Dealing with algebra learning, students need to be skillful in some basic conceptual standing. The learning condition of fluency has been described in mathematics education to indicate the importance of using fundamental understanding of basic concepts in mathematics to solve mathematical problems. Hence, it is essential for students have broad knowledge in whole number concepts, fraction concepts and particular aspect of geometry and measurement. In whole number concepts, students should have mastered the number sense theory by the end of Grade 6. They should be able to grasp basic binary operation definition and define their relationship with each other to solve mathematics problems. Students need to able estimate the result of computation and can estimate the scale of dimension like guess height from floor to the ceiling without measuring tools. Then, fluency in fraction, students should be able to differentiate denominator and numerator. In addition, students should easily place fraction in line number when they have correct concepts on positive and negative fractions. They also can transform decimal to fraction and vice versa. When both fluency are properly taught, students can easily undertand symbolic notation and concepts in algebra. For geometry and measurement, algebra concepts have been widely use in formulas. Students should able to calculate perimeter, area and volume of 2D shapes like rectangle and 3D shapes like cuboid.
In Malaysia education, algebra is an important topic in the school mathematics syllabus as it covered almost one third of secondary school curriculum. It has been known as a gatekeeper subject for advancing to higher level. Algebra has been exposed to form one students as they start to learn algebraic expressions. While during preschool and primary school, students have learned arithmetic using box and empty spaces. Experts agreed that many students facing obstacles during transition from arithmetic to algebra. Many studies on algebraic understanding by exploring students’ difficulties in algebra topic have been conducted and recorded. Yet, there are more research in linear equations than in algebraic expression. Since, learning process in mathematics is hierarchy process, students need to master the early stage of algebra firstly before advancing to next level. In fact, students’ conceptions after learning process is crucial, as there is interaction between previous students’ knowledge with the recent knowledge.
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Algebraic Expression and Equation
Algebraic expressions consists of symbols represent the variables, constants and binary operations. Hence , students need to master the concepts of variables and definition of algebraic terms or operation in order to solve the algebraic expressions question correctly. Algebraic equation is mathematical statement where in two expressions are set equal to each other.
Key for algebraic reasoning is understanding concept of equivalence and variables. The omnipresent of equal sign in Mathematics shows its important role like lead actor in a drama. According to Booth & Koedinger , they highlighted that students should have adequate understanding of the feature (e.g. equal sign, negative sign, variables like a term, etc.) in the equation and how changing the location of the feature are the keys to comprehend in solving the algebraic problem. The participants in their study is 49 high school students that have learnt Algebra I. In their research, they have been stressed about students need to have a decent conceptual understanding as a key to fully mastery the instructional information.
Knuth, Stephen, McNeil, & Alibali have conducted a resesearch on middle schools students from grade 6 to grade 8. As a results, they found that students should define the equal sign in relational interpretations and not in operational. This is because when they can define relationally the equal sign in this statement ‘4m + 10 = 70’ as the expression on the left side is equal to the expression on the right side, then they can solve the equation in algebra way. Students who define the equal sign in operational definition like ‘ the total of left expression’ or ‘ how much the number added together equal to’ will usually use unwind strategy because they will use arithmetic solution than algebra strategy. For example, they will subtarct 10 from 70 and then divide by 4 to get the value of ‘m’. Eventhough, they can get correct answer, they still have misconception in equal sign definition. Although these misinterpret definition equal sign do not be a problematic during primary school level where they only need to solve simple equations like ‘a + b = ‘, they will be facing obstacle when solving question in higher level when the misconception still exist.
Based on the research conduct by Zakaria, he analysed the errors made by Form 3 students in solving quadratic equations and he found that the common errors made by them are transformation errors and process skills errors. Many students can transform into algebraic equation but they still do not understand the equivalence concept. He also emphasized the importance to understand the basic concept and skills in early stages as the learning process in mathematics is a hierarchical process. Thus, it is essential for students to master the basic level so that they will not having difficulties at advanced and higher level in the future.
For variables, few students believed that literal symbol is represent the unknown number with fixed value and variables cannot have more than one value. Most students attempted to believe that literal symbol as an object or label of itself. Students will make variable error when they were facing complex algebraic equations. Students should have solid foundation in concept of variables so that it will not become an obstacle in learning algebra. Besides that, students also had ‘concatenation problem’ with the use of literal symbols in algebraDuring primary school, students learnt that concatenation represents addition operation, e.g. 65 equals 60 add 5 and 3 1⁄2 means 3 plus ½. In contrast, in algebra concepts, concatenation define multiplication. For example, ‘5a’ means 5 multiply ‘a’ and when the question asks to add a number like 2 into 5a, usually students will write 7a as final answer. This is because they saw the question in arithmetic concepts. Chalouh and Herscovics reported that students would use algebraic concepts when the questions request specifically.
Then, some students still had a misconception about like terms and unlike terms. They thought that ‘5ab’ and ‘4ba’were unlike terms because they do not have the basic properties of commutative law as they compare their coefficient instead their variables. They also always saw transparent coefficient when solving the equation ‘4a + 3 = 7a’. There is misconception occur as they see symbolism as procedure for calculation.
Besides that, students also had misconception on negative sign since most of them believed that negative sign as binary operation subtraction and do not modify the terms. In Vlassis J. research, he stated that there are two conceptual change for negative sign, which have faced by students; their efforts to clear up arithmetical beliefs about natural number and the algebraic rules that need to work with negative sign. There are still more students that have difficulties in understanding negative sign interpretation. However, limited research conducted on negative sign as this feature do not emphasized in Mathematics syllabus.
Sakpakornkan and Harries reported that major difficulties that faced by students in simplifying the algebraic items are negative signs in multiplication questions and mutiplying out the brackets. In Seng research, he outlined that there are two types of error students made when doing bracket expansion problems. They tend to multiply the first algebraic term only and forget the second algebraic term which locate in the same bracket. Moreover, if the question like this ‘ 2 (3a + 2) + (3 + 4a), they will answer 6a + 4 + 6 + 8a. This is wrong as they multiplied the second bracket with ‘2’ too. Lastly, they also forget the characteristics of negative sign that attached to its pre-multiplier when multipy the algebraic terms in bracket with its pre-multiplier.
Summary
This chapter reviewed literature regarding students’ difficulties in learning algebra from previous research . There are still many misconception and errors made by students during learning algebra. Even in the basic algebra like algebraic expression, they still do not master and have adequate knowledge about the terms, variable, and operation. If this error is still continue, then they will lose their interest and motivation in learning Mathematics not only algebra since algebra has cover almost all the Mathematic syllabus in secondary schools. This will decrease the number of students to further their studies in Mathematics and Science course. Even though the errors made by students will not be obstacles now, they will face difficulties in the advanced level.
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