Learner errors and difficulties in algebra

The field of research in algebra is wide. Usiskin (2004, p. 150) argues that “algebra needs to turn on students rather than turn them off”. To this effect, literature has suggested ways in which teachers could assist learners make sense of school algebra and eventually develop conceptual understanding. These strategies include introducing algebra in the early years of schooling, use of representations, use of technological tools, and focus on relationships between quantities before number is introduced (Kendal & Stacey, 2004; Kieran, 2004; Kilpatrick, et al., 2001; Usiskin, 2004; Watson, 2009). These are not dominant parts of the school algebra curriculum in Zambia, and they are not in focus here. Although with regard to representations, some algebraic problems are evident in the textbooks in form of word problems which require translation into symbolic expressions before manipulation.

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Of interest for my study are the errors and difficulties learners experience in the learning of school algebra. Many researchers have attributed the difficulties that learners experience in learning school algebra to the teaching approaches used in that they are more traditional rather than activity based (Kilpatrick, et al., 2001; MacGregor, 2004; Usiskin, 2004; Watson, 2009). They argue that learners are required to memorise facts, rules and techniques and apply them in different algebraic tasks in a routine manner without developing underpinning understanding of why they had to carry out the manipulations in the way they did. They further argue that many textbooks have focused on rules to be followed in manipulating symbolic expressions and equations rather than on the concepts that support those rules or give meaning to the expressions or equations being manipulated. This suggests, in Skemp’s (1976) terms, teaching for instrumental and not relational understanding. For example, Usiskin (2004, p. 150) states:

Instead of being taught as a living language with a logical structure and many connections between its topics and other subjects, algebra is taught as a dead language with a myriad of rules that seem to come from nowhere, and within applications that are viewed as puzzles, like chess problems.

The problem has been exacerbated by the reason that even where teachers have tried to use reform based approaches such as engaged learners in discussion on a complex task; the traditional approach in manipulating the tasks has remained the focus. This includes the orientation of the tasks the learners have to engage with. As a result, learners tend not to develop conceptual understanding of the algebraic concepts involved.

Because of the way algebra is approached in the classroom, i.e. rule-based instructional approaches, learners are faced with many difficulties. These difficulties result from, in a broader sense, dominance in manipulation over reasoning; applying learnt procedures; inability to read and use algebraic symbolism in a meaningful way; failure to distinguish data from conclusion; and tendency to write arithmetical at the expense of algebraic expressions (Bell, 1995; Watson, 2009). Watson (op cit) argues that over-reliance on rules result in learners misapplying them, or misremembering them, or might not think of the appropriate situation in which to apply them. Such rules could include, guess-and-check used when solving equations; and Bracket of Division, Multiplication, Addition and Subtraction (BODMAS) used when simplifying expressions with different operations.

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In referring to the tendency by learners to misapply arithmetical meanings to algebraic expressions, Watson (2009) argues that reasons for this include: seeing algebra as calculations rather than relations; not understanding the inverses and operations in solving the unknown values; not seeing that some situations need to be transformed into algebraic form before solving; not treating letters and numbers as symbols in a structure in the event that they are used together [2(3 + b) has a different structure from 6 + 2b]; and seeing the equal sign as ‘calculate’ in an attempt to express an answer and not ‘is equal to’ or ‘is equivalent to’. Seeking closure to obtain one term is common in this situation, since learners might not realize that for example in the notation a + b = ab, ab mean ‘multiply’. A desire for an answer is also evident when learners ‘solve’ an expression as if it is an equation (Ryan & Williams, 2007; Wagner & Parker, 1993). Watson (2009) argues that this is an indication of how problematic the aspect of interpreting is while Nickson (2000) argues that learning arithmetic is not necessarily the same as learning algebra and therefore needs careful attention.

