CHAPTER I THE PROBLEM AND ITS BACKGROUND Introduction
“Mathematical problems are really difficult to be solved”. Students do not know how to do it. That is why they do not like the subject especially problem solving as a skill, the more that students hate mathematics when they are given a test in problem solving. However, they cannot avoid this challenge being an important skill to be acquired. In fact, problems are spices in life. S o, it is a “must” for us to solve problems to make on life meaningful and fruitful. Problem solving in mathematics is not a topic but a process underlie the whole mathematics programmes which contextually helped concepts and skills to be learned (Ibrahim 1997). It is true that problem solving in mathematics is not a topic but a process on how to solve or deal with mathematical problems. This mathematical problems serves as a guide on how to apply its process in solving our daily life problems. Many mathematical skills are involved in problem solving. Mathematical difficulties are frequently characterized by cognitive deficits such as ineffective problem solving strategies, lack of comprehension and computational fluency. Leongson (2003) reveals that Pilipino students excel in knowledge acquisition but fare considerably low in lessons requiring higher order thinking skills. This disappointing condition is evident in the performance of students in national and international surveys on mathematics and science competencies. The Third International Mathematical Science Study (TIMMS, 2000) examined patterns of student’s student’s achievement in mathematics and found out that the school effectiveness 1
and teachers competency impact learning and promote higher level of achievements. In addition, the quality of instruction and effective instructional design are necessary to alleviate problems related to teaching and learning mathematics (Dursun & Dede,2004). In the 21 st century, one shifting paradigm in education is abou t teacher’s role and competencies. Competent mathematics teachers provide a roadmap to guide students to an organized understanding of mathematical concepts, to reflective learning, to critical thinking, and ultimately to mathematical achievement. Problem solving is the real test in mathematical abilities. It concerns understanding of the problem which practically demands reading comprehension. It also requires the appropriate formula that is needed to get what is required. It also requires the proper operation of the numbers/ expressions involved. These are some of the common steps in solving problem. It must be noted that in every problem there is a solution. It is then a challenge for every problem solver to find some ways and measures to find the solution of the problem. Any student who lacks knowledge and skills in solving problem in mathematics in general (Algebra, Arithmetic, Geometry, etc.) will find difficulties in order to arrive the correct solution, thus, this study.
Statement of the Problem 2
The purpose of this study is to identify students’ difficulties in solving mathematical word problems through a self-made test whose validity and reliability are determined. This study also investigates what teaching strategies are used to foster success to students difficulties in solving mathematical word problem. Specifically this study intends to answer the following questions: 1. What is the demographic profile of the student as described in: 1.1 age; 1.2 gender; and 1.3 grade 2. What is the demographic profile of the teachers as described in: 2.1 age; 2.2 gender; 2.3 civil status; 2.4 highest educational attainment; and 2.5 teaching position 3. What are the difficulties of the students in solving mathematical word problems? 4.What are the factors that affect students’ difficul ty in solving mathematical word problem? 5. What teaching practices that the teacher’s employ in teaching mathematical word problems? 6. What are the strategies used by the students in solving mathematical word problems?
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Hypothesis of the Study
Ho 1: There is no significant difference in the extent of causes in solving problems when the respondents are grouped according to their profile variables.
Significance of the Study
The result of this study will be very beneficial to school administrators, teachers, students, parents, researchers and to the researcher herself. To the school administrators, the findings can give them insights to find
some measures to improve or enhance teacher’s competence other than themselves. To the teachers, the findings can guide them in determining strategies for the
development of their skills in problem solving to be taught/ learned by the students. To the students, the results of this study are their guide to acquire more skills
in problem-solving. To the researchers, the result of this study may serve as tool or guide in their
research studies. To the parents, this study will serve as a guide to help their children become
expose in different world problems to eliminate fears in facing it.
Scope of the Study
4
The research will be conducted to the selected secondary institutions in the Municipality of Siquijor. Particularly the researcher will get ten percent (10%) of the total population of the student-respondents and one hundred percent (100%) of the total population of the teacher-respondents both in the selected public and private secondary education in the Municipality of Siquijor.
