4. CHAPTER FOUR
4.2 R ESULTS AND D ISCUSSION FOR ANA Q UESTION P APERS
4.2.2 Patterns, Functions and Algebra
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qualities for it to be an algorithm and test procedural fluency (Kilpatrick et al., 2001).
Additionally, other desired qualities of an algorithm, the computational speed must be fast, as is in this algorithm (efficiency), and then it may be programmed in machines (NCTM, 2000). And, since the algorithm must be used and learned by humans, (Bass, 2003) its effective use of calculating simple interest does not lead to high frequency of error (ease of use), and the steps of the problem advance calculations of simple interest (transparency).
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(DBE, 2013c: 7). The question is a familiar factorisation (SC1) which required the recall of knowledge of algebraic factors (SC3). There were no computations in finding the factors or the use of an algorithm (SP).
Table 4.2: Codes of SMP in Patterns, Functions and Algebra
ANA examination questions
Codes of SMP in Patterns, Functions and Algebra
Conceptual Procedural Total
2012 ANA 2CU1-SC1-SC3 13SP-SC1-SC3
1SP-SC1-SC2-SC3 12PF1-SC1-SC3 1PF2-SC1-SC3
29
2013 ANA 3CU1-SC1-SC3 6SP-SC1-SC3
1SP-SC1-SC2-SC3 1SP-SC1-SC3-AR1 1SP-SC1-SC3-AR2 7PF1-SC1-SC2 2PF1-SC1-SC2-SC3 1PF2-SC1-SC3
22
2014 ANA 8CU1-SC1-SC3 14SP-SC1-SC3
1SP-SC1-SC3-AR1 2SP-SC1-SC2-SC3 8PF1-SC1-SC3 1PF1-SC1-SC2-SC3 1PF2-SC1-SC3
35
Totals Percent
13 15.1
73 84.9
86 100
For the second category, an example that was coded SP-SC1-SC2-SC3 is question 7.1.1 which had an additional ‘SC2’ code and was as follows; “Write down the coordinates of the points A,B and C in the table.” DBE, 2013c: 11). The table and the graph were given (SC2), but no computations were required, just reading from the graph (SP) and the graph was familiar to the grade (SC1) which required learners to recall of knowledge of linear functions (SC3).
For the third category, an example was question 5.2 from the 2013 ANA, coded SP-SC1-SC3-AR1 (Table 4.2), which was as follows: “Write down the general term Tn of the above sequence.” (DBE, 2013c: 9). For learners to write the general term of the sequence, they do not need computations (SP), the question is routine (SC1) and requires recall of knowledge of sequences (SC3). However, writing the general term needs learners to base reasoning on the given sequence (AR1).
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COMPUTATIONS (PF1)
Again in this content area computations emerged in ANA questions, a category for procedural fluency (Kilpatrick et al., 2001). Consequently, these computations are in number patterns, algebraic expressions, equations and graphs. There were two categories of computations that emerged during the analysis of ANA questions and were coded PF1-SC1-SC3 and PF1-SC3-SC2-SC3 (Table 4.2). An example of the first category is question 5.2.2 in the 2012 ANA which was coded PF1-SC1-SC3 and the question was as follows: “The lines intersect at T. Show by calculation that the co- ordinates of T are 𝑥 = 1 and 𝑦 = 1 or (1; −1).” (DBE, 2012c: 12). This question is a follow-on from the graph drawn in question 5.2.1 (SC2). The solution requires learners to compute the co-ordinates of the point of intersection (PF1) by equating given equations (SC1) to show knowledge of intersecting lines (SC3)
An example of the second category is question 7.2.1 in the 2013 ANA which was as follows: “Draw the graphs defined by 𝑦 = −2𝑥 + 4 and 𝑥 = 1 on the given set of axes. Label the graph and clearly mark the points where the lines cut the axes.” (DBE, 2013c: 12). The solution to this problem requires learners to write the domain (x co- ordinate) for both functions then compute (PF1) the range (y-co-ordinates) of the given equations (SC1) using knowledge of linear equations (SC3). Subsequently, using those values, draw the graphs (SC2).
ALGORITHMS (PF2)
In patterns, functions and algebra, there were questions that had a sequence of steps with specialised qualities of an algorithm (Bass, 2003). An example was question 4.4 in the 2012 ANA which was coded PF2-SC1-SC3. This question follows on from the conjecture in question 4.3 and using that conjecture is generic in computing any number of terms (generality). When correctly used it will always produce the desired number of terms (accuracy), then it qualifies as an algorithm (PF2). In addition, effective use of the conjecture may not lead to high frequency of error (ease of use) because it requires the reproduction of known calculations of required term (SC3).
The familiar steps (SC1) advance calculations of the required term (transparency).
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Hence it may be used by humans. Last, the algorithm is easy and fast to compute the required term (efficiency) then it may be programmed in machines.
CONCEPTUAL CONNECTIONS (CU1)
In this content area, conceptual connections that emerged from the analysis refers to a connection of concepts in a single ANA question (NCTM, 2000). Concepts are discursive meanings that learners ascribes to a mathematical term (Khashan, 2014).
As a consequence, for learners to be proficient in mathematical concepts, they need to exhibit quality connections and in the conceptual aspect of mathematics (Mhlolo et al., 2012; Mwakapenda, 2008). There was one category of conceptual connections, coded CU1-SC1-SC3 (Table 4.2) that emerged from the analysis of the 2012, 2013 and 2014 ANA questions in this content area and an example is question 3.3 from the 2014 ANA which was as follows: “Simplify each of the following expressions. The denominators in the fractions are not equal to zero. 𝑥2−4𝑥
𝑥2−2𝑥−8.” (DBE, 2014b: 6). The solution to this question required learners to factorise (SC3) a familiar (SC1) numerator which was a binomial, factorise a familiar denominator to Grade 9 (DBE, 2011) which is a trinomial and divide like terms, three distinct concepts comprehended (CU1) in one problem (Kilpatrick et al., 2001).