variables.
Polynomials. These are algebraic expressions with
integer exponents.
Real numbers. These are the numbers composing of
rational and irrational numbers.
Sets. These are collection of distinct objects.
24
Venn diagrams. These are diagrams proposed by the
mathematician A. Venn, which are used to show relationships among
sets.
Factoring patterns. These include the topics in factoring
given a polynomial. These include common monomial factor, perfect
square trinomial, general trinomial, factoring by grouping and factoring
completely.
Linear equations in one unkown. This includes topics on
equations with one variable such as 2x- 4 = 10 and 5x - 2x = 36. The
main thrust of this topic is for an unkown variable to be solved in an
equation.
Rational expressions. These are expressions involving two
(2) algebraic expressions, whose denominator must not be equal to zero.
The topics included are simplifying and operating on rational
expressions.
Special product patterns. These topics include the patterns
in multiplying polynomials easily. These patterns include the sum and
difference of two identical terms, square of a binomial, product of two
binomials, cube of a binomial and square of a trinomial.
Systems of linear equations in two unknowns. This topic
discusses how the solution set of a given system is solved. The methods
25
that are used in this certain topics include graphical, substitution and
elimination methods.
Constraints. These refer to a performance with a descriptive
equivalent of fair performance and below.
Error analysis. It is a diagnostic procedure aimed at determining
specific inaccuracies of the students in College Algebra. The analysis is
made using the Newmann Error Analysis tool (1977).
Error categories. These are the classes of inaccuracies according
to Newmann (1977). These error categories are reading, comprehension,
transformation or ―mathematising‖, process and encoding.
Encoding errors. These are committed when someone
correctly worked out the solution to a problem, but could not express
this solution in an acceptable written form. In some case, if the answer is
not in its accepted simplified form and does not indicate the unit of
measurement.
Comprehension errors. These are committed when someone
had been able to read all the words in the question, but had not grasped
the overall meaning of the words; thus, can only indicate partially what
are the given, what are unknown in the problem
Processing errors. These are committed when someone
identified an appropriate operation, or sequence of operations or the
26
working equation, but did not know the procedures necessary to carry
out these operations or equation accurately
Transformation errors. These are committed when someone
had understood what the questions wanted him/her to find out but was
unable to identify the operation, or sequence of operations or the working
equation needed to solve the problem
Reading errors. These are committed when someone could
not read a key word or symbol in the written problem to the extent that
this prevented him/her from writing anything on his solution sheet or
from proceeding further along an appropriate problem- solving path.
Higher Education Institutions (HEIs). This refers to the twelve
(12) respondent academic colleges and universities, public or private, in
La Union offering College Algebra for the school year 2013-2014.
Instructional intervention plan. This plan contains the teaching
approaches that address dismal performance. It is composed of the
background, the general objectives, the specific topics, the error
categories and causes, the sample error, the intervention and the
assessment strategy. This serves as the output of the study.
27
CHAPTER II
METHOD AND PROCEDURES
This chapter presents the research design, sources of data, data
analysis, the parts of the instructional intervention plan and ethical
considerations.
Research Design
The descriptive method of investigation was used in the study. This
design aims at gathering data about the existing conditions. Leary (2010)
defines such design as one that includes all studies that purport to
present facts concerning the nature and status of anything. This design
is appropriate for the study since it is aimed at gathering pertinent data
to describe the performance and errors of students in College Algebra.
Further, the quantitative research approach was also used.
Hohmann (2006) defines quantitative research approach as a component
of descriptive design making use of numerical analysis. It is aimed at
analyzing input variables using quantitative techniques such as
averages, percentages, etc. This approach is apt for this study since it
makes use of quantitative techniques to show the performance and
errors of the students in College Algebra.
28
Sources of Data
Locale and Population of the Study. The population of this
study was composed of College Algebra students enrolled in the Higher
Education Institutions (HEIs) of La Union for the first semester, school
year 2013-2014.
The total population of 5,849 students was pre-surveyed in this
study; however, since the population reached 500, random sampling was
employed.
To generate the sample population, the Slovin’s formula (Leary
2010) was used.
n = 𝑁
1+𝑁(𝑒2)
where:
n = the sample population
N = the population
1 = constant
e = level of significance @ .05
Using the Slovin’s formula, a total of 374 students distributed
among the 12 respondent Higher Education Institutions of La Union
constituted the respondents of this study.
Table 1 reveals the distribution of the sample population.
29
Table 1. Distribution of Respondents
Respondent HEIs N n
Institution A 78 5 Institution B 482 31 Institution C 230 15
Institution D 900 58 Institution E 609 39 Institution F 1349 86
Institution G 65 4 Institution H 196 13
Institution I 51 3 Institution J 1536 98 Institution K 170 11
Institution L 183 12
Total 5849 374
Instrumentation and Data Collection
A pre-survey was conducted to gather the contents of the syllabus
in College Algebra in each of the HEIs. The researcher was able to meet
the math instructors, department heads/chairs and academic deans who
gave data pertinent to the scope of College Algebra. The conglomerated
topics indicated in all the syllabi served as basis in the topics specified in
the research tool. (Please see appended table of specifications)
To gather the data pertinent to the level of performance and the
error categories, a researcher-made test was made. The researcher-made
test is an all-word-problem 20-item test, 5 points per item, covering all
the topics in College Algebra. Most of the questions were based on the
word problems from College Algebra books. All problem questions were
30
aligned along the synthesis-evaluation/evaluating-creating level under
the Bloom’s Taxonomy. As such, the questions dug into the overall
conceptualization and utilization of algebraic concepts and principles to
be able to carry out such problem. Hence, an item combined several
related subtopics to ensure that the scope of the course was still covered.
The whole test was administered by the math instructors handling
the classes through the permission of the presidents or concerned
authority in the HEI. The test was good only for one hour and did not
allow the use of calculators.
Validity and Reliability of the Questionnaire. To ensure the
validity of the research tool, it was presented to the members of the panel
and to experts in the field of mathematics. The experts are professors of
mathematics. Further, the suggestions made by the validators were
incorporated in the test (see suggestions in the appendix). The computed
validity rating was 4.32, interpreted as high validity (please see
appended computation). This means that the research tool was able to
measure what it intended to measure.
Moreover, to establish its reliability, it was pilot-tested to thirty (30)
students of Saint Louis College. The thirty (30) students were not
included as respondents of the study. The internal consistency or
reliability was determined using the Kuder-Richardson 21 formula. The
formula is (Monzon-Ybanez 2002):
31
𝐾𝑅21 = 𝑘
𝑘−1 1 −
𝑥 𝑘−𝑥
𝑘𝜎2
where:
k = number of items
𝑥 = mean of the distribution
𝜎2= the variance of the distribution
Thus, the computed reliability coefficient was 0.72 (please see
appended computation). This means that the test was highly reliable,
which pinpoints that the test was internally consistent and stable.
Administration and Retrieval of the Questionnaire. With the
necessary endorsement from the Dean of the Graduate School
(Dr. Rosario C. Garcia) of Saint Louis College, City of San Fernando,
La Union, the researcher sought permission from the president or head
of the different twelve (12) respondents-institutions to float the
questionnaire. The copies of the questionnaire was handed to the
deans/program heads of the various college institutions who were also
requested to administer the said questionnaire to the respondents of
which the answered questionnaires were retrieved on a specified date as
it was scheduled by the deans/program heads of the various
respondents-institutions.
32
Tools for Data Analysis
The data gathered, collated and tabulated were subjected for
analysis and interpretation using the appropriate statistical tools. The
raw data were tallied and presented in tables for easier understanding.
For problem 1, frequency count, mean and rate were utilized to
determine the level of performance in College Algebra. The formula for
mean is as follows (Ybanez, 2002):
M = ∑x
N
Where: M – mean
x – sum of all the score of the students
N – number of students
For problem 2, the capabilities and constraints were deduced
based on the findings, particularly on the level of performance in College
Algebra. An area was considered a capability when it received a
descriptive rating of satisfactory and above; otherwise, the area was
considered a constraint.
For problem 3, the Newmann Error Analysis Tool (1977) was used
to identify the errors and error categories of the students. (Please see the
error categories in the definition of terms.) Moreover, frequency count,
average and rate were used to determine the error categories of the
students.
33
The MS Excel Worksheet and StaText were employed in treating
the data.
Data Categorization
For the scoring/checking of the test, the scheme below was used:
Point Assignment Error Category
0 Reading Error
1 Comprehension Error
2 Mathematising Error
3 Processing Error
4 Encoding Error
5 No Error
For the level of performance in each topic in College Algebra, the
following scale systems were utilized.
Elementary Topics/ Factoring
Score Range Level of Performance Descriptive Equiva-
lent Rating
16.00-20.00 Outstanding Performance (OP) Capability
12.00-15.99 Satisfactory Performance (SP) Capability
8.00 -11.99 Fair Performance (FP) Constraint
4.00-7.99 Poor Performance (PP) Constraint
0-3.99 Very Poor Performance (VPP) Constraint
34
Special Products and Patterns/Rational Expressions/Linear Equations in One Variable
Score Range Level of Performance Descriptive Equiva-
lent Rating 12.00-15.00 Outstanding Performance (OP) Capability
9.00-11.99 Satisfactory Performance (SP) Capability
6.00-8.99 Fair Performance (FP) Constraint
3.00-5.99 Poor Performance (PP) Constraint
0.00-2.99 Very Poor Performance (VPP) Constraint
Systems of Linear Equations in Two Variables
Score Range Level of Performance Descriptive Equiva-
lent Rating 8.00-10.00 Outstanding Performance (OP) Capability
6.00-7.99 Satisfactory Performance (SP) Capability
4.00-5.99 Fair Performance (FP) Constraint
2.00-3.99 Poor Performance (PP) Constraint
0-1.99 Very Poor Performance (VPP) Constraint
Exponents and Radicals
Score Range Level of Performance Descriptive Equiva-
lent Rating
4.00-5.00 Outstanding Performance (OP) Capability
3.00-3.99 Satisfactory Performance (SP) Capability
2.00-2.99 Fair Performance (FP) Constraint
35
Score Range Level of Performance Descriptive Equiva- lent Rating
1.00-1.99 Poor Performance (PP) Constraint
0-0.99 Very Poor Performance (VPP) Constraint
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
Table 3 shows the performance of the students in College Algebra
along special product patterns. It reveals that the students had a mean
score of 7.41 or 49.40%, a fair performance in special product patterns.
This means that the students could not correctly perform special
42
Table 3. Level of Performance of Students in Special Product Patterns
Subtopic Mean Score Rate Descriptive
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
Overall 7.41 49.40% Fair
product patterns implying that the students failed to master the skills
along special products. Further, it reveals that the students had fair
performance along product of two binomials. This implies that the
students could not productively use the FOIL method in getting the
product of binomials, implying that they cannot multiply and simplify
two alike or different binomials. Also, they had fair performance along
the square of a trinomial. This entails that the students cannot use the
(F + M +L)2= (F2 + M2 + L2 + 2FM + 2FL + 2ML) pattern reasonably.
Moreover, they also had fair performance along the cube of a binomial.
This indicates that the students cannot use the (F ± L)3= (F3 ± 3F2L ±
3FL2 ± L3) pattern correctly. Since the performance was within the fair
level only, it can be construed that the students had not attained to the
fullest the skills along the utilization of such patterns.
43
The findings of the study adhere to the study of Wood (2003)
emphasizing that the students performed fairly in College Algebra,
especially in special product and factoring patterns. He mentioned that
the students’ level of performance dug into a level of 39% and below.
The findings of the study also corroborate with the study of Pamani
(2006) stressing that the students had moderate competence in special
products. She mentioned that the students failed to master to the fullest
the needed skills in all the special product patterns.
Further, the study jibes with Oredina (2011) stating that the
students had moderate competence in special products. This means that
the students can handle special product patterns but had not fully
mastered the desired competencies. The students had very low
competence in squaring a binomial, low competence in monomial
multiplier, low competence in sum and difference of 2 binomials, high
competence in product of 2 different binomials but very high competence
on cube of a binomial and square of a trinomial.
Further, the study also agrees with the study of Bucsit (2009)
stating that the students had poor performance in special products. She
stated that this very dismal performance pointed out to the fact the
students could not really perform multiplication using polynomials. She
further explained that the students had not very well understood the
concepts and processes involved in special products.
44
Factoring Patterns
Table 4 illustrates the performance of the students in College
Algebra along factoring patterns. It shows that the students had a mean
score of 8.03 or 40. 15%, interpreted as a fair performance. This means
that the students could perform, to a restrained extent, factoring
patterns, pinpointing that the students failed to master, to the fullest, all
the skills along factoring.
It also shows that the students had poor performance in difference
of two perfect squares. It can be inferred that the students could not
distinguish and factor correctly polynomials of the form (x2-y2). Further,
the students had fair performance in perfect square trinomial. This
stresses that the students could not optimally recognize and factor
patterns of the form (F2 ± √2FL + L2). It also reveals that the students had
fair performance in factoring general trinomials. This means that they
were deficient along the required skills. It also reveals that the students
had poor performance in factoring by grouping. This implies that the
students failed to distinguish expressions within a polynomial that can
be grouped together for the purposes of simplification through factoring.
The study harmonizes with Gordon (2008) emphasizing that the
students had dismal performance in concepts involving algebraic
expressions, factoring and special product patterns.
45
Table 4. Level of Performance of Students in Factoring Patterns
Subtopic Mean Score Rate Descriptive Equivalent
Difference of 2 Perfect Squares (5)
1.05
21.00%
Poor
Perfect Square Trinomial (5) 2.64 52.80% Fair
General Trinomial (5) 2.67 53.40% Fair
Factoring by Grouping (5) 1.67 33.40% Poor
Overall 8.03 40.15% Fair
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
Table 5 shows the performance of the students in College Algebra
along rational expressions. It shows that the students had a mean score
of 4.73 or 31. 53%, interpreted as a poor performance in rational
expressions. This pinpoints that the students failed to correctly simplify
and perform operations involving rational expressions or expressions
involving fractions.
Further, it reflects that the students had fair performance in
simplification of RAEs. This means that the students could not simplify
competently rational expressions to their simplest form by performing
cancellation and reduction. It also mirrors that the students had poor
performance in operations of RAEs. The students could not proficiently
add, subtract, multiply and divide rational algebraic terms or
expressions.
It also shows that the students had very poor performance in
simplification of complex RAEs. This means that the students failed to
perform procedures and algorithms pertinent to the simplification of
complex fractions.
The findings of the study run parallel to the study of Laura (2005)
stressing that students’ performance in College Algebra was in crisis. He
explained that the cohort of students passing College Algebra was only
about 33.33%. He pinpointed that factoring and rational expressions
47
Table 5. Level of Performance of Students in Rational Expressions (RAEs)
Subtopic Mean
Score
Rate Descriptive
Equivalent
Simplification of RAEs (5)
2.43
48.6%
Fair
Operations of RAEs (5) 1.52 31.40% Poor
Simplification of Complex RAEs (5) 0.78 15.60% Very Poor
Overall 4.73 31.53% Poor
were the most difficult for the students.
The findings jibe with the study of Bucsit (2009) revealing that her
respondents had poor performance along rational or fractional
expressions. She stressed that the students had deficient skills as
regards performing operations and simplifying involving rational
expressions. The students were not able to deal with finding the correct
LCDs to simplify correctly the expressions.
Contrary, the findings do not relate to the study of Oredina (2011)
showing that the students had moderate competence in rational
expressions. This means that the students had not fully acquired the
needed competence along the indicated areas. It was stressed that the
students could not correctly manipulate rational expressions, simplify
such and operate using the fundamental operations.
48
Linear Equations in One Variable
Table 6 shows the performance of the students in College Algebra
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
Subtopic Mean Score Rate Descriptive
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
Overall 3.29 21.93% 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
Table 7 shows the performance of the students in College Algebra
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
Subtopic Mean Score Rate Descriptive
Equivalent
Applied Problems on fare (5)
1.28
25.60%
Poor
Applied Problems on numbers (5) 2.27 45.40% Fair
Overall 3.55 35.50% Poor
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.
Exponents and Radicals
Table 8 unveils the performance of the students in College Algebra
along exponents and radicals. It discloses that the students had a mean
score of 0.39 or 7.80%, a very poor performance. This means that the
students had not mastered the needed skills for them to deal with
exponential and radical expressions competently. They were deficient in
manipulating expressions and equations involving exponents and
radicals. They were not able to correctly treat data inside the radical
symbols and express correctly the square of certain expressions.
52
Table 8. Level of Performance of Students in Exponents and Radicals
Subtopic Mean
Score
Rate Descriptive
Equivalent
Exponential and Radicals (5)
0.39
7.80%
Very Poor
Overall 0.39 7.80% Very Poor
The findings corroborate with the study of Li (2007) stating that
students had difficulty in dealing with exponents and radicals. He
explained that the students did not master the mathematical
principles behind simplification of such concepts. This dismal
performance points out to the fact that mastery was not attained.
In addition, the findings also jibe with the study of Pamani (2009)
showing that the students had fair performance in exponential and
radical expressions and equations. It was stressed that students failed to
understand the rudiments of these algebraic concepts.
Summary on the Level of Performance of Students in College Algebra in the HEIs in La Union
Table 9 shows the summary of the level of performance of students
in College Algebra. It can be clearly gleaned from the table that generally,
the students had a mean score of 36.08 or 36.08%, interpreted as poor
performance. This implies that students did not really achieve to the
53
Table 9. Summary Table on the Level of Performance of Students in College Algebra
TOPIC Mean
Score
Rate Descriptive
Equivalent
Elementary Concepts (20) 8.69 43.45% Fair
Special Product Patterns (15) 7.41 49.40% Fair
Factoring (20) 8.03 40.15% Fair
Rational Expressions (15) 4.73 31.53% Poor
Linear Equation in One Variable (15) 3.28 21.93% Poor
Systems of Linear Equations (10) 3.55 35.50% Poor
Exponents and Radicals (5) 0.39 7.80% Very Poor
Overall 36.08 36.08% 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.
Special Product Patterns
Table 12 unveils the error categories of the students in special
product patterns. It can be seen from the table that 151.33 or 40.46%
errors were committed along processing, 78.33 or 20.94% were along
reading, 47.67 or 12.75% were along Mathematising, 36 or 9.63% were
along encoding and 16.67 or 4.46% were along comprehension. It is also
good to note that 44 or 11.76% were not considered as errors. This
means that majority of the students committed processing errors in
special product patterns. They were able to read, understand and set up
the working equation but failed in proceeding to the correct solution
path, leaving incorrect answers.
Specifically, the table shows that 201 errors in product of 2
binomials were committed along processing errors. Students incorrectly
multiplied (3x2-5) to (3y+4) and (2x2+45) to (5y+2). Others committed
errors in evaluating (3x2-5); instead of writing ―(3(10)2-5 = 295)‖, they
wrote ―900-5 = 895‖. They also failed to multiply the measure of the lot
64
Table 12. Error Categories in Special Product Patterns
Subtopic Error Categories
R C M P E N
Product of Two Binomials
64 16 15 201 26 52
Square of a Trinomial
86 18 82 146 30 12
Cube of a Binomial 85 16 46 107 52 68
Average
78.33
16.67
47.67
151.33
36
44
Rate
20.94%
4.46%
12.75%
40.46%
9.63%
11.76% Legend: R- Reading Error C- Comprehension Error M- Mathematising Error P- Processing Error E- Encoding Error N- No Error
by its respective price, leaving the solution process incomplete. This was
due to lack of critical thinking and deficient skill. Moreover, 64 errors
were along reading. The students left the item unanswered. This implies
that the students did not know what to do. This was caused by poor
mastery of content. It can also be gleaned that 16 errors were along
comprehension and 15 errors were along Mathematising. The students
failed to get the gist of the problem. The students, due to their
misunderstanding of the focus of the problem, failed to craft the working
equation or remember the formula suited to the problem.
