Research Proposal
Research Proposal
Research Proposal
A Research Proposal
Submitted
by
Submitted
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DR.TERESITA V. DE LA CRUZ
Chapter 1
INTRODUCTION
understanding influences decision making in all areas of life—private, social, and civil.
young people, but today, as in the past, many students struggle with mathematics and become
that we understand what effective mathematics teaching looks like—and what teachers can
do to break this pattern.( by Glenda Anthony and Margaret Walshaw in their book entitled
Problem solving is a key subject in Standards and Focal Points. Learning how to solve
story problems involves knowledge about semantic construction and mathematical relations
as well as knowledge of basic numerical skills and strategies. Yet, word problems pose
difficulties for many students because of the complexity of the solution process . Because
problem solving, in particular word problems as a process, is more complex than simply
extracting numbers from a story situation to solve an equation researchers and educators must
afford attention to the design of problem-solving instruction to enhance student learning for
grasping concepts and ideas pupils are learning. Problem solving skills develop fast if the
solver gets new and new experience with the activity. Pupils’ performance in problem solving
improves if they repeatedly meet the same type of problem or if they can make use of their
A teacher’s attitude and the teaching strategies he/she uses significantly influence
representations and topics, and between mathematics and everyday experiences.( Glenda
The role of the teacher is simpler, he/she only has to detect the place where pupils
make mistakes and assess correctness of their solutions. That is the reason why teachers often
choose problems in whose case the search for the appropriate algorithm is easy and also often
hint at the suitable solving procedure. This means pupils instead of solving a problem simply
apply some
algorithm chosen according to the signals from the assignment or the teacher. Then they fail
if they are to solve non-standard problems whose assignment does not contain elements they
are used to, elements that serve as indicators for selection of the right solving strategy. They
feel helpless if they face an atypical, unusual problem or a problem set in an unknown
context. Needless to say that this often happens in case of application problems, where pupils
are expected to use mathematics for solution to problems from everyday life. One of the
indicators telling a teacher whether a pupil understands the subject matter is the pupil’s
ability to come up with new, original solving procedures when solving a new problem. But
this is something a teacher cannot teach directly. He/she can expect this approach from their
pupils, he/she can ask for it, support them in it but he/she cannot teach it (Sarrazy and
Novotná, 2013). This is one of the key concepts of didactics of mathematics, the didactical
teachers’ attitude and approaches to mathematics education, (Tichá and Hošpesová, 2006).
We often ask adults to write their math autobiographies, and their stories would
make a grown man cry. They struggled through a labyrinth of incomprehensible symbols and
rules, memorizing facts and procedures. They remember their panic when called upon to go
to the chalkboard to compute 212 divided by 512. “Ours is not to reason why; we just invert
and multiply.” Many of these adults are now parents, and they not-so-subtly send a message
to their children: “Math is hard. I never could understand it. Gee whiz, I can’t even balance
my check book.” Is mathematics so inherently difficult that only a few who are “wired” for
math can understand it? Unfortunately, most people in the United States would say yes. This
erroneous view of mathematics has been prevalent for decades. Many come to believe that
they are incapable of doing math. As they progress through the grades, fewer and fewer
students understand and enjoy math, leaving only a handful to continue with capability and
confidence.
When students find they can use mathematics as a tool for solving significant problems
in their everyday lives, they begin to view it as relevant and interesting. Effective teachers
take care that the contexts they choose do not distract students from the task’s mathematical
purpose. They make the mathematical connections and goals explicit, to support those
students who are inclined to focus on context issues at the expense of the mathematics. They
also support students who tend to compartmentalize problems and miss the ideas that connect
Good problem solving skills empower students in their educational, professional, and
personal lives. Nationally and internationally, there is growing recognition that if education is
to produce skilled thinkers and innovators in a fast-changing global economy, then problem
solving skills are more important than ever. The ability to solve problems in a range of
encourages them to use content knowledge in innovative and creative ways and promotes
deep understanding(Crebert, G., Patrick, C.-J., Cragnolini, V., Smith, C., Worsfold, K., &
Webb, F. (2011)).
BACKGROUND OF THE STUDY
The study will be conducted in the four secondary schools in the district of Padre
Burgos, Padre Burgos, Quezon. These schools are, Hinguiwin national High School,
Danlagan National High School, Lina Gayeta-Lasquety National High School and San Isidro
National High School . The first three mentioned secondary schools are situated near or
along the provincial road while the fourth one is almost two kilometers along the road. These
secondary schools has almost have the same number of students, with two sections per grade
teaching and letting problem solving be done accurately by the students. There are times that
the researcher experienced that almost 100% of the students don’t have the ability to solve
problems specially to the section with slow learners. Also, she observed that, this part of
teaching Mathematics is the least interested part for the students, as years passed, there is a
sudden decreased of the schools Mean Percentage Score in Mathematics. Because of these
This study intended to find out different strategies used by the Grade 7 Mathematics
1.) What difficulties were encountered by the Grade 7 Mathematics teachers in Padre
Burgos in making their student understand and solve questions involving problem solving?