Theoretical Framework
This study is guided by the Situated Learning Theory of Jean Lave and Etienne Wenger (1990) which is learning in the same context in which concepts and theories are applied. It hinges on the belief that learning is embedded in or connected to the context in which knowledge and skills are developed (Santrock 2004). Snowman and Biehler (2006) maintain that learning is made meaningful when it is anchored on a realistic context because learning cannot be separated from daily routine that keep on acquiring new information every day in life.
According to Aquino (2015), the perspective of situated learning theory is anchored on the idea that knowledge acquisition is focused on problem-solving skills. Situated learning engages students in tasks that are related to real world situations. It stresses the context and application of knowledge instead of memorizing facts and ideas. In addition, Piaget’s foremost contributions was to enhance the understanding of mental development. This would lead to the activities that involve thinking, perceiving and problem-solving. 5
Constructivist approach views learning as a process of constructing meaning which is greatly influenced by the learners’ experiences and understanding . Therefore, constructivist teacher should make the learners construct their own knowledge by connecting lesson to their past experiences
Conceptual Paradigm
INPUT
Student’s Profile
Teacher’s Profile
Test result in problem
PROCESS
Student’s Strategies in Solving Mathematical
Administration of Test and Questionnaire
solving
OUTPUT
Tallying
Treatment of Data
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Intervention
Program
Definition of Terms
Figure 1 Conceptual Framework of the Study
Definition of Terms
The following terms are defined as the researcher uses it in the study: Difficulties
Refers to the obstacles that cause error or mistakes made by students when dealing with mathematical word problems. Mathematical Word Problem
7
Refers to mathematical exercises that present relevant information on a problem as text, rather than in the form of mathematical notation Mathematical Skill
Refers to the ability to solve mathematical problems Lack
Refers to the insufficient skills in solving mathematical problems Average
Refers to the normal skills of students in solving mathematical word problems
CHAPTER II RELATED LITERATURE AND STUDIES
Problem-solving is the ability to identify and solve problems by applying appropriate
skills
systematically
(Jabberwocky).
It
provides
students
with
opportunities to use their newly acquired knowledge in meaningful, real-life activities and assists them in working at higher levels of thinking.
8
Problem solving has been and will be a necessary skill not only in Mathematics but in everyday living. Part and parcel of problem solving is to translate word problems into mathematical equation. However, students especially in grade school have difficulties in analyzing and interpreting word problems. Students most especially in grade school can easily perform an indicated operation but when this is given verbal forms, students need to first identify what operation is involved by translating this into a mathematical sentence before actually performing the operation and arriving at the correct answer (Dela Cruz and Lapinid, 2014).
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 analyzed to find the real cause. Lack of mathematical skills caused difficulties in solving problem. Students are required to apply and integrate many mathematical concepts and skills during the process of making decisions and problem-solving. The lacked of mathematical skills among students are varied (Hill 2008).
These mathematical skills are:
i) number fact skill (proficiency of number facts, tables and mathematics principal);
9
ii) arithmetic skill (accuracy and logarithm in computational and mathematical working-procedure);
iii) information skill (expertise to connect information to a concept, operational, and experience as well as the expertise to transfer information and transform problems into mathematical sentences.);
iv) language skill (proficiency of terms and relevance of mathematical information);
v) visual spatial skill (skill to visualize mathematical concepts, manipulate geometrical shape and space meaningfully.)
Incomplete mastery of number facts, weakness in computational, inability to connect conceptual aspects of math, in efficiency to transfer knowledge, difficulty to make meaningful connection among information, incompetency to transform information mathematically, incomplete mastery of mathematical terms, incomplete mastery of mathematical language and difficulty to comprehend and visualizing mathematical concept might results to difficulties (Nathan et al. 2002). In addition, lack of commitment to solving problem, misinterpreting the problem, lack of knowledge of the techniques and processes involved in problem solving, inability to use techniques ineffectively, using a method inappropriate to the particular problem and insufficient or inaccurate information lead to the difficulties in solving mathematical word problem and making various errors and confusion in the process of problem-solving.