Further, 146 of the committed errors in square of a trinomial were
along processing errors. The students failed to correctly square a
65
trinomial. Most of them answered (2x-4y+6z)2 as (4x2+16y2+36z2), worse
(4x2-8xy2+12y2) instead of 4x2+16y2+36z2+16xy+24xz-48yz. Others also
wrote 4x2+16y2+36z2–8xy +12xz -24yz. Others performed correctly the
pattern but failed to employ the rules of signs. This was caused by
deficient mastery of the subject matter.
It is also noted that 86 errors were along reading. This means that
some students left the item unanswered. The students had not
understood fully the problem or did not really know how to deal with the
problem. This was caused by poor competence. Also, 82 errors were
along Mathematising. The students failed to write the correct formula.
Instead of writing A= ∏r2, most of them wrote A= 2∏r, and A= 2∏r2. This
was misalignment of formulas. Others also were not able to write any
formula or working equation. This was caused by deficient recall. 30
errors were also committed along encoding errors. Most of them failed to
write the unit of measurement of the final answer. Others also committed
parenthetical error, a kind of encoding error. Instead of writing
(4x2+16y2+36z2–16xy +24xz -48yz)∏ cm2, they wrote 4x2+16y2+36z2–16xy
+24xz -48yz∏ cm2 . This was due to carelessness and lack of critical
thinking. Lastly, 18 errors were along comprehension. The students
failed to completely identify all the given from the data. They just listed
(2x-4y + 6z). Others even wrote (2x+4y+6z). This was due to carelessness
among students.
66
The table also shows that 107 errors in cube of a binomial were
along processing errors. The students failed to correctly cube the
binomial (2x+4). Most of them just wrote (8x3+63) or worse (8x3+12) and
(6x3+12) and (8x+64). The students failed to apply the pattern of (F+L)3 =
(F3+3F2L+3FL2+L3). This was caused by poor competence. In addition, 85
errors were along reading. The students left the items unanswered. They
did not know what to do to be able to arrive at the correct answer. This
was caused by poor mastery.
It can also be noted that 46 errors were along Mathematising errors.
The students failed to write the correct formula, V = s3. The students
wrote s2 or (s)(s). Some also wrote V= 3s3 and V= 4s. This was due to
poor retention of formulas taught to them even in the elementary. Also,
52 errors were along encoding errors. Students failed to write the final
answer with the correct unit of measurement. Others wrote cm, cm2 or
none at all. This was due to lack of criticality and carelessness among
students. Lastly, the 16 errors were committed along comprehension.
The students failed to write completely the given data. Instead of writing
(2x +4), some wrote (2x-4), (2+4), (x+4). This was due to carelessness.
The findings of the study corroborate with the study of Egodawatte
(2011) divulging that most students committed transformation and
processing errors along word problems involving algebraic expressions,
factoring and special products. He explained that the students failed to
67
remember and apply perfectly the special product and factoring patterns.
He further stressed that the students committed these kinds of errors
because the students had difficulty in carrying out several steps involved
in the mathematical process. He specifically itemized that the students
were poor in simplification, performing operations, exponential laws as
applied in factoring and product patterns, incorrect distribution and
invalid cancellation.
Also, the study of Allen (2007) harmonizes with the finding of the
study revealing that most students committed processing errors when
dealing with special products and factoring. He stressed that students
did not apply the correct rules in simplification of polynomials, algebraic
expressions, special products and factoring. He showed that many
students expanded (x+3)2 as x2+9 or worse x+6. Many of the errors were
caused by poor mastery of the mathematical principles in the said topics.
Factoring Patterns
Table 13 exposes the error categories of students in factoring
patterns. It shows that the students committed 128.25 or 34.29%
reading errors, 78 or 20.85% Mathematising errors, 60 or 16.17%
encoding errors, 39 or 10.42% processing errors and 25.58 or 6.75%
comprehension errors. It also shows that 43 or 11. 50% were not
considered errors. This implies that majority of the students failed to
68
Table 13. Error Categories in Factoring Patterns
Subtopic Error Categories
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: R- Reading Error C- Comprehension Error M- Mathematising Error P- Processing Error E- Encoding Error N- No Error
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,
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Moreover, it can also be gleaned from the table that majority of the
errors along perfect square trinomial were along encoding. The students
failed to indicate the correct unit of measurement of the answer. Instead
of writing (2x-5) m, the students wrote simply (2x-5). Others wrote the
incorrect unit such as ―m2‖ and ―cm‖. This was due to carelessness and
lack of critical thinking. In addition, 88 errors were along reading. This
means that the students left the items unanswered. The students had
not understood the meaning of the problem which led them to leave the
item unanswered. They did not know how to hurdle such applied
problem. This was due to poor performance.
Additionally, 53 errors were along Mathematising. The students did
not write the formula or the working equation of the problem. Some
incorrectly wrote the formula. Instead of writing A = s2, they wrote A = 4s.
Others had incorrect derivation of the formula for ―s‖. Instead of writing s
=√A, they wrote s = A/2. This was caused by poor recall and poor
competence. Also, 37 errors were along processing errors. They failed to
get the factored form of the PST (4x2-20x+25). They divided the
expression by 2 instead of performing factoring. Lastly, 29 errors were
along comprehension. Students failed to completely understand what the
problem is asking them. They also incorrectly copied the given data. So
instead of writing (4x2-20x+25), some wrote (4x2+20x+25) and (4x-
20x+25).
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Likewise, it is also reflected in the table that 100 errors in factoring
general quadratic trinomial were along encoding. They were able to
correctly get the answer (x+5) but failed to write the correct unit of
measurement, cm. This was due to lack of reflection among the students.
In addition, 95 errors were along reading. Students never wrote
something that leads to the solution of the problem. This implies that the
students did not know how to deal with the problem.
Further, 57 errors were along Mathematising. Students failed to
correctly write the working equation. Some did not write any formula
while the others wrote an incorrect one. The students wrote (x2+3x-40) -
(x-8) instead of (x2+3x-40)/ (x-8). This was in spite of the presence of the
word ―divide‖ in the problem. This was due to poor competence and
analytical thinking. Also, 34 errors were along processing. Students
failed to correctly factor (x2+3x-40) leaving it unfactored and unsimplified
with the denominator. Students also incorrectly cancelled x2 with x and
40 with 8 in their equation, (x2+3x-40)/ (x-8). This was invalid
cancellation. This implies that the students really did not know how to
factor trinomials of this form. This was due to poor competence and
mastery. Lastly, 12 errors were along comprehension. Students failed to
correctly write the two given data correctly. Instead of writing (x2+3x-40)
and (x-8), students wrote (x2+3x-40) and (x+8) or (x2-3x-40) and (x-8).
This was due to carelessness. Others wrote the number ―2‖ as an
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important detail in the problem solution besides the fact that it only
details the equal measurements of the string when divided into two;
writing a given as (x2+3x-40)/2.
Additionally, the table also shows that 148 errors in factoring by
grouping were along reading errors. The students did not understand
what the problem is asking them to do. Others really did not know the
answer. Students even ignored a problem when prompted with series of
algebraic expressions such as x2+2xy+y2+x+y. Pamani (2006) stressed
that students with high anxiety and poor mathematical performance
often ignore expressions which were lengthy and contain complex
expressions and exponents. The errors were caused by high anxiety and
poor exposure to such kind of problem.
Also, 91 errors were along Mathematising. Students were not able
to write any working equation to solve the problem. Others performed
subtraction instead of division despite the implication of ―2 equal shares‖
in the problem; the working equations used were ―x2+2xy+y2+x+y – x+ y‖
and‖ x2+2xy+y2+x+y-xy‖. This was caused by poor understanding and
mastery.
Likewise, 68 errors were along processing. Students failed to factor
correctly and completely x2+2xy+y2+x+y. Others invalidly cancelled ―x+y‖
in (x2+2xy+y2+x+y) with (x+y), resulting in an incorrect answer x2+2xy+y2.
This was a clear reflection of misuse of cancellation rules. Others also
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wrote the correct common factor (x+y) but failed to correctly factor the
remaining expressions. This was caused by poor mastery of factoring by
grouping.
Further, 35 errors were along encoding. Students were able to
correctly factor the given expressions but failed to write the correct unit
of measurement. Lastly, 14 errors were along comprehension. Students
did not completely and accurately analyze what the problem wanted
them to do. Students incompletely wrote the given while the others wrote
additional unnecessary data such as ―2‖ resulting in a data
(x2+2xy+y2+x+y)/2.
The findings of the study corroborate with the study of Egodawatte
(2011) divulging that most students committed transformation and
processing errors along word problems involving algebraic expressions,
factoring and special products. He explained that the students failed to
remember and apply perfectly the special product and factoring patterns.
He mentioned that students generated incorrect factored forms of x2+x,
which were x(x+x) and worse, x(1). He stated that the students
―oversimplified‖ the answer. They lacked critical analysis as to when and
how to end the factoring process correctly. He also explained that
―overdoing‖ existed as he pointed out to incorrect cancellation of
expressions.
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This also agrees with the study of McIntyre (2005) revealing that
his respondents had misconceptions in writing final answers in algebraic
expressions and factoring patterns. The answer ―x+y‖ was still reduced to
xy. He explained that in factoring patterns and algebraic expressions,
students never leave an answer with an addition symbol present; the two
variables must be physically conjoined. According to him, students felt
that x+y can still be combined through the indicated operation. This
error, according to him, was caused by misassociation of arithmetic
principles; ―7+3= 10‖ is misassociated to ―x+y = xy‖.
Rational Expressions
Table 14 shows the error categories of students in rational
expressions. It can be gleaned from the table that 165.33 or 44.21% of
the errors were along reading, 90 or 24.06% were along Mathematising,
41 or 10.96% were along processing, 22 or 5.88% were along encoding
and 19.67 or 5.26% were along comprehension. It is also worthy to note
that 36 or 9.63% were not considered errors. This means that majority of
the students committed reading errors in simplifying rational algebraic
expressions. This implies that the students left the item unanswered.
They had not understood clearly and comprehensively the problem that
hindered them to write even a single data from the problem. This was
caused by the lack of exposure to such kinds of problems. According to
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Table 14. Error Categories in Rational Expressions
Subtopic Error Categories
R C M P E N
Simplification of RAEs
105 18 51 84 46 70
Operations of RAEs 138 17 161 20 17 21
Simplification of
Complex RAEs 253 24 58 19 3 17
Average
165.33
19.67
90
41
22
36
Rate
44.21%
5.26%
24.06%
10.96%
5.88%
9.63% Legend: R- Reading Error C- Comprehension Error M- Mathematising Error P- Processing Error E- Encoding Error N- No Error
some reactions of professors after retrieving the questionnaires, the
students failed to recognize the operations or the mechanical procedures
when expressions were converted to word problems. Blakelock (2013)
agrees with this observation of the professors when she mentioned that
when students just learned direct operation, direct cancellation or
simplification in the class, students would be hard up dealing with such
kind of expressions when written in word problems.
It can also be seen from the table that 105 errors in simplification
of rational algebraic expressions (RAEs) were along reading. This means
that most of the students left the item unanswered. This implies that the
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students failed to write any data given by the problem. This means that
they had not understood what the problem is all about. It also means
that the students were not interested to solve problems involving
fractions or fractional expressions. Hall (2007) emphasized that most
students had difficulty dealing with exponents, fractions and radicals.
Most students, who find difficulty with these, often abandon solving such
problems. Further, 84 errors were along processing errors. This implies
that the students failed to correctly solve the given problems. Most of
them performed incorrect cancellation in (12x4y6/7xy) and (21/6x3y5).
Others placed the incorrect exponents in the denominator instead of in
the numerator such as (2/3xy). Others incorrectly placed the cancelled
form of 21/7 as 1/3 instead of 3/1 or 3.
Additionally, 51 errors were along Mathematising errors. Students
failed to write down the correct working equation of the problem. Others
wrote the incorrect working equation such as (12x4y6/7xy) ÷ (21/6x3y5)
or (12x4y6/7xy) - (21/6x3y5) or (12x4y6/7xy) = (21/6x3y5). This is due to
poor analytical skills. Further, 46 errors were along encoding. This
means that the answers were not written in a correct form. Others did
not write the unit of measurement. Others did not simplify 6/1 pesos.
Lastly, 18 errors were along comprehension errors. Students failed to
fully understand the given in the problems. Others wrote only partial
given such as (12x4y6/7xy) alone or (21/6x3y5) alone. Others wrote
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(12x4y6/7xy) and (21/6x3y5) but without their corresponding units. This
was due to carelessness.
The table also reflects that 161 errors in operation of RAEs were
along Mathematising. The students failed to correctly write the working
equation of the problem. It is surprising that even if the students came
from different schools with different instructors, the students commonly
wrote the equation (1/2x)(8x/2) instead of (5/2x)(80x/2). This means
that the students failed to transform verbal expressions to numerical
expressions correctly. This was due to poor mathematical skills.
Also, 138 errors were along reading. The students failed to write
any data from the given problem. This means that the students failed to
understand the given problem which impeded them to deal with the
problem. Further, the 20 errors were committed along processing. The
students failed to correctly perform the mechanical procedures in solving
the given problem. Others placed (5/2x) ÷ (80/2x) instead of (5/2x) x
(80x/2). Others evaluated the value of 5 in (1/2x) and 10 in (8x/2). This
was due to carelessness and poor performance. Additionally, 17 errors
were committed along comprehension and encoding errors. This means
that they incompletely wrote the data, excluding 5 and 10 pesos as vital
in the solution of the problem. This also implied that the students left the
final answer without the correct unit. This was due to carelessness.
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The table also exposes that 235 errors in simplification of complex
RAEs were along reading. The students left the items unanswered. This
means that they were not interested in solving the problems especially so
that the problem involves fractional expressions. They also forgot how to
deal with interest problems involving fractional items. This is due to their
low performance. Also, 58 errors were along Mathematising. This means
that the students failed to write the formula for interest, I = PRT. Others
did not write the formula and just multiplied the given. Others wrote the
formula I = 1 + PRT and I = PR. This agrees with the number of errors
along reading. Additionally, 24 errors were along comprehension. The
students failed to correctly indicate all the data in the problem. They had
not written correctly (1- 1/3) and wrote only 1/3 instead. Most of them
did not indicate a representation for time, which should had been ―x
years‖. This is due to insufficient critical analysis. Also, 19 errors were
along processing. The students failed to correctly compute the answer to
the given problem. Others incorrectly substituted the given to the
formula such as P = I/RT as (1/6 x 12,000) =P/[(1- 1t/3)]. This was due
to deficient mastery. Lastly, only 3 errors were along encoding. The 3
errors failed to write the expression ―t‖ in the final answer. The students
felt that an answer with a variable was still not the accepted final
answer. This was due to an incorrect thinking of oversimplification.
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The finding of the study corroborates with Egodawatte (2011)
divulging that most students committed transformation and processing
errors along word problems involving algebraic, polynomial and rational
expressions. He explained that these errors were committed since the
problems were too symbolic and the most challenging part for students
was to find the correct method of solution or algorithm and making use
of the algorithm to produce a correct answer. He further stressed that
students had to choose the correct method from a wide range of possible
strategies which include but were not limited to determining common
denominators, common factors for cancellation, expansions using the
patterns, building up expressions, simplifications and comparisons.
Many of the incomplete answers of his students that were observed bear
evidence that the students could not select and apply the correct
strategy. He also explained that most students committed ―exhaustion
errors‖ when dealing with rational expressions and simplifying answers
in algebraic equations. Exhaustion errors are errors which were not
made at the beginning of the problem where an opportunity for its
commission existed. This type of Mathematising error may had existed
due to the incomplete concept recall of the students. This error can also
be attributed to the misleading background of the students pertaining to
the subject at hand. This error can also be caused by misapplication of
the algorithm learned.
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It also agrees with Hall (2007) when he said that deletion and
cancellation errors were prevalent among the respondents of the study as
regards working on arithmetical equations and expressions, algebraic
and rational equations and fractional expressions. He explained that
―overgeneralizing‖ was the main cause of this type of error. He also added
that when students solve equations, they commit transposing errors
such as forgetting the change in signs of quantities.
Linear Equations in One Variable
Table 15 reflects the error categories of the students on linear
equations in one variable. It shows that 172.33 or 46.07% errors in
linear equations in one variable were along reading, 99.33 or 26.56%
were along comprehension, 55 or 15.71% were along Mathematising,
14 or 3.74% were along encoding and 10.67 or 2.85% were along
processing. It can also be seen that 22.67 or 6.06% had completely
answered the items with correct final answer. This implies that majority
of the students committed reading errors. This means that the students
failed to write any given data from the table; they left the item
unanswered in linear equations. This was caused by poor mastery of the
subject matter.
It also shows that 186 errors in distance problem were along
reading errors. This means that the students left the item unanswered.
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Table 15. Error Categories in Linear Equations in One Variable
Subtopic Error Categories
R C M P E N
Distance Problem
186
89
53
6
17
23
Money Problem 175 100 46 22 4 27
Age Problem 156 109 66 4 21 18
Average 172.33 99.33 55 10.67 14 22.67
Rate 46.07% 26.56% 14.71% 2.85% 3.74% 6.06%
Legend: R- Reading Error C- Comprehension Error M- Mathematising Error P- Processing Error E- Encoding Error N- No Error
They failed to write even a single data from the problem. This was due to
their lack of interest towards the problem. Blakelock (2013) asserts that
students’ interest in math is high when they were still toddlers, but when
they get older, this interest lowers down due to their experiences. This is
the reason why most college students do not bother solving problems,
especially so when such do not relate to their future profession. Further,
89 errors were along comprehension. This means that the students failed
to fully understand the thrust of the problem. Most of them incompletely
wrote the given data. Most of them did not present the data in a more
comprehensible format, such as using a table. This was caused by poor
skills. In addition, 53 errors were along Mathematising. This means that
the students were able to present the data but failed to write the correct
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formula, D = RT. Others wrote Vf2= Vo
2 + 2 fusing Physics and College
Algebra. Others wrote 440-220= 220 as their working equation. This was
due to poor recall and deficient mastery of the subject matter. Also,
17 errors were along encoding. The students failed to write the correct
unit of the final answer. They wrote 240 and 200 as their final answers.
This was due to insufficient criticality and carelessness. Lastly, 6 errors
were along processing. These were committed because of the
incompleteness of the answers. The students failed to substitute the
value of x, which was 2, to the data presentation for the covered
distance. This was due to lack of criticality among the students.
Moreover, 175 errors in money problem were committed along
reading. This implies that most students left the item unanswered. This
means that the students did not know what to do. This also means that
the students failed to fully understand what the problem is talking
about.
Also, 100 errors were along comprehension errors. They failed to
fully understand the problem; only partially indicating what the problem
is giving. They also failed to understand that the problem data need to be
presented in a more organized way, such as using a table or column. In
addition, 46 errors were along mathematising. This implies that the
students failed to correctly write the formula or working equation needed
to solve the problem correctly, D x N = A or denomination multiplied to
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the number of bills is equal to the OVERALL amount. Others added 1
and 20 and 1 and 50; then applied guess and check method for the two
numbers. This means that the students cannot transfer their ideas into
mathematical expressions. This is due to deficient mastery.