2). What are the different teaching strategies used by Grade 7 Mathematics teachers in
3). Were the strategies used by the Grade 7 Mathematics teachers in Padre Burgos
has a positive effect in the achievement of their students’ ability to perform problem
solving?
HYPOTHESIS
There is a significant relation between strategies used by the Mathematics teachers in the
Grade 7 Mathematics teachers in the district of Padre Burgos, for they will be able to
identify which among the strategies of the teachers can be used as an effective strategies in
teaching problem solving to the students. Also, being grade 7 as the basis of the result,
teacher’s teaching Mathematics in the higher grade level will also be benefited since ,
developing students ability to solve problem solving in the start of their secondary education
will be a continuous achievement for the learning process, provided efficient guidance and
facilitation of the teachers teaching Mathematics in their students in the higher level will be
intact and continuous too. On the other hand, teachers will also be able to identify approaches
that should be discard since it has no positive effect in the problem solving ability of the
students.
Future researchers who will conduct the same study about problem solving would be
Lastly, the students because being the center of the educational system, they are the one
who will benefit with these effective strategies in teaching problem solving, since they will
be provided with techniques that will make them an excellent problem solver.
The main purpose of the study is to find out the different effective teaching strategies
used by the Grade 7 Mathematics teachers in the district of Padre Burgos. It was limited to
the Grade 7 students and teachers teaching Grade 7 Mathematics in the four secondary
DEFINITION OF TERMS
Conceptual definitions:
Problem Solving- Finding solution to a problem or puzzle that requires logical and
Effective Strategies – a general plan or set of plans intended to achieve something that
Positive Effect- The outcome attained parallel to the intended result of the activity.
Grade 7 Students – Students enrolled in Grade 7 level of the K-12 curriculum in the four
Grade 7 mathematics.
Operational definitions:
the operation to be used and which is being asked in order to solve and find the
desired solution.
Difficulties – These refers to the problems encountered by the teachers in Padre Burgos
Padre Burgos that shows they can solve problem solving correctly.
Grade 7 Students – Students enrolled in Grade 7 level of the K-12 curriculum in the four
Grade 7 Mathematics.
Chapter II
This chapter presents the related literature about, problem solving, difficulties in solving
problems, strategies in solving mathematical problems. It also shows the previous study of
researchers related to the different strategies in teaching problem solving, the research
PROBLEM SOLVING
It is a truth universally acknowledged that problem solving forms the basis for
refine and
problem solving is the most important learning outcomes in most contexts. He believes that
the most important cognitive goal of education (formal and informal) in every educational
context (public schools, universities and corporate training) is problem solving. Therefore,
the instructional design and technology community should learn how to design problem-
solving instruction. In order to support those efforts, instructional design and technology
researchers should conduct high quality research on learning to solve problems that will
First, problem solving is the most authentic and therefore the most relevant learning
activity that students can engage in. Karl Popper (1999) wrote a book of essays that claimed
that “all life is problem solving.” In everyday contexts, including work and personal lives,
people solve problems constantly. No one in personal and professional contexts is rewarded
problems in both conventional and innovative ways and to identify and ask significant
questions that clarify various points of view and lead to better solutions
(http://www.21stcenturyskills.org).
Second, research has shown that knowledge constructed in the context of solving
problems is better comprehended, retained, and therefore more transferable. When solving
problems, students must think more critically. Also, the learning is situated in some authentic
context, which makes it more meaningful. Third, problem solving requires intentional
learning. Learners must manifest an intention to understand the system or context in which
problems occur in order to solve problems effectively. Meaningful learning cannot occur
until and unless learners manifest an intention to learn. All human behavior is goal-driven.
The clearer our goals are for learning, the more likely we are to learn meaningfully and
mindfully. Fourth, knowledge that is recalled and not used in some authentic tasks is too
quickly forgotten, cannot be effectively applied, and in most disciplines becomes obsolete in
a short time. Therefore, the primary purpose of education should be to engage and support
learning to solve problems. (Reif et al., 1976; McDermott, 1991; Heller et al.,1992) stated
that, one of the fundamental achievements of education is to enable students to use their
Problem solving has become the means to rejoin content and application in a learning
environment for basic skills as well as their application in various contexts. (Hiebert, 1996).
component of the curriculum. The need for learners to become successful problem solvers
has become a dominant theme in many national standards (AAAS, 1993; NCSS, 1997;
NCTE, 1996; NCTM, 1989, 1991). For example, the 1989 Curriculum Standards of the
mathematics instruction and an integral part of all mathematical activity. Problem solving is
not a distinct topic but a process that should permeate the entire program and provide the
context in which concepts and skills can be learned” (National Council of Teachers of
Problem solving is a key skill , and it is one that makes a huge difference to your career.
