PROBLEM SOLVING SKILLS OF SHS STUDENTS IN GENERAL MATHEMATICS
A Research Proposal Presented to the Candon National High School Senior High School Department In partial fulfilment of the requirements for the subject Research 2 (Quantitative Research)
By: BAWAS, JHOANCEL CAGAS, CHRISTOPHER LLOYD O. FELICIANO, MARJORIE A. GABOR, ANGELO RAVEN P. GALUTAN, JOHN JOSHUA T. GANNAPAO, JOHN LESTER M. GARNACE, ALLEAH T. MOLINA, MARK JOSEPH G.
Feljone G. Ragma Ed.D (Research Adviser)
2
Republic of the Philippines Region 1 Candon National High School Senior High School Candon City, Ilocos Sur
INDORSEMENT This is to certify that the researchers who conceptualized the study “Problem Solving Skills of SHS Students in General Mathematics” are ready for Oral Examination.
Feljone G. Ragma, Ed.D. Adviser Republic of the Philippines Region 1 Candon National High School Senior High School Candon City, Ilocos Sur
APPROVAL This
is
to
certify
that
the
abovementioned
study
has
SUCCESSFULLY PASSED the Oral Examination on October, 2017 before the following, whose signatures are accordingly affixed.
MR. FILOMENO VALDEZ Panel Member
3
ACKNOWLEDGEMENT In appreciation to the support given to this quantitative research study entitled: PROBLEM SOLVING SKILLS OF SHS STUDENTS IN GENERAL MATHEMATICS, the researchers would like to thank the people behind the success of the study. Specifically, the researchers would like to thank the following people: Dr. Feljone G. Ragma, the researchers adviser, co-author of the said research, as well as the chairperson of the defense panel for nurturing the researchers and supporting the study which led to its completion. Mr. Filomeno Valdez, member of the defense panel, for their contextual criticisms, in-depth evaluation, and immeasurable assistance for the finalization of the study. The grade 12 Senior High School students, for being part of the study by taking the test to bring out necessary data for the research. Lastly, the researchers would like to thank the respondents for sharing their knowledge with the researchers. The study cannot be accomplished without the help of the respondents.
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DEDICATION This study is dedicated to our teacher in Research 2, Dr. Feljone G. Ragma, who inspired us to persevere in accomplishing this research and acted as a model and a friend who is always there to help us in accomplishing this research. He taught us how to conduct a research specifically a Quantitative Research. He was always there to help us on the things that we do not understand about research. He installed to our minds that time is precious and procrastination is something that must be avoided. He was facilitating us and ensuring that we reach the peak of success in this research.
5
ABSTRACT Title: Problem Solving Skill of SHS Students in General Mathematics Researchers:
BAWAS, JHOANCEL CAGAS, CHRISTOPHER LLOYD O. FELICIANO, MARJORIE A. GABOR, ANGELO RAVEN P. GALUTAN, JOHN JOSHUA T. GANNAPAO, JOHN LESTER M. GARNACE, ALLEAH T. MOLINA, MARK JOSEPH G.
Adviser:
Feljone G. Ragma, Ed.D.
Abstract: The research aimed to determine the problem solving skills and competency of senior high school students of Candon National High School in General Mathematics. The quantitative research made used of descriptive and comparative approaches and utilized problem solving test to gather the data. The gathered information was deduced through Frequency and Kruskal-Wallis H-Test to identify the skills of SHS students. There were 100 respondents from senior high school along different strands: Science Technology Engineering and Mathematics (STEM), Accountancy and Business Management (ABM), Information Communication Technology (ICT) and Humanities and Social Sciences (HUMSS). The test was composed of 10 items that covers different topics in General Mathematics namely: Functions, Rational functions, equations and inequalities, Inverse functions, Exponential functions, equations and inequalities, Logarithmic Functions, equations and inequalities, Simple and compound interests and Consumer and Business Loans. After the conduct of the study, the researchers found out that the level of problem solving skills of SHS students in General Mathematics is not satisfactory. The respondents have not mastered the topics/competencies in General Mathematics thereby reflecting a very poor problem solving ability. The researchers also found out that there is no significant difference among the different strands when it comes to their problem solving skills. It recommended, among others, that concerned authorities should make use of the research data to create proper action to lessen or to avoid the very low problem solving ability of SHS students in General Mathematics.
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TABLE OF CONTENTS Page Number TITLE PAGE………………………………………………………….……….
i
INDORSEMENT……………………………………………………….…….
ii
APPROVAL SHEET………………………………………………………….
ii
ACKNOWLEDGEMENT…………………………………………………….
iii
DEDICATION…………………………………………………………………
iv
ABSTRACT……………………………………………………………………
v
TABLE OF CONTENTS..................................................................
vi
FIGURE........................................................................................
viii
TABLE…………………………….……………………….........................
ix
CHAPTER I
1
II
Introduction…………………………………….………….. Background of the Study………………....……………
1-6
Framework of the Study.......................................
6-12
Statement of the Problem…………..………………....
6-13
Hypothesis…...……………………………………..……
13
Scope and Delimitation...…………………………..….
14
Importance of the Study…………………………..…..
14-15
Definition of Terms…………..……………………..…..
15-17
Review of Related Literature and Studies………….
18-36
Method and Procedures…..………………………..……
37
Research Design…….……………………….…...…….
37-38
Sources of Data….……………………………….……..
38
7
Locale and Population of the Study……...….…………… Instrumentation and Data Collection…….....…...........
III
IV
38-40 40-41
Validity and Reliability...............................................
41
Tools for Data Analysis...............................................
41-42
Data Categorization....................................................
42
Ethical Considerations..............................................
43
Findings and Discussion…………………………………..
44
Level of Problem Solving Skills....................................
44-47
Comparison of the Level of Problem solving skills of SHS students according to strands.............................
47
Comparison of Strands...............................................
48-49
Summary, Conclusions and Recommendations......
50
Summary...................................................................
50
Findings.....................................................................
50
Conclusions...............................................................
51
Recommendations......................................................
51-52
Bibliography……………………………………………………………………
53
Appendices A......................................................................................
56
B.....................................................................................
60
C.....................................................................................
64
Curriculum Vitae.........................................................................
69
8
FIGURE Figure 1
Page Bloom’s Hierarchy Theory .............................................
11
9
TABLE Table
Page
1
Population Distribution .................................................
39
2
Level of Problem Solving Skills of SHS Students............
45
3
Comparison of Strands………………………………………..
48
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CHAPTER I INTRODUCTION Background of the Study In order for man to reach his goal, making things in order and preventing chaos would probably help. People are aligned with different professions such as being a mechanic, an engineer, a doctor, a scientist, a musician and even just being a living thing. Every single day, people develop their power of reasoning, creativity, abstract or spatial thinking, critical thinking, problem-solving ability and even effective communication skills through Mathematics. Mathematics is a part of the accumulated knowledge of the human race. It is a subject that deals with problems which involve a process of analysis, computation and other mental skills.