Likewise, 22 errors were along processing errors. Students
committed errors in multiplying 50 (27-x). Instead of writing 1350 – 50x,
others wrote 1350 – x or 135 – 50x or worse, 135 –x. This was due to
poor mastery and carelessness. Lastly, four errors were along encoding
errors. These 4 errors were along writing the correct unit. Instead of
writing km, they wrote kph; others left the answer with no unit. This was
due to carelessness and lack of reflective ability to verify if the final
answer is in its accepted form.
It can also be seen that 156 errors in age problem were along
reading. This means that majority of the students left the item
unanswered. This implies that they did not know what to do to be able to
get the correct answer needed by the problem. This is saddening since
high school mathematics had taught them topics on applied problems in
linear equations which started in first year, reinforced in the second year,
enforced in their 4th year and repeated in their tertiary year. This was
due to deficient skills in algebraic expressions and applied problems.
Also, 109 errors were along comprehension. The students failed to
completely present the data into tables. Others wrote in tables but failed
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to represent the two time zones involved in the problem, the present and
the future. This was due to lack of criticality.
Also, 66 errors were along Mathematising errors. The students were
unable to write the correct equation x+20+10 = 2(x+10). Others wrote 2x
= 30 +10 and 30 + x = 2x. Others stopped when the data were already
presented in correct tables. Others tried to solve using trial and error
method by trying 2 numbers that fit the given categories. This was
caused by poor mathematical skills. Lastly, 4 errors were committed
along processing errors. The errors were along multiplication of
constants and variables and transposition. Others wrote x+2x = 30+10
instead of x-2x=30-20. Others wrote x+20+10 = 2x +10 instead of x +
20+10 = 2x (20). This was incomplete distribution. This was due to
deficient skills in handling algebraic expressions.
The findings of the study run parallel to Clement (2002) divulging
that most students’ errors on linear equations fall along transformation.
He stressed that his respondents had difficulty in translating words to
algebraic equations. He also expressed that analytical thinking falls short
among his students which led them to an incorrect process.
The findings of the study also run parallel to the study of
Egodawatte (2011) revealing that in linear equations and systems of
linear equations, most of the students got the correct answer; however,
some committed transformation and processing errors. Students failed to
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produce a correctly transformed equation. Students failed to form correct
equations. The others failed to use correctly the methods of substitution,
elimination and the working backward methods. He explained that
students were unable to carry out these methods due to insufficient
skills on the procedures. Students failed to use the standard
mathematical practices. He also added that the number one problem of
his students is on variables. The students misinterpreted the product of
two variables. The students were not able to apply the laws of exponents.
He explained that the students misjudged the magnitudes of the
variables; he pointed out that the students lack the understanding of
variables.
Also, the findings agree with the study of Allen (2007) stressing
that students had trouble solving such items. He stressed that students
need to be skilled on fundamental principles pertaining to equalities. It
can be deduced that insufficient background causes the predicament.
Systems of Linear Equations in Two Variables
Table 16 presents the errors of the students along linear equations
in two variables. It shows that 119.5 or 31.95% errors in systems of
linear equations in two variables were along reading, 95 or 25. 40% were
along Mathematising, 72.5 or 19.39% were along comprehension, 14.5 or
3.88% were along encoding and 10 or 2.67% were along processing. It is
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Table 16. Error Categories in Systems of Linear Equations in Two Variables
Subtopic Error Categories
R C M P E N
Problems on
fare/price
160 89 71 5 13 36
Problems on number relation
79 56 119 15 16 89
Average
119.5
72.5
95
10
14.5
62.5
Rate
31.95%
19.39%
25.40%
2.67%
3.88%
16.71% Legend: R- Reading Error C- Comprehension Error M- Mathematising Error P- Processing Error E- Encoding Error N- No Error
also good to note that 62.5 or 16.71% were not considered errors.
Further, the errors imply that majority of the students committed
reading errors. The students failed to fully understand the problem
thereby leaving the item undealt. This further means that the students
do not had the know-how in dealing with the given problems. This was
due to poor mastery of the expected competencies. This is really
saddening since this is not their first time to encounter such systems of
linear equations. They were able to deal with these even during their
secondary school days.
The table also points out that 160 errors in applied fare/price
problems were along reading. This means that majority of the students
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left the items unanswered. The students failed to write even a single data
deduced from the given problem. This is caused by insufficient exposure
to such problem. It is true that when teachers facilitate the topic on
systems of linear equations, majority of them focused on the methods of
solving the value of x. Since this is one of the last topics offered in the
course syllabus, most teachers fail to teach how such systems were
transformed to applied problems due to lack of time.
Also, 89 errors were along comprehension. This means that the
students failed to completely and correctly understand the problem. They
failed to completely present the data into a more fathomable way, using a
tabular format. Also, they failed to correctly represent values for x and y.
This is due to lack of organization and criticality. Further, 71 errors were
along Mathematising. The students failed to correctly write the needed
working equations 8x + 10y = 200 and 3x + 10y = 150. Others simply
guessed and checked for 2 numbers that can satisfy the given
conditions. This clearly pointed out to the fact that the students cannot
transfer their ideas into mathematical expressions. Others did not write
any working equation. This is due to poor mastery of the subject matter.
Moreover, 13 errors were along encoding errors. The students
forgot to write the answers in an unacceptable written form. Most of
them failed to indicate the unit. The others were able to get the values for
x and y, but failed to pinpoint which among the two values answer the
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question of the problem. Lastly, 5 errors were along processing. Students
failed to correctly apply substitution in the solution of the problem.
Instead of writing (3(200-10y)/8) + 10y = 150, they wrote ((200-10y)/8) +
10y = 150. Others did not proceed with their solutions when they
finished writing their working equation. This means that the students did
not know how to deal with the formulated system of linear equations.
This was due to carelessness and poor mathematical abilities.
It is also reflected in the table that 119 errors in number relation
were along Mathematising errors. The students were unable to correctly
write the formula or the working equation. Others wrote ―x +x = 100 and
x-x = 20‖ as their working equations. Most of the students applied trial
and error in solving the correct 2 numbers. This means that the students
were unable to correctly transform their ideas into mathematical
expressions. They were able to guess and check their answers to the
problem but find it hard to create a working solution to be able to get
their ―theorized‖ answers. This is not surprising since Ashlock (2006)
revealed the same finding in his study that students can jump into the
answers without any working solution. They had their solution in their
head but cannot write their solutions. Most instructors, even the
researcher, often meet students who can give the answers right away but
when asked of their solutions, fail to present any.
89
Further, 79 errors were along reading. This means that the
students left the item unanswered. This implies that the students did not
know what to do. Also, 56 errors were along comprehension. The
students failed to fully understand the given problem. They failed to
indicate representation of the given problem. This was due to lack of
critical and analytical ability. Further, 16 errors were along encoding.
The students were able to get the values for x and y but failed to indicate
a final sentence to be able to correctly answer the thrust of the
problem.
Lastly, 15 errors were along processing errors. Students failed to
solve the problem using a correct solution path. The students failed to
substitute correctly the derived equation to the other equation such as
y=100-x to x-y = 20. Others committed transposition errors in
transposing y in x +y = 100. This is due to carelessness and low mastery
of the subject matter. It is also good to note that 89 answered correctly
the given problem. This means that some students correctly and
completely answered the given problem. This contributed much on their
level of performance, a satisfactory performance.
The findings of the study agree with the study of Clement (2002)
stressing that most of his students committed transformation and
processing errors on systems of equations. He explained that these errors
were caused by insufficiency of skill or knowledge pertaining to how
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certain variables were handled or how certain equation algorithms were
processed. He showed, too, that generally, students got the correct
answers but failed to simplify the answers in problems that need
simplification of answers. Others forgot to correctly indicate the unit for
the answers to be accepted. These errors were due to carelessness. He
explained that students forget to analyze their final answers. They did
not verify their answers by some accepted means.
Also, it agrees with Ashlock’s study (2006) divulging that students
can even produce the correct answer even if the solution is incorrect.
This situation abounds in problems involving numbers and number
relations. With this situation at hand, teachers do not only need to
correct the final answer but the process on how the answer is derived. He
stated further that students commit what he calls as ―overgeneralizing‖.
Students ―overgeneralize‖ data by jumping into the conclusions without
adequate data at hand. This overgeneralizing error leads them to
incorrect approach and answer. This error abounds in vast areas of
mathematics especially on number problems, arithmetic and
simplification problems. With this, he remarked that the students lacked
the needed computational fluency.
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Exponents and Radicals
Table 17 presents the error categories in College Algebra,
specifically along exponential and radical expressions. It shows that 297
errors were along reading. This means that the students did not
understand the thrust of the problem. They really did not know what to
do to be able to answer the problem. They left the item unanswered. This
is so since the students had not touched this last topic of the course
syllabus. Many schools had festivities on their foundation, intramurals,
founder’s day and the like which limited the number of contact days for
discussion. This means that the students failed to write the working
equation. Others incorrectly wrote the working equation such as
√(2x+7) + 3x = 90 despite the fact that the problem indicated the word
―angle bisector‖ and ―equal parts‖. Others combined trigonometric
functions in the formula, including Sin x and Tan x. So, their working
equation is A = ½ (√(2x+7) – Sin 3x)r2. This clearly pointed out that the
students mixed up their concepts on College Algebra and Plane and
Spherical Trigonometry. This is misassociation of concepts of two
branches of College Mathematics. Further, they also wrote
90 = (√(2x+7)+3x, which pinpoints that the students jumped into the
incorrect conclusion that the angle is a right angle even if there is no
indication in the problem that the angle measures 90 degrees. Moreover,
29 errors were along comprehension errors. The students failed to
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Table 17. Error Categories in Exponents and Radicals
Subtopic Error Categories R C M P E N
Exponential and
Radical expressions or equations
297 29 35 8 1 4
Average 297 29 35 8 1 4
Rate 79.41%
7.75%
9.36%
2.14%
0.27%
1.07%
Legend: R- Reading Error C- Comprehension Error M- Mathematising Error P- Processing Error E- Encoding Error N- No Error
indicate all the given data in the problem. Others wrote 2x- 7 only.
Others wrote √(2x+7)/2and 3x/2 infusing the number 2 in the problem,
a clear sign of misinterpretation. Also, 8 errors were along processing.
The students just deleted the radical symbol in √(2x+7) = 3x without
performing the squaring process. Others squared 3 but not x, resulting
in the expression 9x instead of 9x2. Others chose the value -7/9˚ over 1˚
as the correct answer. This was due to carelessness and no criticality.
Lastly, only one error is committed along encoding error. One did not
indicate the unit for degrees ( ˚ ) in the final answer. This was due to
carelessness.
The findings agree with the study of Boon (2003) stressing that the
high occurrence of errors in exponents and radicals is due to over-
generalization. This over-generalization was due to carelessness and
insufficient practice. It also appeared that such error existed due to
93
misconceptions that students had actively construed when they use their
existing schema to interpret new ideas. He also explained that this error
may be brought by deficient mastery of concepts, rules and pre-requisite
skills which can be overcome by practice. He specifically stressed that
most students misconnect the rule on √4 = 2 to be true to √16 =8 or
worse, √6 =3. This was well explained by Allen (2007) when he
enumerated some of the errors of the respondents of the study on error
analysis in radical expressions and equations. He pointed out that
students had incorrect interpretation and representation of radicals,
especially on square roots. Students tend to divide the numbers when
getting the square of 16. So, instead of 4, the students wrote 8. This was
due to misalignment of rules. They applied the rule in √4 = 2 as true to
all numbers being extracted.
Summary on the Error Categories in College Algebra
Table 18 shows the summary on the error categories in College
Algebra. It reveals that 146.96 or 39.29% of errors in College Algebra
were along reading, 69.38 or 18.55% were along Mathematising, 47.42 or
12.68% were along comprehension, 45.86 or 12.26% were along
processing, and 30.29 or 8.10% were along encoding. This means that
majority of the students committed reading errors. This means that most
94
Table 18. Summary Table of Error Categories in College Algebra
Subtopic Error Categories
R C M P E N
Elementary Concepts 68 69.50 85 61 64 26.50
Special Product
Patterns
78.33 16.67 47.67 151.33 36 44
Factoring
128.25 25.25 78 39 60.5 43
Rational Expressions
165.33 19.67 90 41 22 36
Linear Equation in One Variable
172.33 99.33 55 10.67 14 22.67
Systems of Linear Equation
119.5 72.5 95 10 14.5 62.5
Exponents and
Radicals
297 29 35 8 1 4
Average
146.96
47.42
69.38
45.86
30.29
34.10
Rate 39.29% 12.68% 18.55% 12.26% 8.10% 9.12%
Rank 1 3 2 4 6 5
Legend: R- Reading Error C- Comprehension Error M- Mathematising Error P- Processing Error E- Encoding Error N- No Error
of the students failed to critically understand the problem which led
them to leave the items unanswered. They never attempted to answer the
given items. This is due to their fair performance. It can also be deduced
and construed that students hardly can formulate the working equation
or remember the formula to solve the given problem.
95
It is also reflected that only 9.12% completed and correctly solved
the given problems in College Algebra. This means that majority of the
students really committed errors on the different categories.
The findings run parallel to Hall (2007) divulging the following
errors of his respondents in College Algebra: Computational Constraint.
Many students while they understand mathematical concepts are
inconsistent at computing mainly because they misread signs or carry
out numbers incorrectly or may not write numerals in the correct
column; Difficulty in transferring knowledge. Many students experience
difficulty in mathematics because of their inability to connect abstract or
conceptual aspects of mathematics with reality. Understanding what
mathematical symbols represent in the physical world proves to be
difficult to most students and this makes it common to find that some
students cannot visualize an equilateral triangle; Making Connections.
Some students cannot comprehend the relationship between numbers
and the quantities they represent and this makes mathematical skills not
to be anchored in any meaningful manner, making it harder for them to
recall and apply mathematical knowledge in new situations; incomplete
understanding of the language of mathematics.
Further, for some students, mathematical disability is as a result
of problems with the language of mathematics. Such students had
difficulty with reading, writing and speaking mathematical terminologies
96
which normally were not used outside the mathematics lesson. They
were unable to understand written or verbal mathematical explanations
or questions and cannot relate mathematical knowledge to physical
world; Difficulty in comprehending the visual and spatial aspects and
perceptual difficulties. Many students had the inability to visualize the
mathematical concepts. This makes students to memorize mathematical
formulae and facts - the difficulty in applying such knowledge in solving
unfamiliar mathematical problem.
Validated Instructional Intervention Plan in College Algebra
Rationale
Mathematics has always been regarded as a very essential element
in education for it does not only provide higher training for the human
mind but it is life, itself. Everyone, whether consciously or not, uses
mathematics in his daily life.
College Algebra, one branch of Mathematics, deals with elementary
topics, special products and factoring, rational expressions, linear
equation in one unknown, systems of equations in two unknowns and
exponents and radicals. It provides avenues for students to recall
important concepts learned in the secondary school. It also provides a
good foundation of readiness for students to hurdle the demands of
97
higher mathematics such as Trigonometery, Advanced Algebra, Geometry
and the like.
The noted dismal performance in this subject is caused by different
factors such as negative attitude, misconceptions, misapplications,
misalignmnet of rules, lack of criticality among others. With these
presents, it is apt to look into the reasons behind these. One good
mechanism that can address such dismal performance is an
instructional intervention plan.
The validated instructional intervention plan is based upon the
identified students’ level of performance, their capabilities and
constraints and the different error categories in College Algebra. All the
error categories are addressed in the plan since all of them were
considered constraints; but, more emphasis is given to a course content
with a very poor to poor performance level - these were the areas on
rational expressions, linear equations, systems of linear equations and
radicals and exponents. The instructional plan also gives emphasis on
addressing two (2) foremost error categories, reading and
mathematising. Further, there are some instructional interventions that
address two error categories.
Further, the instructional intervention plan details the specific
objectives; topics; level of performance, error categories and theoized
causes (arranged according to degree of error); error samples;
98
interventions, process, activities; and assessment strategy. The contents
serve as a comprehensive guide for teachers to improve performance and
check on errors.
General Objectives
The instructional intervention plan is formulated to:
1. Improve performance in all the topics in College Algebra; and
2. Address the different errors of students in solving problems in
College Algebra.
Matrix of the Instructional Intervention Plan
The Instructional Intervention Plan in College Algebra is detailed in
matrix form in the succeeding pages (see pages 99 – 295).
99
Instructional Intervention Plan in College Algebra
Specific Objectives
Topics Level of Perform-
ance
Error Cate-gories and
Theorized Causes
(arranged
according to degree of
error)
Sample Error Interventions, Process, Activities
Assessment Strategy
To use
Venn Diagrams in representing sets and set relationships
To utilize
Venn Diagrams in solving
applied problems.
To present complete
and accurate solutions
involving
A. Elemen-
tary Topics
Sets and the Venn Diagrams
Poor Mathema-
tising (dismal performance, insufficient recall)
Incorrect
working diagram- using tables
as solution diagram.
―250-160-150-180 = x‖
as the working equation.
Visual-Spatial
Processing This is a skill-based intervention that
emphasizes the skill on visualizing the given problem. It makes use
of diagrams and illustrations to show to
students how a certain problem is translated into an illustration or
diagram for easier understanding. It uses
direct instruction and the instructor models how problems are
illustrated. After, the students are given handouts and sample
The instructor
can check students’ learning during
the solving process of the students. The
learning is further
assessed when the students explain their
answers on the board.
100
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
sets and
Venn Diagrams
problems for them to
demonstrate how such problems are
illustrated and solved. Procedures: 1. The instructor
presents a given problem and uses the Venn Diagram to
illustrate. He has to emphasize why the
Venn Diagram is the correct strategy to be used.
2. After the instructor models, he gives each
student a handout that contains an empty Venn Diagram where
students can write their answers.
101
Specific
Objectives
Topics Level of Perform-
ance
Error Catego-
ries and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
3. The students are
still guided by the instructor as he roams
around the classroom. The instructor checks students’ answers.
4. Students who are done with their answers and have
presented their correct answers can be
assigned to students who need assistance. 5. Presentation of
problems will be done after. Priority of
presentation is given to those who are assisted to check
102
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Think-Pair-Share-
Explain Activity This is an interesting
cooperative-learning-based activity for students who struggle
to come up with correct answers. Procedures:
1. After the instructor finishes discussing the
lesson, the instructor pairs the students. The pairing is done
strategically pairing the fast and the
struggling ones. 2. The students will be given a problem to
solve. They will be given a problem to solve. They will help
Students’
answers
103
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
each other in reading,
analyzing and solving the problem. They are
required to discuss the problem until the two are convinced of the
solution. 3. The instructor monitors the students’
activity and checks for their answers. The
struggling ones will be required to explain the solution to the
instructor.
Model-Matching Activity Sheets This is an interactive
instructional activity that will lead students to match problems to
The students explain their answers to the
instructor and to the class.
104
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
their accomplished
Venn Diagrams or to match Venn Diagrams
to their corresponding working equations.
Procedures: 1. Instructors provide activity sheets
containing a matching type assessment.
2. The students match the items in column A
to Column B.
3. After, they will be asked to craft their own solution without
the models.
105
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Reading (insufficient recall, deficient mastery, poor exposure)
The students left the items
unanswered
*Note: If instructors do
not want activity sheets, he can write
the items to be match on the board and let a matching exist among
students. Using this approach, the instructor can even
ask the students to explain how the
matching of concepts is done.
