Stage 2 : Refining the framework

The content validity of the framework was determined by a panel of seven experts through a focus group discussion. The experts were from various areas :

• 2 on mathematical content (functions and derivative) (MC1, MC2) • 2 on visual reasoning (VR1, VR2)

• 2 on Cartesian graph (CG1, CG2)

• 1 on mathematics educations (problem solving) (ME)

The experts did not only examined and confirmed each category of both the encoding and decoding processes, but they had also systematically scrutinised the framework in parallel with the proposed instruments to ensure that it fully reflects the potential solution methods by the students. The technique of assessing, through both the encoding and decoding processes, the conceptual ideas on functions and derivatives,

consideration in order to structure a visual setting of reasoning. Questions were set to guide the experts on the topics of discussion (Appendix G). The summary of their responses are as listed in Table 4.6.

The first question requested for some ideas on visual reasoning related to mathematics. Four of the experts referred visual in mathematics to be tasks or information on non-word problems while the other three of the experts treated them as those tasks or questions that are posted in other forms than algebraic expressions or numbers. All of them categorised visual in mathematics to be other than both texts and numbers, such as graphs, diagrams, images, pictures or any 2-dmensional or 3- dimensional geometrical figures. In terms of how they employed visual to reason their mathematical understanding, four of them made use of the information provided in the graphs while the other three would draw, or at least sketch, graphs related to the problems in the contexts.

The second question sought the experts’ opinions on the use of Cartesian graphs in the learning of functions and derivatives. Three of the experts admitted that students did not make use or draw graphs as their solution or parts of the method in solving mathematical word problems while the others stated that students would refer to graphs if only they have strong understanding on the graphs or the relationships between the algebraic and graphical representations. Six of the experts asserted that students would relate the mathematical concepts to their graph representations through the properties of graphs and functions. Six of the experts agreed that it is possible for the students to achieve the correct solutions when employing graphs although most of them would struggle throughout.

Table 4.6: Responses from the experts in the focus group discussion

Question Responses Expert No. (%)

1(a)

Not word problems MC2, VR1, VR2, ME 4 (57) Other than algebraic expressions /

numbers MC1, CG1, CG2 3 (43)

1(b)

Graphs (all types) MC2, VR1, CG1, CG2 4 (57) Diagrams / images / pictures MC1, ME 2 (29)

Geometry VR2 1 (14)

1(c) Use information in graphs / diagrams MC1, VR1, VR2, CG1 4 (57) Draw related graphs / diagrams MC2, CG2, ME 3 (43) 2(a) No, need more exposure VR2, CG2, ME 3 (43)

Yes, if understand graphs VR1, CG1 2 (29)

Maybe, depending on understanding

the relationship MC1, MC2 2 (29)

2(b) Understand properties of graphs VR1, VR2, CG1 3 (43) Understand properties of functions MC1, CG2, ME 3 (43)

Knowledge on slope MC2 1 (14)

2(c) Yes, but mostly with struggle MC2, VR1, VR2, CG1,

CG2, ME 6 (86)

Yes if strong basic knowledge on

functions and derivatives MC1 1 (14)

3(a) Understand the relationships between algebraic/symbolic & graph

MC2, VR1, VR2, CG1,

CG2 5 (71)

Understand the relationship between

variables MC1, ME 2 (29)

3(b) Strong understanding on relationships between functions/derivatives & graphs

MC1, MC2, VR1, VR2,

CG2, ME 6 (86)

With help from algebraic expressions

or equations (if given) CG1 1 (14)

3(c) Looking at patterns of graphs MC1, MC2, VR1, ME 4 (57) Understand properties of graph of

The third question needed for the experts to analyse on how the students would read and interpret data or information that are embedded in graphs. All of them (except one) agreed that students need to comprehend the relationships between the algebraic expressions and their graphical representations or between the independent and dependent variables in order to be able to read and interpret graphs efficiently. When extracting information that are not shown on graphs and when interpolating or extrapolating the graphs for hidden information or specific patterns of the characteristics on the functions, again, almost all of them highlighted that students must have very strong knowledge on the relationships between the functions and their derivatives and between the algebraic expressions and graphical representations.

Comments based on Question 4 to refine the framework were gathered for improvement :

1) Encoding :

a. The addition of with correct solution and with incorrect solution to each of the categories Draw correct graph, Draw incorrect graph and Algebraic solution.

b. Elaboration on the No answer/Not attempted to indicate the possible skills and knowledge of the students

c. The use of consistent terminologies among Draw correct graph/Draw incorrect graph and Algebraic method to Correct/Incorrect and No graph since students may produce any other method than algebraic manipulations. d. Further elaborations on the descriptions for all categories

2) Decoding :

a. The inclusion of with valid reason, with invalid reason and with no reason to the category of Correct solution

b. The inclusion of invalid reason and no reason to the category Incorrect solution.

c. Elaboration on the No answer/Not attempted to indicate the possible skills and knowledge of the students

d. The elaborations on the descriptions for all categories

The refined framework is as shown in Table 4.7.