Fong’s (1993) schematic model for error analysis is based on the schematic approach for analysing students’ strategies in solving both computational and word problems. Fong defined a schema as a network of interrelationships between different sets of knowledge that constitute a concept and schemata as data structures that represent the generic concepts stored in memory. According to Santrock (2010) schemas are actions or mental representations that organise knowledge. In Piaget’s theory, older children have schemas that include strategies and plans for solving problems and by the time they reach adulthood, they have constructed enormous number of diverse schemas (Santrock, 2010). The review of previous studies on error analysis that preceded Fong’s schematic model for error analysis led the scholar to conceptualise and classify errors in to two levels. Fong (1993) argues that the classical way of mathematical error analyses on pupils’ written solution (level 2) seem to lack analysis of children’s cognitive process in greater detail. The two levels model provides researcher with a further insight into pupils’ errors from different perspective. Basically, according to Fong, the principle behind the classification of errors into two levels is whether there is any schematic error or specific error classified by classical approach. The first level is categorised in terms of strategic schema. According to the Fong, the problem solver must first overcome this level in order solve problems successfully. With respect to this, the following five error categories are identified: No solution, using irrelevant procedure, incomplete schema with no errors, incomplete schema with errors, and complete schema but with errors. The second level, according to Fong (1993), is categorised in terms of classical way of classifying errors. With respect to this, errors are categorised into four categories: (a) language, including reading and comprehension, (b) operational, including encoding and transformation, mathematical themes, such as basic facts, algorithms and concepts, and (d) psychological factors, including motivation and carelessness. With regard to the fourth error category of the second level (psychological factors), Jiang (2013) maintains that despite this error category being always an important factor that affect students’ problem-solving activities, they are difficult to identify from students’ written solutions. Fong (1993) further pointed out that errors in the second level are subsumed under the first level as indicated in figure 3 below. For example, “complete schema but with errors” is a category of level 1 errors. Within this class of errors, a set of level 2 type of errors such
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as language, operational, mathematical thematic and psychological knowledge is subsumed under it.
Figure 3 Network of two levels of error of problem solving: Adapted from Fong (1993)
In the paragraph that follows, definitions and characteristics of the five first-level error categories (schematic errors) are briefly outlined.
“No solution” (first category of error) refers to a solution which has no written responses (Fong, 1993). According to Fong (1993), In terms of schematic explanation, the problem solver is unable to connect or relate any available schemas to the information obtained from the question. As a result of this, no solution is presented. The second category of errors “using irrelevant procedure”, the problem solver is unable to retrieve any relevant knowledge or information and apply it to work out the solution. According to Fong, any knowledge or information which is retrieved has no connection or link to the question, although the problem solver may assume that those pieces of information retrieved are the best possible solutions. Jiang (2013), illustrate this error category by the following example taken from pupil’s solution:
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According to Jiang, the second level of error is in the comprehension and mathematical theme categories because the student took speed of the third part as the total distance, which is erroneous.
The problem to the above learner solution was: Mike made a journey from City P to City Q. In the first hour, he covered 1
3 of the journey. In the second hour, he covered 1 5 of the whole journey. Finally, he took 2 hours to finish the remaining journey at a speed of 42 km/h. Calculate his average speed for the whole journey.
According to Jiang, the following five skills are involved: (a) apply fraction and part whole concepts to find the fraction of the distance of the third part (D3) to the total
distance (TD), i.e, D3 𝑇𝐷 ( D3 𝑇𝐷 = 1 - D1 𝑇𝐷 - D3
𝑇𝐷 ), (b) using the concept of speed to find D3,
(c) applying the concept of fraction to find TD (TD = D3 ÷ D3
𝑇𝐷 ), (d) applying the part-
whole concept to find the total time TT (TT = T1 + T2 + T3), and (e) using the concept of average speed to find the answer. The solution is coded a-b-c-d-e, which is called strategic path (Fong, 1993). According to Jiang, the correct solution to this problem is:
The third error category is “incomplete schema but without errors”. In this type of error, students present some, but not all, of the correct steps in the solution and no actual error is made other than incomplete retrieval of schema leading to a solution (Fong, 1993). The problem solver has a limited or insufficient schema or is unable to connect all the relevant information that leads to the solution. Jiang (2013) illustrates this error category with following example taken from pupil’s solution:
From learner’s solution above, no actual error was committed. However, the solution to the problem is incomplete. According to Jiang (2013) the student is not able to connect all the relevant information that leads to the solution. In addition, Jiang contends that there are no second level errors. The fourth error category “incomplete schema but with errors” differs from above categories in that the student makes errors such as computation and/or encoding errors in addition to demonstrate incomplete
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schema or inability to connect all relevant information (Fong, 1993). The last error category “complete schema but with errors”, According to Fong (1993), this error category arises when an error is made in computation or encoding information although a problem solver is able to connect all relevant schemas to problems’ requirement. Jiang (2013) illustrates the error type “complete schema but with errors” with the leaner’s solution below: