Measurement

4. CHAPTER FOUR

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

4.2.4 Measurement

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 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).

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In this content area, again the use of SMP which posits that SMP are intertwined (Kilpatrick et al., 2001) hence the codes are represented in strands (Table 4.4).

Subsequently, four categories of codes emerged: 1) simple procedures, 2) computations, 3) algorithms, and 4) computational connections. These are explored with illustrations below.

Table 4.4: Codes of SMP in Measurement

ANA examination questions

Codes of SMP in Measurement

Conceptual Procedural Total

2012 ANA 1SP-SC1-SC3

1SP-SC1-SC2-SC3 3PF2-SC1-SC2-SC3

5

2013 ANA 4PF1-SC1-SC2-SC3

3PF1-SC1-SC2-SC3-AR1

7

2014 ANA 1CU2-SC1-SC3 1SP-SC1-SC3

5PF1-SC1-SC2-SC3

7 Totals

Percent

1 5.3

18 94.7

19 100

 SIMPLE PROCEDURES (SP)

In this content area, there were procedures that did not need computations or algorithms that emerged from the analysis of ANA question papers (see Table 4.4).

There were two categories of SMP coded SP-SC1-SC3 and SP-SC1-SC2-SC3 respectively (Table 4.4). For the first category, an example is question 1.7, a multiple choice question from the 2012 ANA and the question was as follows; “The volume of a cube with side of length 7 cm is, A) 49cm³, B) 28cm³, C) 343cm³, D) 14 cm³.” (DBE, 2012c: 3). The question does not necessarily require learners to compute the volume (SP). Learners must recall known formula (SC1) and identify the value of seven cubed (SC3).

For the second category, an example is question 8.1 from the 2012 ANA which was as follows: “Complete the table by filling in the name of the 3-D figure, the number of faces, the number of vertices, the number of faces and the shape of the faces.”

(DBE, 2012c: 19). A table was provided with the shape with ‘SC2’ and required learners to recall knowledge of a cylinder and use this to identify its properties (SP).

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 COMPUTATIONS (PF1)

Again, from the analysis of question papers using SMP, there were computations that emerged and were in two categories. The first category of computations was coded, PF1-SC1-SC2-SC3. An example is question 10.2.3 from the 2013 ANA phrased as follows: “Hence calculate the area of PQR.” (DBE, 2013c: 19). The question requires learners to calculate the area of a triangle (PF1) using already calculated values (SC3) and knowledge of calculating area (SC3) with the aid of a given triangular prism (SC2).

The second category of computations in measurement was coded PF1-SC1- SC2-SC3-AR1. An example is question 10.1.1 from the 2013 ANA which was as follows: “Show that the area of the shaded ring is equal to 𝜋(𝑅²− 𝑟²).” (DBE, 2013c:

18). The question required learners to derive (PF1) the known conjecture (SC1) with the aid of a given diagram (SC2). Subsequently, the question required learners to show that the area of the shaded is equal to the difference of the area of the outer circle and the inner circle (SC3) using the formulae (AR1).

 ALGORITHMS (PF2)

In this content area, there was one category of algorithms that was coded PF2-SC1- SC2-SC3. An example is question 8.2 from the 2012 ANA which was as follows:

“Calculate the total surface area of the rectangular prism with length= 7.2m breath = 5m and height = 3.32m. Give your answer correct to 2 decimal places.” (DBE, 2012c:

20). In the marking guideline, they use the formula, 2(𝑙 × 𝑏) + 2(𝑙 × ℎ) + 2(𝑏 × ℎ) to substitute the given values to compute the surface area. The formula is generic and may be used to calculate the surface area for any given dimensions (SC1) of a prism (generality). Subsequently, the formula, when correctly used, yields the desired surface area (SC3) of a prism (accuracy). Additionally, the use of a diagram enhances sense making and does not affect the complexity of the question (SC2). Hence, it is an algorithm because it qualifies for generality and accuracy (PF2). Furthermore, substituting given values advances calculation of the desired surface area

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(transparency). Since the formula is aimed to be used by learners, its effective use of results in reduced error proneness (ease of use). Additionally, computational speed of using the algorithm is low, then it may be programmed in machines.

 CONCEPTUAL CONNECTIONS (CU2)

In measurement, one category of conceptual connections emerged from the analysis of ANA questions using SMP coded, CU2-SC1-SC3. An example is question 11.3 from the 2014 ANA which was as follows: “The circumference of a circle is 52 cm.

Calculate the area of the circle correct to 2 decimal places.” (DBE, 2014b). The question required learners to compute the area of a circle by first computing the radius, then use it compute the area of the circle (CU2). The question required recall of familiar knowledge (SC1) of calculating area of a circle and circumference (SC3).