Direct Instruction with Paired reading
This is a type of instruction that focuses on the
essentials or the specific skill that needs to be targeted.
Students’ scores in the
activity sheets.
106
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Procedures:
1. The instructor focuses on how applied
problems on sets and Venn Diagrams are understood or solved.
2. He can use technology in presenting the problem
or hand-outs. 3. The instructor pairs
the students for reading of the item assigned to them.
4. During the discussion of the item
assigned to the students, the instructor asks
questions on how students understood the problem. The
107
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Comprehen-sion (poor exposure, lack of skills)
Students incompletely indicated all
data necessary to
the solution of the problem
students can switch to
the vernacular when not comfortable in
using English when explaining. Other students are asked to
give comments regarding the understanding of the
presenters.
Conceptual Processing This standards-based
mathematical inter-vention emphasizes the
need to build a deeper understanding of concepts. This involves
making ideas, facts and skills reflecting upon and refining
Students explain their understanding of
the problem.
Students are asked to give comments.
108
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
one’s own under-
standing. It utilizes concept-builder
materials such as diagrams and other manipulative.
Procedures: 1. After an interactive
discussion, the instructor asks
students to indicate all data from the problem. 2. After that, the
students are asked to explain the meaning of
each data, how the data must be sorted or how such data must
be treated.
109
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Processing
(lack of practice, poor mastery, carelessness
Incorrect
difference of 180- 175 =10
Incorrect placement of
data and difference in the Venn
Diagrams.
3. The instructor gives
redirection or gives clarifying questions if
students are mislead. Trio Timed Drill
This is a variant of group learning that creates groups of 3
students.
Procedures: 1. The instructor assigns a student
leader in a group of 2 students, making them
a trio. 2. The instructor gives math worksheets that
the students will solve. 3. The leader facilitates the solution process of
Students
explain their understanding of the problem.
110
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Encoding (carelessness, lack of criticality)
Students just left 5 inside
the Venn diagram; thus, there is
the given problem. The
leader is given a copy of the correct answer
for him to verify if his answer is correct and to check whether his
group mates get the correct answer. * Note, the instructor
gives only the copy of the answer if the
leader gets the correct answer. 3. The student leader
directs and redirects the students under
him. Self-Check This is a strategy that
directs and redirects students to check their personal work.
Students’ answers on the
drill exercises. Students’ answers on the
111
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
To
visualize
correctly real numbers
Real Number
System
Fair
Encoding (carelessness, lack of criticality)
no indication
of final answers.
Students just indicated 5,
instead of 5 people
Students just wrote 11
instead of 11 units as the final answer.
Procedures:
1. The students are given worksheets with
directing questions which include: Is your working solution
correct? Is your final answer in its simplified form? Does your final
answer address the question of the
problem? Does it have a unit? 2. The instructor
checks on the students’ answer.
Answer-switch-verify This interactive activity
asks for students to compare, contrast and give comments to the
self-check
exercises.
Students’ answer to the
given problem Seat works
112
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
using the
number line. To
perform operations involving
real numbers.
To solve
applied problems on
real numbers.
Incorrect
distance.
Incorrect Counting
answers of their fellow
students.
Procedures: 1. The instructor asks the students to answer
a given problem. 2. The instructor sets the time for all the
students to answer the given problem.
3. After the given time, the students exchange solution sheets with
each other. The students give or write
comments as to the completeness of the final answer, etc.
4. After the comment period, the students address the comments.
113
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Processing
(lack of practice, poor mastery, carelessness
Students
failed to indicate the
correct formula or the working
equation. They wrote ―9 + (-2) = 7‖
and ―10 + 8 = 18‖; thus the
formula they used was D = P1 +P2.
Others did not indicate
any formula. Students left
the item unanswered.
Gallery Walk
This is a post-teaching instructional strategy
that assesses how students solve a given problem.
Procedures: 1. The instructor
divides the class into smaller learning
groups. 2. Each group is assigned an item to
solve. They are also given manila paper
and markers to present their solutions. 3. The students are
required to solve the items individually. They are only allowed
Student
solution presentations
114
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Mathemati-sing (dismal performance deficient recall)
Students incompletely
wrote all the given in the problem
to write their answer
on the manila paper once everyone has
solved the problem at hand. 4. The students are
asked to present their answers to the class. The other groups can
give reactions to their answers.
Formula Match This is a strategy that
involves the formula used in solving items.
Students’ answers to the
activity Recitation
115
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Reading (insufficient recall, deficient mastery, poor exposure)
Number line
pinpointing to the distance
of 10 and -2, as the largest and smallest
coordinates from among the 4
coordinates
Procedures:
1. The instructor presents the formulas
or the working equation and the different problems.
2. The students are asked to match the needed formula to the
respective problems. 3. The students will be
asked to explain their choices. 4. The class is free to
give comments.
Round Robin Reading This is a reading improvement strategy
that successively calls on students to read aloud a given problem.
Student reads
the given problem.
Answered hand-outs
116
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Comprehen-sion (poor exposure, lack of skills)
Procedures:
1. The teacher gives handouts on different
word problems or mathematical expressions.
2. The students will be asked to read on a round-robin basis.
3. After the reading sessions, the students
are asked to explain what they read. The students are free to
use the vernacular.
Comprehension Checker This is self-check
strategy that focuses on students’ understanding.
Understanding checker sheets
117
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Procedures:
1. The instructor gives a work sheet that
contains word problems and comprehension
question item checklist. 2. The students read
the problem and check the item that
corresponds to their understanding. 3. The instructor
checks the items. If he sees that the students
have low scores, he gives direct instruction or assigns him to
someone with a perfect score. The instructor again gives another set
118
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
To translate verbal
expression to numerical
expression and vice versa.
To perform
operations on algebraic expression
To simplify all answers
Algebraic Expres-sions
Poor
Comprehen-sion (poor exposure, lack of skills)
Students did not understand
well the term ―the height is
(x+9) cm more than the base.
Students just wrote (x+9)
instead of (x+9) + (2x-5).
of comprehension
checker work sheets to the students to check
on improved understanding.
Explicit Instruction It is a dynamic, structured and
systematic methodology for
teaching academic skills. It is characterized by
learning guides or scaffolds, whereby
students are guided throughout the learning process.
Procedures: 1. Focus instruction on critical content, com-
Seatwork Assignments Student
Board work
119
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
prehension, analysis,
problem-solving strategies.
2. Break down the content on specific targets.
3. Tell students of what they need to learn before starting
instruction. 4. Review prior
knowledge and provide learning supports or guides for students to
learn the rudiments of the lesson.
5. Break the class into smaller learning groups to check on the
extent of attainment of the instructional objectives.
120
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Comprehen-sion (poor exposure, lack of skills)
Students left the items
unanswered. They do not know what to
do.
6. The rote classroom
activities can be done to assess learning
Systematic Instruction It
means breaking down complex skills into smaller, manageable
―chunks‖ of learning and carefully
considering how to best teach these discrete pieces to
achieve the overall learning goal.
Procedures: 1. Sequence learning
chunks from easier to more difficult and providing scaffolding,
Seatwork Assignments
Student Boardwork
121
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
or temporary supports,
to control the level of difficulty throughout
the learning process. 2. Teachers break
down a complex task, like analyzing and solving a math
problem, into multiple steps or processes with
manageable learning chunks and teach each chunk to mastery
before bringing together the entire
process. 3. In turn, the
students do the same process independently or by pair.
122
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Reading
(insufficient recall, deficient mastery, poor exposure)
Students
failed to write the formula
for the problem.
Others wrote A = bh instead of A =
½ bh.
Sustained Silent
Reading (SSR) SSR is reading
instructional strategy that gives students instructional time to
read and analyze the problem.
Procedures: 1. Students are given
problem sets to be read silently. 2. The instructors give
the instructional item for them to analyze
and read the problem 3. After the SSR period, the teacher
asks questions that students will answer.
Seatwork
Assignments Student Board
work
123
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Mathemati-sing (dismal performancedeficient recall)
Wrong addition of (2x+5) and (x-
9). Instead of writing (3x-4); others wrote
The questions focus on
how the students understood the
problem, how they can deal with the problem, the strategy and the
like. 4. The teacher again gives another problem
using the SSR method, but the difference is
the students will explain their understanding and
method on the board.
Quick Write It introduces a concept and connects this concept with prior
knowledge or experiences and allows students to
Students’ sharing of prior knowledge and
responses.
124
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
3x +4.
discuss and learn from
each other
Procedures: 1. Introduce a single word, phrase formula
to the class. 2. Students copy the concept on index
cards. 3. Students are given
two minutes to write whatever comes to their minds relative to
the concept. They may write freely using
single words, phrases, sentences, etc. 4. After time is called,
students may volunteer to share their thoughts on the
125
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Processing (lack of practice, poor mastery, carelessness
Others overdid their solutions.
They wrote 2x+5 <0 and
3x<4 Others failed to write the
unit of measurement
.
subject.
5. The teacher gives direction, clarify or
affirm the student’s answers
Solve and React This allows students to solve whether
independently or independently. The
students will be asked to comment on the solutions to be
presented as regards the procedures of the
solution. Procedures:
1. The students will be asked to solve different items.
Students’ answers and reactions.
126
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Encoding
(carelessness, lack of criticality)
Others performed
incorrect simplification of final
answers.
2. A student will be
asked to present solution on the board.
3. The students who are seated will be asked to comment on
the solution procedures as regards their correctness.
4. The students take note of this for future
use. Say Something
This is a variant of solve and react that
asks students to comment on the answers of the
students.
Students’
group work sheets
127
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Procedures:
1. The students will be grouped into several
small learning groups (LG). 2. They will be solving
specific problem. 3. They will be exchanging and
commenting on the answers of the
students. 4. The teacher guides the students in the
correct examination and scrutiny of the
solution. Reflection Sheets
Teachers provide reflection sheets that ask the following:
Answered
Reflection sheets
128
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
To perform
operations on polynomials
To simplify polynomials accurately
To solve problems involving
polynomials
Polyno-mials
Fair
Mathemati-sing (dismal performance, insufficient recall)
Incorrect working
equation such as ―(x4-1)-(x+1)
No written
working equation
1. Is my answer in its
acceptable form? 2. Is my final answer
simplified correctly? 3. Does my answer contain unit? 4. Does my answer have the correct unit of measurement?
Five Word/
Formula Prediction Its purpose is to
encourage students to make predictions about text, working
equation or solution, to activate prior
knowledge, to set purposes for reading, and to introduce new
vocabulary
Quizzes Seat works Assignment
Recitation
Group work
129
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Processing (lack of practice, poor mastery, careless-ness)
Incorrect factoring ―(x4-
1)‖ such as ―(x3)(x-1)‖
Incorrect cancellation (x4-1)/(x+1)
Procedures:
1. Select five key math words/ working equa-
tion from a set of problems that students are about to read.
2. List the words in order on the chalk-board.
3. Using Socratic Method, Clarify the
meaning of any unfamiliar words.
Carousel Brainstorm Purposes: This
strategy can fit almost any purpose intended, especially when
students find difficulty in understanding
Class presenta-tions and
reactions to solutions
130
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
problems or presenting solutions to problems.
Procedures:
1. Teacher determines what problems will be placed on chart paper.
2. Chart paper is placed on walls around the room.
3. Teacher places students into groups of
four. 4. Students begin at a designated chart.
5. They read the prompt, discuss with
group, and respond directly on the chart. 6. After an allotted
amount of time, students rotate to next
131
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
chart.
7. Students read next prompt and previous
recordings, and then record any new discoveries or
discussion points. 8. Continue until each group has responded
to each prompt. 9. Teacher shares
information from charts and conversations heard
while responding. 10. Students will be
asked to clarify points in the solution of the problem.
** This strategy can be modified by having the chart ―carousel‖ to
132
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Encoding (carelessness, lack of criticality)
Incorrect copying of
signs in the final answer; but correct
solution
No unit.
groups, rather than
groups moving to chart.
Say Something This encourages
students to react on one’s work and then eventually to react on
other’s work.
Procedures: 1. Instructor asks the students to solve
different problems. 2. The instructor gives
direction and time frame for students to solve.
3. After the specific time, the instructor reminds the students
Solution sheets
133
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Reading
(insufficient recall, deficient mastery, poor exposure)
Students left
the items unanswered.
to finalize their
answer. 4. After 2 problems,
the students can exchange solution sheets and say
something about the solution and final answer.
GIST (Generating
Interactions between Schemata and Text) It directs students’
reflection on the content of the lesson
and leads them to summarize the problem and strategies
to differentiate between essential and non-essential information.
Students’ GIST
sheets
134
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Procedures:
The task is to write a summary of the
keywords, the problem-solving strategy in groups. The
words, the notes and strategies capture the ―gist‖ of the text.
1. The instructor
models how to solve a certain problem. 2. Instructor models
the procedures by drawing blanks or
columns on the board. 3. Instructor thinks aloud as (s)he begins
to facilitate the intervention activity.
135
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Comprehen-sion
(poor exposure, lack of skills)
Incorrect copying of
given data as to the signs
Incomplete representa-
tion of data
4. Students work with
a group or partner to complete a GIST for
the next chunk of problem. Students will eventually be asked to
create independent GISTs.
Copy-Solve-Cover It arouses students’
keen observation and comprehension about a certain text or
problem.
Procedures: 1. Instructor sets the objectives of the class.
2. The instructor demonstrates how certain data are
Solution sheets
136
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
organized and how
certain mathematical expressions are
properly understood. The students just copy.
3. On the succeeding items, the instructor covers the other half of
the item, then students will continue.
They will also be asked to explain their answers.
4. Gradually, students will do the same task
independently.
137
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
To get
the product of two
polynomials To solve
problems
involving product of two (2)
polyno-mials.
B. Special
Product Patterns
Product of two (2)
polyno-mials
Fair Processing
(lack of practice, poor mastery, carelessness
Incorrect
multiplication of (3x2-5) to
(3y+4) and (2x2+45) to (5y+2)
Incorrect evaluation in
―(3(10)2-5 = 295)‖, they
wrote ―900-5 = 895
Strategic Teaching
It is a teaching focused on specific lesson
contents. It is done after a diagnostic assessment is done.
Procedures: 1. Administer a
diagnostic test. For this study, the
research tool served as the diagnostic assessment.
2. From the results of the assessment, plan
or strategize the teaching based on the results. For this study,
the focus is on product of two (2) polynomials.
Quizzes
Worksheets Recitation
Group Presentation
138
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Reading (insufficient recall, deficient mastery, poor exposure)
Item unanswered
3. The teacher teachers
using different approaches; then
assesses after the instructional time.
The Directed Reading-Thinking Activity (DRTA )
The DRTA is a discussion format that
focuses on making problems more understandable. It
requires students to use their background
knowledge, make connections to what they know, make
predictions about the text, set their own purpose for reading,
Activity Sheets
139
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
use the information in
the text and then make evaluative judgments.
It can be used with nonfiction and fiction texts.
FOCUS: Comprehension
Strategies: Prediction, Inference and Setting
Reading Purpose Procedures (begin by
explaining and modeling):
1. The teacher divides the reading assign-ments into meaningful
segments and plans the lesson around these segments.
140
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
2. In the class
introduction, the teacher leads the
students in thinking about what they already know about
the topic. (―What do you know about ...? What connections can
you make?) 3. The teacher then
has the students preview the reading segment examining the
illustrations, headings and other clues to the
content. 4. The teacher asks students to make
predictions about what they will learn.
141
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
5. Students may write
individual predictions, write with a partner or
contribute to an oral discussion creating a list of class
predictions. 6. Students then read the selection and
evaluate their predictions. Were their
predictions verified? Were they on the wrong track? What
evidence supported the predictions?
Contradicted the predictions? 7. Students discuss
their predictions and the content of the reading.
142
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Encoding (carelessness, lack of criticality)
Comprehen- sion
(poor exposure, lack of skills)
Incomplete
solution.
Correct solution but
incorrect generalization
8. The teacher and
students discuss how they can use this
strategy on their own and how it facilitates understanding and
critical thinking. 9. The teacher and students repeat the
process with the next reading segment that
the teacher has identified.
Self-Verification Procedures:
1. The teacher guides the students in the reflection of final
answers.
Students’
comments and reactions
Students’ work solutions
143
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
2. The students rectify
their solutions based on the reflection
directions. 3. The teacher gives another item to check
on understanding Question-Answer
Relationship (QAR) FOCUS:
Comprehension Strategies: Determining
Importance, Questioning and
Synthesizing QAR is a strategy that
targets the question ―Where is the answer?‖ by having the
144
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
classroom teacher and
eventually the students create
questions that fit into a four-level thinking guide. The level of
questions requires students to use explicit and implicit
information in the text: • First level: ―Right
There!‖ answers. Answers that are directly answered in
the text. • Second level: ―Think
and Search.‖ This requires putting together information
from the text and making an inference.
145
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
• Third level: ―You and
the Author.‖ The answer might be found
in the student’s background know-ledge, but would not
make sense unless the student had read the text.
• Fourth level: ―On Your Own.‖ Poses a
question for which the answer must come from the student’s own
background knowledge
Procedure (begin by explaining and modeling):
1. The teacher makes up a series of QAR questions related to
QAR Chart
146
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
the materials to known
to the students and a series of QAR
questions related to the next reading assignment.
2. The teacher introduces QAR and explains that there are
two kinds of information in a book
explicit and implicit. 3. The teacher explains the levels of questions
and where the answers are found and gives
examples that are appropriate for the age level and the content.
A story like Cinderella that is known by most students usually works
147
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
well as an example,
even in high school classes.
4. The teacher then assigns a reading and the QAR questions
he/she has developed for the reading. Students read, answer
the QAR questions and discuss their answers.
5. The teacher and students discuss how they can use this
strategy on their own and how it facilitates
understanding and critical thinking. 6. After using the QAR
strategy several times, the students can begin to make up their own
148
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
QAR questions and in
small groups share with their classmates.
7. The teacher closes this activity with a discussion of how
students can use this strategy in their own reading and learning.
The ultimate goal of this activity (and most
of the activities presented here) is for students to become
very proficient in using the activity and
eventually use the activity automatically to help themselves
comprehend text.
QAR Chart
149
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Incomplete
encoding of data from the
problem No formula or
working equation written
KWL Chart and
Demonstration The know/want-to-
know/learned (KWL) chart guides students’ thinking as they begin
reading and involves them in each step of the reading process.
Students begin by identifying what they
already know about the subject of the assigned reading topic,
what they want to know about the topic
and finally, after they have read the material, what they have learned
as a result of reading. The strategy requires students to build on
KWL Chart
150
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
past knowledge and is
useful in making connections, setting a
purpose for reading, and evaluating one’s own learning.
FOCUS: Comprehension
Strategies: Activating Background
Knowledge, Questioning, Determining
Importance
Procedure (begin by explaining and modeling):
1. The teacher shows a blank KWL chart and explains what each
151
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Mathemati-sing (dismal performance, insufficient recall)
Incomplete
encoding of data from the problem
No formula or
working equation written
column requires.
2. The teacher, using a
current reading assignment, demonstrates how to
complete the columns and creates a class KWL chart.