At work, problems are the center of what many people do everyday. The problem you face
can be large or small, simple or complex, and essay or difficult to solve. Regardless of the
nature of the problems, a fundamental part of every manager’s role is finding ways to solve
http://www.mindtools.com)
In Germany, problem solving has important roots that date back at least to the
beginning of the twentieth century. However, problem solving was not primarily an aspect of
classroom, problem solving is, “the heart of Mathematics,” did not attract the interest it
deserved as a genuine mathematical topic. There are some evidences that this situation may
change. In the past few years, nationwide standards for school mathematics have been
process-oriented standard that should be part of mathematics classroom through all grades.
This article provides an overview on the problem solving in with reference to psychology and
mathematics (http://www.springer.com).
In particular, problem solving is examined from perspectives of research, curricula and
instructional practice, and assessment. We identify three key themes underlying observed
the Language and construction of the curricula and in related policy documents have
the assessment research in Australia has demonstrated the need for alignment of the
curriculum, instruction and assessment, particularly in the case of the complex performances
such as mathematical problem-solving and student work samples that illustrate such complex
global term and taken in the context of Mathematics. It leads to the consideration of problem
solving processes from a variety of perspectives. Problem solving ignores the imaginative
aspects of real life doings. This is an innovative approach in considering problems from a
mathematical perspective and is the one which stresses that problems need not specifically be
concerned with the application to be real but can also arise within Mathematics itself .
Involving the Four Fundamental Operations”, and she stated that the difficulties in solving
word problems depend on the pupil’s background in the lower level of Mathematics. She also
found out that encouraging the pupils, the teacher should share first the love and interest for
the subject. Montague (2006) states that, students who have difficulty representing math
motivation, that is, student’s willingness to persist in solving problem. Goal orientation,
motivational variable, explains reasons why students engage in the activity because they want
Even basic problem solving skills are scarce in the work force, as well. The 1993
National Adult Literacy Survey (NALS) found that more than half of employed adults had
difficulty with completing various problem solving tasks—even simple ones. One such task
tested to see if adults could solve problems such as determining the correct change using
information from a menu. Another test involved answering a caller’s question using
information from a nursing home sign-out sheet (Kirsch, Jungeblat, Jenkins, &
Kolstad,1993).
1980's, researchers found that attempts to teach abstract, generalized problem solving skills
proved ineffective (Beyer, 1984). They found that mastery of generalized problem solving
skills did not differentiate well between good and poor problem solvers. In fact, researchers
concluded that knowledge of context was the most critical feature of skill in problem solving.
Thus, current research supports problem solving as a situational and context-bound process
Gagne(1985), states that, instruction in problem solving needs to focus on two distinct
the context knowledge mentioned above. A common error is to teach only declarative
knowledge, and assume that learners who have mastered declarative knowledge can solve
problems in a domain. Conversely, attempts to teach problem solving alone, without teaching
believed that building adequate problem representation, goal-directed planning, inference and
elaborating by using one's world knowledge, testing hypotheses, applying heuristics and
comprehension monitoring, are seen as basic operational building blocks of problem solving,
as well as thinking skills. Nakamura (2006), on the other hand categorized those mistakes
into two different forms such as language problems and the ongoing problem solving process.
These categorization proposed by them is actually the continuity of what Newman had said.
This understanding can identify the influence of language factor on learning mathematics and
its corresponding remedial efforts taken in the teaching and learning process.
performance with mathematical word problems is a trend that I became aware of very early
on in my teaching career and one that an interest has been taken in by many who are involved
mathematical word problems, I was able to gain a more detailed insight into the causes of
children’s difficulties. Using the evidence from existing research, I formulated five categories
of difficulties that presented as: Reading and Understanding the Language Used Within a
Word Problem Difficulties in this category involve children not being able to decode the
words used in a word problem, not comprehending a sentence, not understanding specific
having confidence or the ability to concentrate when reading. (Ballew and Cunningham 1982:
Shuard and Rothery 1984: Cummins et al 1988: Bernardo 1999). Recognising and
Imagining the Context in Which a Word Problem is Set. These difficulties arise when
children cannot imagine the context in which a word problem is set or their approach is
altered by the context in which the word problem is given. (Caldwelland Goldin 1979: Nunes
1993). Forming a Number Sentence to Represent the Mathematics Involved in the Word
Problem. Children appear to find it harder to form a number sentence for some word
problems structures than others. These difficulties can result in children not being able to
1998). Carrying Out the Mathematical Calculation. Difficulties can occur here with
children’s selection of, and aptitude with calculation strategies (for example formal
algorithms, pencil and paper methods and calculators). The context in which a word problem
is given and the size of numbers involved can impact on children’s choice of a calculation
Interpreting the Answer in the Context of the Question. Children have been shown to not
consider real-life factors and constraints when giving an answer to word problems which can
result in giving an answer that is impossible in the context and therefore incorrect.
