QUANTITATIVE ANALYSIS

The quantitative analysis adopted the error analysis of Haghverdi et al. (2011) which is as follows:

 Linguistic knowledge;

 Comprehension knowledge (a) semantic (b) structural and (c) intuitional;

 Communicational knowledge; and

 Calculation knowledge.

Linguistic knowledge: the problem solvers use this knowledge to read the text in the word problem. The lack of linguistic knowledge at the beginning of the problem solving process disenable the learners’ progress to solve the problem. The learners’ mathematical learning and linguistic learning are two sides of the same coin (Barwell, 2005). This determines whether the learners have the basis, which Bloom taxonomy explains as knowledge level. This implies that any learners who have the basic ability to read, will demonstrate the knowledge level (to recall formula).

Comprehension knowledge means the knowledge learners acquire when they read to understand algebraic word problems. Comprehension is the process of reading and understanding a word problem. Cercone, Naruedomal & Supap, (2011) explain comprehension as identifying the problem situation, the characteristics of the problem and the problem type. Comprehension knowledge enhances the ability to construct the equation for the word problem. This

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comprehension knowledge consists of three knowledge types, namely semantic knowledge, structural knowledge and intuition knowledge.

Semantic knowledge. This knowledge enables problem solvers to understand word problems, such as getting the meaning of what the text question requires. The presence of semantic knowledge allows for data and mathematical expressions not to be seen literally anymore. Semantic knowledge helps learners to form meanings of the word problem. Supap et al (2011) describe semantic knowledge as the ability to have the meaning of the word and the meaning of the word order in a sentence; such as to be able to interpret these words as mathematical concepts. Semantic knowledge enhances learners’ understanding of what the problem requires and the ability to interpret the problem correctly. For example, learner A in Question 11 could not interpret the problems correctly. Question 11 stated: A rectangular parking area has a dimension of 50 𝑚 by

120m. If the parking area is doubled….’ Lack of semantic knowledge let the learner misinterpret the wording and have an incoherent problem representation of the word problem (Cummins, 1988; 2006). For example, learner A here doubled the length and doubled the width; he had a misconception of the question. The learner had no understanding of how to apply the method to solve the problem. Therefore, he lacked semantic knowledge of the question.

Intuition knowledge refers to the knowledge such as the learner’s formal and informal education, past knowledge, objective experiences, and environment, as well as the learner’s capabilities. This knowledge also deals with the significance of problem-related data and information (Burton & Jarrette, 1999). The learner, after reading and solving the word problem, uses intuition knowledge (also called common sense) to examine the answer’s correctness or incorrectness. Polya (1973) refers to this knowledge as looking back. Take the example of Question 10. The question informs that the age of a learner is half times less than that of the elder brother. Intuition knowledge helps

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the learner to know he is wrong when calculating the answer and finding it is different from the expected or realistic answer given the information.

Structural knowledge refers to schemas of mathematical concepts existing as internal representations stored in the memory. Schemas are data structures for representing the generic concepts stored in the memory (Rumelhart & Norman, 1985). Similarly, Fischbein (1999) opines that schemes provide the strategy for solving problems, such as schemes for solving quadratic equations. A case in point is the scheme used for completing the square is different from the one required for solving using a quadratic formula. The schemata are knowledge structures, which help learners to classify problems and enable them to find the appropriate solution. Therefore, schemata (also the structural knowledge of mathematical concepts) are given to learners, or constructed by learners themselves. Schemata or structural knowledge helps learners to select a proper method or pattern for their solutions when solving word problems. Nesher and Herskovits (2003) find in their research that schemata are significantly more ordered and more available. In addition, the meaning of structures is more easily accessible by expert solvers than by slow solvers when solving word problems. The majority of the learners had difficulties in Question 1 to figure out the algebraic equations to be constructed, such as to identify the variable, operation and equations to use for the phase “consecutive of odd numbers”. Learners were found to lack structural knowledge.

Communicational knowledge is a kind of knowledge, which links the words in the problem representation to mathematical concepts and structures. Learners encounter challenges to link expressions in the word problem with the mathematical symbols and concepts. In Question 11, which requires translating the word into an algebraic equation, learner C, for example, could only write the double of the area is 6000, but could not continue to write the required algebraic equation

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which then would have to be factorised to get the value of the increase in the length and breadth of the area. Learner C lacks communication knowledge to understand this question.

Calculation knowledge is mathematical knowledge, which relates to computation, operation and solving algebraic word problems. Hiebert & Carpenter (1992) explain calculation knowledge as “any individual task that can be performed correctly without understanding” (p.89). Many of the learners are doing or solving the mathematics questions given in the research test correctly, but do not really understand what they are doing. Take learner G, for example. In Question 1 he arrives at answers 25 and 27, the consecutive numbers, without actually knowing the meaning of ‘consecutive odd numbers’ and how to construct these words into algebraic terms or equations. However, Majid and his team assert that learners’ lack of linguistic knowledge results in poor semantics knowledge. Similarly, Greeno (1985) emphasizes that lack of learners’ linguistic knowledge leads to poor performance in word problems. They also highlighted errors learners make when solving word problems as a result of a lack of linguistic and comprehension knowledge, in particular the comprehension, comprising semantic and structural knowledge. In this study, the researcher used 150 learners. Among them are 90 learners who scored zero; the remaining 60 learners are analysed for errors. If 150 learners are 100%, then 60 learners will be 40% who write the MSWPT set according to the CAPS requirements for Grade 11(Appendix I, Topic 2., Point 2 & 4: quadratic equations by factorization and quadratic formula; equations in two unknowns, one linear and other quadratic). The tests written are analysed for error types. Following the analyses of algebraic errors by Mamba (2012) and Haghverdi et al., (2011), the error types found are a lack of the following knowledge: linguistic and comprehension (consisting of the following knowledge types: semantic and structural, intuitional, communicational and calculation).

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3.6.2 Qualitative analysis: In the following section, eight learners’ interviews were analysed,