K W L
• For the know column: As students brainstorm
background knowledge, they
should be encouraged to group or categorize the information they
know about the topic. This step helps them get prepared to link
KWL CHART
152
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
what they know with
what they read. • For the want-to-
know column: Students form questions about the
topic in terms of what they want to know. The teacher decides
whether students should preview the
reading material before they begin to create questions; it depends
on the reading materials and
students’ background knowledge. Since the questions prepare the
students to find information and set their purpose for
153
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
reading, previewing the
material at this point often results in more
relevant questions. Students should generate more
questions as they read. • For the learned
column: This step provides students with
opportunities to make direct links among their purpose for
reading, the questions they had as they read
and the information they found. Here they identify what they have
learned. It is a crucial step in helping students identify the
154
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
important information
and summarize the important aspects of
the text. During this step, students can be reflective about their
process and make plans. 3. The teacher on the
next reading assignments can ask
students individually or in pairs to identify what they already
know and then share with the class, create
questions for the want to-know column either individually or in pairs
and share with class, and finally after reading, complete the
155
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
To square a
trinomial correctly
To
solve applied problems using
Square of a
trinomial
Fair
Processing (lack of practice, poor mastery, carelessness
Incorrect squaring of
(2x-4y+6z)2 as (4x2+16y2+36z2)
learned column.
4. The teacher closes
this activity with a discussion of how students can use KWL
charts in their own reading and learning. 5. The teacher
demonstrates the process in formulating
working equations and deriving formulas.
Error Bull’s-eye It directs students to
target specific errors in presented solutions.
Procedures: 1. The instructor presents different
List of errors culled out from
the solution. Students’
presentation of correct answers.
156
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
squaring a
trinomial
Reading
(insufficient recall, deficient mastery, poor exposure)
No answer
solutions with errors.
2. The students will be given instructional
time to study the solutions. 3. The students will be
asked to identify the errors. They will be asked to explain why
that certain part of the solution is wrong.
4. They will present the correct solution afterwards.
Group Reading with
Guide Sheets It directs reading comprehension by
giving questions that cull out student understanding.
Answers in
Reading Guide sheets
157
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Mathemati-sing (dismal performance, insufficient recall)
Incorrect formulas as
A= 2∏r, and A= 2∏r2
Instead of writing
Procedures:
1. The instructor groups the students
and gives guide sheets in interpreting the applied problems.
2. The instructor checks the answers in the guide sheets.
3. The teacher gives comments and
redirections if necessary.
Comparison Matrix FOCUS:
Comprehension Strategies: Recognizing Similarities and
Differences
Answers in the Comparison
Matrix
158
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
(4x2+16y2+36
z2–16xy +24xz -48yz)∏ cm2,
they wrote 4x2+16y2+36z2–16xy +24xz
-48yz∏ cm2
(parenthetical error)
Procedures (begin by
explaining and modeling):
1. The teacher writes the subjects/categories/to
pics/etc. across the top row of boxes. 2. The teacher writes
the attributes/characterist
ics/details/etc. down the left column of boxes.
3. Use as few or many of rows and columns
as necessary; there should be a specific reason students need
to recognize the similarities and differences between the
159
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
provided topics and
details. 4. Explain to and
model for students what each
column/row of the matrix requires.
Expressions
Given Mathemati-
cal expresions
Ans-wers
Operations
Related
vocabulary
Patterns
160
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Encoding
(carelessness, lack of criticality)
Instead of
writing (2x-4y + 6z), others
even wrote (2x+4y+6z).
Response Notes
FOCUS: Comprehension
Strategies: Questioning, Inferring, Activating Background
Knowledge Procedures (begin by
explaining and modeling):
1. The teacher introduces the response notes and
models how to respond to open-ended
questions, share understanding, make connections to
background knowledge, share feelings, justify
Students’
answers on their response
notes
161
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
To
correctly
perform cubing of a binomial.
Cube of a binomial
Fair
Comprehen-sion
(poor exposure, lack of skills)
Processing (lack of practice, poor mastery, carelessness)
Incorrect cubing of
(2x+4). Their answers were (8x3+63) or
pinions, etc.
2. Students then read and create their own
responses in their notebooks or journals. 3. The teacher then
asks students to share with the class and/or collects the notes.
4. The teacher and students discuss how
they can use this strategy on their own and how it facilitates
understanding and critical thinking.
Solution Theater This will present
different answers of students and will let them select the correct
Presented Solution
162
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
To
apply the correct
process of cubing a polynomial
in word problems.
worse
(8x3+12)
solution.
Procedures:
1. The students will be presented with a problem. They will be
asked to present solutions on the board. 2. After, the students,
by group, shall be watching or observing
(like in theater) all the solutions. 3. After, they will select
which is a wrong solution and which is
correct. 4. The students will explain the error of the
solution and to correct the error they found out,
163
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Encoding
(carelessness, lack of criticality)
Reading (insufficient recall, deficient mastery, poor exposure)
No unit
No answer
Self-Verification
(Please look at the details of the strategy in the earlier cells)
Listening Teams – prior to the lesson, the class is divided into 4
groups/sectors of the class:
FOCUS: CULLING OUT UNDERSTANDING OF
WORD PROBLEMS Procedures:
1. The teacher classifies students into:
* Readers – responsible for reading the applied
problem
Students’ work
sheets
Student
responses
164
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
* Strategists –
responsible for coming up with a solution
strategy. * Questioners – responsible for coming
up with 2 questions they have about the topic
*Agreers – responsible for coming up with 2
points they agree about on the topic *Nay Sayers – 2 points
about the lecture that they disagree with
*Example Givers – 2 examples that are applicable to the topic.
*Listeners – responsible to listen and list down key
165
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Mathemati-sing (dismal performance, insufficient recall)
Incorrect formulas:
V= 3s3 and V= 4s
ideas.
2. The instructor facilitates the
presentation. 3. After some time, students do it alone.
Think-Alouds/ Metacognitive
Process
STRATEGY FOCUS: Comprehension Strategies: Monitoring
for Meaning, Predicting, Making
Connections Procedures (begin by
explaining and modeling): 1. The teacher chooses
Students’ answers in the instructional activity.
166
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
applied problems.
2. The teacher reads the text aloud and
thinks aloud as he/she reads. 3. Read the text slowly
and stop frequently to ―think-aloud‖ — reporting on the use of
the targeted strategies — ―Hmmm….‖ can be
used to signal the shift to a ―think-aloud‖ from reading.
4. Students underline the words and phrases
that helped the teacher use a strategy. 5. The teacher and
students list the strategies used.
167
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Comprehen-
sion (poor exposure, lack of skills)
Incorrect
copying of the given (2x +4);
some wrote (2x-4), (2+4), (x+4)
6. The teacher asks
students to identify other situations in
which they could use these strategies. 7. The teacher
reinforces the process with additional demonstrations and
follow-up lessons.
COMPREHENSION CHECKER It helps teachers to
check whether students have correct
comprehension or not. It is a variety of anticipation-reaction
guide.
Student
responses Recitation
168
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Procedures:
1.The instructor provides checklist on
the key words of the applied problems.
The students check the expressions that correspond to their
understanding.
The instructor directs the checking and redirects students who
are misled.
Daily re-looping of previously learned material
It is a process of always bringing in previously learned
169
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
material to build on
each day so that students have a base
knowledge to start with and so that learned structures are
constantly reinforced. This is for a topic that uses the same content area: linear equations, systems or rational expressions. Procedures:
1. Before beginning discussion, the teacher
elicits prior knowledge on the previous but related topics.
2. The students will be directed to relate the lesson at hand to the
170
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
To
perform factoring involving
difference of 2 perfect
squares, To
answer
applied problems
involving factoring of difference of
2 perfect squares.
C. Factor-
ing Patterns
Difference of 2
Perfect Squares
Pooor
Reading
(insufficient recall, deficient mastery, poor exposure)
Item left
unanswered
previous knowledge.
3. Discussion begins after the above-cited
processes. Structured Language
Experiences It is a well-structured learning activity where
students have abundant
opportunities to use language to describe their mathematical
understanding. It directs Students can
verbally explain/describe their math understanding,
they can write out their understanding, or
Student
presentations
171
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
they can draw pictures
and then explain.
Procedures: 1. Select a math concept/skill for which
students have received prior instruction and for which they have
demonstrated at least initial acquisition.
2. Develop a structured activity in which students can
describe their math understanding. The
activity should clearly relate the math concept/skill to the
language activity (e.g. students should clearly "see" the relationship
172
Specific
Objectives
Topics Level of Perform-mance
Error Cate-
gories and Theorized Causes
(arranged according to degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
student elaborations in
a systematic way, ensuring that every
student receives feedback regarding their explanations (e.g.
for smaller groups, the instructor does this individually; for larger
groups, peer tutors evaluate each other’s
explanations while instructor monitors tutor pairs).
6. Instructor has opportunity to evaluate
at least one explanation/description for every student.
7. Review activity by modeling an accurate description of the math
173
Specific
Objectives
Topics Level of Perform-mance
Error Cate-
gories and Theorized Causes
(arranged according to degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
concept/skill,
providing appropriate cueing (e.g. "think
alouds," visual, auditory, kinesthetic, tactile modalities).
Metacognitive strategy
It is a memorable "plan of action" that
provides students an easy to follow procedure for solving a
particular math problem. It is taught
using explicit teaching methods. It includes the student's thinking
as well as their physical actions.
174
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
1. Provide ample
opportunities for students to practice
using the strategy. 2. Provide timely
corrective feedback and remodel use of strategy as needed.
3. Provide students
with strategy cue sheets (or post the strategy in the
classroom) as students begin independently
using the strategy. Fade the use of cues as students demonstrate
they have memorized the strategy and how (as well as when) to
175
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
use it. (*Some students
will benefit from a "strategy notebook" in
which they keep both the strategies they have learned and the
corresponding math skill they can use each strategy for.)
4. Make a point of
reinforcing students for using the strategy appropriately.
5. Implicitly model using the strategy
when performing the corresponding math skill in class.
176
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Mathemati-
sing (dismal performance, insufficient recall)
Incorrect
formula for area of a
square: A = s2 (A=2s)
Paired Think tank
It is variant of partner learning that uses
recalling of formulas encountered or taught.
Procedures: 1. The instructor asks the students to pair,
pairing must be according to degree of
mastery. 2. The teacher directs the recall of the
formulas in math. 3. The students, in
pair, will list down all the recalled formulas. 4. Checking of answers
will be done afterwards.
Student
responses
177
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Comprehen-
sion (poor exposure, lack of skills)
Incorrect
copying of data from the
given
Reading for Meaning
Students become curious about printed
symbols or mathema-tical expressions once they recognize that
print, like talk, conveys meaningful messages that direct, inform or
entertain people. One goal of this curriculum
is to develop fluent and proficient readers who are knowledgeable
about the reading process. Effective read-
ing instruction should enable students to eventually become self-
directed readers who can:
construct meaning
Students
answers and responses
178
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Processing (lack of practice, poor mastery, carelessness)
Incorrect and incomplete
factoring of (2x2-162). They factored
from various types of
print material; recognize that there
are different kinds of reading materials and different purposes for
reading; select strategies appropriate for
different reading activities; and,
develop a life-long interest and enjoy-ment in reading a
variety of material for different purposes.
Independent Study (Using Learning
Activity Packages) This is a form of a seat work, using learning
Students answers and
responses
179
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
activity packages as
learning materials.
Procedures: 1. Students will be given some work to do,
based on prepared learning activity packages or skill book.
2. The students will be asked to check on their
answers by comparing with the answer key. 3. The students will be
asked to continue solving. The target is
for them to solve at least 5 items.
180
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Problem solving
instruction: explicit instruction in the steps
to solving a mathematical problem including
understanding the question, identifying relevant and irrelevant
information, choosing a plan to solve the
problem, solving it, and checking answers.
Procedures: 1. The teacher
presents certain problems and how these items are solved
with different solution strategies.
Answers Sheets
181
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
To factor
PST.
To differentiate
a PST from other trinomials
To solve applied problems
Perfect Square
Trinomial
Fair
Encoding (carelessness, lack of criticality)
No unit in the answer.
Wrong unit of measurement
indicated: cm2, m instead of cm
2. The students chose
which among the strategies they should
use. 3. They solve individually but can
compare answers with their seatmates. They discuss their answers,
especially when the items
Self-help and self-correcting materials
Students practice a math concept/skill
using materials that provide them both math concept/skill
prompts (e.g. questions, math equations, word
Students’ answers
Quizzes
182
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
involving
PSTs.
problems, etc.) and the
solutions to each prompt.
Procedures: 1. Identify appropriate
math skill for student practice. 2. Incorporate
materials that include the features listed in
Critical Components. 3. Model how to perform the math skill
using each self-correcting material.
4. Ensure that students clearly understand how to use
the self-correcting material. Be especially sensitive to individual
183
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
students who have
difficulty with particular verbal or
nonverbal response modes that are required when using
each self-correcting material. Be especially sensitive to individual
students who have difficulty with
particular verbal or nonverbal response modes that are
required when using each self-correcting
material (e.g. for students who have significant writing
problems, then materials that require writing responses may
184
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
produce student
frustration and therefore would not be
appropriate). 5. Periodically monitor students who
are using self-correcting materials, providing them
feedback about appropriate or
inappropriate use of self-correcting materials.
6. Provide students with a way to record
their responses (e.g. a sheet of paper on which they record their
responses; have students record responses with dry-
185
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
erase-marker on
laminated response cards/sheets that
contain each math skill prompt). 7. Evaluate student
responses by examining student response sheets.
Provide students with corrective feedback
regarding their performance as soon as possible.
8. Do not grade student performance
using self-correcting materials! Grading performance will
detract from the motivation self-correcting materials
186
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
can elicit from
students and grading will inhibit student
willingness to "take risks," a crucial behavior for learning.
Scaffolding Instruction
It provides students who have learning
problems the crucial learning support they need to move from
initial acquisition of a math concept/skill
toward independent performance of the math concept/skill.
Also referred to as "guided practice."
187
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Procedures:
1. Begin scaffolding after you have first
directly described and modeled the skill at least three times.
2. Perform the skill or learning task while asking questions aloud
and answering them aloud (questions
should pertain to specific essential features for specific
problem solving steps). Choose one or two
places during the problem solving process to question
your students.
188
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
3. Provide immediate
and specific feedback as well as positive
reinforcement with each student response. 4. When students
answer incorrectly, praise the student for his/her risk-taking
and effort while also describing and
modeling the correct response. When students answer
correctly, always provide positive
reinforcement by specifically stating what it is they did
correctly.
189
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
5. As your students
demonstrate success in responding to one or
two questions, then ask for an increased number of student
responses with the next example. (Corrective and positive
feedback continues as indicated by student
responses). 6. When your students demonstrate increased
competence, continue to fade your direction,
prompting students to complete more and more of the problem
solving process. Eventually, you only ask questions and
190
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
your students provide
all the answers. 7. As your students
demonstrate success in responding to one or two questions, then
ask for an increased number of student responses with the
next example. (Corrective and positive
feedback continues as indicated by student responses).
8. When your students demonstrate increased
competence, continue to fade your direction, prompting students to
complete more and more of the problem solving
191
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
process. Eventually,
you only ask questions and your students
provide all the answers. 9. When you are
confident that your students understand the problem-solving
process, invite them to actively problem-solve
with you (students direct problem-solving students ask question,
then both students and you respond).
10. Let student accuracy of responses and student nonverbal
behavior guide your decisions about when to continue fading your
192
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Reading
(insufficient recall, deficient mastery, poor exposure)
No indication
of solution
direction.
Brain Storming
(Formulas) This is done by using learning circles.
Procedures: 1. Students will be
given different applied problems.
2. The task of the students is to give the corresponding working
equations or formulas that are needed for the
problems to be solved. 3. Presentation and critiquing of students’
answers will follow.
Student
responses Recitation
193
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Mathemati-
sing (dismal performance, insufficient recall)
Incorrect
Formulas: A = 4s
S = A/2
Curriculum-Based
Probe It directs students to
solve 2-3 sheets of problems in a set amount of time
assessing the same skill. Instructor counts the number of
correctly written digits, finds the median
correct digits per minute and then determines whether
the student is at frustration,
instructional, or mastery level.
Students’
responses Recitation
194
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Procedures:
1. The instructor assesses the students’
mastery level through the quiz or seat work. 2. Based on the
results, the instructor gives students differentiated student
exercises based on their mastery level.
3. The instructor can focus on teaching the students under the
frustration level. The students in the
mastery level can facilitate drill for the instructional level.
195
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Processing
(lack of practice, poor mastery, carelessness)
Incorrect
factoring of PST. They
simply divided by two.
Assigned Questions
(as Assignments) Focus: Reading,
Comprehension, Content
Procedures: 1. Students give assignments to
students to read at home.
2. When they enter the class, the instructor asks them to present
their work. 3. Other students will
be asked to give reactions. 4. Discussion on
critical concerns follow.
Student
Responses
196
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
To factor General Quadratic
Trinomials To solve
applied problems using
general quadratic trinomials.
General Quadratic Trinomial
Fair
Comprehen-
sion (poor exposure, lack of skills)
Encoding (carelessness, lack of criticality)
Reading (insufficient recall, deficient
mastery, poor exposure)
Wrong
indication of sign of the
copied data
No unit of measurement
No solution
Catching Signs
This is a strategy patterned sign
mnemonics.
Apply Self-Correcting Materials (See procedures above)
Adjusted speech: instructor changes speech patterns to
increase student comprehension. It includes facing the
Student
responses Solution Sheets
Recitation Students’
Answers Recitation Students’
Answers
Student responses Recitation
Recitation
197
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
students, paraphrasing
often, clearly indicating most important ideas,
limiting asides, etc Procedures: (This is
simply a variant of language switching) 1. Instructor can ask
the students to read. 2. When the instructor
directs students to understand the problem, he can switch
to the students’ mother tongue to stress the
essentials and to clarify vague thoughts, especially on
mathematical expressions.
198
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Mathemati-
sing (dismal performance, insufficient recall)
Encoding
Incorrect
working equation:
(x2+3x-40) - (x-8) instead of (x2+3x-40)
/ (x-8)
No unit
Puzzle Game
This is a variant of instructional game, or
another form of interactive worksheets.
Procedures: 1. Instructor gives an activity sheet that has
formulas and empty cells.
2. They match the formula and the letters to guess the magic
word.
Structured Peer Tutelage It is a well planned/
structured practice activities where students problem
Solution sheets
Activity sheets
Students’ answers on the activity sheets
199
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
solve in pairs
Procedures:
1. Determine goals for each peer tutelage activity.
2. Target specific math skills to be practiced.
3. Select appropriate materials that match
learning objectives and that can be implemented within a
peer tutoring format (i.e. provide both a
prompt sheet that contains problems to be solved and an
answer key that can be easily used by your students).
200
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
4. Design and teach
procedures/behaviors for tutoring.
5. Review classroom rules and teach new rules when
appropriate. 6. Pair students of varying achievement
levels. 7. Practice peer-
tutoring procedures before implementing them with academic
tasks. 8. Divide peer-
tutoring time into halves so each player has equal time as
coach and as player. 9. Signal students when it is time to
201
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Processing
(lack of practice, poor mastery, carelessness)
Incorrect
cancellation in (x2+3x-40)/
(x-8;x2 an x were immediately
cancelled.
switch roles.