The Merriam-Webster
dictionary (2016) defines Mathematics as the science of numbers and their operations,
interrelations,
abstractions
and
of
space
combinations, configurations
generalizations, and
their
and
structure,
measurement, transformations, and generalizations. Mathematics therefore must be taught and learned comprehensively and with much depth. It should be given great importance because of its infinite applications in almost all walks of life. In a world advanced by
11
mathematical and technological breakthroughs; neglecting it would definitely cause a downfall in our society. The Implementation of the new K to 12 Basic Education Program through the Enhanced Basic Education Act of 2013 puts emphasis on the study of Mathematics. According to the Department of Education or DepEd (2015) the new curriculum provides a solid foundation and deeper understanding in Mathematics to students. More importantly, it prepares students for Global future and provides them the best possible tools and career choices after High School. It aims to provide necessary concepts and life skills needed by Filipino learners as they proceed to the next stage in their life as life-long learners and as citizens of the country. Under the SHS there are only two branches of mathematics as core subjects to be undertaken - Statistics and Probability and General Mathematics (Commission on Higher Education K-12 Transition Program, 2016). General Mathematics focuses on the use of mathematics to solve problems in contexts (University of Calgary, 2017). It aims to provide students with the knowledge of mathematics in its breadth by drawing its courses from the different mathematical divisions: Pure and Applied Mathematics, Statistics and Actuarial Science. In addition to acquiring basic mathematical skills, students are free to explore the manifold areas
12
of this discipline to gain exposure to a variety of modern mathematical subjects. Thus, General Mathematics enhance the set of valuable skills of students reasoning,
that
include
practical
calculation
and
techniques,
probabilistic
logical
(inductive)
(deductive)
reasoning
and
formulation of mathematical problems from real-world scenarios. Studying this discipline develops students’ natural ability to work with numbers, shapes and figures, reason in a logical manner and be engaged with problem solving. Problem solving has been and will be a necessary skill not only in Mathematics but in everyday living. Polya (1981) stated that problem solving is a process starting from the moment the students are faced with the problem until the end when the problem is solved. According to Ibrahim (1997) there are two main procedural steps in problem solving; i) transforming
the
problem
into
mathematical
sentences;
and
ii)
computation of the operational involved in the mathematical sentences. According to Stendall (2009) the abilities to give good concentration, to make meaningful perceptions, to think logically and to use memory effectively are important factors in learning skills and solving problems. Conceptual understanding and procedural knowledge are essential skills in problem solving (Geary 2014). These Problem solving skills should be
13
supported by cognitive systems that control focus and interference in information processing. Apart from that, language and visual-spatial skills are also important to interpret and manipulate information effectively in the working memory. However, many students struggled to accomplish Mathematics especially in problem solving. In Malaysia, studies showed that students felt
difficult
in
problem
solving
because
they
had
difficulty
in
understanding and retrieving concepts, formulas, facts and procedures (Zahrah et al. 2013) and lacked the ability to visualize Mathematics problems and concepts (Tarzimah 2005). In Thailand, the results of the National Examination on Mathematics achievement of nationwide sixth grade students between 2007-2009 shows that most students are in an improvement-needed level (NIETS, 2012). In the results of Program for International Student Assessment (PISA) in the year 2012 when it comes to Mathematical Problem Solving the countries of Montenegro (0.8%), Colombia (1.2%), Uruguay (1.2%), Bulgaria (1.6%) and Brazil with fewer than 2% of students perform at level 5 or 6; and all of these countries share low achievers and perform well below the OECD average. Despite the importance of Mathematics, studies revealed an alarming performance in the Philippine schools in terms of Science and Mathematics both in the national and international. According to a global
14
survey the Philippine ranked 115th out of 142 countries in perceived quality of Math and Science education. These results are based on WEF or World Economic Forum’s Global Competitiveness Report for 2011-2012. According to Jalmasco (2014), the Philippines ranked lowest among 10 countries even with only the science high schools participating in the Advanced Mathematics category in 2008. In the Locality of Candon City, home of the researchers, low performance of students in Mathematics was observed. Students were not able to cope with their daily lessons. As a result, they are having hard time in applying what they have learned in their daily life. And because of that, during the performance tasks, particularly in Mathematical Problem solving they face difficulties which would possibly result to a failing grade. In Candon National High School, General Mathematics was already taken by the SHS students particularly in Grade 12. One of the problems faced is that, are there lessons retained on them and have they really understand something. This Research study aims to determine the capabilities and competence
of
SHS
students
in
solving
Mathematical
problems
particularly in General Mathematics. Also, this aims to test if they really understood their lessons by applying it on a Mathematical Problem
15
solving. And lastly, this aims to know the difficulties faced by the students in solving Mathematical problems. Framework of the Study The framework of this study is aligned to the following underlying learning theories, principles, and concepts. Problem Solving is not a topic but a process underlies the whole mathematics programmes which contextually helped concepts and skills to be learned (Ibrahim, 1997). Thus, the K-12 Basic Education Curriculum targets to develop and improve the knowledge and ability of students to improve their knowledge particularly their skills in solving mathematical problems. According to the curriculum guide of K-12 Basic Education in General Mathematics (2016), there are three contents to be taken by the students namely: Functions and their Graphs, Basic Business Mathematics and Logic. The different contents include learning competencies that students need to learn and develop. The first content of the curriculum guide (2016) covers the different functions and their graphs which include the different competencies such as (a) Functions (b) Rational functions, equations and inequalities (c) inverse functions, equations and inequalities (d) exponential functions, equations and inequalities; and (e) logarithmic functions, equations and inequalities. The second content covers the basic business mathematics
16
which includes (a) simple and compound interests (b) stocks and bonds; and (c) Business and Consumer Loans. And lastly, the third content covers (a) Simple and Compound Propositions (b) Syllogisms and Fallacies; and (c) Proof and Disproof. These contents and competencies target to improve the students’ mathematical abilities such as their problem solving skills. In addition, Problem solving is an important means of developing mathematical thinking as a tool for daily living because it lies at the heart of mathematics (Cockcroft, 1982). This is parallel to Theory of Learning in Constructivism (Piaget, 1936), which has a special importance in learning and teaching mathematics. It is an important component of mathematics education because it is the single vehicle which seems to be able to achieve at school level all three of the values of mathematics; logical, functional and aesthetic (National Council of Teachers of Mathematics, 2006). Ideas are constructed or made meaningful when children integrate them into their existing structures of knowledge (Clements and Battista, 1991). Thus, the theory of learning in constructivism is a psychological theory of knowledge, which argues that human generate knowledge and meaning from their experiences. It also enhances students’ logical and conceptual growth (Piaget, 1936). It argues that people produce their knowledge and form meaning through their experiences. This is where to
17
test their knowledge on Mathematical problem solving based on what they have learned and understand. In connection to the study, the learners are provided with activities that are integrated with their existing knowledge in which they can detect the connection between what they have learned. For instance, when the teacher gives an assessment like quizzes, activities or performance tasks to their students regarding Mathematical Problem solving, it enables the educator to measure the learning competency of a student. Furthermore, According to Thorndike (1926), learning becomes more effective when one is ready for the activity, practices what he has learned and enjoys the learning experience. As applied to Mathematical competency of students they cannot solve mathematical problem solving as an application to what they have learned if they did not understand and enjoyed the learning process. This is anchored to Thorndike’s law of effect (1905) which states that, in learning, the more frequently a stimulus and response are associated with each other, the more likely the particular response will follow the stimulus. The law implies that one learns by doing or application such as Mathematical problem solving and not by just merely learning and listening.
18
Moreover, the philosophy of John Dewey, Charles Pierce, and William James, advocates that the aim of education is the total development of the child either through experience, self – activity or learning by doing. Learning is indeed done by doing. The learners should be active participants in the teaching – learning process. They should be provided with activities that will test their capabilities and competency which were considered in this study. This is further supported by Caine and Caine’s Brain – Based Learning theory which states that learning is a sense-making activity and that new knowledge is acquired relative to existing knowledge (Caine, 2012). Caine further argues that brain research confirms that multiple complex and concrete experiences are essential for meaningful learning and teaching. They add that the brain is designed as a "pattern detector" and that the function of educators should be to provide students with the kind of experiences that enable them to perceive "the patterns that connect." In connection to the study, students must have a personally meaningful challenge. Such challenges stimulate a student’s mind to the desired state of alertness such as of solving mathematical problems. In order for a student to gain insight about a problem, there must be
19
intensive analysis of the different ways to approach it, and about learning in general. Another theory that supports this study is Bloom’s Theory which was proposed by Benjamin Bloom in 1913. According to Bloom (1956) the cognitive domain involves the knowledge and the development of intellectual skills of a human being. Thus, his theory is about mastery of learning and higher level of thinking. This includes the recall or recognition of specific facts, procedural patterns, and concepts that serve in the development of intellectual abilities and skills. According to Bloom’s theory (1913) there are six major categories of cognitive process, starting
from
the
simplest
to
the
most
complex;
knowledge,
Comprehension, Application, Analysis, Synthesis and Evaluation. Bloom’s Taxonomy is essentially a hierarchy, with knowledge as the first level and evaluation as the sixth level. According to Bloom (1913) Knowledge is about recalling information and answering comprehension questions, Comprehension is about interpreting information, Application is about using knowledge gained to solve problems, Analysis is about breaking down concepts or ideas to understand the relationship of the parts to the whole, Synthesis is about putting together something original from learned information and Evaluation which is about judging something against specific criteria.
20
. Figure 1 Bloom’s Hierarchy Theory In application to the study, Bloom’s theory will help the educators in Mathematics to encourage students to pose their own questions and enhance their critical thinking skills. Critical thinking was defined as the ability to engage in reflective and independent thinking (Elena Caceres, 2014). Thus, presenting knowledge (first level of Hierarchy) will help the student to acquire concepts about Mathematics, Comprehension will enable the students to understand what they have learned on the knowledge presented to them by the educators. Application is one purpose of the learning process wherein the educators will be able to measure the knowledge acquired by the students by giving them quizzes, activities etc. Analysis will help the students to analyze or evaluate the learning they have acquired for examples are the formulas in problem solving, they will be able to solve the specific mathematical problem and perform different steps in solving it. Synthesis is synonymous to
21
Application but only it is higher form wherein the students are not just applying what they have learned but also they will share it to others and lastly. Evaluation (last level of hierarchy) in this stage the students judgment or interpretation on the knowledge they have acquired is the important tool for the educators to test how deep their understanding on the lesson about problem solving. In conclusion, Problem solving is the process of finding solutions to difficult questions or issues (Caceres, 2014). It is a higher order and active learning task that is important for students to develop. To learn and use mathematics, it requires mastery of computation. To master a skill of computation requires constant practice, repetition and drill (Ragma, 2011). Therefore, Mathematics must be taught and learned comprehensively and with much depth. Statement of the Problem This study intends to determine the Problem solving skills of SHS students in General Mathematics. Thus, the study seeks to give answers to the following questions.
1. What is the level of Problem solving skills of Grade 12 learners in General Mathematics along: a. Functions;
22
b. Rational Equations and inequalities; c. Rational Functions; d. Inverse Functions; e. Exponential functions; f. Exponential Equations and Inequalities; g. Logarithmic Functions; h. Logarithmic Inequalities; i. Compound Interest; and j. Business and Consumer Loans (Amortization and Mortgage)?
2. Is there a significant difference on the level of problem solving skills of Grade 12 learners when grouped according to their strands?
