Use of Rules of Algebraic Manipulation to Solve Equations

As mentioned in Subsection 4.2.2, students are required to apply a succession of transformation rules in their manipulation of letters, symbols or variables involved in the process of simplifying algebraic expression or solving an algebraic equation. In the process, students frequently fail to realise that formulas in mathematical symbol systems have an intrinsic structure where expressions are structured explicitly by the use of parentheses, and implicitly by assuming conventions for order in which we perform arithmetic operations. Students commit errors as a result of their inattention to the structure of expressions and equations. Among the misconceptions identified were: (1) order of operation is unimportant; (2) parentheses don’t mean anything; and (3) over-generalisation of some mathematical rules while ignoring others.

4.2.3.1 Order of operation is unimportant

The order of operation is very important when simplifying expressions and equations. The failure to use the order of operation can result in a wrong answer to a problem. For example, in Item 20, to find the value of 3 2, without the order of operation one might decide to simplify the problem working left-to-right. The resulting answer would be 6y. In contrast, the correct answer is 3 2 .

Alarmingly, there were 41 students (out of 78) choosing to ignore the need for a rule in this item (Item 20, Table 5).These students simply did not see a need for rules presented within the order of operation. Instead, they attempted to perform multiple- operations in any order which appears possible – they will attempt any part of an expression which they think they can do, and in any order. Hence, solving the expression based on how the items are listed, in a left-to-right fashion, consistent with their cultural tradition of reading and writing.

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4.2.3.2 Parentheses don’t mean anything

Students are found neglecting to use brackets when needed. Brackets are an essential element of mathematical notation in both arithmetic and algebra. Algebra, however, requires students to have a much more flexible understanding of brackets. In arithmetic, brackets are generally used as a static signal telling students which operation to perform first (or to specify the order of operation). Although symbolising the grouping of two terms (in an additive situation) is one important usage, students need to understand that brackets can also be used as a multiplicative operator (Linchevski, 1995). Although this dual usage is vital in understanding algebraic equivalence, Gallardo (1995) found that students were unable to correctly apply both usages of brackets to disapprove the number sentence: 20 – (7 – 8) = (20 – 7) – 8.

Many students, however, accept the need for brackets in arithmetic but the rule: ‘do the the brackets first’ often is impossible in algebra. Therefore, students simply ignore it. To some students, parentheses don’t mean anything in algebra. This is related to the misconception that working is always from left to right; the way we read. For example, simplify the expression 2 . Responses such as 2 and 2 are not uncommon instead of the correct answer 2 3 . The results of Item 22 clearly indicated that there were approximately 17% of the students who selected the response such as 2 and roughly 11% of them selecting the response such as 2 , indicating that they were either compelled to either ‘close’ their answer or to ignore the parentheses and work from left to right.

Another question concerning the non-use of parentheses was selecting the incorrect expression representing the total number of newspaper Lee delivers in 5 days (Item 16). Approximately 9% of the students (selected option A, 5 3 instead of 5 3 were either ignorant of the use of brackets or chose to ignore the use of brackets mainly because they considered them unnecessary and took it for granted that operations are performed from left-to-right.

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4.2.3.3 Over-generalisation of mathematical rules

Chua and Wood (2005) argued that among the common misconceptions students have in algebra, the most significant were errors due to over-generalisation of concepts and rules. These authors attribute these errors to misconceptions that students have actively constructed when they use their existing schema to interpret data which are grounded in faulty understanding. Students’ misconceptions of over- generalisation and/or inability to generalise mathematical rules discussed in this subsection are: (a) to remove a term from the equation, students subtract it from both sides of the equation; and, (b) dividing the larger number by the smaller rather than respecting the order of inversing.

4.2.3.3 (a) A common strategy students have for solving equations is that if they

want to remove a term from the equation, they subtract it from both sides of the equation irrespective of the adjoining operator symbol ( . This works just fine for removing 3 from the equation 3 9. However, when they encounter equations like 3 9, many students still try to subtract 3 from both sides to solve the problem. Booth and Koedinger (2008) attribute this misconception to students’ inability to process the fact that the negative sign modifies the 3 and is a necessary part of the ‘term’ they are trying to remove. Students were found to often ignoring it to the detriment of their goal of solving the problem.

Results from Item 15 "What is "′ ′ 12 4?" revealed the presence of this misconception. A significant proportion (56%) of the students tried to remove – 12 from the equation ended up with 12 12 4 12 resulting in 8. By selecting option ‘A’, these 44 students appeared to be unable to process the fact that the negative sign modifies the 12.

4.2.3.3 (b) The misconception of dividing the larger number by the smaller rather

than following the order of inversing is related to students’ inablility to generalise because of a lack of understanding of arithmetic operations. It also assumes that frequently students do not know which operation will undo another operation. For

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example, results from Item 17 concerning this misconception as “If 5 9 , then . " Twenty six students answered this question wrongly by selecting option B, 9 5, instead of 5 9.

Here, it is obvious that students perceived the need to isolate the variable, but were unsure which operation was necessary to inverse the one given. Thirty three percent of the students selecting option B appeared to be aware of the need to perform the operation of division but exhibit a lack of understanding of the arithmetic operator. They did not consider that the inverse operation for multiplying by 9 was dividing by 9. It was simply seen that it is easier to divide a smaller number into a larger one, and so this was done. Generalisation from early experiences that you can’t divide smaller numbers by larger ones is thought to be the cause of this misconception.

Misconceptions due to students’ inablility to use knowledge of algebraic structure and syntax to generate equations are discussed next.

4.2.4 Formulation of Algebraic Expression or Equation from Information