10. Set goals for tutoring pairs and
provide positive reinforcement for tutoring pairs that
meet goals. 11. Provide response record sheets so you
can evaluate the performance of
individual students. Authentic Contexts
The purpose of Teaching Math
Concepts/Skills within Authentic Contexts is to explicitly connect
the target math
Students’
answers on the activity sheets
202
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Comprehen-sion (poor exposure, lack of skills)
Adding of superfluous data in the
given (x2+3x-
40)/2 instead of
concept/skill to a
relevant and meaningful context,
promoting a deeper level of understanding for students
Procedures: 1. Instructor chooses
appropriate context within which to teach
target math concept/skill. Refer to the assessment
strategy
Dynamic Assessment, for information about how to collect
information about students' interests and to use this information
Students’ answers
203
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
simply
(x2+3x-40)
to create authentic
contexts for assessment and
teaching Mathematics Student Interest Inventory
2. Instructor activates student prior knowledge of
authentic context, identifies the math
concept/skill students will learn, and explicitly relates the
target math concept/skill to the
meaningful context. 3. Instructor explicitly models math
concept/skill within authentic context.
204
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
4. Instructor involves
students by prompting student thinking
about how the math concept/skill is relevant to the
authentic context. 5. Instructor checks for student
understanding. 6. Students receive
opportunities to apply math concept or perform math skill
within authentic context. Instructor
monitors, provides specific corrective feedback, remodels
math concept/skill as needed, and provides positive reinforcement.
205
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
7. Instructor provides
review and closure, explicitly re-stating
how the target math skill relates to the authentic context and
remodeling the skill. 8. Students receive multiple opportunities
to apply math concept or practice math skill
after initial instruc-tional activity. Incor-porating the instructor
instruction strategies, Building Meaningful
Student Connections, Explicit Instructor Modeling, & Scaffold-
ing Instruction when teaching within
206
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
To apply the correct procedure in
factoring by grouping To use the
correct procedure in
answering word problems.
Factoring by grouping
Poor
Reading (insufficient recall, deficient mastery, poor exposure)
Item left unanswered
authentic contexts can
be very effective.
Focusing on "Big Ideas" or the essentials It facilitates student understanding by concentrating student attention on key concepts and procedures. The linkages and connections between math concepts are made explicit by linking previously learned big ideas to new concepts and problem solving situations. By emphasizing the big ideas in each lesson,
instructors can build students' acquisition
Student responses
207
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
and use of key
conceptual knowledge across lesson content.
Procedures: 1. Choose math big
ideas that are foundational to the lesson and that
represent understandings that
can be applied across lessons (e.g. formula, mathematical
expressions). 2. Explicitly teach the
math big idea, linking it to previously learned information.
3. Explicitly teach the target math skill within the context of
208
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
4. The math big idea.
5. Provide multiple practice opportunities
for students using the Big Idea with the new math skill you taught.
6. Apply the math big idea to the target math skill using a variety of
problem solving situations.
7. Pair a visual cue with each math big idea (e.g. a picture of
an array for the Big Idea of "area").
8. Post the visual cue along with one sentence describing
why the big idea is important.
209
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Mathemati-
sing (dismal performance, insufficient recall)
Incorrect
working equation
(x2+2xy+y2+x+y) - (x+y) instead of
(x2+2xy+y2+x+y) / (x+y);
Structured
Cooperative Learning Groups
Students practice math concepts/skills they have previously
required with peers in teams or small groups.
Procedures: 1. Determine goals for
each cooperative learning activity. 2. Target specific
academic skills to be learned/practiced.
3. Select appropriate materials that match learning objectives.
4. Design and teach procedures/behaviors for team members to
Recitation
210
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
help each other.
5. Review classroom rules and teach new
rules when appropriate. 6. Assign students of
varying achievement levels to the same team.
7. Practice cooperative group procedures
before implementing them with academic tasks.
8. Set team goals and provide positive
reinforcement for teams that meet goals. 9. Evaluate success of
cooperative learning activity.
211
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Processing
(lack of practice, poor mastery, carelessness)
Invalid
cancellation in
(x2+2xy+y2+x+y) / (x+y); the expression
―x+y‖ was immediately cancelled.
Think-Pair-Share
Think-Pair-Share is a strategy designed to
provide students with "food for thought" on a given topics enabling
them to formulate individual ideas and share these ideas with
another student. It is a learning strategy
developed by Lyman and associates to encourage student
classroom participation. Rather
than using a basic recitation method in which a instructor
poses a question and one student offers a response, Think-Pair-
Students’
answers
212
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Share encourages a
high degree of pupil response and can help
keep students on task. Procedures:
With students seated in teams of 4, have them number them
from 1 to 4. Announce a
discussion topic or problem to solve. (Example: Which
room in our school is larger, the cafeteria
or the gymnasium? How could we find out the answer?)
Give students at least 10 seconds of think time to THINK
213
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
of their own answer.
(Research shows that the quality of
student responses goes up significantly when you allow
"think time.") Using student
numbers, announce
discussion partners. (Example: For this
discussion, Student #1 and #2 will be partners. At the
same time, Student #3 and #4 will talk
over their ideas.) Ask students to PAIR
with their partner to
discuss the topic or solution.
214
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Finally, randomly
call on a few students to SHARE their ideas
with the class. Instructors may also
ask students to write or diagram their responses while doing
the Think-Pair-Share activity. Think, Pair, Share helps students
develop conceptual understanding of a
topic, develop the ability to filter information and draw
conclusions, and develop the ability to
consider other points of view.
215
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Encoding
(carelessness, lack of criticality)
No indication
of unit.
Structured
Controversy Using structured controversy in the classroom can take many forms. In its most typical form, you select a specific problem. The closer the problem is to multiple issues central to the course the better. This strategy involves providing students with a limited amount of background information and asking them to construct an argument based on this information. This they do by working in groups.
Solution sheets
216
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Procedures:
Choose a discussion topic that has at
least two well documented positions.
Prepare materials: o Clear expectations
for the group task.
o Define the positions to be
advocated with a summary of the key arguments
supporting the positions.
o Provide reference materials including a
bibliography that support and elaborate the
217
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
arguments for the
positions to be advocated.
Structure the controversy: o Assign students to
groups of four. o Divide each group
into dyads who are
assigned opposing positions on the
topic. o Require each
group to reach
consensus on the issue and turn in
a group report on which all members will be evaluated.
218
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Comprehen-sion (poor exposure, lack of skills)
Incorrect data. Adding unnecessary
data. Instead of
(x2+2xy+y2+x+y) only, they wrote
(x2+2xy+y2+x+y) / 2
Conduct the
controversy: o Plan positions.
o Present positions. o Argue the issue. o Reverse positions
and argue the issue from those perspectives.
o Reach a decision.
Explicit Instructor Modeling The purpose of explicit
instructor modeling is to provide students
with a clear, multi-sensory model of a skill or concept. The
instructor is the person best equipped
Students’ responses
219
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
to provide such a
model.
Procedures: 1. Ensure that your
students have the
prerequisite skills to perform the skill.
2. Break down the
skill into logical and learnable parts (Ask
yourself, "what do I do and what do I think as I perform
the skill?"). 3. Provide a
meaningful context for the skill (e.g. word or story
problem suited to the age & interests of your students).
220
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
4. Provide visual,
auditory, kinesthetic
(movement), and tactile means for illustrating
important aspects of the concept/skill (e.g. visually display
word problem and equation, orally cue
students by varying vocal intonations, point, circle,
highlight computation signs
or important information in story problems).
5. "Think aloud" as you perform each step of the skill
221
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
(i.e. say aloud what
you are thinking as you problem-solve).
6. Link each step of the problem solving process (e.g. restate
what you did in the previous step, what you are going to do
in the next step, and why the next
step is important to the previous step).
7. Periodically check
student understanding with
questions, remodeling steps when there is
confusion.
222
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
To
simplify correctly RAEs
To apply simplification of RAEs in
D. Ration-
al Expres-sions
Simplifi-cation of RAEs
Fair
Reading
(insufficient recall, deficient mastery, poor exposure)
Item
unsolved.
8. Maintain a lively
pace while being conscious of
student information processing difficulties (e.g. need
additional time to process questions).
9. Model a
concept/skill at least three times
before beginning to scaffold your instruction.
Assigned Questions
(as Seat work) Focus: Reading and Formulas
Students’
homework
223
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
applied
problems.
Procedures:
1. Students give assignments to
students to read. They can be grouped and mixed according to
degree of ability. 2. When they enter the class, the instructor
asks them to present their work.
3. Other students will be asked to give reactions.
4. Discussion on critical concerns
follow.
224
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Processing
(lack of practice, poor mastery, carelessness)
Incorrect
cancellation in
(12x4y6/7xy) and (21/6x3y5).
Incorrect
placing of simplified
form. Instead of 3 in the numerator, it
was placed in the
denominator.
Planned Discovery
Activities The purpose of
Planned Discovery Activities is to provide students who have
learning problems the opportunity to make meaningful
connections between two or more math
concepts for which they have previously received instruction
which they have previously mastered. It
is important to remember that this is a student practice
strategy and it is not intended for initial instruction.
Students’
solutions
225
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Procedures:
1. Determine two or more concepts that
have a mathematical relationship which
students would benefit from understanding.
These concepts must have already
been taught and must have been already mastered by
the students. 2. Develop a well
organized/structured activity that provides students
the opportunity to understand the desired mathema-
226
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
tical relationship
between the selected math
concepts. 3. Provide explicit
directions for
completing the activity.
4. Develop and provide
to students a structured learning
sheet or other appropriate prompt that guides
students toward the learning objective.
5. Monitor students as they participate in the activity.
Circulate the classroom, provide specific corrective
227
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
feedback, model
appropriate skills as needed, prompt
student thinking, and provide positive reinforcement.
6. At the conclusion of the activity, provide students with the
solutions to the structured learning
sheet/prompt and explicitly state/ illustrate the
desired mathema-tical relationship(s).
7. How Does This Instructional Strategy Positively
Impact Students Who Have Learning Problems?
228
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Mathemati-
sing (dismal performance, insufficient recall)
Incorrect
working equation such
as (12x4y6/7xy) ÷ (21/6x3y5) or
(12x4y6/7xy) - (21/6x3y5) instead of
(12x4y6/7xy) x (21/6x3y5)
Experiential Learning
(focus: solving math problems)
Experiential learning is inductive, learner centered, and activity
oriented. Personalized reflection about an experience and the
formulation of plans to apply learning to other
contexts are critical factors in effective experiential learning.
The emphasis in experiential learning is
on the process of learning and not on the product.
Experiential learning can be viewed as a
Students’
answers Quizzes
Recitations
229
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
cycle consisting of five
phases, all of which are necessary:
experiencing (an activity occurs);
sharing or
publishing (reactions and observations are shared);
analyzing or processing (patterns
and dynamics are determined);
inferring or
generalizing (principles are
derived); and, applying (plans are
made to use learning
in new situations).
230
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Encoding
(carelessness, lack of criticality)
No unit.
Answers were
not simplified:6/1
Procedures:
1. Instructor presents solved problems.
2. Instructor leads the students to analyze the solved problems, to
direct the students to analyze solution patterns.
3. Students are given items to solve. They
can clarify misconceptions if necessary.
Graphic organizers:
visual displays to organize information from the problem. They
help students to consolidate informa-
Organizer
sheets
231
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Comprehen-
sion (poor exposure, lack of skills)
Incomplete
data
tion into meaningful
whole and they are used to improve
comprehension of stories, organization of writing, and
understanding of difficult concepts in word problems.
Procedures:
1. Instructor presents problems. 2. Instructor
Demonstrates how graphic organizers are
used. 3. Understanding of students will be
checked based on the teaching techniques of
232
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
To
perform the basic operations
on RAEs To apply
operations on RAEs to applied
problems
Operation
on RAEs
Poor
Mathemati-
sing (dismal performance, insufficient recall)
Incorrect
working equation (1/2x)(8x/2)
instead of (5/2x)(80x/2)
the teachers. Follow-
up questions that will probe into the in-depth
understanding of the students must be structured.
4. Students use the organizers independently.
Think-Pair-Share
Think-Pair-Share is a strategy designed to provide students with
"food for thought" on a given topics enabling
them to formulate individual ideas and share these ideas with
another student. It is a learning strategy
Students’
responses
233
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
developed by Lyman
and associates to encourage student
classroom participation. Rather than using a basic
recitation method in which a instructor poses a question and
one student offers a response, Think-Pair-
Share encourages a high degree of pupil response and can help
keep students on task.
Procedures: With students seated in teams of 4,
have them number them from 1 to 4.
234
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Announce a
discussion topic or problem to solve.
(Example: Which room in our school is larger, the cafeteria or the
gymnasium? How could we find out the answer?)
Give students at least 10 seconds of
think time to THINK of their own answer. (Research shows that
the quality of student responses goes up
significantly when you allow "think time.") Using student
numbers, announce discussion partners.
235
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
(Example: For this
discussion, Student #1 and #2 will be
partners. At the same time, Student #3 and #4 will talk over their
ideas.) Ask students to PAIR with their partner
to discuss the topic or solution.
Finally, randomly call on a few students to SHARE their ideas
with the class.
Instructors may also ask students to write or diagram their
responses while doing the Think-Pair-Share activity. Think, Pair,
236
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Share helps students
develop conceptual understanding of a
topic, develop the ability to filter information and draw
conclusions, and develop the ability to consider other points
of view.
Learning Partners: discuss a document, interview each other
for reactions to a document or
presentation, critique or edit each others’ work, recap a lesson,
develop a test question together, compare
237
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Reading
(insufficient recall, deficient mastery, poor exposure)
Items
unanswered
notes, stump your
partner.
Procedures: 1. Instructor provides problems after the
students are paired. 2. The students are asked to recall the
correct formulas. 3. Critiquing of
answers is done. Continuous
Performance Charting The goal of continuous
monitoring and charting of student performance is twofold.
First, it provides you, the instructor, information about
Student
responses
Recitation
238
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
student progress on
discrete, short-term objectives. It enables
you to adjust your instruction to review or re-teach concepts or
skills immediately, rather than waiting until you've covered
several topics to find out that one or more
students didn't learn a particular skill or concept. Second, it
provides your students with a visual represen-
tation of their learning. Students can become more engaged in their
learning by charting and graphing their own performance.
239
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Procedures:
1. Determine a specific instructional
objective (classify objects according to color, size, shape,
pattern; add two digit numbers without regrouping, solve story
problems with + and - fractions, select the
relevant information in a story problem). 2. Design a
"curriculum slice" using the C-R-A
assessment strategy (see Additional Information for an
example of a curriculum slice below.) Choose
240
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
3. appropriate items
that accurately reflect the target math skill at
the appropriate level of understanding (concrete,
representational, abstract) and that can be administered in a
short time period (perhaps a 1-3 minute
timing). Include more items than you think the student can
complete within the designated time period
so that you get an accurate indication of their optimal
performance.
241
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Procedures:
1. Determine a specific instructional
objective (classify objects according to color, size, shape,
pattern; add two digit numbers without regrouping, solve story
problems with + and - fractions, select the
relevant information in a story problem). 2. Design a
"curriculum slice" using the C-R-A
assessment strategy (see Additional Infor-mation for an example
of a curriculum slice below.) Choose appro-priate items that
242
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
3. accurately reflect
the target math skill at the appropriate level of
understanding (con-crete, representa-tional, abstract) and
that can be administered in a short time period (perhaps a
1-3 minute timing). Include more items
than you think the student can complete within the designated
time period so that you get an accurate
indication of their optimal performance. 4. Administer and
score the assessment. 5. Have students plot incorrect and correct
243
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
responses on a graph
(see Additional Information for an
example of a graph/chart). 6. Discuss and draw
goal lines on graph. 7. Repeat process. 8. Determine success
of your instruction based on the "learning
picture" depicted on the student's chart/ graph (see Additional
Information for examples of different
learning pictures and what they mean). 9. Make appropriate
instructional decisions based on the student's learning picture.
244
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Processing
(lack of practice, poor mastery, carelessness)
Others wrote
(5/2x) ÷ (80/2x)
instead of (5/2x) x (80x/2)
Drill & Practice
As an instructional strategy, drill &
practice is familiar to all educators. It "promotes the
acquisition of knowledge or skill through repetitive
practice." It refers to small tasks such as
the memorization of spelling or vocabulary words, or the
practicing of arithmetic facts and may also be
found in more sophisticated learning tasks or physical
education games and
Student
responses
Recitation Performance
Chart
245
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
sports. Drill-and-
practice, like memorization, involves
repetition of specific skills, such as addition and subtraction, or
spelling. To be meaningful to learners, the skills built through
drill-and-practice should become the
building blocks for more meaningful learning.
Procedures:
1. Students will be given exercises to solve independently, by pair
or by small groups. 2. The instructor roams around to check
246
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Comprehen-sion (poor exposure, lack of skills)
Incomplete data. They
missed out 5 and 10 for the denomination
.
No unit.
on students’ answers.
3. The instructor assists students who
are experiencing difficulty. 4. Students are asked
to present their solutions.
Drill & Practice As an instructional
strategy, drill & practice is familiar to all educators. It
"promotes the acquisition of
knowledge or skill through repetitive practice." It refers to
small tasks such as the memorization of spelling or vocabulary
Student responses
Recitation
Solution sheets on drills
247
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
words, or the
practicing of arithmetic facts and may also be
found in more sophisticated learning tasks or physical
education games and sports. Drill-and-practice, like
memorization, involves repetition of specific
skills, such as addition and subtraction, or spelling. To be
meaningful to learners, the skills built through
drill-and-practice should become the building blocks for
more meaningful learning.
248
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Encoding (carelessness, lack of criticality)
Procedures:
1. Students will be given exercises to solve
independently, by pair or by small groups. 2. The instructor
roams around to check on students’ answers. 3. The instructor
assists students who are experiencing
difficulty. 4. Students are asked to present their
solutions.
Didactic Questions This is variant of partner learning that
focuses on students’ understanding. It focuses on how
Student’s responses
Recitation
249
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
To
simplify complex
RAES. To use
simplificatio
n of RAEs in solving
applied problems.
Simplifi-
cation of Complex
RAEs/ fractions
Very
Poor
Reading
(insufficient recall, deficient mastery, poor exposure)
Items left
unanswered
students should
understand applied problems. They will
focus on how answers are solved and simplified for
acceptance. Apply SSR
(see mechanics above)
Response journal:
Students record in a journal what they
learned that day or strategies they learned or questions they have.
Students can share their ideas in the class,
Student
responses
Recitation Journals
250
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Mathemati-sing (dismal performance, insufficient recall)
Incorrect formulas: I =
1 + PRT and I = PR
with partners, and
with the instructor.
Procedures: 1. Students are given assignments. They are
asked to write their questions pertinent their reading
assignment. 2. The questions shall
be shared and discussed in class. 3. The questions shall
serve as the starting point of the instructor.
Didactic Questions This is variant of
partner learning that focuses on students’ understanding. It
Student responses
251
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Comprehen-sion
(poor exposure, lack of skills)
Time (t) was not
represented and indicated.
focuses on how
students should understand applied
problems. They will focus on how answers are solved and
simplified for acceptance.
Instructional Game The goal of each
student practice strategy in this program is to provide
students who have learning problems
multiple opportunities to respond to a particular learning
task. This is certainly true for Instructional Games as well.
Recitation
252
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Instructional Games
can also make practicing math skills
fun due to its game format.
Procedures: 1. Determine math skill(s) for which target
students have received prior instruction and
which they can perform with at least moderate success.
2. Select a student age/interest appro-
priate game context in which the target math skill can be performed.
3. Develop game procedures that allow for many math-skill
253
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
practice opportuni-
ties (complexity of game procedures
should not detract from skill practice).
4. Provide students
with necessary materials to play the game.
