Errors from a constructivist/sociocultural perspective

Constructivists view errors as systematic wrong answers because they are applied regularly in the same circumstances and are symptoms of the underlying conceptual structures (Olivier, 1989, p. 3). This means that misconceptions form part of learner’s conceptual structures which interact with new concepts, and influence new learning, mostly in a negative way: misconceptions generate errors (Olivier, 1989, p. 3). Moreover, from a sociocultural perspective, errors are described as superficial behavioural results of actions performed on a task and that correcting the error disregarding reason for the error committed is detrimental to a leaner’s intellectual mathematical development (Ryan & Williams, 2007). In Ryan and Williams’ (2007) terms systematic errors result from ‘conceptual limitations’ or ‘identifiable misconceptions’ and that diagnosis of such errors are linked to modelling, prototype,

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overgeneralization, and process-object reification. The issue of cognition in terms of learner conceptions is implicated in how errors are described from a constructivist and sociocultural perspective.

Modelling error occurs when learners model a situation contrary to the rules of the mathematical game in the academic context of school. An error is diagnostic of prototypical thinking if it is as a result of a culturally ‘typical example’ of the concept. Error as a result of overgeneralization arises when generalisations that make sense to a set of cases are inappropriately extended (Ryan & Williams, 2007). The extension can either be forward or backwards. This can be as a result of what Lima and Tall (2008, p. 6) term “met-before” and “met-after” in that old experiences can affect new learning and similarly new learning can affect the remembering of previous knowledge, respectively (Lima & Tall, 2008). An example of met-before would be learners’ experience in arithmetic affecting their conceptions of algebra, that is, in arithmetic learners have learned that 8 + 9 = 17, and when it comes to algebra, they do the following 5x + 3 = 8x when asked to simplify. An example of met-after would be the learners’ experiences in algebra such as 5x4 x 3x2 = 15x6 can be misapplied in arithmetic in that if asked to simplify 54 x 32, they will give 156.

When learners fail to navigate between process and object in the learning of mathematics then the error is as a result of lack of completion of the process-object reification (Ryan & Williams, 2007, p. 25). This suggests learners’ inability to have an operational as well as structural conception of the mathematical tasks (Sfard & Linchevski, 1994). For my study, overgeneralization and process-object conception emerged as the two possible dominant explanations for sources of learner errors in school algebra from the analysis of each of the six scenarios prior to analysing student-teachers’ talk. The question for my study then is how these explanations for sources of learner errors manifest in student-teachers’ engagement with tasks.

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Table 3: Errors and possible social constructivist/sociocultural descriptions

Error Possible constructivists description

Modelling error:

6 ÷ ½ = 3 or “division makes smaller” “Division makes smaller” is only appropriate for whole _{numbers where the numerator is bigger than the denominator.} This could also serve as an overgeneralization.

Prototypical error:

Failing to identify that a square or a rectangle with different orientations is a rectangle

As a result of a culturally ‘typical example’ of the concept in that a rectangle lies flat with its longest side horizontal. Error as a result of overgeneralisation:

‘Multiplication always makes bigger’

It is appropriate for whole numbers and it becomes an overgeneralization when applied to all rational numbers, thus including proper fractions or decimal fractions between 0 and 1.

Inadequate ‘process’ conception of the equal sign

A learner saying 548 is the answer in the problem

2000 1452

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It is argued that in order to understand misconceptions, you have to appeal to a theory of learning (Olivier, 1989; Sfard, 2008; Peng & Luo, 2009) and that not doing so renders the task of error analysis ineffective (Peng & Luo, 2009). Error analysis is regarded as an important aspect of teaching mathematics although viewed as challenging for teachers. Constructivists and Social constructivists view prior knowledge as the primary source for acquiring new knowledge, hence viewing error as a natural stage in knowledge construction (Hatano, 1996; Nesher, 1987; Peng & Luo, 2009; Smith, et al., 1993). For example, when confronted with new knowledge, learners’ prior knowledge can constrain a range of possible operations and answers as well as their understanding of mathematical entities, such as how to formulate a given problem mathematically (Hatano, 1996, p. 209). This process results in learners creating misconceptions which from the constructivist perspective are viewed as part of learning. Prior knowledge, though constraining, plays a foundational role in developing understanding (Smith, et al., 1993).

This view is unlike earlier research where errors and misconceptions, from a behaviourist perspective, were rendered redundant in that learners’ prior knowledge was not considered relevant to learning as cited in Olivier (1989). Errors and misconceptions were viewed as

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though something has gone wrong in the computer’s memory, that is – if we don’t like what is there, it can simply be erased or written over, by telling the pupil the correct view of the matter (Strike, 1983). To be more elaborate, Gagne (1983 p. 15) states:

The effects of incorrect rules of computation, as exhibited in faulty performance, can most readily be overcome by deliberate teaching of correct rules ... This means that teachers would best ignore the incorrect performances and set about as directly as possible the rules for correct ones.