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Difficulties faced among students were more noticeable during the first procedural step in solving problem compared to the other. Polya (1981) stated that problem-solving is a process starting from the minute students is faced with the problem until the end when the problem is solved. There are many problem solving models.
Problem-solving models Polya (1981)
Krulick&Rudnick (1996)
4- hierarchy phase
5-hierarchy phase
i) understanding problem;
i) reading and thinking;
Zalina (2005)
3- hierarchy phase i) understanding problem;
ii) planning;
ii) analyze and planning;
ii) solving the problem;
iii) performing the plan;
iii) organizing strategy;
iii) stating the answer;
iv) confirmation of the
iv) getting the answer;
answer
v) confirmation of the answer
Each phase involved a different combination of mathematical skills and different cognitive abilities. 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. These abilities vary among students.
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According to Dendane (2009), mathematical problem solving related to counting and daily transactions of commerce has been present from the earliest days of human experience. On the other hand, geometry was also widely used in land measurement [1] Mathematics is now used to quantify numerically and spatially natural as well as man-made situations. It is used to solve problems and with the advent of the computer technology, it has helped in making social, economical and technological advances that few decades ago would have been unthinkable.
Learning mathematical facts and contents is important but is not enough. Students should learn how to use these facts to develop their thinking skills and solve problems. Mathematics educators have accepted the idea that the development of problem solving ability deserves special attention [2] and that one of the most important components in any mathematics curriculum or program is genuine mathematical problem solving.
If well facilitated, mathematical problem may help
students:
1. develop and improve the generic ability to solve real life problems, 2. develop critical thinking skills and reasoning, 3. gain deep understanding of concepts, 4. work in groups, interact with and help each other.
Mathematical problem solving is a process that involves a set of factors and tasks to achieve a defined goal.
It depends on many skills and factors which
therefore makes it challenging both to learn and to teach.
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Mathematical application is no longer an intellectual exercise for the new nor limited to scientist alone.
It has become an essential part of man’s working
equipment, which provides the best hope for coping effectively with problems in everyday area of life (Parami, et. al., 2016).
According to Uthai in Phonapichat, et. al. (2013), difficulties affecting mathematical problem solving can be classified as: 1) students cannot understand the whole or some parts of the problem due to the lack of imagination and experience needed to consider the problem. 2) Students have difficulties in reading and comprehension, unable to understand what important information is in a problem and organize it accordingly. Thus they cannot invert the text into mathematical symbols. (3) Students lack interest in solving mathematical problems due to the length and complexity of the problems, which is demotivating. 4. Teachers do not present daily life matters as a problems very often.
5) Teachers are likely to make students
memorize “keywords” in the problems to use in f ormulae. 6) Teachers focus on following examples given in textbooks rather than teaching the principles behind each problem. 7) Teachers teach without concern with thinking process orders.
Related Studies Foreign Studies
Meese (2001) said that teachers need to understand students’ potential, problems and learning difficulties in order to implement effective teaching strategy and to produce meaningful learning among students. If learning approaches and teaching strategies applied did not fulfil the intellectual needs of the students these 13
could lead to students’ difficulties in learning mathematics. Aside from that, weakness in understanding concepts, logic-thinking and lacking of strategic knowledge caused errors in problem-solving (Tay Lay Heong 2005). Phonapichat, et. al. (2013) asserts that mathematics teaching is to enable students to solve problems in daily life. It reflects that students have difficulties in comprehending mathematical problems affecting the process of problem-solving. Therefore, in order to allow teachers to establish a proper teaching plan suitable for students’ learning process.
This study aimed to analyze the difficulties in
mathematical problem solving among elementary school students.
The results
suggest that there are several difficulties in problem solving, namely 1) students have difficulties in understanding the keyword appearing in problems, thus cannot interpret them in mathematical sentences. 2) Students are unable to figure out what to assume and what information from the problem is necessary to solving it, 3) whenever students do not understand the problem, they tend to guess the answer without any thinking process; 4) Students are impatient and do not like to read mathematical problems, and 5) Students do not like to read long problems. Therefore, the results found in this research will lead to the creation and the development of mathematical problem solving diagnostic tests for teachers, in order to improve students’ mathematical problem solving s kills. Angateeah (2017) conducted a study where it explored the cognitive processes undergone by Mauritian students who have difficulties in solving word problems.
Based on the findings, all students could read the problems.
Achievers (HA) are wrong due to careless errors.
High
While HA demonstrate good 14
problem solving skills, some exhibit overconfidence. Average Achievers suffer from procedural errors while Low Achievers face difficulties mainly in visualizing and representing the problem.
Tambychika, Subahan and Meerah (2010) asserts that problem solving is one of major aspect in mathematics curriculum which required students to apply and to integrate many mathematical concepts and skills as well as making decision.
The findings showed that respondents lacked in many mathematical skills such as number-fact, visual-spatial and information skills. Information skill was the most critical. The deficiency of these mathematics skills and also of cognitive abilities in learning inhibits the mathematics problem-solving. This understanding on how the deficits influenced the problem-solving is expected to give effective guide lines in preparing diagnostic instruments and learning modules in order to develop the mathematics skills.
Local Studies
Miranda (2006) stated that children might experience difficulties in thinking and learning when they demonstrated difficulty in giving attention, describing orientation of shape and space, making perception and understanding mathematical language. As a result, students struggle in different phases in the process of problem-solving.
Dela Cruz and Lapinid (2014) examined the students’ difficulties and level of performance in translating worded problems into mathematical symbols. A 20-item
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problem solving test involving the four fundamental operations was given during the third quarter of the school year 2012-2013 to 2014 Grade 5 students. Scores in this test measured their performance level in translating worded problems while interpretation of their mistakes identified their difficulties in translating word problems. Results indicate that 40% of the respondents are below the satisfactory level in translating worded problems. Carelessness, lack of comprehension, interchanging values, and unfamiliar words are some of the common difficulties encountered by the respondents in translating word problems.
Ganal and Guiab (2014) focused on the problems and difficulties encountered by Bachelor of Elementary Education sophomore students towards mastering learning competencies in mathematics. The problems and difficulties are categorized into personal problems, emotional problems, problems on teacher’s instruction, problems with school adjustment, problems in adjusting to classmates and boardmates, and problems arising from over-extended schedule/workloads for practice in different competitions. The study reveals the findings that in general, the respondents encountered personal problems relating to school expenses, lack of interest and negative attitude towards the subject.
The emotional problems
encountered are excessive stress in doing academic tasks and low self-esteem or not believing in one’s capabilities.
On problems relating to teacher’s instruction, there
are no effective motivation and introduction, and not creative enough to adapt his/her method to the learner’s capability. As to problems with school adjustment, the most frequent are difficulty in adjusting to life/role of a college student, and not doing the tasks well.
The problems in adjusting to classmates/boardmates are how to be 16
accepted by classmates and boardmates, and working effectively with different kinds of
classmates.
With
regard
to
problems
arising
from
over-extended
schedule/workloads for practice in different competitions, the most common problems are too many academic tasks and project assigned, and studying and reviewing too many subjects everyday. Ambut, et. al. (2008) conducted a study on reading and mathematics tutorial based on intervention. The findings show that the respondents were performing very low in reading. After the intervention, out of 26, 6 or 23.1 percent were on frustration level, 13 or 50 percent were on instructional level and 7 or 26.9 percent were on independent level. CHAPTER III RESEARCH METHODOLOGY
This chapter presents the description of the research design, research respondents, and data gathering procedure, validation of questionnaires, research instruments, and statistical tools.
Research Design
This research will utilize the descriptive-quantitative method. This study will utilize the descriptive method to know the difficulties of the students in solving mathematical word problems. The descriptive research involves describing, analyzing, and interpreting data. The quantitative method is utilized to measure to
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what extent causes of the difficulties of the students in solving mathematical word problem. Research Locale
The research will be conducted to the selected private and public schools in the Municipality of Siquijor. The researcher will get ten percent (10%) of the total population of the student-respondents in the selected private and public secondary institutions in the Municipality of Siquijor and one hundred percent (100%) of the total population of the Mathematics teacher-respondents in the selected private and public secondary institution in the Municipality of Siquijor.