(Verschaffel, De Corte and Lasure 1994; Wyndham and Säljö1997; Cooper and Dunne
2000).
Most researchers working on problem solving (Dewey, 1910; Newell & Simon,1972
etc.) agree that a problem occurs only when someone is confronted with a difficulty for which
a problem because it depends on the solver’s knowledge and experience (Garrett, 1986; Gil-
Perez et al., 1990). So, a problem might be a genuine problem for one individual, but might
not be for another. In short, problem solving refers to the effort needed in achieving a goal or
finding a solution when no automatic solution is available. Therefore, many researchers find
that their students do not solve problems at the necessary level of proficiency (Van Heuvelen,
1991; Reif, 1995; Redish et al.,2006). To help improve the teaching and learning of physics
problem solving,
studies began in the 1970’s (McDermott & Redish, 1999). Reif & Heller (1982) discussed
this view of problem solvers by comparing and contrasting the problem solving abilities of
inexperienced and experienced problemsolvers. Their findings showed that the principal
difference between the two was in how they organize and use their knowledge about solving
a problem. Experienced problem solvers rapidly re-describe the problem and often use
qualitative
Inexperienced problem solvers rush into the solution by stringing together miscellaneous
not necessarily have this knowledge structure, as their understanding consists of random facts
and equations that have little conceptual meaning. This gap between experienced and
inexperienced problem solvers has been well studied with an emphasis on classifying the
differences between students and experienced problem solvers in an effort to discover how
Most high school students take the minimum number of math classes needed to graduate.
By college, only a small percentage of our nation’s students elect to major in mathematics.
Others take only the minimum courses required, despite the fact that many careers depend
upon mathematical knowledge ( Steven Zemelman, Harvey Daniels, and Arthur Hyde). We
can't solve problems by using the same kind of thinking we used when we created them.
(Einstein,Albert) The vertical thinker says: 'I know what I am looking for.' The lateral thinker
says: 'I am looking but I won't know what I am looking for until I have found it.'” (Edward de
Bono)
STRATEGIES IN SOLVING MATHEMATICAL PROBLEMS
Nesher, Hershkowitz and Novotná(2003) stated that the concept of “problem solving” is a
very loosely defined notion, a kind of umbrella term for a number of different theoretical
approaches. If we admit that solving a genuine problem is not just a matter of following a
particular algorithm, we have to define heuristic strategies used for their solution.
it is about solving them. The ability to shape and solve Mathematical problems is the essence
of constructing mathematical meaning. Adults can help to pose problems engaging pupils to
find mathematical solutions, giving real understanding and purpose to Mathematics lesson.
However, problem finding helps children to become enthusiastic about problems. In this way,
they also develop the sound learning dispositions which will support problem solving
throughout life and across subjects. As stated, Mathematics appears to be beneficial to work
Švec (2012) stated that a teacher’s attitude and the teaching strategies he/she uses
significantly influence educational outcomes. Observations from Czech schools suggest that
pupils as well as teachers prefer problems in whose case the algorithm suitable for their
solution is apparent, in whose case there are no doubts about the choice of the suitable
algorithm (Novotná,2000).
For decades, education in the US endured a silent and gradual revolution in goals and
methods used to increase the awareness toward creative problem solving and creative
experiences. However, many educators have not yet realized that such changes occurred
toward more creative education. Similarly, little significant change occurred in teaching
methods and teacher-student relationships (Hamza &Alhalabi, 1999; Smith & Ragan, 2000;
problem solving can earn higher grades in subjects like mathematics, science and technical
subjects. The skill set for problem solving in all these areas have the same
components(http://www.ehow.com)
Educational and business leaders want today’s student both to master school subjects
and to excel in areas such as problem solving, critical thinking and communication—abilities
often referred to by such labels as “ deeper learning” and “ 21st century skills”. In contrast to
the view that these are general skills that can be applied across a range of tasks in academic,
workplace, or family settings, new report from the National Research Council found that 21 st-
century skills are specific to content knowledge and performance within a particular subject
area. The report describes how this set of key skills relates to learning mathematics, English,
and science and in education, and work. Goals for deeper learning and 21st –century
competencies are found in the new Common Core State Standards for mathematics and
English language arts and the National Research Council’s Framework for K-12 Science
Education. All three disciplines emphasize the development of cognitive competencies such
as critical thinking, problem solving, and argumentation, but differ in interpretation of these
competencies( http://phys.org/knowledge-skills-success.html).