Hypothesis There is no significant difference on the level of problem solving skills of Grade 12 learners when grouped according to strands.
Scope and Delimitation The respondents of the study are the SHS students particularly the Grade 12 students of different strands: STEM, ABM, HUMSS, ICT in Candon National High School. The study will be conducted from the month of September until the last day of 1st semester in October 2017.
23
The researchers will be using a Problem Solving Test to determine the Problem solving skills of SHS students in General Mathematics.
Importance of the Study This study will greatly benefit the people, DepEd, teachers, student body, researchers and future researchers. This study will lend a hand to the people by providing information regarding problem solving skills and applying these as their daily basis. Also, this encourages people to believe in their ability to think mathematically. This study will help the DepEd (Department of Education). With the help of this study, DepEd will gain information about the update of the students in Candon National High School which will be of great contribution for them to prepare for better learning materials. This study will also benefit the teachers for they will know the weaknesses and strengths of their present students when it comes to solving mathematical problem. With the aid of the study, they shall be guided in improving their teaching skills. The student body of the said school will have an even greater understanding on how to solve mathematical problems. Using the data, they can also augment these information that will serve as their prior knowledge or advance knowledge in General Mathematics.
24
The researchers that are found and working in the said school, they will be able to give information and share their knowledge basing on this study. At the same time, their mindset about Mathematics shall be widened through the information stated in the study. The future researchers will have a head start with the aid of this study. If ever they shall be studying regarding General Mathematics, then the data indicated in the study shall be of great help in their related studies. It will also serve as a guide or basis for their future studies/researches.
Definition of Terms General Mathematics. It is a core subject taught in Senior High School as part of the new K-12 curriculum program. It aims to strengthen the knowledge of students in Mathematics by supporting the knowledge they acquired during Junior High School. Level of Problem Solving skills. It refers to the manifestation of the capacity/capability of students in solving Mathematical problems. It is also the main objective of the study. Functions – It refers to a relation from a set of inputs (domain) to to a set of possible outputs (codomain) where each input is related to exactly one output, according to Nykamp (2009).
25
Rational Equations and inequalities – Rational Equation refers to an equation that contains one or more rational expressions. Thus, Rational Inequality refers to one or more rational expressions, according to Mc Dougal (2013). Rational Functions – it refers to functions that contain the ratio
of
two
polynomials
(cannot
be
zero).
(https://www.mathsisfun.com, 2017) Inverse Functions – it refers to functions that are denoted by 𝑓 −1 that are pronounced as “f inverse”. It is not necessary as the reciprocal of function. (https://people.richland.edu) Exponential Functions – it refers to functions of the form 𝑓 (𝑥 ) = 𝑏 𝑥 in which the input variable x occurs as an exponent, according to Goldstein and Schneider (2006). Exponential Equations and inequalities – Exponential equations refer to equations that contain variables on the exponent according to Chegg Inc., (2013). Thus, Exponential inequalities refer to inequalities in which one (or both) sides involve a variable exponent, according to Katz, et al., (2017). Logarithmic Functions – it refers to functions (such as 𝑦 = 𝑙𝑜𝑔𝑎 x or 𝑦 = 𝑥) that is the inverse of an exponential function (such as 𝑦 = 𝑎𝑥 or 𝑦 = 𝑒 𝑥 ) so that the independent variables appears in logarithm, according to Merriam Webster (2017).
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Logarithmic inequalities – it refers to inequalities in which one (or both) sides involve a logarithm. Just like exponential inequalities, it is also useful in analyzing situations involving repeated multiplication, such as interest and exponential decay, according to katz, et al. (2017) Compound Interest – it refers to an interest calculated on the initial principal and also on the accumulated interest of previous periods of a deposit or loan. (https://investopedia.com) Business and Consumer Loans – Business loans refer to the money lent specifically for a business purpose. It may be used to start a business or to have a business expansion. Thus, Consumer loans refer to the money lent to an individual for personal or family purpose, according to Commission on Higher Education or CHED (2016).
Review of Related Literature and Studies This part of the study presents the significant readings including literature and studies that were surveyed from database documents and from different scholarly reading materials such as books, journals, theses, and dissertations.
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Level of Problem Solving Skills According to University of Kent (2013), Problem solving skills involve both analytical and creative skills which vary, depending on the problem. Also, analytical ability, literal thinking, initiative, logical reasoning and persistence are keys to problem-solving. Analytical and critical thinking skills help to evaluate the problem and to make decisions. A logical and methodical thinking approach is best in some circumstances. In other situations, using creativity or lateral thinking will be necessary to come up with ideas for resolving the problem and find fresh solutions. According to ITS Education Asia (2005), Problem solving requires two distinct types of mental skill, analytical and creative. Analytical or logical thinking includes skills such as ordering, comparing, contrasting, evaluating and selecting. It provides a logical framework for problem solving and helps to select the best alternative from those available by narrowing down the range of possibilities (a convergent process). Analytical thinking often predominates in solving closed problems, where the many possible causes have to be identified and analysed to find the real cause. Van Merrienboer and Jeroen (2013) investigated the perspectives on problem solving and instruction. It was found that problem solving should
28
not be limited to well structured problem solving but be extended to real life problem solving. Sungur and Tekkaya (2006) investigated the effectiveness of problem based learning and traditional instructional approaches on various facts of students self regulated learning, including motivational and learning strategies. Results revealed that problem based learning students had higher levels of intrinsic goal orientation, critical thinking, meta-cognitive, self regulation and peer learning compared with control group students. Sunitha (2004) made a study on effectiveness of problem solving approach on achievements and problem solving ability at higher secondary level. It is concluded that the problem solving approach is more effective than the conventional text book approach. Faux (1992) investigated the extent of relationship among creative thinking, critical thinking, intelligence and problem solving ability. It was found that critical thinking and intelligence have relation with problem solving ability. Penner and Voss (1983) compared the problem solving processes of experts and non experts and the results indicated that experts did not use a one solution process, rather, their processes differed with respect to problem decomposition into sub problems and in the way they close to represent the problem statement.
29
Leti (2015) accentuated that the results of a number of researches in Mathematics competencies of Filipino students showed that most of them did poorly in Mathematics. Pupils likewise developed negative attitudes towards this discipline. Much has been done to improve the ability of the students but the problems still remain. It is then the major concern of teachers, administrators, curriculum planners, as well as parents to improve the Mathematics performance of learners. In the study of Alicna (2016), she disclosed that the level of problem solving skills along knowing and understanding of Grade 7 students of Diocesan Schools of La Union is poor. This means that the students still lack in getting the meaning of the terms used in the problem. She added that their lack of skills in the said area added to their difficulty identifying what needs to be answered in a given problem. Correspondingly, Putil (2014) determined the capabilities and constraints of Mathematics instruction along attainment of objectives, adequacy of materials, competency of Mathematics teachers in the use of different methods and techniques, and the effectiveness of the utilization of the various evaluative techniques in the intermediate grades of Suyo District, Division of Ilocos Sur. From the constraints found, he developed an intervention program aimed to improve the Mathematics instruction in his district.
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Functions According to curriculum guide presented by the Department of Education (2015), a function is a relation where each element in the domain is related to only one value in the range by some rule. The elements of the domain can be imagined as input to a machine that applies a rule so that each input corresponds to only one output. A function is set of ordered pairs (x and y) such that no two ordered pairs have the same xvalue but different y-values. A function relates an input to an output. It is like a machine that has an input and an output. And the output is rekated somehow to to the input. The classic way of writing a function is f(x). In functions there are three main parts of the input, the relationship, and the output. (Drlik, 2015) In 1838, Lobachevsky gave a definition of a general function which is still required it to be continuous. A function of x is a number which is given for each x and which changes gradually together with x. The value of the function could be given either by an analytic expression or by a condition which offers a mean for testing all number and selecting one from them or lastly the dependence may exist but remain unknown. In 1882, Fourier says that the function f(x) represents a succession of values or ordinates each of arbitrary. An infinity of values being given of
31
the abscissa x, there are an equal number of ordinates f(x). All have actual numerical values, either positive or negative. We do not suppose these ordinates to be subject to a common law; they succeed each other in many matters whatever, and each of them is given as it were a single quantity. According to Fourier (1882), his studies about function moves away from analytic expressions. However, when he begins to prove theorems about expressing an arbitrary function as a Fourier series, he used the facts that his arbitrary function is continuous in the modern sense. According to the study of (Özalkan, 2010) wherein his study is subjected on the difference between the students taught by Problem Solving Method and those taught by Traditional Method in regards to Function. He found out that in understanding Functions, the instructor can actually use different methods depending on the students’ capability and level of understanding. He found out also that the level of problem skills of students along functions vary on the method that are used by the instructor. Another study conducted by (Drlik, 2015) talks about the relationship of Functions to students’ success on Calculus. The results showed student’s ability to work with functions would translate into roughly the appropriate level of success in calculus (working with functions at the process level would translate into a high rate of success
32
while working with functions at the pre-action levels would translate into a lower rate of success). Another study conducted by Valdez (2016) the learners obtained a mean score of 14.98 basing on the 1st quarterly examination he conducted with an equivalent rate of 59.92% in Functions. He concluded that the level of competence of students in General Mathematics along functions is satisfactory. However, based on his findings the topic about functions considered as weakness of the students and it was based on the mean score of 14.98 and a rate of 59.92%.
