# Makalah Aritmatika & Aljabar

THE RELATIONSHIP
BETWEEN ARITHMETIC ABILITY
AND ALGEBRA ACHIEVEMENT OF THE SMP STUDENTS

This paper is used to fulfill the requirement of Research Methods
in Mathematics Education Course (MT 504)

Written by:
NIM : 1005382

Mathematics Education Department
Faculty Mathematics and Science Education
Indonesia University of Education
2013

CHAPTER 1
INTRODUCTION

A.    Background of Study (Rationale)
Humans life always have relationshipwith mathematics. Mathematics is a field of knowledge which is a tool to think, communicate, and solve problem for all practical things in humans life. Without inventions in mathematics, humans life won’t be as modern as now.
Mathematics grows and develops because of thinking process, then logic is a base of mathematics construction. At the beginning, branches of mathematics that people was found are arithmetic, algebra, and geometry. Then calculus was found and has function as a base of another more complex branches such as statistics, topology, algebra (linear, abstract, set), geometry (geometry system, linear geometry), vector analysis, etc. (Suherman, Erman; 2003)
All of concepts in mathematics have relationship one and another. It is started from the simple one and continued to the complex one. Students have to understand the previous lesson so they can understand the next lesson. There are such structural steps in learning mathematics until the goal of learning is achieved.
One of problem in mathematics education is mathematics phobia, which is exhibited by many students, is the persistent, illogical, intense fear of not succeeding in math. It is the belief that one is unable to handle the difficulty associated with learning mathematics. (Narins, Brigham: 2001). This kind of phobia, especially in arithmetic, even started since the children were young, while they were in primary or elementary school and it will take effect to their next life when they are growing up.
Young children like numbers. They love “to count”. They are quick to compare numbers of candies, pencils, marbles, and the like. They chant number rhymes and delight it in number games. For some children artihmetic is ever a pleasant pursuit. But for others what once was sweet turns bitter. Arithmetic brings them annoyance and frustation, as it is converted from a charming pastime to a hateful and dreaded discipline. Mental tastes differ, to be sure. No subject is universally liked or universally disliked. Yet it would seem that mathematics draws more than its fair share of pupil aversion. This is a sad and serious state of affairs. We live in a highly complex society. Man or woman, a citizen who has not mastered the basic arithmetical skills is a drag on the rest of us and is himself gravely handicapped. To dislike arithmetic, in our modern era, is to visit upon oneself continual frustation. (Swain, R. L: 1963)
Algebra cannot be separated from arithmetic because algebra’s properties of mathematical operations is just the same as arithmetic. One of algebra’s characteristic is using symbols as substitute of real numbers. But the rule of algebra is the same as arithmetic.  Arithmetic is learned by students since they were at kindergarden and primary or elementary school, but algebra is learned by students when they are at junior high school. Actually, the government organized the curriculum based on the development of the students’ cognitif as the Piaget’s theory. But in the real conditions, there are many students still feel hard or difficult to learn algebra at junior high school and it takes effect until they are in high school or college. Algebra is one of mathematics material in junior high school, which also may be one of the most difficult lesson for the students.
A large share of the difficulties which students encounter in their study of algebra may be traced to the fact that it presents them a radically new and different approach to the study of quantitative relationships, characterized by a new symbolism, new concepts, a  new language, a much higher degree of generalization and abstraction than they have encountered previously, and an essential dissociation of many of its parts from intuition and concrete experience. (Butler, C. H., and Wren, F. L: 1960)
The problems about students’ difficulties in learning arithmetic and algebra are being an interesting topic for writer’s research. This kind of research is very important because arithmetic ability is developed since children were young and algebra achievement will be very important to solve problems in a lot of fields in our life, such as economy, physics, chemistry, etc. The result of this research hopefully can give information for us – both teachers as educators, parents, even students – about relationship between arithmetic ability and algebra achievement of SMP (junior high school) students.

B.     Purpose and Advantage of Study
The purposes of this study are:
1.      To identify the existence of relationship between arithmetic ability and algebra achievement of the SMP students
2.      To identify and describe relationship between arithmetic ability and algebra achievement of the SMP students
The advantages of this study are:
1.      To give information to elementary or primary school teachers that students’ arithmetic ability will take effect to the next mathematics lesson in junior high school
2.      To give information to junior high school teachers that there is a relationship between arithmetic ability and algebra achievement of the SMP students
3.      To give motivation of learning-algebra to junior high school students so they can apply it in other field of study

C.    Research Problems/Questions
1.      Is there a relationship between arithmetic ability and algebra achievement of the SMP students?
2.      How is the relationship between arithmetic ability and algebra achievement of the SMP students?