Polya (1945) is cited for his four steps problem solving strategy. The first step is
Understanding the Problem, by identifying the unknown, the data and the condition, and
then drawing a figure and introducing a suitable notation. The second step is Devising a
Plan, in which the solver seeks a connection between the data and the unknown. If an
immediate connection is not found, the solver considers related problems or problems that
have already been solved, and uses this information to devise a plan to reach the unknown. In
the third step, Carrying out the Plan, the steps outlined in part two are carried out, and each
step is checked for correctness. In the final step Looking Back, the problem solution is
Reif et al. (1976) tried to teach students a simple problem solving strategy consisting
of the following four major steps: Description, which lists clearly the given and wanted
information. Draw a diagram of the situation. The next step, Planning, selects the basic
relations suitable for solving the problem and outline how they are to be used. The
Implementation step performs the preceding plan by doing all necessary calculations. The
final step is Checking, which ensures that each of the preceding steps was valid and that the
Tolga GOK(2010) in his research presents the selected and modified three steps in
the problem solving strategy based on the problem solving strategies reported by the
researchers mentioned before. The developed IPSS (Integrated Problem Solving Strategy
I. Identifying the Fundamental Principle(s): In the first and most important step, a student
should accurately identify and understand the problem. A student should examine both the
qualitative and quantitative aspects of the problem and interpret the problem in light of
his/her own knowledge and experience. This enables a student to decide whether information
is important and what other information may be needed. In this step students must: (i)
simplify the problem situation by describing it with a diagram or a sketch in terms of simple
physical objects and essential physical quantities; (ii) restate what you want to find by
naming specific mathematical quantities; and (iii) represent the problem with formal concepts
and principles.
II. Solving: A student uses qualitative understanding of the problem to prepare a quantitative
solution. Dividing the problem into subproblems is an effective strategy for constructing the
solution. Thus, the solution process involves repeated applications of the following two steps:
(i) choosing some useful subproblems and (ii) carrying out the solution of these subproblems.
These steps can then be recursively repeated until the original problem has been solved. The
decisions
needed to solve a problem arisen from choosing sub problems. The two main obstacles can
be: (i) lack of needed information and (ii) available numerical relationships that are
potentially useful, but contain undesirable features. These choices are promoted if there are
only few reasonable options among which a student needs to choose. An effective
organization of knowledge has crucial importance in making easy the decisions needed for
problem solving. The organization done after applying the particular principle is facilitated
by all of a student’s previously gained technical knowledge. The final step contains plugging
in all the relative quantities into the algebraic solution to determine a numerical value for the
III. Checking: In the final step, a student should check the solution to assess whether it is
correct and satisfactory and to revise it properly if any shortages are detected by following
this checklist: (i) Has all wanted information been found?; (ii)are answers expressed in terms
of known quantities?; (iii) are units, signs or directions in equations consistent?; (iv) are both
magnitudes and directions of vectors specified?; (v) are answers consistent with special cases
or with expected functional dependence?; (vi) are answers consistent with those obtained by
another solution method?; (vii) are answers and solution as clear and simple as possible?;
and (viii) are answers in general algebraic form? Those IPSS are expected to eliminate the
new design (IPSS) of computer-based problem solving systems was investigated with a pilot
study that is detailed below. Also, the perceptions of the volunteer students who attended this
Students need opportunities to practice what they are learning,whether it be to improve their
development can often be incorporated into “doing” mathematics; for example,learning about
perimeter and area offers opportunities for students to practice multiplication and fractions.
Games can also be a means of developing fluency and automaticity. Instead of using them as
time fillers, effective teachers choose and use games because they meet specific mathematical
purposes and because they provide appropriate feedback and challenge for all participants.
solution or to new learning. During the course of this problem solving, teachers
further encourage students to make conjectures and justify solutions. The communication that
occurs during and after the process of problem solving helps all students to see the problem
from different perspectives and opens the door to a multitude of strategies for getting at a
solution. By seeing how others solve a problem, students can begin to think about their own
thinking ( meta cognition) and the thinking of others and can consciously adjust their own
strategies to make them as efficient and accurate as possible. (Fosnot & Dolk, 2001)
Egodawatte( 2011) stated that Algebra is a very useful tool that can be utilized in
solving different kinds of problems that every people encounter in their daily living. Learning
which are likely applicable real-life situations. Knowledge of the said matter can contribute
greatly to the competence of an individual to cope up with the modern world. Set of
circumstances are basically needed to provide the chances to see how they can deal to this by
Resnick (1998) as cited in Pound and Lee (2011) noted that if they are to engage students
in contextualized mathematical problem solving, they must find ways to create classroom
situations of sufficient complexity and engagement that they become mathematically engaged
in their own right. They should also permit students to develop questions and not only to
solve problems posed by others. Problems do often enhance and help the learners develop
their critical thinking skills and recognize the purpose of Mathematics lesson and how it
would be beneficial on the learner’s own problem. However, it would depend on the
pedagogy of the strategies and techniques they employ and the learning environment they
create. The teacher, school, and the learning environment have something to do with this.