Rational Function, Equation and Inequalities According to Peter Kosek (2003), he defined Rational function as a function that is also similar to fraction that has the property of polynomial. Thus, according to the Curriculum guide (2015), a rational function is a function of the form
𝑝(𝑥) 𝑞(𝑥)
where p(x) and q(x) are polynomial functions, and
q(x) is not the zero function, q(x) ≠ 0. The domain of f(x) is all values of x where q(x) ≠ 0. In other words, R(x) is a rational function if R(x) = p(x) /q(x) where p(x) and q(x) are both polynomials. In addition, a polynomial is any function of the form f(x)= 𝑎0 + 𝑎1 (𝑥 ) + 𝑎2 (𝑥 ) + ⋯ + 𝑎𝑛 where 𝑎0 , 𝑎1 … . 𝑎𝑛 are all real numbers and the exponents of each x is a non-negative integer.
33
𝑥 2 +4𝑥−1
For example, the function R(x)=3𝑥2 −9𝑥+2 is a rational function since the numerator, 𝑥 2 + 4𝑥 − 1, is a polynomial and the denominator 3𝑥 2 − 9𝑥 + 2 is also a polynomial. A rational equation or inequality can be solved for all 𝑥 values that satisfy the equation or inequality. Whereas we solve an equation or inequality, we do not “solve” functions. Rather, a function (and in particular, a rational function) expresses a relationship between two variables (such as 𝑥 and 𝑦), and can be represented by a table of values or a graph. In
solving
Rational
Equations
you
have
to
a.)
Eliminate
denominators by multiplying each term of the equation by the least common denominator (LCD) and b.) Note that eliminating denominators may introduce extraneous solutions. Check the solutions of the transformed equations with the original equation (Curriculum Guide, 2015). A Rational Inequality is an inequality in which the two expressions are rational expressions (Bales, 2012). A rational inequality just only mean a rational expression combined with a ≤, ≥, >, < or sign. Solving Rational Inequalities requires the same initial steps as solving quadratic equations; first, get all the terms on the left side of the inequality sign and have zero on the right side of the inequality sign. Second, Once all terms are on the
34
left side of the inequality and lastly, make sure there is only one single rational expression. In solving rational inequalities a.) Use addition or subtraction to rewrite the inequality as a single fraction on one side of the inequality symbol and 0 on the other side b.) Determine over what intervals the fraction takes positive and negative values. First, Locate the x-values for which the rational expression is zero of undefined (factoring the numerator and denominator is a useful strategy). Second, mark the numbers found on a number line. Use a shade circle to indicate that the value is included in the solution set, and a hollow circle to indicate that the value is excluded. These numbers partition the number line into intervals. Third, select a test point within the interior of each interval. The sign of the rational expression at this test point is also the sign of the rational expression at each interior point in the aforementioned interval. And lastly, summarize the intervals containing the solutions. A study conducted by Valdez (2016) along Rational Functions, Equations, and Inequalities, based on the examination results of students they obtained a mean score of 12.33 with an equivalent rate of 61.65%. He found out that the competence level of students along Rational functions, equations and inequalities is very satisfactory. He also found out that topic about Rational Functions, Equations, and Inequalities is considered as a
35
strength of the students because of the mean score of 12.33 and a rate of 61.65%
Inverse Functions An Inverse function is a function that undoes the action of the other another function. A function g is the inverse of a function f if whenever 𝑦 = 𝑓 (𝑥 ) then 𝑥 = 𝑔(𝑦). In another words, applying f and the g is the same thing as doing nothing (Nykamp, 2010). A function f has an inverse function only if for every y in its range there is only one value of x in its domain for which 𝑓 (𝑥 ) = 𝑦. This inverse function is unique and is frequently denoted by 𝑓 −1 and called “f inverse”. In finding the inverse function first First, replace f (x) with y. This is done to make the rest of the process easier. Second, Replace every x with a y and replace every y with an x. Third, solve the equation from Step 2 for y. This is the step where mistakes are most often made so be careful with
this
step.
replacing y with
Fourth,
managed
to
find
the
inverse
by
𝑓 −1 (x) and lastly, verify your work by checking that
(𝑓 ᴏ 𝑓 −1 )(𝑥 ) = 𝑥 and (𝑓 −1 ᴏ 𝑓 )(𝑥 ) = 𝑥 are both true. A function says that for every x, there is exactly one y. That is, y values can be duplicated but x values can not be repeated.
36
If the function has an inverse that is also a function, then there can only be one y for every x. A one-to-one function is a function in which for every x there is exactly one y and for every y, there is exactly one x. A oneto-one function has an inverse that is also a function. There are functions which have inverses that are not functions. There are also inverses for relations. For the most part, we disregard these, and deal only with functions whose inverses are also functions. If the inverse of a function is also a function, then the inverse relation must pass a vertical line test. Since all the x-coordinates and ycoordinates are switched when finding the inverse, saying that the inverse must pass a vertical line test is the same as saying the original function must pass a horizontal line test. If a function passes both the vertical line test (so that it is a function in the first place) and the horizontal line test (so that its inverse is a function), then the function is one-to-one and has an inverse function. According to the study Valdez (2016) based on the examination results of his students they obtained a mean score of 7.57 with an equivalent rate of 50.47%. He concluded that his students’ level of competence along Inverse function is satisfactory. However, the topic on Inverse function is considered as weakness based on the mean score of 7.57 with a rate of 50.47%. His findings show that the competence level of
37
Grade 11 learners in General Mathematics is not that impressive. Basically, the competence level of the learners as manifested in the test results for the first quarter shows that the learners can satisfactorily deal with most of the problems, but not all. They can answer problems of average difficulty yet they find difficulty with complex problems. Exponential Functions, Equations and Inequalities Exponential functions occur in various real world situations. Exponential functions are used to model real-life situations such as population growth, radioactive decay, carbon dating, growth of an epidemic, loan rates, and investments. An exponential function with base b is a function of that form f(x) = bx or y = bx, where b > 0, b ≠ 1. A transformation of an exponential function with base b is a function of the form g(x) = a ∙ bx-c + d where a, c, and d are real numbers. An exponential expression is an expression of the form a ∙ bx-c + d, where b > 0. Exponential equation is an equation involving exponential expression. Exponential inequality is an inequality involving exponential expressions. An exponential equation or inequality can be solved for all x values that satisfy the equation or inequality. An exponential function is not solved rather it expresses a relationship between two variables, and can be represented by a table of values or a graph.
38
There are two main property of exponential inequalities. First, if b > 1, then the exponential function y = bx is increasing for all x. This means that bx >by if and only if x < y. And second, if 0 < b < 1, then the exponential function y = bx is decreasing for all x. This means that bx>by if and only if x < y. According to Valdez (2016) his learners obtained a mean score of 4.01 with an equivalent rate of 40.10% in Exponential Functions, Equations, and Inequalities. He concluded that the level of competency of his students along this topic is satisfactory. However, this topic is still considered as a weakness based on the mean score obtained by his students of 4.01 and a rate of 40.10%.
Logarithmic Functions, Equations and Inequalities Logarithmic functions are the inverses of exponential functions. The inverse of the exponential function y = a
x
is x = a
y
. The logarithmic
function y = loga x is defined to be equivalent to the exponential equation x = a
y
.y=
loga x only
under
the
following
conditions: x = ay , a > 0 , and a≠1 . It is called the logarithmic function with base a, according to Dawkins (2012). The domain of a logarithmic function is real numbers greater than zero, and the range is real numbers.
39
The graph of y = loga x is symmetrical to the graph of y = a
x
with respect
to the line y = x . This relationship is true for any function and its inverse. In addition, natural logarithmic function is a logarithmic function with base e . f (x) = loge x = ln x , where x > 0 . ln x is just a new form of notation for logarithms with base e . Most calculators have buttons labeled "log" and "ln". The "log" button assumes the base is ten, and the "ln" button, of course, lets the base equal e . The logarithmic function with base 10 is sometimes called the common logarithmic function. It is used widely because our numbering system has base ten. Natural logarithms are seen more often in calculus. Two formulas exist which allow the base of a logarithmic function to 1
be changed. The first one states this: loga = 𝑙𝑜𝑔
𝑏(𝑎)
. The more famous and
useful formula for changing bases is commonly called the Change of Base Formula. It allows the base of a logarithmic function to be changed to any positive real number ≠1. It states that 𝑙𝑜𝑔𝑎 =
𝑙𝑜𝑔𝑏 𝑥 𝑙𝑜𝑔𝑏 𝑎
. In this case, a , b ,
and x are all positive real numbers and a, b≠1 . Thus, Logarithmic Inequalities are inequalities in which one (or both) sides involve a logarithm. Like exponential inequalities, they are useful in analyzing situations involving repeated multiplication, such as in the cases of interest and exponential decay, according to Katz et. al
40
(2017). A logarithmic equation or inequality can be solved for all x values that satisfy the equation or inequality According to study of Valdez (2016) his students obtained a mean score 3.87 with an equivalent rate of 38.70%. He concluded that the level of competency of his students on logarithmic function, equations and inequalities fairly satisfactory. Based on his findings, he found out that the topic about logarithmic functions, equations and inequalities are considered as a weakness of his students and it was based on the mean score of 3.87 and a rate of 38.70%.