5. Model the math skill(s) to be practiced
at least once in isolation and at least once within the game
context before the game is played.
6. Provide explicit directions for playing the game and model
game procedures.
254
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
7. State behavioral
expectations and model essential game-
playing behaviors (e.g. turn-taking, respond-ing appropriately when
I am not the winner, etc.) 8. Invite several
students to model playing the game
before game begins. Provide an opportunity for students to ask
questions and to clarify misconceptions.
9. Monitor students as they play the game, providing specific
corrective feedback, modeling skills when appropriate, and
255
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Processing (lack of practice, poor mastery, carelessness)
Incorrect substitution:
P = I/RT as (1/6 x
12,000) =P/[(1- 1t/3)].
providing positive
reinforcement. Demonstrate
enthusiasm for game as students play Provide a way for
students to show their work so that you can evaluate their
performance after the game is completed.
Procedures: Model-lead-test
strategy instruction (MLT): 3 stage process
for teaching students to independently use learning strategies: 1)
instructor models correct use of strategy; 2) instructor leads
256
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Encoding (carelessness, lack of criticality)
No indication of ―t‖ in the final answer.
No unit.
students to practice
correct use; 3) instructor tests’
students’ independent use of it. Once students attain a score
of 80% correct on two consecutive tests, instruction on the
strategy stops.
Apply self-help
257
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
To solve
applied problems on
distance using linear equations.
To solve the value of x
E. Linear
Equations in One
Unknown Applied
Problems on Distance
Poor Reading
(insufficient recall, deficient mastery, poor exposure)
No answer.
Explicit vocabulary
building through random recurrent
assessments: Using brief assessments to help students build
basic subject-specific vocabulary and also gauge student
retention of subject-specific vocabulary.
Procedures: 1. Instructor asks
students to read certain problems.
2. They will be asked to share their ideas and even problems
regarding the problem. 3. Discussions follow.
Student
responses
Recitation
258
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Comprehen-
sion (poor exposure, lack of skills)
Incomplete
indication of data.
Native language
support: providing auditory or written
content input to students in their native language.
Procedures: 1. Using GLCs,
students will be redirected to
understand the problem. 2. They are allowed to
explain using the vernacular.
Student developed glossary: Students
keep track of key content and concept words and define them
Student
responses
Recitation
259
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Mathemati-
sing (dismal performance, insufficient recall)
They failed to
write the correct formula, D =
RT. Others wrote Vf
2= Vo2
+ 2 fusing Physics and College
Algebra. Others wrote
440-220= 220
in a log or series of
worksheets that they keep with their text to
refer to. Correcting formula
mismatch A variant of formula matching
Procedures:
1. Students will be presented with columns of problems
and formulas. The items are already
matched, but incorrectly done. 2. The instructor will
then give the task to correct the incorrect matching.
Student
responses Recitation
Quizzes
260
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Encoding
(carelessness, lack of criticality)
No or
incorrect unit.
Unit Matching
This will let students match the conditions
of the problem to its corresponding unit of measurement.
Procedures: 1. Instructor provides
activity sheets that contain a column for
problem conditions and a column for unit of measurement.
2. After answering, discussion of answers
follows. 3. Correction and redirection of
misconceptions shall be a follow-up teaching procedure.
Student
responses
Recitation
261
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Processing
(lack of practice, poor mastery, carelessness)
Incomplete
solution. After getting the
value of x, they did not go back to the
tabular repre-sentation.
Columnar Battle
This is an instructional game that integrates
group learning circles (GLCs)
Procedures: 1. Students will be grouped into 3-4.
2. Leaders and assistant leaders will
be assigned. 3. A representative per group shall be called to
solve given items. 4. Students who are
seated will also be asked. Monitoring by the instructor and
assistant leader must be done.
Student
responses
Recitation
262
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
To
represent data
correctly To
interpret
problems correctly
Applied
Problems on
Money/ Amount
Poor
Reading
(insufficient recall, deficient mastery, poor exposure)
Not answered.
5. This continues until
majority of students have gone to the
board. 6. Discussion of misconceptions will be
done. Timed Reading
This is a variant of structured reading
strategy. Procedures: 1. The instructor gives
students applied problems to solve.
2. After several minutes or the instructional time for
reading, the instructor gives guide questions for students to be led
Student
responses
Recitation
263
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Comprehen-
sion (poor exposure, lack of skills)
Partial
indication of data.
towards the correct
understanding. 3. The students are
asked to present their answers highlighting keywords and
translations. Student-led
discussions This is a variant of
cooperative learning groups. Its focus is to hasten reading,
comprehension and mathematical skills.
Procedures: 1. Using clustered
groups, a leader will be assigned to facilitate the analysis and
Seatwork
264
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Mathemati-sing
(dismal performance, insufficient recall)
Others added 1 and 20 and
1 and 50 as the working equation.
solution of certain
problems. 2. The leader will ask
his group members to read. The leader, in turn, will give
directions as to how the problem should be hurdled.
3. Checking of answers will be done.
4. Critiquing will also be done.
Table Completion This is a form of data
collection strategy.
Completed Table
Student responses
Recitation
265
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Processing (lack of practice, poor mastery, carelessness)
Instead of writing 1350 – 50x, others
wrote 1350 – x or 135 –
Procedures:
1. The instructor provided supplemental
materials on applied problems and tables where headings for all
required data are indicated. 2. After silent reading,
the students will be asked to fill in the
table. 3. The teacher checks and redirects students
when necessary.
Equation Generator This will ask students to present solutions on
the board.
Generated equations Quiz
266
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
50x or worse,
135 –x
Procedures:
1. After teaching, students will be called
to go in front to generate equations or to recall formulas.
2. Students will be asked to write answers on the board. Students
who are seated will be asked to verify written
formulas. 3. The students will explain whether a
generated equation is correct or not. It
should be clear when an equation is correct or not.
4. A seat work may follow for assessment.
267
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Encoding
(carelessness, lack of criticality)
No unit.
Incorrect unit of KPH
instead of KM only.
Quiz bees
This is a form of instructional activity
that tests students’ mastery of the subject matter.
Procedures: 1. The instructor
divides the class into three. The instructor
provides flaps for students to answer. 2. The flaps will be
raised once the bell is signaled.
3. Checking of answers follow. 4. If an item is not
solved by majority of the groups, the teacher stops temporarily the
Student
responses
Recitation Answers
268
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
game and discusses
with the class the correct mechanical
procedures. Fishbowl of Units
This is another form of Q&A technique.
Procedures. 1. The teacher provides
a bowl/ box where units of measures are indicated.
2. Students will be called to pick a strip
paper from the bowl/box. 3. The students are
asked to explain when they should use the unit of measure.
Students’
explanation
269
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
To
formulate working
equations on age problems
To find the value of x
To represent
data correctly
To solve
age problems
Applied
Problems on Age
Poor Reading
(insufficient recall, deficient mastery, poor exposure)
Item
unanswered.
Reciprocal peer
tutoring (RPT) to improve math
achievement This is a general strategy in improving
performance. Focus: How to deal with problems -
1. Have students pair, choose a team
2. Explain to the students the goal of the activity.
3. Let the fast learners tutor on math
problems, and then individually work a sheet of drill problems.
Students get points for correct problems and work toward a goal.
Student
responses
Recitation
270
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Send me a problem
This is a reading game that leads students to
read and solve certain problems.
Procedures: 1. 10 students shall be selected to write
certain mathematical expressions or
problems. 2. The problems or mathematical
expressions shall be sent to certain
students. 3. The students, in turn, will solve the
problem and will give the answer to the students who gave
271
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Comprehen-sion
(poor exposure, lack of skills)
No represen-tation for the
age of two persons
involved.
them the question. The
student-sender determines if the
answer is correct or not. If incorrect, the sender will give guides.
Question Generation This lets students to
write questions and give their
corresponding understanding as regards the given and
the requirement of the problem.
Procedures: 1. Students are asked
to create five types of questions from a reading assignment,
Generated Questions
272
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
with each question
moving to a higher level of thinking. Place
the questions on note cards to be passed and discussed or handed
in. 2. Students are then asked to write their
opinions regarding the thrusts of the problem:
the given and the required. 3. The instructor
checks and reinforces topics not understood
by majority of the class.
273
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Mathemati-
sing (dismal performance, insufficient recall)
No written
formulas or working
equations. Others wrote
30 +x = 2x as the working equation.
Concentric Circles –
small circle forms inside a larger one,
smaller circle discusses while the larger circle listens and
then roles are reversed.
Procedures: 1. The students shall
form 2 concentric circles. 2. On the first round,
the inner circle shall compose a problem.
The outer circle shall write the corresponding
equation or formula.
Student
responses
Recitation
274
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Encoding (carelessness, lack of criticality)
No unit.
3. On the 2nd round,
the tasks are reversed. 4. Discussion of
answers is a follow-up procedure.
Deck of Cards This is an interactive seatwork.
Procedures:
1. Students are asked to fold a sheet of paper, creating a card.
2. On the left part, a problem is written.
3. The students will exchange cards. 4. After, they will be
solving the given problems. The solutions will be placed
Student responses
Recitation
275
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
To
formulate expressions
To apply
concepts on linear equation in
Applied
Problems on Fare
Poor
Processing
(lack of practice, poor mastery, carelessness)
Reading
(insufficient recall, deficient mastery, poor exposure)
Incorrect
transposition; Others wrote x+2x = 30+10
instead of x-2x=30-20
Items
unanswered.
on the right portion of
the card. 5. Then, the card shall
be given back to the owner for checking. 6. The students will
determine if the answers are correct or not.
7. Students who did not get the answer
correctly shall be helped by the person who gave the problem.
Re-teaching
This is a form of instruction where the instructor re-teaches
the topic but with a different strategy.
Student
responses Recitation
Seatwork
276
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
word
problem
Procedures:
1. The teacher, after assessing students’
difficulties, plans for re-teaching.
2. The teacher has to alter the strategies like using math websites or
instructional games.
3. The target of the intervention is to refocus the skill and
target the difficulties.
277
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Comprehen-
sion (poor exposure, lack of skills)
No tabular
representa-tions or
representation of the missing data.
Idea Spinner
This is an interesting way to elicit students’
knowhow and knowwhy of the subject matter
Procedures: 1. instructor creates a
spinner marked into 4 quadrants and labeled:
Predict, Explain, Summarize and Evaluate.
2. After new material is presented, the
instructor spins and asks a student to respond accordingly.
3. Redirection is done when necessary.
Student
responses
Recitation Seat work
278
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Mathemati-
sing (dismal performance, insufficient recall)
No working
equation. Majority
applied guess and check.
Comprehension
Builder This is an instructional
strategy that helps students to structure their understanding of
the applied problems. Procedures:
1. Table utilization shall be demonstrated
to the class – how it is filled up completed. 2. After, the students
will be asked to do it independently or by
pair. 3. Redirection is done when necessary.
Completed
tables
279
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Encoding
(carelessness, lack of criticality)
No unit.
Find the Rule –
students are given sets of examples that
demonstrate a single rule and are asked to find and state the rule.
Procedures: 1. Students will be
presented with PowerPoint
presentations on algebraic expressions, formulas and working
equations per each problem.
2. The instructor models how each equation is derived for
the working equation. 3. After ample items, the students are asked
Student
responses
280
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Processing
(lack of practice, poor mastery, carelessness)
Incorrect
substitution; Instead of writing
(3(200-10y)/8) + 10y
= 150, they wrote ((200-10y)/8) + 10y
= 150.
to do it themselves.
4. Units of measure can also be included in
the presentations. Solution Inspector
This is an instructional activity that will lead students to be critical
about presented problems.
Procedures: 1. The instructor
presents the objectives of the activity and to
elicit from the class the job of an inspector. 2. The instructor
presents to the class solutions to applied problems.
Inspectors’
reports
281
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
To
formulate equations involving
verbal expressions
To solve problems in number
relation
Applied
Problems on Number
Relation
Fair
Mathemati-
sing (dismal performance, insufficient recall)
Incorrect
Working Equation. Others wrote
―x +x = 100 and x-x = 20
as the working equation.
3. As inspectors, they
will look into the errors of the solutions. They
will explain why that is an error and where did it start.
4. The students are asked to provide the necessary corrections.
Jumbled Equations
This activity elicits students’ prior knowledge on formulas
and working equation.
Procedures: 1. Instructor provides a randomly ordered set
of equations.
Student
responses Recitation
282
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Reading
(insufficient recall, deficient mastery, poor exposure)
No working
equation. Others
applied trial and error
No solution.
No
representa-tions for x
and y.
2. The students will be
asked if the equation is correct or not.
3. They will be asked to reorder the jumbled formulas and
equations based on the dictates of the word problems.
4. They will be asked to explain their
answers. Apply Structured
Reading Variable Basket
This is a variant of fishbowl method for formula selection but
used in improving comprehension.
Student
responses
283
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Comprehen-
sion (poor exposure, lack of skills)
They
committed transposition
errors in transposing y in x +y = 100
No unit. No indication of final
answer in the conclusion.
Procedures:
1. The instructor presents the objectives
of the instructional activity. 2. In the basket are
variables and the expressions where the variables are culled
out. 3. They will explain if
the representing variables are correctly written or not. They
will be asked to explain their answers and
correct the representations when necessary.
Recitation
284
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Processing
(lack of practice, poor mastery, carelessness)
Encoding
(carelessness, lack of criticality)
Problem solving
instruction: explicit instruction in the steps
to solving a mathematical problem including
understanding the question, identifying relevant and irrelevant
information, choosing a plan to solve the
problem, solving it, and checking answers.
Procedures: 1. The teacher
presents certain problems and how these items are solved
with different solution strategies.
Student
responses
Recitation Solution sheets
285
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
To
simplify exponents and radicals
correctly
G. Expo-
nents and Radicals
Very
Poor
Reading
(insufficient recall, deficient mastery, poor exposure)
Item not
answered.
2. The students chose
which among the strategies should they
use. 3. They solve individually but can
compare answers with their seatmates. They discuss their answers,
especially when their answers are different.
4. Board work can be a good assessment strategy.
Direct Instruction
with Paired reading This is a type of instruction that
focuses on the essentials or the specific skill that needs
Student
responses Recitation
286
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
To solve
exponential and radical
equations To solve
problems
involving radical and exponential
expressions and equa-
tions.
to be targeted.
Procedures:
1. The instructor focuses on how applied problems on sets and
Venn Diagrams are understood or solved. 2. He can use
technology in presenting the problem
or hand-outs. 3. The instructor pairs the students for
reading of the item assigned to them.
4. During the discussion of the item assigned to the
students, the instructor asks
287
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Mathemati-sing (dismal performance, insufficient recall)
Incorrect formulas/
working equation: √(2x+7) + 3x =
90 A = ½ (√(2x+7) – Sin
questions on how
students understood the problem. The
students can switch to the vernacular when not comfortable in
using English when explaining. Other students are asked to
give comments regarding the
understanding of the presenters.
Graffiti This is an instructional
activity that elicits students ideas on a certain problem.
Student responses
Recitation
Board work
288
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
3x)r2
Procedures:
1. An issue/question/ problem is indicated on flipchart paper and
there may be many in the room on tables. 2. As individuals or
groups (with different colored markers) the
students visit each station and write their opinions/answers/que
stions. 3. Sharing of ideas is
done. Redirecting and processing of answers will also be done.
289
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Comprehen-
sion (poor exposure, lack of skills)
Incorrect data
representation
Others wrote √(2x+7)/2 and 3x/2 instead
of √(2x+7) and 3x only.
Reading Corners
This is a variant of reading stations or
centers. Procedures:
1. Students are presented with word problems and different
options of solving the problem. Theses sets
are placed on the corners of the room. 2. Groups or batches
of students go to the corners and read the
problem. They will also select the best strategy listed
in the options column.
Student
responses
Recitation
290
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Processing (lack of practice, poor mastery, carelessness)
Incorrect deletion of the radical
symbol in √(2x+7) = 3x
without squaring.
3. After, the students
will be asked to share answers.
4. Discussion of answers will be done
Formula Derivation and Analysis This is a form of direct
instruction.
Procedures: 1. The instructor directly teaches
students on how to generate working
equation out of the given applied problems.
2. Demonstration and chunking of data will be done.
Student work sheets
291
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Encoding (carelessness, lack of criticality)
No unit Wrong
selection of the value of x.
3. Instructor asks the
students the area of the process where they
find difficulty. 4. Essentials will be stressed on the area of
difficulty. 5. Analysis and derivation of formulas
will be done. 6. Pairing can be done
so the students can help each other. 7. Activity sheets are
good supplemental materials.
Using data diagram This is a form of
tabular data collection.
Completed diagram
292
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Procedures:
1. Instead of tables, the teacher
demonstrates how to use diagrams in collecting data from
the given problem. 2. The teacher explain how the items are
translated into workable expressions.
3. The students are asked to do it.
Policy Recall
This is an interactive strategy that will lead students to critique
their own work.
Student
responses Recitation
Student responses
293
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
Procedures:
1. The students will be presented with rules
when square root is done, deleting radical symbols, squaring, etc.
2. The students will also be presented with correct and incorrect
application of policies. 3. In the correction of
policies, students will be asked to give comments on what
policy is violated. 4. They can also be
paired for supplement or assistance. 5. The students will be
given task to perform
Recitation
Quiz
294
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
operations involving
radicals and exponents.
Direct Instruction focusing on the
essentials This is an integrative approach of teaching
students emphasizing on the essential
aspects, specifically their point of error.
Procedures: 1. Using the results of
the assessment, the instructor plans direct instruction focusing on
key elements in data
295
Specific
Objectives
Topics Level of Perform-
ance
Error Cate-
gories and Theorized
Causes
(arranged according to
degree of
error)
Sample Error Interventions,
Process, Activities
Assessment
Strategy
selection, conclusion,
simplification and unit utilization.
2. The students will be given the task to solve
something. 3. The instructor
monitors and checks immediately on
students’ errors.
Two-day Seminar-Workshop on the Utilization of the Instructional Intervention Plan
I. Rationale
The challenges of teaching mathematics to students of the 21st
century are not easy. One challenge is the lowering performance of
students in Mathematics. But, the most pressing according to
Egodawatte (2010) are the causes why students really fail in their
performance in mathematics.
Educational experts contend that teachers, for them to be effective,
must not only know what to teach; they must also know how to teach.
Instructional strategies and principles of teaching are very necessary in
the life of the teacher. But, how can these instructional strategies or
interventions be known and mastered by the instructors, especially if
they are not sent to seminars and are attending graduate school, more so
with instructors who are not graduates of a teaching course? Indeed,
seminar-workshop is necessary.
Seminar-workshop is the process of acquiring specific skills to
perform a better job. It helps people to become familiar with essential
tools and elements necessary for them to be more effective. Through
seminar-workshop, people’s behavior towards a task becomes modified.
Such modified behavior contributes to the successful attainment of goals
and objectives.
296
297
The proposed two-day seminar-workshop is based upon the
foremost constraints and error categories of the students in College
Algebra. Their foremost error categories are along reading and
mathematising.
II. General Objectives
1. Improve instructional pedagogical competencies; and,
2. Apply and adopt the different instructional interventions.
III. Seminar-Workshop Course Contents
Instructional Interventions on the different error categories
IV. Methodologies
Participative Lectures and demonstration will be the main
methodologies of the seminar-workshop.
V. Facilitators
The facilitators for the proposed seminar-workshop were chosen
based on their extent of involvement in the research, qualifications,
trainings and seminars attended and organized.