They generally affect the meaningful learning of the learners and the application of this
The more knowledge they have, the more they can think of ways to develop and to enhance
Kieran (1992) as cited in Nickson (2004) suggested that Algebra is seen by pupils as
memorizing rules and procedures . Algebra is a part of Mathematics which is particularly rich
in this respect. When a child learns Algebra, he is considerably enriching his/her problem-
solving repertoire. However, although it may seem to be at a level of abstraction that some
may consider inaccessible to all children, the simplest modeling procedure used in solving
word problems employ basic ideas of Algebra that are immensely powerful. Therefore,
identifying these levels at which the word structure helps to clarify the source of some
life problems are usually presented in words or in text form. They need to formulate English
phrases and sentences into mathematical phrases in sentences. Solving problems involve
applying previously required knowledge and some problems using strategies. Problems may
pattern of solution. The use of common sense, guessing and estimation are of great help in
solving non-routine problems are as follows: drawing a diagram, guessing and checking,
making a table, making an organized list, looking for patterns, working backward, making a
In the book of Salandanan (1996), Rondilla mentioned that the problem solving is order
ability. It is most difficult to test. Selection of items depends largely on student’s background.
Rondilla mentioned also in the book of Lardizabal, et. Al. (1991), that the processes of
problem solving and learning are highly unique and specific. Each individual has his own
unique style of learning and solving problem. As individual becomes exposed to alternative
models used by their individuals, they can refine and modify their personal learning style so
Morris( 1992) cited that in order to be useful in learning, an individual must experience
the common properties together with the four conceptual structures or scheme. For teachers,
this means that the structures of Mathematics must be clear in the children before they can
According to Bayer (2001), problem solving can be introduced in the 4-step sequence of
identifying the problem, choosing a solution plan, executing plan, and checking the answer.
Then, the teachers reinforce this strategy and elaborate it as student’s progress through
According to Cook (2001), one of the challenges of Math at any level is dealing with
five(5) misunderstanding that often arise; Math is essentially computation, the important
outcome in Math is the right answer, Math problems have only one right answer, there is only
one right way to solve a problem and the teachers and the books should not be questioned.
Wade and Tavris (1998), said that Asians believe that mathematical ability is innate. They
thought that if you “have” it, you don’t have to work hard , and if you don’t have it, there’s
no point in trying. They believe that the most important factor in Math performance is to
study hard.
school. The value of Mathematics lies not only in it’s practical use in every life but also as a
tool in learning other branches. Mathematics is the combination of sciences that treat to the
relations between quantities and the operations, which demonstrate those relations by the use
within in the classroom most closely approaches the mathematical activity of the world
outside. It involves pupils in learning about problem solving focusing on the learning and
problem solving processes, i.e., understanding the problem, devising a plan, carrying out the
significant improvement has shown on the level of 4th graders. The decreasing achievement
of Czech pupils occurred also in PISA between 2003 and 2009. That is why it is important to
look for ways of improving the situation (Mullis et al., 2012; OECD, 2010).
Schoenfeld (1987) pointed out that the knowledge of meta-cognitive and cognitive
skills will help students build a thinking plan which involves strategy, skills and procedures
to solve the given problems. This new thinking plan is connected to the students’
understanding of the relevant mathematical concepts that will be used. While solving the
problems, students will go through two phases such as interpretation of the mathematical
language and the calculation process . Newman (1977) also postulated that both language and
mathematical acumen are necessary for the successful solution of mathematical exercises .
"strategies", which are mathematical content knowledge that learners need to bring with them
to a mathematical task together with the ability to interpret and comprehend mathematical
jargons and semantics in order to successfully comprehend and solve mathematics problems.
represent a
complex mental activity consisting of a variety of cognitive skills and actions. Problem
solving included higher order thinking skills such as "visualization, association, abstraction,
be 'managed' and 'coordinated'" (Garofalo & Lester, 1985) Problem solving also includes
so, and they have to believe they can. Motivation and attitudinal aspects such as effort,
confidence, anxiety, persistence and knowledge about self are important to the problem
student should be able to translate the concrete to the abstract and the abstract to the concrete.
Therefore the mathematical word problem exam is more unique and challenging task than the
processes and strategies. Montague (2006) defined mathematical word problem solving as a
process involving two stages: problem "representation" and "problem execution". Both of
them are necessary for problem solving successfully. Successful problem solving is not
representation indicates that the problem solver has perceived the problem and serves to
guide the student toward the solution plan Mathematical problem solving also requires "self-
regulation" strategies . Mayer (2003) divided mathematical word problem solving into four
"cognitive phases": translating, integrating, planning and execution . Thus, students normally
find difficulty in solving word problems firstly from translating the word representations into
images, invoking images of formulae from memory". Hegarty et al (1995) argued that we
contrast two general approaches to understanding mathematical word problems that have
been introduced by previous researchers: "a short-cut approach" and a "meaningful approach"
that is based on an elaborated problem model. Bruner (1964) believed that one factor of
problem solving process makes to indicate real comprehension of words and concepts in
problem solving.
Obviously the necessary condition for teaching mathematics via problem solving
teachers’ solid
knowledge of mathematics, their own experience with creative approach to problem solving,
but also sufficient information and materials ready for use in the classroom. Important is the
so called specialized content knowledge (Ball, Thames and Phelps, 2008); this knowledge
includes identification of key mathematical concepts and possibilities in the given activity,
Fan and Zhu (2007) list among heuristic strategies also the following strategies: “Draw
a diagram”, “Guess and check”, “Look for a pattern”, “Make a systematic list”, “Use
beforeafter conception”. Eisner (1982), Sanford (1985), Kaufmann (1985) state that it is
The developed heuristic strategies are the author’s modification of strategies published
in (Kopka, 2013) and (Polya, 2004). Strategy of analogy: Analogy is a type of similitude. If
we are to solve a particular problem we find an analogical problem, i.e. a problem that will
deal with a similar problem in a similar way. If we manage to solve this similar problem, we
can then apply the method of its solution or its result in the solution to the original problem,
drawing from our experience, make a guess about the solution to the given problem. Then we
check whether the solution meets the conditions of the assignment. The next guess is made
with respect to the previous result. We carry on in this way until we find a solution.
Systematic experimentation: Systematic experimentation is a strategy in which we try to find
several experiments. First we apply some algorithm that we hope will help us solve the
problem. Then we proceed in a systematic way and change the input values of the algorithm
until we find
the correct solution. Problem reformulation: When using this strategy we reformulate
the given problem and make another one which may either be brand new, is easier for us to
solve and whose solution is either directly the solution to the original problem or facilitates
its solution. A specific and very important example of this strategy is translation of a word
problem from one language of mathematics to another. Classical geometrical problems such
as
trisection of an angle were easy to solve when translated to the language of algebra.
Solution drawing: When using graphical representation we usually visualize the problem by
making a drawing. We write down what is given and often also what we want to get. The
drawing we get in this way is called an illustrative drawing as it illustrates the solved
problem. Sometimes we can see the solution of the problem immediately in this drawing.
However,
in most cases we must manipulate with the drawing (e.g. we add suitable auxiliary elements)
and we solve the problem with the help of this modified drawing. We call this drawing the
solution
that what we have to find/prove/ construct holds/exists. Then we try to deduce from this
prove/calculate/construct. Thus we in fact try to get from the end to the starting situation as
within the solving process that it is desirable to introduce functions then it is usually good to
Research Paradigm
Grade 7
Mathematics Grade 7 Students
Teachers
Burgos, Quezon.
solving in daily instruction, teaching using games, puzzles and workbooks, computer assisted
method, translating words into local dialect, using visual representations, using role playing,
cooperative learning method, using multi-media and exposing students to multiple problem
solving strategies. On the other hand, the dependent variable is represented by the positive
effect of these strategies in the problem solving skills of the Grade 7 students shown in their
ability to recognize and articulate mathematical concepts and notation during problem
solving activities, solve problem solving questions with speed and accuracy, and answer real-
life questions involving problem solving correctly. It will also be reflected in their NAT
This also shows that the independent variables can block the intervening variable
which is students’ difficulties in solving problems that can affect their achievement in
mathematics.