D.    Operational Definitions
These are some operational definitions used in this research, based on Oxford Advanced Learner’s Dictionary of Current English (1974)
1.      Ability: (potencial) capacity or power (to do something physical or mental)
2.      Achievement: something done successfully, with effort and skill
3.      Algebra: branch of mathematics in which signs and letters are used to represent quantities
4.      Arithmetic: science of numbers; working with numbers

CHAPTER 2
FRAMEWORK OF STUDY
(LITERATURE REVIEW)

A.    Arithmetic Ability
Arithmetic is one branch of mathematics. Generally, students start to learn arithmetic in primary or elementary schools to develop their numeracy skills in calculating, because arithmetic ability will be widely used to solve various problems in mathematics and in their daily life. There are four mathematical operations used in arithmetic: addition, substraction, multiplication, and division.
Oxford Advanced Learner’s Dictionary of Current English (1974) defined arithmetic as science of numbers; working with numbers.
Taylor, E. H., and Mills, C. N. (1961) noted that: “Arithmetic is a method of thinking in which we neglect all aspects of experience ... Arithmetic is fundamental in education, both for its development of the higher mental processes and for its value as a means of computation and problem solving.” They also divided teaching of arithmetic into three phases: (1) teaching the meaning of number, (2) teaching the number facts and number process, (3) teaching the use of number process in solving problems in everyday life.
Badan Standar Nasional Pendidikan (2007) or Comitee of National Standard of Education noted that basic competences of arithmetic ability at first semester of seventh grade are:
1.      Doing calculation operations of round number and fraction
2.      Using calculation operations of round number and fraction to solve problems

B.     Algebra Achievement
Algebra is learned by the students at the middle of first semester at seventh grade and at the beginning of eighth grade. Algebra is used in a lot of fields of study, such as economy, physics. etc. So, algebra is one of important material that has to be understood by the students.
Oxford Advanced Learner’s Dictionary of Current English (1974) defines algebra as branch of mathematics in which signs and letters are used to represent quantities.
Picciotto, Henri (1995) said, “Many teachers and textbooks describe algebra as the ‘natural’ extension and generalization of arithmetic.”
Wahyudin (2004) noted that “Algebra is such a way and strategy to understand and solve problems.”
Algebra achievement of students depends on their algebra ability. Badan Standar Nasional Pendidikan (2007) or Comitee of National Standard of Education noted that basic competences of algebra ability at first semester of seventh grade are:
1.      Recognizing algebra’s form and its elements
2.      Doing operation of algebra’s form
3.      Solving one-variabled linear equation
4.      Solving one-variabled linear non-equation
5.      Making mathematical models from problems related with one-variabled linear equation and one-variabled linear non-equation
6.      Solving mathematical models from problems related with one-variabled linear equation
7.      Using algebra concepts to solve simple social-arithmetic problems
8.      Using propotions to solve problems

C.    Relationship Between Arithmetic and Algebra
Butler, C. H., and Wren, F. L. (1960) explained about arithmetic in elementary school and algebra in junior high school as: “It is the responsibility of the elementary school to lay the fundamental groundwork of basic concepts, principles, and skills upon which the arithmetical structure must be built. The broader mathematical obligations of the junior and senior high schools, however, are to strengthen and increase the working vocabulary of arithmetical terms; to clearer understanding of basic concepts, relationships, and principles; to develop further facility in the fundamental skills; to shape a more mature concept of the basic structure of the number system of arithmetic; and to emphasize the abstraction of arithmetical processes to problem situations. A large share of the difficulties which students encounter in their study of algebra may be traced to the fact that it presents them a radically new and different approach to the study of quantitative relationships, characterized by a new symbolism, new concepts, a  new language, a much higher degree of generalization and abstraction than they have encountered previously, and an essential dissociation of many of its parts from intuition and concrete experience. Also, in contrast to arithmetic, algebra is more concerned with the concious examination and study of processes and basic structure than with particular answers to particular problems.”
Taylor, E. H., and Mills, C. N. (1961) noted about arithmetic and algebra in junior high school, “Arithmetic should begin and end with problem solving. Good teaching uses problems both to introduce and to fix new ideas and processes. Since the teacher of arithmetic in the seventh and eighth grades needs to be prepared to introduce the pupils to algebra and geometry, introduction to algebra is given here through the use of the formula, the simple equation, and the graph. Seventh- and eighth-grade pupils should be shown that algebra is a useful tool in solving problems.”
We can conclude that arithmetic – which learned since students in elementary school – has relationship with algebra. Arithmetic is a fundamental material to learn the next materials, include algebra because basic concepts, principles, and skills of arithmetic will be used in algebra. Algebra cannot be separated from arithmetic because algebra’s properties of mathematical operations is just the same as arithmetic. One of difference between algebra and arithmetic is algebra uses symbols as substitute of real numbers. But the rule of algebra is the same as arithmetic.

