Space and Shape (Geometry)

4. CHAPTER FOUR

4.2 R ESULTS AND D ISCUSSION FOR ANA Q UESTION P APERS

4.2.3 Space and Shape (Geometry)

125

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).

126 Table 4.3: Codes of SMP in Geometry

ANA examination questions Codes of SMP in Space and Shape (Geometry)

Procedural Total

2012 ANA 2SP-SC1-SC3

2SP-SC1-SC2-SC3 1SP-SC1-SC2-SC3-AR2

1SP-SC1-SC3-AR1 6SP-SC1-SC2-SC3-AR3

3PF1-SC1-SC2-SC3 7PF1-SC1-SC2-SC3-AR1

22

2013 ANA 6SP-SC1-SC2-SC3

5SP-SC1-SC2-SC3-AR1 2SP-SC1-SC2-SC3-AR3

1PF1-SC1-SC3

14

2014 ANA 5SP-SC1-SC3

4SP-SC1-SC2-SC3-AR1 4SP-SC1-SC2-SC3-AR3 3PF1-SC1-SC2-SC3-AR1

16

Totals Percent

52 100

52 100

 SIMPLE PROCEDURES (SP)

In this content area, simple procedures have emerged from the analysis of ANA questions. As stated in the previous content areas, simple procedures tested only a quick recall and mention of procedures without computations or algorithms, hence again they were categorised as simple procedures. There were six categories of simple procedures in geometry. The first category is coded SP-SC1-SC3, and an example is question 9.1.1 from the 2014 ANA which was as follows; “𝐷̂ and 𝐹̂ are complementary angles if_______” (DBE, 2014b: 13). The question requires learners to recall knowledge of complementary angles (SC1) and state that their sum is 90⁰ (SC3) without computations (SP).

The second category is coded SP-SC1-SC2-SC3 An example is question 9.2 from the 2013 ANA and phrased as follows; “Write down the coordinates of 𝐵^′, the image of B” (DBE, 2013c: 17). This question required learners to read from the graph (SC2), using basic knowledge of the Cartesian plane (SC1) the coordinates of the image (SC3).

The third category is coded SP-SC1-SC3-AR1, with an example, question 6.2 from the 2012 ANA and was as follows: “State which triangle is congruent to ∆𝐴𝐵𝐶.”

127

(DBE, 2012c: 14). The question required learners to use their basic knowledge of congruency to state (SP), giving reasons and infer (AR1), using given triangles (SC2) that the triangles are congruent (SC3).

The fourth category is coded SP-SC1-SC2-SC3-AR1. An example is question 8.1.1 from the 2013 ANA which was phrased as follows; “Calculate with reasons: The size of 𝑇̂1.” (DBE, 3013c: 13). The question required learners to state the size of the angle (SC1), without calculations (SP), and giving reasons recalling the knowledge (SC3) of the relationship of angles of an isosceles triangle (AR1) with the aid of the given diagram (SC2).

The fifth category of simple procedures is coded SP-SC1-SC2-SC3-AR2. An example is question 6.3.4 from the 2012 ANA, which questioned as follows; “Hence, state the relationship between AE and BC.” (DBE, 2012c: 16). The question required learners to infer (AR2) on how two lines relate in a given diagram (SC2) based on proofs (SC3) in prior questions (SC1).

The last category of simple procedures in geometry is coded SP-SC1-SC2- SC3-AR3. An example was question 8.3 from the 2013 ANA, which was as follows;

“Prove with reasons that ∆𝐾𝑁𝑄 ≡ 𝑀𝑃𝑄” (DBE, 2013c: 15). The question required learners to use knowledge of congruency (SC3) to identify (SP) with reasons, relations of corresponding angles and sides (AR3), in a given (SC1) pair of triangles (SC2) to prove that they are congruent. This was an analogy, and analogical reasoning refers to the ability to identify similar structural commonalities of objects (Amir-Mofidi et al., 2012; Lee & Sriraman, 2011; Whitacre et al., 2017) and mostly it is regarded as moderate reasoning strategy due to the low level of rigour that is without computations. Analogical reasoning (Markovits & Doyon, 2011) is an essential recipe for bridging the gap between concrete and abstract reasoning. Proving is an essential part of mathematics to convince oneself that inferences when made when solving mathematical problems, such as proving that theorems are in fact true (Bleiler- Baxter, 2017).

128

 COMPUTATIONS (PF1)

In this content area, as in the previous content areas, there were three categories of computations that emerged from the analysis of the ANA questions coded PF1-SC1- SC3, PF1-SC1-SC2-SC3 and PF1-SC1-SC2-SC3-AR1 respectively.

An example for the first category is question 1.9, coded PF1-SC1-SC3 and from the 2013 ANA which was as follows: “In the figure below, side 𝐷𝐹 of ∆𝐸𝐷𝐹 is produced to 𝐶. Calculate the size of 𝐸̂ in terms of 𝑥.” (DBE, 2013c: 4). The question required learners to compute (PF1) using the given diagram (SC2) and the recall of knowledge of properties of a triangle (SC3) the value of 𝑥 (SC1).

An example for the second category is question 7.3, coded PF1-SC1-SC2-SC3 and from the 2012 ANA which was as follows: “The length of each side of figure P is halved. Calculate the perimeter of the new figure.” (DBE, 2012c: 18). The question required learners to divide each side (SC1) of a given figure (SC2) and compute (PF1) the perimeter using their knowledge of perimeter (SC3).

An example for the third category is question 9.3 coded PF1-SC1-SC2-SC3- AR1 and from the 2014 ANA was as follows; “In ∆𝐴𝐵𝐶, 𝐴𝐵 = 𝐴𝐶 and 𝐶̂ = 𝑥⁰. Determine the size of 𝐴̂ in terms of 𝑥.” (DBE, 2014b). The question required learners to compute (PF1) the size of an angle in terms of a variable, giving reasons (AR1), using a given diagram (SC2) and recall of knowledge of properties (SC1) of an isosceles triangle (SC3).