Chapter III
METHODOLOGY
This chapter deals with the research design, the research site, population and sampling
procedure, the instrument used in the data gathering procedures, and the statistical treatment
utilized for the reader’s understanding of how the study was completed and how the
The study was conducted in the four secondary schools in Padre Burgos, Quezon. These
schools are, Hinguiwin national High School, Danlagan National High School, Lina Gayeta-
Lasquety National High School and San Isidro National High School . The first three
mentioned secondary schools are situated near or along the provincial road while the fourth
Respondents
The respondents of the study were the 7 Grade7 Mathematics’ teachers, 2 from
Hinguiwin National High school, 2 from Danlagan National High School, 2 from San Isidro
National High School and 1 from Lina Gayeta-Lasquety National High School. It also
involves 75 Grade 7 students from the two sections in Hinguiwin National High School, 64
Grade 7 students from the two sections of Danlagan National High School, 65 Grade 7
students from San Isidro National High school and 60 Grade 7 students from Lina Gayeta-
Lasquety National High School with a total of 264 Grade 7 students. Necessary actions to
preserve confidentiality of the informations about the participants is applied through giving
Non-randomized sampling was used in this study since all the Grade 7 Mathematics
teachers and Grade 7 students in each school was used as the participants.
Research Instrumentations
identify what strategies they are applying in teaching problem solving to their students, at
least, since the start of the present school year. Another questionnaire was constructed by the
researcher for the students participant which is composed of 15 problem solving questions
divided into three category: 5 easy, 5 average and 5 difficult problem solving questions.
Teacher’s name:___________
A. Below are different teaching strategies that can be applied in teaching problem solving
to Grad 7 students. Please check the box if the strategies opposite to it is one of
your/your strategy/strategies utilized in teaching problem solving.
____________________________________________________________
(Questionnaire for students participants)
Name:_________________
1. During the rat campaign, Team 1 turned in 72 rat tails on Monday, 36 on Tuesday, 48 on
Wednesday, 27on Thursday, 15on Friday and 97 on Saturday. Team II turned in 84 rat tails
on Monday, 56 on Tuesday, 22 on Wednesday, 13 on Thursday, 7 on Friday and 89 on
Saturday. How many rat tails did the two teams turned in all?
2. During a Plant a Tree Week, Freddie was able to plant 14 saplings, Manny 18 saplings,
Bertie 27,Totie 29 and Eddie 28. How many trees did the five boys plant in all?
3. At Php 136 per meter, how much will a roll of oil cloth containing 144 meters cost?
4. Mrs. Lapus set aside Php 1 680 for her four children’s weekly allowance. How much is
each child’s weekly allowance if the amount is equally divided among them?
5. Mrs. Torres ordered 3750 hollow blocks. The delivery truck can carry at most 625 hollow
blocks. How many trips did the truck make to deliver the order?
6. Christopher got the following grades in his short quizzes in Mathematics I for the first
grading period: 75, 83, 92, 85, 78, 82, 90 and 95. Find the average grade in the short tests.
7. You are one of the twelve students requested to transfer 336 books from the storeroom to
the library. If you will divide the books equally among yourselves, how many books will
you carry?
8. 80% of the student population of a certain university are from 18-23 years old. If the
student population is 65 000, how many are in this age range?
9. Mr. Reyes placed Php 50 000 in the money market for 3 months at 12% a year. How much
did his money earn?
10. Ifugao farmers now get 90 cavans per hectare as against 60 cavans that they used to get
because they now used good yielding seeds suitable for highland climate. What was the
rate of increase in the yield?
11. The width of a rectangle is 5 dm. The length is 2 dm. more than twice the width. Find the
area of the rectangular region.
12. Andet is going to cement our backyard. He will use the ratio 1:3:5. He has already
measured 10 level wheel borrows of gravel and 6 level wheel borrows of sand. How many
wheel borrows of cement will he needed for the mixture.
13. A boy started on a trip across a lake by a motorboat. After he had traveled and 15 km.,
the motor failed and he had to use a banca for the remaining 6 km to his destination. His
average speed by motor was 4 kph faster than his average speed while rowing a banca.
If the entire trip took 5 ½ hours, what was his average speed while rowing?
14. Renato is 3 years younger than Domingo. Six years ago, Renato’s age was two-thirds of
Domingo’s age. What are their present age?
15. A photocopy machine can finish 500 pages in 1 minute. How many book of 250 pages
can the machine copy in 20 minutes?
Data Gathering Procedure
Data were gathered, first, by asking the permission of the District Supervisor, Ms.
Alicia V. Gonsalez, through letter of request, to conduct my study in the four secondary
schools in the district of Padre Burgos. Upon her approval and recommendation, I
approached and asked the permission of the principals of the four secondary schools to
for my study. I assured them that confidentiality regarding details that can be gathered from
Pre-interview with the Grade7 Mathematics teachers and handling them the
questionnaire were done in the second visit. The researcher also asked them the schedule
convenient to them and for the students to answer the questionnaire intended for them. In the
third visit, the students were given the prepared test involving problem solving questions.
Final interview with the mathematics teachers were done in the fourth visit, informing
them also the result of the test. The researcher thanked them for being so helpful and
cooperative, and assured them that every information gathered will be used well and will be
carefully treated.
Statistical Treatment