Simple and Compound Interest According to Nickolas (2017) Interest is the cost of borrowing money, where the borrower pays a fee to the owner for using the owner's money. The interest is typically expressed as a percentage and can be either simple or compounded. Simple interest is only based on the principal amount of a loan, while compound interest is based on the principal amount and the accumulated interest. Simple interest is the amount calculated on the original principal amount, or on an amount left unpaid from the principal amount. Simple interest holds great importance in our daily life, which is why it's one of the most important topics in mathematics. It is usually the basis of many
41
money transactions, especially in depository areas, where interest rate defines the earnings, according to Surbhi (2015) Simple interest is calculated by multiplying the principal amount by the interest rate and the number of periods in a loan. Generally, simple interest paid or received over a certain period is a fixed percentage of the principal amount that was borrowed or lent. For example, a student obtains a simple interest loan to pay one year of her college tuition, which costs $18,000, and the annual interest rate on her loan is 6%. She repaid her loan over three years and the amount of simple interest she paid was $3,240 = $18,000 x 0.06 x 3. The total amount she repaid was $21,240 = $18,000 + $3,240. While Compound interest, is interest on interest (Nickolas, 2017). It is calculated by multiplying the principal amount by the annual interest rate raised to the number of compound periods. As opposed to simple interest, compound interest accrues on the principal amount and the accumulated interest of previous periods. For example, if the student introduced above obtained a compound interest loan for college. The amount of compound interest that would be paid is $18,000 x ((1.06)3- 1) = $3,438.29, which is higher than the simple interest of $3,240. This is because unlike the simple interest, the compound interest accrues on both the principal and the accumulated interest.
42
When interest rates are calculated using compound interest, they calculate interest for the whole mean period and it is added to principal, and the resulted amount now serves as the principal for the next period of time. During the next period, the interest is calculated on the new principal, which is then added to it again. We use equation for compound interest to calculate interest for the whole mean period Thus, according to Valdez (2016) based on the study he conducted. He found out that his students obtained a mean score of 6.65 with an equivalent rate of 66.50% on Simple and Compound Interests. He concluded that the level of competency of his students along this topic is very satisfactory. Basing also on the result, his students consider Simple and Compound Interests as a strength. Business and Consumer Loans Major and local banks, credit unions and other financial institutions offer both personal and business/commercial loans, but usually under separate departments. These institutions may offer good interest rates, but major banks reject about 80% of small business loan applications. For business loans, the Small Business Association (SBA) works with banks and other financial institutions to help provide funding as do local government and trade organizations in states and larger cities. The Federal Housing Administration (FHA) provides insurance to lenders
43
to help consumers get mortgages and other related housing financing. But government agencies do not provide assistance for other types of personal loans. The number of alternative lenders has substantially increased in recent years for business loans. Lenders, such as BFS Capital, generally have more flexible requirements than most banks, but interest rates are typically higher. There are alternative lenders for personal loans as well, which also charge more interest than standard banks loans. Business loans must be repaid in a shorter time frame than personal loans. Major banks will frequently require collateral for business loans, such as inventory or real estate, while alternative lenders are often more flexible (Fleetwood, 2017). The process for a business loan from a bank is very long and detailed with an extensive application. Lending limits are often higher for business loans, compared with personal loans. However, alternative lenders such as BFS Capital have a much shorter and more flexible process for business loans as it bases approvals on average monthly gross business volume. BFS Capital will make loans ranging from $4,000 to $1 million.
On the other hand, Consumer loans are unsecured, but lenders will usually require some form of income verification, along with proof of other
44
assets worth at least as much as the individual is borrowing. The application process is far easier from banks for personal loans and decisions are often made more quickly.
Consumer loans have significantly lower monthly payment costs because repayment is spread out over a longer amount of time.
A
Consumer loan doesn’t usually require a guarantor to sign the loan. But for business loans, owners usually have to sign as guarantors. It should be considered carefully as business owners are putting both personal and business assets at risk if the loan is not repaid (Fleetwood, 2017).
Thus, according to Valdez (2016) his study described the topic on Business and Consumer loans as satisfactory basing on the mean score of 2.45 with an equivalent rate of 49.00% obtained by his students. He also concluded that the topic about business and consumer loans is considered as a weakness of his students.
His findings imply that the Grade 11 learners do not have remarkable competence level in General Mathematics. This shows that the learners have not yet mastered and attained the desired competencies in the said subject. The findings seemed parallel with that of Damilig (2014) when she found out that the level of performance of Bachelor of Science in Business
45
Administration (BSBA) students along sets and real numbers, algebraic expressions, special products and factoring, rational expressions, exponents and radicals, and solving linear equations is at satisfactory level. Furthermore, these findings
are also in consonance with the
findings of Rodriguez (2010) when she found out that the performance of the freshmen students in Fundamentals of Mathematics is at a moderate level.
46
CHAPTER II METHODS AND PROCEDURES
This chapter comprises the following parts: research design, sources of data, locale, and population of the study, instrumentation and data collection, tools for data analysis, data categorization and ethical consideration. Research Design This
study
utilized
the
descriptive
comparative
method
of
quantitative research. The descriptive comparative method is classified into two namely: Descriptive and comparative approaches. The descriptive method of research was defined by Polit & Hungler (1999) a method that involves the collection of data that will provide an account or description of individual, groups or phenomenon. This type of research also describes what exists and may help to uncover new facts and meaning. Hence, this research design will be used to describe the level of problem solving skills of SHS students in General Mathematics. Moreover, the comparative method of research according to Hantrais (1995) enables the researchers to identify and explore the similarities and differences between chosen phenomena or groups with use of comparison. Richardson (2001) added the comparative studies can be used to increase
47
understanding between cultures and societies and create a foundation for compromise and collaboration. Thus, this research design will be a help to discuss the comparison of the level of problem solving skills of SHS students when grouped according to strands. Also, this study utilized the quantitative approach. According to Babbie (2010) quantitative approach focuses on gathering numerical data and generalizing it across groups of people or to explain a particular phenomenon. Hence, this approach was suited for this study since it will be using quantitative technique particularly a Problem solving test to determine the level of problem solving skills of SHS students in General Mathematics. Sources of Data Locale and Population of the study The population of the study is mainly composed of Grade 12 students, studying at Candon National High School who took General Mathematics during Grade11. They are the ones best suited for the study in order to achieve the aim of the study, which is to determine the level of problem solving skills particularly in General Mathematics. The researchers utilized the Quota sampling and there will be a total of 100 students from STEM, ABM, ICT, and HUMSS that are subjected as
48
respondents in the study. Also, the researchers utilized the stratified random sampling technique. The table below shows the strata of SHS students per strand and section: Table 1. Population Distribution Strand
N
n
36 36 37
10 10 11
STEM 1 2 3 ABM 1 2 3 ICT 1 2 HUMSS
45 43 45
13 13 13
49 33 21
14 10 6
Total
345
100
As shown on the table, the total population of Grade 12 students which is 345 is subdivided according to strands and their section. It is also shown the computed sample size of each section and the total number of students that answered the test. Furthermore, the systematic sampling is used. According to Investopedia (2017), it is a type of probability sampling technique in which sample members from a larger population are selected according to a random starting point or a fixed periodic interval. In order to administer the interval, the following formula was used.
49
𝑲= Where:
𝑵 𝒏
K- Interval N- Population n- Sample population
Instrumentation and Data Collection To gather relevant data and information the researchers formulated Problem Solving Test. It is composed of 10 items that covers different topics in General Mathematics namely: Functions, Rational functions, equations and inequalities, Inverse functions, Exponential functions, equations and inequalities, Logarithmic Functions, equations and inequalities, Simple and compound interests and Consumer and Business Loans. The questions that are formulated are actually based on the Curriculum Guide (2015) presented by the Department of Education in General Mathematics to ensure its validity. Also, a table of specification was made based on the said source which includes the coverage of the topics that are subjected on the Problem solving test. Moreover, the distribution of the Problem solving test is conducted by the researchers themselves. They distributed and retrieved the test after the respondents answered it.
50
Validity and Reliability Since the Problem solving test has been copied from the Curriculum Guide in General Mathematics presented by the Department of Education (2015) and the researchers have ensured that the said source is reliable. Therefore, the validity and reliability has been established and there is no need to conduct a validity and reliability testing. Tools for Data Analysis The data that are collected, collated, and tabulated are subjected for analysis and interpretation using the appropriate statistical tools. The raw data were tallied and presented in tables for better understanding. For problem no. 1, the frequency is used to determine the correct responses of the respondents every topic/competencies. Also, it is used to describe the level of problem solving skills of grade 12 learners in General Mathematics. For problem no. 2, Kruskal-Wallis H-Test is used to determine the comparison of level of problem solving skills of SHS students in General Mathematics when grouped according to strands. According to Andale (2013) it is a rank-based nonparametric test that can be used to determine if there are statistically differences between two or more groups of an independent variable on a continuous or ordinal dependent variable. If the
51
H-critical Value is greater than .05, there is no significant difference and vice versa. If there is significant difference, post-hoc test will also be conducted to determine which among the groups are statistically different. For more organized and accurate conclusions, the data are tabulated and treated using MS Excel Worksheet and the MegaStat which is present in the said application. Data Categorization To describe the level of problem solving skills of SHS students in General Mathematics, the following scale for frequency was utilized to describe the correct responses obtained by the respondents: Scale of Frequency
Descriptive Equivalent Rating
81-100
Outstanding (0)
61 - 80
Very Satisfactory (VS)
41 – 60
Satisfactory (S)
21 – 40
Fairly Satisfactory (FS)
0 – 20
Not Satisfactory (NS)
Ethical Considerations To establish and safeguard ethics in conducting this research, the researcher strictly observed the following:
52
The student’s names will be confidential, only the researchers will know the names and the full detail of the students involved. The students were emotionally and physically harmed just to be a respondent of the study. Proper document sourcing or referencing of materials will be done to ensure and promote copyright laws and to avoid plagiarism. A communication letters with the approval letter from the principal will be sent to the advisers of the students as well as the students themselves.