Data Gathering Procedure
The researcher will write a letter to the Superintendent in the Division of Siquijor, and the School Director of the Catholic Schools asking permission to conduct a survey about one hundred (100%) of the total population of the Mathematics teacher-respondents in the selected basic education institutions in the Municipality of Siquijor and ten percent (10%) of the total population of the studentrespondents in both selected private and public schools in the Municipality of Siquijor. The researcher herself will conduct the survey and retrieve the questionnaires, tabulate, compute, analyze, and interpret the data.
Validation of Questionnaires
18
Research questions will be formulated by the researcher based on the 21 st Century Mathematics, by Henry N. Adorna, et. al., Math Power
(Intermediate
Algebra) by A.O. Reyes and Next Century Mathematics Geometry by Jesus P. Mercado, et.al.,(2008). The research questions validated by the experts in the field like the adviser of the researcher. The research questions were checked and were revised. To achieve content-related validity, the researcher will show the research instruments for comments and improvements to experts. The researcher hopes that the comments of the experts will be very helpful for the improvement of the research instruments.
Research Instruments
The survey questionnaires have two (2) parts. The first part is on the demographic profile of the student-respondents (which include age, gender, and year level) and teacher-respondents (which include age, gender, highest educational attainment, civil status, length of service, and teaching position). The second part is the survey questionnaire proper. It composed of stu dents’ survey questionnaire and teachers’ survey questionnaire. Statistical Tool
The following statistical tools will be applied in the presentation, analysis, and interpretation of the data gathered. 19
1. Percentage
The researcher will use the percentage formula (P=
∑
X 100) in getting the
percentage of the demographic profile of the respondents where P= Percentage ∑= Summation f = Frequency
n= Total number of respondents
The researcher will use the percentage ( P=
X100) in getting the percentage
of the scores of the respondents in the questionnaires where: P= Percentage TS= Total score TI= Total number of Items To measure the students’ difficulties when solving mathematical word problems, numerical scaling is provided below.
Scale
Verbal Description
96-100
Excellent
Meaning
Students show
excellent skills in
solving mathematical word problem starting problem,
from
understanding using
the
appropriate
20
strategies/methods,
visualizing
the
problem, familiarizing problem context, computational skill and the ability to identify the formula to be used to solve the problem. (Score of 5/5) 90-95
Very Good
Students show a very good skills in solving mathematical word problem starting
from
problem,
understanding using
strategies/methods, problem,
the
appropriate visualizing
familiarizing
the
problem
context/computational skill to solve the problem. (Score of 4/5) 85-89
Average
Students
show
average skills
in
solving mathematical word problem, using appropriate strategies/methods, visualizing the problem, familiarizing the
problem
content
and
computational skill. (Score of 3/5) 80-84
Good
Students
show
understanding appropriate
good
the
skills
problem,
in
using
strategies/methods,
visualizing the problem, familiarizing 21
problem context, and has a good computational
skill
to
solve
the
problem. (2/5) 75-79
Lack of Understanding
Students show lack in understanding the
problem,
using
strategies/methods,
appropriate
visualizing
the
problem, familiarizing problem context, and lack in computational skill to solve the problem. (Score of 1/5) Below 74
Poor
Students are poor in understanding the
problem,
using
strategies/methods, problem,
appropriate
visualizing
familiarizing
the
problem
contexts and poor in computational skill to solve the problem. (Score 0/5)
2. Weighted Mean
The researcher will use weighted mean formula ( =
∑
) in getting the
mean of the extent causes of students’ difficulties in solving mathematical word problem to both students and teachers where X w = Weighted mean
∑ = Summation 22
f = Frequency x = Weights
n = Total number of respondents Likert Scaling
The researcher will use Likert Scaling in identifying the extent causes of students’ difficulties in solving mathematical word problems will be rated in the scale of 1 to 5, where 1 is the lowest and 5 is the highest with accompanying verbal description of 5 as strongly agree, 4 as agree, 3 as doubtful, 2 as disagree and 1 as strongly disagree.