The indication here is that the role prior knowledge plays in expert reasoning, in that it gets refined and reused (Smith et al., 1993) is overlooked. Moreover, Smith et al. (1993) would argue that earlier views of errors and misconceptions as interfering with learning (they must be replaced by expert concepts; and that they resist instruction) is limiting if viewed from the constructivist perspective. Here misconceptions are viewed as faulty extensions of productive prior knowledge in that conceptions that result in erroneous conclusions in one context can be quite useful in others. For example, “multiplication makes bigger” is useful when dealing with natural numbers but this might not always hold for some rational or real numbers. This suggests that “... commonly reported misconceptions represent knowledge that is functional but has been extended beyond its productive range of application” (Smith et al. 1993, p. 152).

Therefore, misconceptions could arise from learners’ overgeneralization of mathematical knowledge from one domain to another, for example, overgeneralization of number and number properties seems to be the major cause of learner misconceptions (Olivier, 1989). Interestingly, misconceptions can result in correct answers, and hence can be very difficult to identify (Nesher, 1987). For example, the conception that “the more digits in the number, the bigger it is” will yield a correct answer in the case of 0,567 being bigger than 0,45 but not necessarily for 0, 567 and 0,67, and yet the mathematical grounds on which it is made are faulty.

Borasi (1987) argues that seeing errors as a sign that something has gone wrong in the learning process and therefore it needs remediation is quite limiting. He acknowledges that researchers who have worked with errors as a powerful tool to diagnose learning difficulties and consequently direct remediation have made a valuable contribution to mathematics education. They have brought to the fore an understanding that learners are different, that they experience difficulties in learning mathematics, and that the strategy of simply

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explaining the same topic over again or giving learners additional practice exercises does not remove the errors. In Arcavi’s (1995, p. 149) terms, the issue of giving learners additional practice exercises suggests “the harmful advocacy of the drill-and-practice rationale of learning mathematics where the concern is: Let’s take care of the drill, meanings will emerge from practice”. Doerr & Wood (2004, p. 175) argue that student-teachers, from their own experience as learners, come to the initial teacher training with the understanding that ‘doing maths’ involves application of discrete rules “best learned through repeated practice”, for example, solving forty mathematical problems of similar focus. This in Doerr (2004) terms could be one of the “dilemma of experience” teacher-educators have to grapple with.

However, the issue of giving more practice exercises as a remediating strategy is contested in mathematics education literature. For example, Arcavi (1995, p. 149) agrees with Sfard that thinking at a structural level is not so straight forward, it requires “a great deal of doing, of practicing, of tolerating and living with partial understandings”. He contends that if practicing is going to be foregrounded, then the question that needs addressing other than just focusing on practicing the solution process to a number of similar mathematical problems is: ‘What kinds of practices, environments, and tasks and projects should be made available to learners if they are to develop this understanding?’ Moreover, these practices should take into consideration learners’ “engagement, perception of usefulness, and construction of meaning (even when partial), and their sense of ownership of the learning process” (Arcavi, 1995, p. 149).

Borasi (1987, p. 2) therefore proposes viewing errors as “springboards for inquiry”, and this is elaborated in Section 3.3.5. Early research has also shown the relevance of focusing on error in that among the early mathematicians, failure to experience success on a particular goal resulted in unexpected and revolutionary outcomes, for instance the role error has played in the history of mathematics (Borasi, 1987). She argues that mathematicians in their search for mathematical knowledge make use of their errors in that wrong conjectures, unjustified guesses, and partial results are all important steps in the creation of new mathematical results, hence growth of the discipline. Therefore, the experience of error as motivational and a means of inquiry in mathematics should not only be felt by the mathematicians but also mathematics learners at their own levels of education. This means that errors play an important role in restructuring a learner’s conceptual understanding.

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The explanation that can be given to misconceptions that are persistent and resistant to change is that their experiential foundations are broad and deep rooted (Smith et al., 1993). Smith et al. (op cit) argue that misconceptions are not always resistant to change in that if interventions are designed appropriately, they can yield rapid and deep conceptual change within the short space of time. Others are resistant because they lack conceivable alternatives; and others because they are part of conceptual systems that contain useful elements whose breadth and utility are not immediately apparent. Thus understanding the strength of a particular conception depends on the characterization of the knowledge systems that embed that element.

Moreover, Smith et al. (1993) assert that viewing misconceptions as something needing replacement is not the norm in the constructivist perspective because learning processes are much more complex than what replacement suggests. They state that knowledge is reused in new contexts, and is refined into more productive forms, and in such a process, you see the reappearance of misconceptions that were thought to have been resolved. This has implications for instruction in that it should provide the experiential basis for complex and gradual processes of conceptual change. As discussed, some of the research is constructivist, and more recent research is social constructivist and sociocultural – and thus indicating the significance of error in mathematics teaching and learning, and that this is part of the learning process.