53
CHAPTER III RESULTS AND DISCUSSION
This chapter presents, analyzes, and interprets the results and discussions of the data gathered based on the major and specific problems of the study. Level of Problem Solving Skills The first problem in this study dealt with the problem solving skills of SHS students in Candon National High School. The table below shows that the level of problem solving skills of SHS students in General Mathematics is not satisfactory. This is shown by the number of students who got the items correctly; it is shown that 16 out of 100 got the correct answer. This means that only 16% of the total test takers have gotten the answers correctly while a bigger portion of the population, 84% got the item incorrectly. This only points out that only a few of the SHS students can answer word problems in General Mathematics. This further indicates that the students have not mastered the skills needed in solving word problems. They have not mastered how concepts are applied to real-life problems thereby reflecting a very poor problem solving ability. From among the listed topics, the highest number of correct responses obtained from the respondents is 39, which is along Functions
54
Table 2. Level of Problem Solving skills of SHS students Topics Functions Rational Equations Rational Functions Inverse Functions Exponential Function Exponential Equation Logarithmic Function Logarithmic Inequalities Compound Interest Business and Consumer Loans Overall Legend : FS – Fairly Satisfactory
Frequency 39 28 39 21 13 20 0 0 3 0 16
Description
FS FS FS FS NS NS NS NS NS NS NS NS- Not Satisfactory
and Rational Functions. This means that only a few among the respondents understood the concepts about these topics thus, can apply their learnings on real-life problem solving. This further indicates that these topics are considered as strength of those respondents who got it correctly, though at some point it needs further improvement. In addition, the second highest among the topics is along Rational Equation with 28 correct responses acquired from the respondents. Still, this implies that only a few among the SHS students mastered the desired competency on this topic. Moreover, the topics along Inverse Functions and Exponential equations got 21 and 20 correct responses from the respondents, respectively. Compared to the first three topics, the number of respondents who got the items correctly declined with almost 10. This may imply that
55
along these topics most of the students are out of focus or not paying attention during discussion. These topics are also considered as weakness of students thus it also needs to be improved. Likewise, Exponential Equation with only 13 correct responses is also considered as a weakness of students among the topics. Thus, it can also be concluded that only a few of the students understood this topic. Meanwhile, the lowest among the listed topics are along Compound Interest with only 3 respondents got the correct answer followed by Logarithmic Functions, Logarithmic Inequalities and Business and Consumer Loans where there are no correct responses obtained from the respondents. This only indicates that the whole population has not mastered the concepts about these topics. Also, this may imply that the teacher’s way of teaching was not compatible with the learning system preferred by the students. As a result, low problem solving ability and competency were reflected based on the number of correct responses of the students especially on these topics. This signifies that these topics are considered as a weakness by most of the students hence, this really needs to be improved. As a corroboration, the Findings seemed parallel to the studies of Zahrah et al. (2013) and Tarzimah (2005) in Malaysia, their results showed that students felt difficult in problem solving because they had difficulty
56
in understanding and retrieving concepts, formulas, facts and procedures and lacked the ability to visualize Mathematics problems and concepts. Comparison of the Level of Problem solving skills of SHS students according to strands The second problem dealt with the comparison of the level of Problem solving skills of SHS students when grouped according to strands. The table below shows the results of the gathered data using Kruskal Wallis H- test. The table below shows that the calculated H-critical value is 0.0630, which is greater than 0.05, indicating that researchers accepted the null hypothesis. This means that there is no significant difference on the level of problem solving skills of SHS students when grouped according to strands. This only shows that the strands have almost the same level of problem solving ability and competency in General Mathematics. Since they have very low scores in all the domains, it can be implied that the students have not gained the necessary skills in all topics in the said subject. This further implies that SHS students in Candon National High School are not competent enough in solving mathematical problem solving. Furthermore, they have not mastered and attained the desired competencies in General Mathematics thereby reflecting poor problem solving skill.
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Table 3. Comparison of Strands Comparative Average Groups Scores STEM 25.70 ABM
24.30
HUMSS
15.25
ICT
16.75
H- Stat 7.299
H-Critical Decision Remarks Value 0.0630 Accept 𝐻0 Not Significant
However, it can be seen on the table that Science Technology Engineering and Mathematics (STEM) strand got the highest average number of correct responses with almost 25.70. This means that most of the respondents who got a correct answer came from this strand. This further implies that among SHS students, STEM constitute a large portion of the population of students who mastered different competencies in General Mathematics thus can solve real-life problems. But considering the significance testing, none among the strands performed better than the others. The second highest among the strands is Accountancy and Business Management (ABM) strand with an average of 24.30 correct responses obtained from the respondents. It can be observed that only 1 point is the difference of this strand among STEM. This means that ABM also contributes to the population of students who mastered the desired
58
competencies in General Mathematics. This further implies that among the strands of SHS students, STEM and ABM have higher competency and problem solving ability when it comes to General Mathematics. Also, it can be concluded that several students in STEM and ABM strands are good in solving real-life problems since mathematics is one of their area of expertise. On the other hand, Information Communication Technology (ICT) strand together with Humanities and Social Sciences (HUMSS) got the lowest average number of correct responses with 16.75 and 15.25 respectively. This means that only few of the students among this strands contributed to a few of those respondents who can solve problems thereby reflecting low competency and problem solving ability in General Mathematics. This further implies that students along these strands do not prioritize General Mathematics since it is not their area of expertise. As corroboration, this study has a connection to the study of Valdez (2016) wherein based on his findings, he found out that Grade 11 learners of Candon National High School do not have remarkable competence level in General Mathematics. This shows that the learners have not yet mastered and attained the desired competencies in the said subject.
59
CHAPTER IV SUMMARY, CONCLUSIONS, AND RECOMMENDATION This chapter presents the summary, the significant findings, conclusion and recommendations. Summary This Quantitative study, which made use of descriptive-comparative approach, focused on the level of problem solving skills of SHS students in General Mathematics. It utilized the Problem solving test to gather pertinent data from a quota of 100 Grade 12 students from senior high school in Science Technology Engineering and Mathematics (STEM), Accountancy and Business Management (ABM), Information Communication Technology (ICT) and Humanities and Social Sciences (HUMSS). The frequency and the Kruskal-Wallis H-test with its corresponding hypothesis testing were utilized in treating the data. Findings The following are the salient findings of the study: 1. The Level of Problem solving skills of SHS students in General Mathematics is not satisfactory. 2. The comparison between the strands of SHS students in Candon National High School when it comes to the level of Problem solving skills in General Mathematics is not significant.
60
Conclusions The researchers arrived at the following conclusions: 1. The respondents have not mastered the topics/competencies in General Mathematics thereby reflecting a very poor problem solving ability. 2. The respondents have almost the same level of Problem solving skills and competency in General Mathematics. Recommendations Based on the findings and conclusions of the study, the following are suggested: 1. The SHS students should pay more attention on their studies and setting aside other stuffs/things that aren’t important. 2.
Parents should inspire their children by giving their full support in all aspects.
3. The Teachers should craft remarkable strategies in their teachings or lectures that will encourage the students to strive harder. 4. The School should conduct activities that would enhance the problem solving ability of the students. 5. The Government or DepEd should provide learning materials or equipment such as books especially in General Mathematics. 6. Further studies should also be conducted for the improvement of the study.
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BIBLIOGRAPHY
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Bibliography Assaf, S., & Hejji, A. (2009). Causes of d Clements, D. H. and M. T. Battista (2009). Constructivist Learning and Teaching. The National Council of Teachers of Mathematics, Inc. Retrieved on September 25, 2017 from file:///E:/751_chapter.pdf Bales, J. W. (2012). Solving Rational Inequalities. United States of America: gnu general public press.
Chayo, R. (1978). Making Practice Fun. United States of America: AddisonWesley Publishing Company
Drlik, D. I. (2015, May). STUDENT UNDERSTANDING OF FUNCTION AND SUCCESS IN CALCULUS. Boise.
Duka, C. D. (1997). Historical, Philosophical, Anthropological, and Sociological Foundations of Education. Quezon City: Phoenix Publishing House, Inc.