D.    Relevant Studies
Research about relationship between arithmetic ability and algebra achievement of the SMP students has relevance with the other researches:
1.      Pengaruh Penguasaan Materi Bilangan Bulat Terhadap Kemampuan Menyelesaikan Soal-Soal Faktorisasi Bentuk Aljabar which was researched by Bukhori Muslim at 2006 and the result is: mastering in round numbers material can influence ability to solve algebra’s form factorization problems, that is 48%.
2.      Pemahaman Aritmatika dan Hasil Belajar Aljabar Siswa SMU which was researched by Lily Suatini at 2002 and the result is: there is a contribution of arithmetic understanding to the result (or outcome) of algebra learning, that is 65,5%.

E.     Hypothesis
Based on research problem and literature review, writer’s hypothesis is “There is a positive relationship between arithmetic ability and algebra achievement of the SMP students”.

CHAPTER 3
RESEARCH DESIGN

A.    Research Plan
This research is a correlational research. Correlational research is non-experimental research with no manipulation of variables in this research, this research attempts to investigate possible relationship among variables.
Independent variable in this research is arithmetic ability of SMP students and dependent variable is algebra achievement of SMP students.
Writer’s research plan is:
R:  O1   O2
with O1 : first observation (arithmetic test)
O2: second observation (algebra test)

B.     Instrumentation
The instruments used to measure the two variables involved in this correlational research are:
1.      Seventh grade transcript of students
2.      Multiple-choices tests (arithmetic and algebra)
3.      Likert attitude scale for students

C.    Validation
Fraenkel, J. R., and Wallen, N. E. (1993) noted, “The quality of instruments used in research is very important, for the conclusion researchers draw are based on the information they obtain using these instruments. ... Validity refers to the appropriateness, meaningfulness, and usefulness of the inferences a researcher makes. Reliability refers to the consistency of scores or answers from one administration of an instrument to another, and from one set of items to another.”
1.      Validity of Instrument
For the multiple-choices test, the validity of instrument can be tested by comparing instrument’s contents with lesson material. To measure validity of instrument, we use the formula of Moment Pearson Product.
2.      Reliability of Instrument
To measure reliability of instrument, we can use Cronbach alpha formula/
Researchers use Cronbach alpha when measures have items that are not scored simply as right or wrong, such as Likert attitude scale which individual may receive a score from 1 to 5 depending on which option was chosen.

REFERENCES

Badan Standar Nasional Pendidikan. 2007. Model Silabus dan Rencana Pelaksanaan Pembelajaran, Mata Pelajaran: Matematika. Jakarta: Departemen Pendidikan Nasional.
Butler, C. H., and Wren, F. L. 1960. The Teaching of Secondary Mathematics. USA: McGraw-Hill, Inc.
Fraenkel, J. R., and Wallen, N. E. 1993. How to Design and Evaluate Research in Education. Singapore: Lane Akers, Inc.
Hornby, A. S. 1974. Oxford Advanced Learner’s Dictionary of Current English. New York: Oxford University Press.
Narins, Brigham. 2001. World of Mathematics Volume 2. USA: The Gale Group.
Picciotto, Henri. 1995. Operation Sense, Tool-Based Pedagogy, Curricular Breadth: A Proposal.
Suherman, Erman. 2003. Evaluasi Pembelajaran Matematika. Bandung: Universitas Pendidikan Indonesia.
Swain, R. L. 1963. Understanding Arithmetic. USA: Holt Rinehart Winston.
Taylor, E. H., and Mills, C. N. 1961. Arithmetic for Teacher-training Classes. USA: Holt Rinehart Winston.
Wahyudin. 2004. Ensiklopedi Matematika untuk SLTP. Jakarta: Tarity Samudra Berlian.

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