3. Chi-square
The researcher will use chi-square formula ( = ∑
(−)
) in getting the
significant relationship between the integration of concepts on history and philosophy by the teachers and their profile where O = Observed frequency E = Expected frequency ∑ = Summation 2 = Chi-aquare
Formula for E is given below: =
() 23
Where GT= Grand Total
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References
Ambut, C.V., Palomo, E., Liza, N. and Uy, E. (2008). tutorial: An intervention.
Reading and mathematics
The Science and Technology Research Journal.
Volume 3 No. 1 ISSN No. 1908-4420.
Angateeah, K.S. (2017). An investigation of students’ difficulties in solving nonroutine word problem at Lower Secondary. International Journal of Learning
and Teaching. Vl. 3, No. 1. Mauritius Institute of Education, Reduit, Mauritius. Aquino, A. M. (2015). Facilitating human learning (2nd Ed.). Manila: REX Printing Company, Incorporated. Corpuz, B., Salandanan, G. (2015). Principles of teaching: with TLE. Lorimar Publishing Incorporated. Curie, M. (2005). The Skills of Problem Solving. Retrieved from http://www.itseducation. asia/the-skills-of-problem-solving.htm Curie, M. (2005). Why People Fail To Solve Problems Effectively. Retrieved from http://www.itseducation.asia/why-people-fail.htm Dela Cruz, J.K.B. and Lapinid, M.R. (2014).
Students’ difficulties in translating
worded problems into mathematical symbols.
Presented during DLSU
Research Congress. De La Salle University, Manila, Philippines. Dendane, A. (2009).
Skills needed for mathematical problems.
Presented paper
at the 10 th Annual Research Conference. UAE University. Fisico, M., Sia. L., et. al., (1995). 21st century mathematics first year . Phoenix Publishing House Incorporated. Fredericks, A. D. (2005). The Complete Idiot's Guide to Success as a Teacher. Retrieved from https://www.teachervision.com/problem-solving/problem
solving.
25
Ganal, N.N. and Guiab, M.R. (2014).
Problems and difficulties encountered by
students mastering learning competencies in Mathematics.
Academic
Journal Article. Jupri, A., Drijvers, P. (2016). Student Difficulties in Mathematizing Word in
Algebra. Eurasia Journal of Mathematics, Science &
Technology
Retrieved from http://iserjournals.com/journals/
ejmste
Problems Education.
Klymchuk, S., Zverkova, T., Gruenwald, N., et.al. (2010). University Students’ Difficulties in Solving Application Problems in Calculus: Student Perspectives. Mathematics Education Research Journal. Retrieved from https://link.springer.com/article/10.1007/BF03217567 Mapile, E. (2005). Intermediate algebra . Marikina City: Academe Publishing House Incorporated. Orines, F., Mercado, J. (2008). Next century mathematics. Geometry (2nd Ed.). Phoenix Publishing House Incorporated. Parami, A.J.P., Yano, R.J.Q. and Padayhag, C.J.B. (2016). module in
problem solving skills in plane and solid geometry for 3 rd
year BEED and BSED-
Mathematics
Thesis Siquijor State College, larena,
An
students.
Undergraduate
Siquijor.
Phonapichat, P., Wongwanich, S. and Sujiva, S. (2013). elementary
Research-based
An analysis of
school students’ difficulties in mathematical problem solving.
Presented paper at 5 th
World
Conference
on
Educational
Sciences.
University of Bangkok, Thailand. Reyes, A. O. (2010). Math Power II: Intermediate algebra . The Library Publishing House Incorporated. Sefi, M., Hagverdi, M., Azizmohamadi, F. (2012). Recognition of Students’ Difficulties in Solving Mathematical Word Problems from the Viewpoint of Teachers. Journal of Basic and Applied Scientific Research. Retrieved from www.textroad.com 26
Tambychik, T., Meerah, T. (2010). Students’ Difficulties in Mathematics Problem Solving: What do they Say?. Procedia Social and Behavioral Sciences 8, 142-151. Retrieved from www.sciencedirect.com
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