Name Position/Extent of Involvement in the Research
Qualifications
Feljone G. Ragma Instructor Proponent
Ed.D
298
Nora A. Oredina Professor Proponent’s Adviser
Ed.D
Lea L. de Guzman Professor
Ed.D
Jovencio T. Balino Professor
External Evaluator
Ed.D.
VI. Participants
All mathematics instructors in the Higher Education Institutions
(HEIs) of La Union
VII. Duration
Two consecutive Saturdays: April 14 and 21, 2014 (refer to the
proposed program of activities)
VIII. Logistics
Registration fee (P500 per participant)
e.g. 50 participants P 25,000.00
Expenses
Honoraria for speakers (2,500/speaker) P 10,000.00
Meals/Snacks for the speakers (P250/speaker) P 1,000.00
Meals/Snacks for the speakers (P250/participant) P 12,500.00
Certificates and kits P 1,500.00
IX. Success Indicator
The mathematics instructors of the Higher Education Institutions
of La Union shall be able to utilize the instructional interventions.
299
SAMPLE FLYER OF THE TWO-DAY SEMINAR-WORKSHOP
300
Level of Validity of the Instructional Intervention Plan
Table 19 shows the level of validity of the instructional intervention
plan. It shows that the level of validity of the intervention plan is 4.51,
interpreted as very high validity. This means that the instructional
intervention plan is very highly functional, acceptable, appropriate,
timely, implementable and sustainable. It further implies that the plan is
a very good material that can address the dismal performance and the
different error categories.
Table 19. Level of Validity of the Instructional Intervention Plan
Criteria Validators Mean
A B C D E
I. Face 3 5 4 4 4 4.0
II. Content
a. Functionality
5
4
5
5
5
4.80
b. Acceptability 4 5 5 4 4 4.40
c. Appropriateness 5 5 5 4 5 4.80
d. Timeliness 3.5 5 5 4 5 4.50
e. Implementability 3.5 5 5 5 4 4.50
f. Sustainability 4 5 5 4 4 4.60
Average 4.0 4.86 4.86 4.29 4.43 4.51
301
CHAPTER IV
SUMMARY, CONCLUSIONS AND RECOMMENDATIONS
This chapter shows the summary, findings, conclusions and
recommendations of the study.
Summary
The study identified and analyzed the error categories of students
in College Algebra in the Higher Education Institutions of La Union as
basis for formulating a Validated Instructional Intervention Plan.
Specifically, it determined the level of performance of the students in
College Algebra along elementary topics, special products patterns,
factoring patterns, rational expression, linear equations in one unknown,
systems of linear equations in two unknowns and exponents and
radicals; the capabilities and constraints of the students in College
Algebra; the error categories of the students along reading,
comprehension, mathematising, processing and encoding errors; and the
validated instructional intervention plan.
The study is descriptive with a researcher-made College Algebra
test as the instrument of the study. The test was administered to 374
students of the HEIs in the province of La Union for 1st semester of the
school year 2013-2014. The data collected were treated using frequency
count, mean, rate and the Newmann’s error analysis tool.
302
Findings
The researcher found out the following:
1. The students had fair performance in elementary topics, special
products and factoring while poor performance in rational expressions,
linear equations and systems of linear equations and very poor
performance in exponents and radicals. The students had a general
performance of poor.
2. The performances of the student in the specified topics were all
considered as constraints.
3. Mathematising and comprehension were the major error
categories of the students in elementary topics, processing and reading
errors in special products, reading and Mathematising in factoring,
reading and Mathematising in rational expressions, reading and
comprehension in linear equations, and reading and Mathematising in
systems of linear equations and exponents and radicals. In general, their
major error categories in College Algebra were along reading and
Mathematising.
4. The instructional intervention plan is very highly valid.
Conclusions
Based on the findings of the study, the following are concluded:
303
1. The students cannot competently deal with elementary
topics, special product and factoring patterns, rational expressions,
linear equations, systems of linear equations and radicals and
exponents.
2. The students are deficient in terms of their skills of the
topics in College Algebra.
3. Majority of the students cannot start the problem-solving
process which leads them not to successfully finish all the stages of
problem solving.
4. The instructional intervention plan is a very good material
that can address the dismal performance and errors of the students.
Recommendations
Based on the conclusions of the study, the following are humbly
recommended:
1. The schools should adopt the Instructional Intervention Plan
and let their mathematics instructors attend the two-day seminar-
workshop.
2. The students should exert more effort in understanding the
different concepts in their College Algebra course. They should spend
more time in dealing with drills and exercises rather than dealing with
social media and entertainment.
304
3. The instructional interventions plan should be used not only
in the province of La Union but in all schools experiencing the same
student error patterns in College Algebra.
4. The mathematics teachers should suit their teaching
strategies based on their students’ needs and interests.
5. The English teachers should intensify the development of the
students’ skill of reading with comprehension in their classes.
6. A study should be conducted to determine the effectiveness
of the instructional intervention plan.
7. A similar study should be conducted in other branches of
Mathematics, applied sciences and English, especially in the subjects
where percentage of failures is high.
305
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APPENDIX A Sample Computations
Sample Computations on the:
Reliability of the College Algebra Test of
College Algebra Test
Validity of College Algebra Test
List of Suggestions Made by the
Validators and the Corresponding
Action/s by the Researcher
313
Sample Computation of Reliability of the College Algebra Test
For the College Algebra Test
Scores:{38,39,29,25,28,38,41,37,33,36,28,9,9,30,34,34,24,29,20,19,41,22,27,27,21,22,19,26,27,24}
Data from StaText:
k=100
k-1=99
𝑥 = 27.87
𝜎2=69.77
𝐾𝑅21 = 𝑘
𝑘 − 1 1 −
𝑥 𝑘 − 𝑥
𝑘𝜎2
𝐾𝑟21 = 100
99 1 −
27.87(100 − 27.87)
100 (69.77)
KR21= 0.71906
KR21 ≈ 0.72;high reliability
314
Sample Computation on the Validity
of College Algebra Test
Criteria Validators Mean
1 2 3 4 5 6
1. The directions are clear
and specific and do not warrant misconceptions among
students.
3 5 5 5 3 4 4.17
2. The sentences are free
from grammatical errors and other construction lapses.
5 4 4 5 2 3 3.83
3. The test items are clearly and specifically
formulated based on student’s level of
understanding.
5 4 4 5 4 4 4.33
4. Mathematical expressions and
equations are encoded clearly to avoid student
misunderstanding.
4 4 4 5 5 3 4.17
5. There are provisions for
students to show their solutions.
4 5 5 4 5 4 4.50
6. The test items cover the
course content as indicated by the table
of specifications.
4 5 5 5 5 4 4.67
7. The test items are
written to cull out the specific errors of students in College
Algebra.
4 5 5 5 4 3 4.33
8. Generally, the test
items represent what they ought to measure.
4 5 5 5 4 4 4.50
Overall 4.13 4.63 4.63 4.88 4.00 3.63 4.32
315
List of Suggestions Made by the Validators and the Corresponding
Action/s by the Researcher
Suggestions Remarks
The 30-item test cannot be accomplished by the students in the specified time frame.
Incorporated The 30 items were reduced to 20
items only; however, the researcher saw to it that the scope of the
College Algebra test still covered the specified scope of the syllabi. For example, instead of separate
items for sets and Venn Diagrams, an item was constructed to deal
with these 2 related topics; instead of separate items for addition, subtraction, multiplication and
division of polynomials and rational expressions, an item that conglomerates the four basic
operations was formulated.
An item is solved by a student in at most 3 minutes.
Provide more space for the students to show their answers.
Incorporated More spaces were provided for the
students to clearly and completely show their solutions.
Delete the line for working equation and illustration since it will take much of the student’s time;
anyways, these will be reflected when they start writing their
preliminary steps for the solutions. This will also give the students the freedom of what specific strategy to
use in solving the given word problems.
Incorporated The provisions for working
equations and illustration were deleted.
316
Suggestions Remarks
Emphasize on the instructions that the students need to show their complete solutions.
Incorporated The instruction on showing the
complete solutions and the non-utilization of calculators was made
bold and of bigger font size.
Check on some lapses on grammar. Incorporated
Grammar lapses were checked.
Add more spaces between and
among numerical coefficients, variables and their exponents for
clarity.
Incorporated
Spaces were provided between and
among the numerals, variables and their exponents.
Some data need to be more realistic.
Incorporated Some data were changed to be
more realistic. Instead of a problem on a concert, a problem on fare in a jeep was written to replace the said
item.
The scoring scheme should be
revised, in consultation with the adviser, so that an item will not
just be 1 point. The points should be distributed along the different levels specified along the error
categories.
Incorporated.
The scoring scheme was revised.
Please see data categorization.
APPENDIX B Research Tool
Letter to Students-Respondents to Administer
the College Algebra Test
The College Algebra Test
Math I - College Algebra Test (Table of Specifications)
317
SAINT LOUIS COLLEGE
City of San Fernando, La Union
GRADUATE SCHOOL
September 2013
My dearest students,
The undersigned is a Doctor of Education Major in Educational
Management (Ed.D-EdM) student of Saint Louis College undertaking the
study entitled, ―Error Analysis in College Algebra in the HEIs in La
Union.‖ It is with this cause that your support is sincerely solicited so
that this study can be carried out and may greatly contribute to the
improvement of the teaching-learning process.
Please lend an hour to answer this word problems set. It may take
much of your precious time but your answers to these problems will
contribute much to the success of this study.
Rest assured that all information obtained herein will be held
strictly confidential. Your immediate attention to this request is highly
cherished.
Thank you so much!
Sincerely yours,
Mr. Feljone G. Ragma
Researcher/ Ed.D. student
318
COLLEGE ALGEBRA TEST
Name (optional)_______________________________School:____________________________________
INSTRUCTIONS: Please read, analyze and solve the problems that follow. Please indicate all
information being asked in the given problems on the test sheets.
PLEASE SHOW ALL SOLUTIONS. NO USING OF CALCULATORS!
1. 250 customers were asked in a survey as to what cell phone brands they like the most. The results
reveal that 160 chose Samsung, 150 chose Nokia and 180 chose iPhone, 75 chose Samsung and Nokia,
100 chose Samsung and iPhone, 90 chose Nokia and iPhone. 20 customers choose all the 3 brands. How
many love other brands?
Given data:
Solution:
2. What is the sum of the distance of 7 from -2 and 10 from 8 on the number line?
Given data:
Solution:
3. The base of a right triangle is expressed as (2x-5) cm and its height is (x+9) cm more than the base,
what is its area in cm2?
Given:
Solution:
319
4. Juan de la Cruz finds out that his money is expressed in (x4-1) pesos. If he wantsto buy (x+1)ice cream,
how many ice cream can he buy?
Given:
Solution:
5. Don Mario is choosing between lots A and B. Lot A is (3x2-5) square meters sold at P (3y+4) per square
meter while lot B is (2x2+45) square meters sold at P (5y+2). If x = 10 and y = 2, which is cheaper?
Given:
Solution:
6. The radius of a circular table is expressed as (2x-4y+6z) cm, what is its area in cm2?
Given:
Solution:
7. The side of a cube measures (2x+4) cm, what is its volume?
Given:
Solution:
320
8. The area of a rhombus is (2x2-162) square units. If one of the diagonals measures (x-9), what is the
measure of the other diagonal?
Given:
Solution:
9. The area of a square garden is expressed as (4x2-20x+25) meters2. What is the measure of its side?
Given:
Solution:
10. A string measuring (x2+3x-40) cm is divided into 2 parts. If one part measures (x+8)cm, what is the
measure of the other part?
Given:
Solution:
11. A truck has (x2+2xy+y2+x+y) loads of stone to be delivered to 2 customers. If the first customer shall be
delivered (x+y) loads only, what is the share of the second customer?
Given:
Solution:
321
12. Aling Maria wishes to buy 12𝑥4𝑦6
7𝑥𝑦 kilo of tomatoes for
21
6𝑥3𝑦5 pesos per kilo. How much will she pay?
Given:
Solution:
13. Agnes has 1
2𝑥 pieces of 5-peso coin and
8𝑥
2 of 10-peso coin. What is the product of the 5-peso and 10-
peso amounts?
Given:
Solution:
14. The interest of an amount invested in a bank at simple interest is 1/6 of 12,000. If the rate is at (1-1/3), how much is the principal investment?
Given:
Solution:
15. Two vehicles travel at the same time but in opposite directions. Vehicle A runs at 120 kph while vehicle B runs at 100kph. After some time, their distance from each other is calculated to be 440 km. What is the distance traveled by each of the two vehicles?
Given:
Solution:
322
16. Feljone has 27 bills consisting of 20-peso and 50-peso bills, If he has a total of 990 pesos, how many 20-peso bills does he have?
Given:
Solution:
17. Lorna is 20 years older than her daughter, Rudylyn. In ten years, she will be twice as old as her
daughter, how old is Rudylyn?
Given:
Solution:
18. The fare for a jeepney was P200 for 8students and 10regular passengers. The fare, on another day,
was P150 for 3students and 10regular passengers. How much was the fare for a regular passenger?
Given:
Solution:
323
19. The sum of 2 numbers is 100 while their difference is 20. What are the two numbers?
Given:
Solution:
20. An angle bisector divides an angle into 2 equal parts. If one of the equal angles measures ( 2𝑥 + 7)˚
while the other measures (3x)˚, what is the measure of one the smaller angles?
Given:
Solution:
MATH 1 - COLLEGE ALGEBRA TEST
TABLE OF SPECIFICATIONS
TOPICS TOTAL
HOURS KNOWLEDGE
COMPREHENSION
REMEMBERING
UNDERSTANDING
ANALYSIS APPLICATION
ANALYZING
APPLYING
SYNTHESIS EVALUATION
EVALUATING
CREATING
ITEM PLACEMENT
TOTAL ITEMS
PRELIMS 15 7 7
Elementary Topics - Sets and Venn Diagrams - Real Number System - Algebraic Expressions - Polynomials
8 4 1-4 4
Special Products and Patterns - Product of 2 polynomials - Square of a Trinomial - Cube of a Binomial
7 3 5-7 3
MIDTERMS 15 7 7
Factoring - Difference of 2 Perfect Squares - Perfect Square Trinomial - General Trinomial
8 4 8-11 4
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313
- Factoring by grouping
Rational Expressions - Simplification of RAEs - Operation on RAEs - Simplification of Complex RAEs/ fractions
7 3 12-14 3
FINALS 15 6 6
Linear Equations in One Variable Applied Problems on: - Distance - Mixture/Money/Coin - Age
6 3 15-17 3
Systems of Linear Equations Applied Problems on: - Fare/Price - Number Relation
6 2 18-19 2
Exponents and Radicals - Exponential and Radical expressions and equations
3 1 20 1
45 30 20
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APPENDIX C Communications
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331
332
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APPENDIX D Sample of Corrected College Algebra Test
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Legend: Guide to Checking 5 pts. – No Error 4 pts. - Encoding Error (EE) 3 pts. – Processing Error (PE) 2 pts. – Mathematising (ME) 1 pt. – Comprehension Error (CE) 0 pt. – Reading Error (RE)
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351
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CURRICULUM VITAE
PERSONAL DATA
Name: Feljone Galima Ragma
Date of Birth: July 31, 1986
Place of Birth: San Isidro, Candon City, Ilocos Sur
Home Address: San Isidro, Candon City, Ilocos Sur
e-mail Address: [email protected]
Civil status: single
EDUCATIONAL ATTAINMENT Pre-Elementary: UCCP
Candon City, Ilocos Sur Graduated 1991 With honors
Elementary: Candon South Central School Candon City, Ilocos Sur Graduated 1997
With honors Secondary: Santa Lucia Academy
Santa Lucia, Ilocos Sur Graduated 2003 With honors
Tertiary: Saint Louis College San Fernando City, La Union Graduated 2007
Bachelor in Secondary Education Cum Laude
Major in Mathematics Recognition Award
Graduate Studies: Saint Louis College
San Fernando City, La Union Graduated 2011
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Master of Arts in Education Cum Laude Major in Mathematics Best in Research
Post-Graduate Studies: Saint Louis College
San Fernando City La Union Graduated 2014
Doctor of Education Magna Cum Laude Major in Educational Best in Research
Management
BOARD EXAMINATION/ CIVIL SERVICE ELIGIBILITY
Licensure Examination for teachers (LET) 2007
P.D. 907 Civil Service Eligible
WORK EXPERIENCE, POSITIONS/SPECIAL ASSIGNMENTS
School/Institution Position Inclusive Dates
Saint Christopher Academy Classroom Teacher 2007-2008 Bangar, La Union
Christ the King College Classroom Teacher 2008-2013 San Fernando City, La Union Subject Area Coordi- 2010-2013
nator
Saint Louis College Instructor I 2013-2014 City of San Fernando, La Union
OTHER WORK-RELATED EXPERIENCES
Adviser and Panelist, Graduate School Researches Saint Louis College
City of San Fernando, La Union
External Evaluator, Undergraduate Researches
Saint Louis College City of San Fernando, La Union
356
Review Facilitator, Civil Service Exam
Dacanay Hall San Fernando City September-October, 2013
TRAININGS/SEMINAR-WORKSHOPS FACILITATED
January 2014
Giving Feedback to Improve Student’s Learning and Behavior Christ the King College City of San Fernando, La Union
2013
Seminar on How to Love and Like Mathematics Saint Louis College City of San Fernando, La Union
Back to Basics of Test Construction
Christ the King College City of San Fernando, La Union June 29, 2012
Understanding by Design and K-12 Christ the King College
May 20, 2012
Problem-Solving Techniques in Secondary Mathematics Association of Private Schools City of San Fernando, La Union
July, 2010
Seminar-Workshop on Creating Gradebooks through MS EXCEL Christ the King College City of San Fernando, La Union
2009 Seminar-Workshop on Creating Interactive Slides through MS
PowerPoint Christ the King College
City of San Fernando, La Union 2009
357
CONFERENCES/ SEMINARS PARTICIPATED
Engaging Learners in the Mathematics Classroom Saint Louis College September, 2013
Sustainability in the Classroom Saint Louis College
September, 2013
Colloquy in Thesis Advising Saint Louis College September, 2013
International Education Conference: How to be an Effective and
Successful Teacher SMX Conventional Hall, Pasay City August, 2012
Formative Assessment in the K-12 Curriculum Christ the King College
August 3-4, 2012
Mathematics Trainer’s Guild Seminar on Singaporean Math Association of Private Schools July 6-7, 2012
Moving Forward with Backward Design: A Deeper look at UBD Saint Louis University Laboratory Elementary School
January, 2011
Understanding and Planning for the 2010 SEC for Mathematics Phoenix Hall, Quezon City
November, 2010
Training Program for Mathematics Teachers University of the Cordilleras September, 2010
358
Seminar on Yoga and Relaxation Christ the King College
August, 2010
Critical Questions to Elicit Critical Thinking Christ the King College July, 2010
Seminar-Workshop on Homeroom Guidance and Counseling Techniques
Christ the King College June, 2010
Utilizing and Interpreting CEM Test Data University of Baguio
May, 2010
In-Service Training and Workshop on Curriculum Programs and Teaching Strategies Christ the King College
November, 2009 Seminar on Innovations in Teaching and Learning Approaches
Christ the King College July, 2009
Understanding and Planning for the SEC Phoenix Hall, Pangasinan
September, 2009
PROFESSIONAL ORGANIZATION
National Organization for Professional Teachers (NOPTI)
PAFTE
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