Ebuk, L. E. and O. O. Bamijoko (2016). The effective management of mathematics workbook: Sure remedy to students’ performance in mathematics. Int. J. Adv. Multidiscip. Res. 3(4): 4653.www.ijarm.com Education, D. o. (2015). Curriculum Guide in General Mathematics. Retrieved from Department of Education: https://www.DepEd.gov.ph
Lardizabal, A. S. et. al. (1999). Principles and Methods of Teaching Third Edition. Quezon City: Phoenix Publishing House, Inc.
Levi, G. et al. (2014). Components of Mathematical Competence in Math Grade of Spanish Universities. Retrieved on September 23, 2017 from http://directorymathsed.net/monTenegro/
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Lipschutz, S. et al. (2005). Theory and Problems of Beginning Finite Mathematics. USA: McGraw-Hill Companies, Inc.
Orines, F. B. et al. (2008). Next Century Mathematics Advanced Algebra, Trigonometry, and Statistics 2nd Edition. Quezon City: Phoenix Publishing House, Inc.
Oronce, O. A. and M. O. Mendoza (2007). e-math Advanced Algebra and Trigonometry. Quezon City: Rex Book Store, Inc.
Özalkan, B. E. (2010, May). The Effects of Problem Solving on the topic functions. Middle East .
Ragma, F. G. (2014). Error Analysis in College Algebra in the Higher Education Institution. Unpublished Dissertation. Saint Louis College, City of San Fernando, La Union Swokowski, E. W. and J. A. Cole (2010). Algebra and Trigonometry with Analytic Geometry, Classic Twelfth Edition. Belmont CA, USA: Brooks/Cole Cengage Learning
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APPENDIX A
65
Table of Specification CONTENT COMPETENCIES FUNCTIONS The learner... 1. represents real-life situations using functions, including piece-wise functions. 2. evaluates a function. 3. performs addition, subtraction, multiplication, division, and composition of functions 4. solves problems involving functions. RATIONAL FUNCTIONS 5. Represents real-life situations using rational functions. 6. distinguishes rational function, rational equation, and rational inequality. 7. solves rational equations and inequalities. 8. represents a rational function through its: (a) table of values, (b) graph, and (c) equation. 9. finds the domain and range of a rational function. 10. determines the: (a) intercepts (b) zeroes; and (c) asymptotes of rational functions 11. graphs rational functions. 12. solves problems involving rational functions, equations, and inequalities. INVERSE FUNCTIONS 1. represents real-life situations using one-to one functions.
APPLIC ATION
1
EVALU ATION
SYNTH ESIS
TOTAL POINTS
1
66
2. determines the inverse of a one-to-one function. 3. represents an inverse function through its: (a) table of values, and (b)graph. 4. finds the domain and range of an inverse function. 5. graphs inverse functions. 6. solves problems involving inverse functions. EXPONENTIAL FUNCTIONS 7. represents real-life situations using exponential functions. 8. distinguishes between exponential function, exponential equation, and exponential inequality. 9. solves exponential equations and inequalities. 10. represents an exponential function through its: (a) table of values, (b) graph, and (c) equation. 11. finds the domain and range of an exponential function. 12. determines the intercepts, zeroes, and asymptotes of an exponential function. 13. graphs exponential functions. 14. solves problems involving exponential functions, equations, and inequalities. LOGARITHMIC FUNCTIONS 15. represents real-life situations using logarithmic functions. 16. distinguishes logarithmic function, logarithmic equation, and logarithmic inequality. 17. illustrates the laws of logarithms.
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18. solves logarithmic equations and inequalities. 19. represents a logarithmic function through its: (a) table of values, (b) graph, and (c) equation. 20. finds the domain and range of a logarithmic function. 21. determines the intercepts, zeroes, and asymptotes of logarithmic functions. 22. graphs logarithmic functions. 23. solves problems involving logarithmic functions, equations, and inequalities. SIMPLE AND COMPOUND INTERESTS 24. illustrates simple and compound interests. 25. distinguishes between simple and compound interests. 26. computes interest, maturity value, future value, and present value in simple interest and compound interest environment. 27. solves problems involving simple and compound interests. STOCKS AND BONDS 33. illustrate stocks and bonds. 34. distinguishes between stocks and bonds. 35. describes the different markets for stocks and bonds. 36. analyzes the different market indices for stocks and bonds. 37. interprets the theory of efficient markets. 38. illustrates business and consumer loans. 39. distinguishes between business and consumer loans. 40. solves problems involving business and consumer loans (amortization, mortgage).
68
APPENDIX B
69
Candon National High School Bagani Campo, Candon City, Ilocos Sur SENIOR HIGH SCHOOL Dear Respondent: We are the SHS student-researchers of CNHS working on a study entitled, “PROBLEM SOLVING SKILLS OF SENIOR HIGH STUDENTS IN GENERAL MATHEMATICS” as part of the requirements in Quantitative Research. Relative to this, we would like to get your SOLUTION/S on the following problems. Name (Optional): Strand and section:
Instructions: Solve the problems completely. SHOW ALL NECESSARY SOLUTIONS. You can use calculator. (Solving problems involving Functions) 1. Edward started selling snacks in the nearby school. In one day, he spends Php 200.00 for rent and Php 25.00 for each snack item he prepares. How much are his expenses if he prepares 100 snack items? Construct the relevant function c(x) and solve.
(Solving Rational equation and Inequalities) 2. In an Inter-barangay basketball league, the team from Barangay San Isidro has won 12 out of 25 games, a winning percentage of 48%. How many games should they win in a row to improve their win percentage to 60%?
(Solving problems involving Rational functions)
3. Suppose that 𝑐(𝑡) =
5𝑡 𝑡 2 +1
(in mg/mL) represents the concentration of a drug in a patient’s
bloodstream t hours after the drug is administered. Construct a table of values for c(t) for t= 1,2,5,10.
70
Round off answers to three decimal places. What is the drug concentration in mg/mL after 15 hrs? Create a table.
(Solving Inverse Functions) 5
4. To convert from degrees Fahrenheit to Kelvin, the function is 𝑘(𝑡) = 9 (𝑡 − 32) = 273.15, where t is Fahrenheit (Kelvin is the SI unit of temperature. Find the inverse function converting the temperature in Kelvin to degree Fahrenheit.
(Solving Exponential Functions) 5. The population of Candon City is approximately 120 000 and its rate increases 4% every year. Find the population after 15 years.
(Solving Exponential Equation and Inequalities) 6. The Half-life of a substance Zn-71 is 2.45 minutes. Initially, there were 𝑦𝑜 grams of Zn-71, but only 1 256
of this amount remains after some time. How much time has passed?
(Solving Logarithmic Functions) 7. The decibel level of sound in a quiet office is 10−6 watts/m2. What is the corresponding sound intensity in decibels using the function of 𝐷 = 10 log
𝐼 . 10−12
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(Logarithmic Inequalities) 8. The 2013 Earthquake in Bohol and Cebu has a magnitude of 7.2, while the 2012 earthquake occurred in Negros Oriental recorded a 6.7 magnitude. How much more energy was released by the 2013 bohol/Cebu earthquake to that by the Negros Oriental earthquake in 2012 by using the formula =
2 E log 4.4 3 10
.
(Solving Problems involving Compound Interests) 9. Find the maturity value and interest if Php 50,000 is invested at 5% compounded annually for 8 years. Find (a.) Maturity value F ; (b.) Compound Interest 𝐼𝑐
(Solving Problems involving Business and Consumer Loans -amortization and mortgage) 10. Mrs. Norris borrowed Php 1,200,000 for the purchase of a car. If his monthly payment is Php 31,000 on a 5-year mortgage, find (a.) the total amount of Interest and (b.) total amount of mortgage.
Thank you for your precious time in answering our test. May God bless you a thousand folds! – Researchers
72
APPENDIX C
73
ANSWERS 1. 𝐶 (𝑥 ) = 25𝑥 + 200
X= 100
= 25 (100) = 2500 + 200 = 𝑃ℎ𝑝 2700 2. Let x represent the number of games they need to win to raise their percentage to 60%. 12+𝑥
The rational equation is 12+𝑥 25+𝑥
25+𝑥
= 0.6
= 0.6
12 + 𝑥 = 0.6 (25 + 𝑥) 12 + 𝑥 = 0.6 (25) + 0.6𝑥 𝑥 − 0.6𝑥 = 15 − 12 0.4𝑥 = 3 𝑥 = 7.5 3. 𝑐(𝑡) =
5𝑡
5(1)
𝑐 (1) =
1+1
𝑐(2) =
5𝑡
5(10)
50
4+1
=
5
=
50 101
or 0.495
or 2
𝑐 (𝑡 ) =
𝑡 2 +1 5(5)
𝑐(5) =
𝑐(15) =
(10)2+1
100+1
5(2)
(2)2 +1 10 10
𝑐 (𝑡 ) =
𝑡 2 +1
𝑐 (10) = 𝑐 (𝑡 ) =
=2 or 2.5
5𝑡
𝑐 (𝑡 ) =
𝑡 2 +1
𝑐(2) =
(1)2+1 5 5
𝑐 (1) = 𝑐 (𝑡 ) =
5𝑡
𝑐 (𝑡 ) =
𝑡 2 +1
𝑐 (5) =
(5)2+1 25 25
= or 0.962
25+1 26
5𝑡 𝑡 2 +1 5(15) (15)2 +1
75 225+1
=
75 226
or 0.332 mg/mL is
the concentration after 15 hrs. T
0
1
2
5
10
15
c(t)
0
2.5
2
0.962
0.495
0.332
74
5
4. 𝑘 (𝑡) = 9 (𝑡 − 32) = 273.15 5
𝑘 − 273.15 = 9 (𝑡 − 32) 9 5
(𝑘 − 273.15) = 𝑡 − 32
9 (𝑘 − 273.15) + 32 = 𝑡 5 9 𝑡 = (𝑘 − 273.15) 5 Therefore, the inverse function is 9
𝑡 = 5 (𝑘 − 273.15) where k is the temperature in Kelvin. 5. 𝑃 = 𝑃0 (1 + 𝑟)2
𝑃15 = 120,000 (1 + 0.04)15 𝑃15 = 120,000 (1.04)15 = 216, 113
𝑃0 = 120, 000 r= 4%or 0.04 n= 15 1
1
6. 𝑦𝑜 = (2) t/2.45 =
256
1
1
𝑦𝑜 = (2) t/2.45 = (2) 8 𝑡 2.45
=8
t= 19.6 𝐼
7. 𝐷 = 10 log 10−12 𝐷 = 10 log
10−6 10−12
= 10log106 log 106 = 6 Thus, 𝐷 = 10(6) = 60 Decibels 8. Let 𝐸𝐵 and 𝐸𝑁 represents the Bohol/Cebu and Negors Oriental Earthquake respectively. We while determine 3
E
Solving for 𝐸𝐵 : 7.2 ( 2 ) = log 10B4.4
𝐸𝐵 𝐸𝑁
.