In Kieran’s (1989, 1992) terms, Watson (2009, p. 15) describes ‘structure’ in algebra as: “(1) surface structure of expression: arrangement of symbols and signs; (2) systemic: operations within an expression and their actions, order, use of brackets, and so on; (3) structure of an equation: equality of expressions and equivalence”. Watson (op cit) also argues for flexibility in how mathematical expressions are acted upon in that they should be seen as an answer to a particular question, an object in itself (for example, 3x + 4), and also an algorithm or process for calculating a particular number. In Gray and Tall (1994) terms, Watson (op cit) refers to the dual meaning of an algebraic expression as proceptual thinking while in Sfard & Linchevski’s (1994) terms, it is referred to as the process-object duality.

Research has also shown learners’ possible interpretations of letters in their early learning of algebra (Hart et al., 1981). This was in the Chelsea diagnostic test instrument involving 2 900, 12 – 16 year old learners. The possible learner interpretations of letters include 6 aspects as described by Hart et al. (1981, p. 104): (1) letter evaluated “where the letter is assigned a numerical value from the outset”; (2) letter not used where the letter is ignored or its existence is acknowledged but without giving it a meaning; (3) Letter used as an object where “the letter is regarded as a shorthand for an object or as an object in its own right”; (4) Letter used as a specific unknown where a letter is regarded as a specific but unknown

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number, and can be operated upon directly; (5) Letter used as a generalized number where “the letter is seen as representing, or at least as being able to take, several values rather than just one”; (6) Letter used as a variable where “the letter is seen as representing a range of unspecified values, and a systematic relationship is seen to exist between such sets of values”. In light of these possible learner actions on letters, Watson (2009, p. 4) argues that “teachers have to understand that students may use any one of these approaches and students need to learn when these are appropriate or inappropriate”.

In seeing the prevalence of similar errors in studies twenty years apart (see Hart et al. (1981 and Ryan and Williams 2007), the argument is that it “is evidence that these are due to students’ normal sense-making of algebra, given their previous experiences with arithmetic and the inherent non obviousness of algebraic notation” (Watson, 2009, p. 19). For my study, of the six possible learner interpretations of letters, the first four descriptions were used in setting up and then analysing the six scenarios, and in describing student-teachers’ discourses of these scenarios. Moreover, in what ways did student-teachers’ explanations of the sources of learner errors in the scenarios resonate with over-reliance on rules, or the tendency by learners to misapply arithmetical meanings to algebraic expressions, or inadequate process- object duality?

As a result learners’ tendencies with learning algebra that I engaged with in the scenarios included: (1) equating an open algebraic expression to zero when asked to simplify (Ryan & Williams, 2007; Wagner & Parker, 1993); (2) conjoin or ‘finish’ open algebraic expressions when asked to simplify (Even & Tirosh, 2002); (3) dividing by x when asked to solve the quadratic equation 2x2 = 6x (Extracted from Learning Mathematics for Teaching project test items meant to measure teachers’ SMK by Hill et al., 2004); (4) distributing the power over the sum of terms in an expression, for example, (x + y)2 = x2 + y2 (Vermeulen, 2007); (5) multiplying numbers only when asked to ‘Multiply n + 5 by 4’ (Hart et al., 1981); and (6) equating each factor to 4 when asked to solve for x in the quadratic equation (x – 1)(x + 2) = 4 (Bell, 1995).

Since my study is more concerned with school algebra, I work with the view that algebra is generalized arithmetic as well as a specialized language for expressing generality. I also work with an understanding that activities of school algebra, as modelled by Kieran (2004), include

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transformational, generational, and global/meta-level, and for my study the algebraic activity is predominantly transformational. The understanding that learners experience difficulties with learning school algebra is not a new phenomenon in mathematics education research. Types of common errors learners make are well documented and some reasons for these errors provided. A selection of these common errors was then used in designing scenarios which I engaged with in my study. Some suggestions on how these errors in algebra could be addressed have also been outlined. My argument with these suggested strategies is that they could also generate errors of their own nature.