75
E
10.8 = log 10B4.4 E
1010.8 = log 10B4.4 𝐸𝐵 = 1010.8 . 104.4 = 1015.2 3
E
Solving for 𝐸𝑁 : 6.7 ( 2 ) = log 10N4.4
𝐸𝑁 = 1010.05. 104.4 = 1014.45
E
10.05 = log 10N4.4 E
1010.05 = log 10N4.4 Thus,
𝐸𝐵 𝐸𝑁
=
1015.2 1014.45
= 100.75≈5.62
The Bohol/Cebu Earthquake released 5.62 times more energy than the Negros Oriental Earthquake. 9. Given: P=Php 50,000 r= 5% or 0.05 t= 8 years. (a.)
F= P(1+r)2 F= (50,000)(1+0.05)8 F=Php 73,872.77
(b.)
𝐼𝑐 = 𝐹 − 𝑃 𝐼𝑐 = 73,872.77-50,000 𝐼𝑐 =Php 23, 872.77
10. Given: P= Php 1,200,000 Monthly Payment= Php 31,000 The total amount paid is given by Total Amount paid = (Php 31,000)(12 months)(5 years) =Php 1,860,000 The Total Interest = Total amount paid – mortgage = Php 1, 860, 000 –Php 1, 200,000 =Php 660,000
76
The Total Mortgage = (Total amount paid + mortgage) – total interest = (Php 1,860,000 + Php 1,200,000)- Php 660,000 =Php 2,400,000
77
CURRICULUM VITAE
78
Curriculum Vitae Name: BAWAS, JHOANCEL Address: Kiblongan, Uso, Suyo, Ilocos Sur Cellphone Number: 09197791245 E-mail Adress:
I.
Personal Information
Nickname: Ancel Birthday: October 27, 1999 Birthplace: Kiblongan, Uso, Suyo, Ilocos Sur Religion: Pentecost Father’s Name: N/A Mother’s Name: Narcisa Bawas II.
Educational Background
Senior High School Junior High School Elementary III.
Candon National High School S.Y. 2017-2018 Candon National High School S.Y. 2015-2016 Uso Elementary School S.Y. 2011-2012
Honors and Awards Received
HONORS Honorable Mention (Elementary) 3rd
Age: 17 Nationality: Filipino Civil Status: Single
AWARDS
79
Curriculum Vitae Name: CAGAS, CHRISTOPHER LLOYD O. Address: Palacapac, Candon City, Ilocos Sur Cellphone Number: 09355385492 E-mail Address:
[email protected]
I.
Personal Information
Nickname: Lloyd Birthday: June 13, 2000 Birthplace: Tala, Bagong Silang, Caloocan City Religion: Pentecost Father’s Name: Jaysen Bataclan Mother’s Name: Jasmin Bataclan II.
Junior High School: Senior High School:
Caloocan North Elementary School Candon National High School Candon National High School
Honors and Awards Received HONORS
8th
Civil Status: Single
Educational Background
Elementary:
III.
Age: 17 years old Nationality: Filipino
honor With Honor With Honor With Honor
AWARDS
80
Curriculum Vitae Name: FELICIANO, MARJORIE A. Address: Langlangca 1st , Candon City, Ilocos Sur Cellphone Number: 09058082530 E-mail Address:
[email protected]
I.
Personal Information
Nickname: Marj Birthday: July 04, 1999 Birthplace: San Fernando, Pampanga Religion: Pentecost Father’s Name: Joel Feliciano Mother’s Name: Marilou Feliciano II.
Junior High School: Senior High School:
Sapang Biabas Resettlement Elementary School Candon National High School Candon National High School
Honors and Awards Received HONORS
4th
Civil Status: Single
Educational Background
Elementary:
III.
Age: 18 years old Nationality: Filipino
honor With Honor With Honor With Honor
AWARDS Feature Writing in English (District Level) 4th Feature Writing in Filipino (Division level) 1st
81
Curriculum Vitae Name: GABOR, ANGELO RAVEN P. Address: San Nicolas, Candon City, Ilocos Sur Cellphone Number: 09057631428 E-mail Adress:
[email protected]
I.
Personal Information
Nickname: Angelo Birthday: July 11,2000 Birthplace: Sta. Lucia, Ilocos Sur Religion: Catholic Father’s Name: Allan M. Gabor Mother’s Name: Ma. Rosie P. Gabor II.
Educational Background
Senior High School Junior High School Elementary III.
Candon National High School S.Y. 2017-2018 Candon National High School S.Y. 2015-2016 Candon South Central School S.Y. 2011-2012
Honors and Awards Received
HONORS Honor ( Grade7 Gumamela) 2nd
Age: 17 Nationality: Filipino Civil Status: Single
AWARDS place in group Math Quiz Bee in Grade 10 2nd
82
Curriculum Vitae Name: GALUTAN,JOHN JOSHUA T. Address: Bagani Campo Candon Cty Ilocos Sur Cellphone Numeber: 09158396697 E-mail:
[email protected] I.
PERSONAL INFORMATION
Nickname: Josh Birthdate: January 10, 2000 Birthplace: Batac City, Ilocos Norte Filipino Religion: Baptist Father’s Name: Jojo Galutan Mother’s Name: Rolalyn Galutan
Age: 17 Nationality: Civil Status: Single
II. EDUCATIONAL ATTAINMENT Senior High Candon National High School S.Y. 2017-2018 School Junior High Candon National High School S.Y. 2015-2016 School Elementary Bagani Elementary School
III.
AWARDS HONORS
With Honor
AWARDS Best in Attendance Best in Performing Arts
83
Curriculum Vitae Name: GANNAPAO, JOHN LESTER M. Address: Bagani Campo, Candon City, Ilocos Sur Cellphone Number: 09973829785 E-mail Adress:
[email protected]
I.
Personal Information
Nickname: Jhon Jhon Birthday: January 29, 2000 Birthplace: Tagudin, Ilocos Sur Religion: Roman Catholic Father’s Name: Orlando Gannapao Mother’s Name: Crispina Gannapao II.
Educational Background
Senior High School Junior High School Elementary III.
Age: 17 Nationality: Filipino Civil Status: Single
Candon National High School S.Y. 2017-2018 Candon National High School S.Y. 2015-2016 Bagani Elementary School S.Y. 2011-2012
Honors and Awards Received
HONORS Grades 3-6 With Honors Grade 10 Completer
AWARDS Artist of the Year
84
Curriculum Vitae Name: GARNACE, ALLEAH T. Address: San Jose, Candon National High School Cellphone Number: 09952240191 E-mail Address: none I.
Personal Information
Nickname: Lleah Birthday: December 14, 1999 Birthplace: San Jose, Candon City, Ilocos Sur Religion: Lutheran Father’s Name: Danny Boy Garnace Mother’s Name: Joselyn Garnace II.
Age: 17 Nationality: Filipino Civil Status: Single
Educational Background
Elementary Candon South Central School Junior High School Candon National High School Senior High School Candon National High School III.
Honors and Awards Received HONORS
7th Honors
AWARDS Most Obedient
85
Curriculum Vitae Name: MOLINA, MARK JOSEPH G. Address: San Nikolas, Candon City, Ilocos Sur Cellphone Number: 09158590703 E-mail Address:
[email protected] I.
Personal Information
Nickname: Mark Birthday: October 19, 1999 Birthplace:Talogtog, Candon City Religion: Roman Catholic Father’s Name: Emanuel Jessie Molina Mother’s Name: Lanei Molina II.
Educational Background
Elementary: Junior High School: Senior High School: III.
Age: 17 years old Nationality: Filipino Civil Status: Single
South Central Elementary School Candon National High School Candon National High School
Honors and Awards Received HONORS
AWARDS
