Step 2: Synthesis

Synthesis involves the process of extracting, clarifying and summarising those ideas and aspects of the theories, models and frameworks to suit the nature of this study. Following are the theories, models and frameworks and their main thoughts that had been taken into consideration for the synthesis.

(a) Theory on representations

Lowrie et al. (2011), partly on the basis that the use of visual and graphic are increasingly taking placed to influence how students make sense of their mathematical concepts, developed a theory to explain how the process of thinking that shapes the students’ mind when dealing with mathematics and mathematical ideas. He alerted on the drastic shift of how mathematical ideas and concepts are being presented and communicated in the last decade although the curriculum had not change much. His theory described the encoding and decoding processes to explain how students composed their own representations based on the textual descriptions and the technique used to employ diagrams provided in order to make sense of situations respectively. He argued that the encoding process was a support system to help students apprehend the reality of problem solving. He decomposed the decoding process into three levels of elementary, intermediate and advanced levels to describe how information is extracted or interpreted from the data in the graphics. He argued against the current practice of providing graphics for the students as compared to letting the students to construct them which will enhanced their thinking skills and understanding. Word problems incline to provide platform for students to practice the encoding techniques in order to understand the mathematical concepts and ideas. On the other hand, various skills of decoding are also required due to different graphics are composed of different elements and structures.

(b) Visual reasoning model

Park and Kim (2007) defined visual reasoning as to progress further than the visual information displayed in two different paths: one is to transform the information based on their conceptual rules or formulae and the other one is to make deductions or implications. The overall process of visual reasoning involved the visual analysis through seeing, the synthesis through imagining and the modelling process through the drawing process. Three activities of visual perception, analysis and interpretation occurred during the seeing process, while another three activities of generation, transformation and maintenance took place during the imagining process. The drawing process involved the evaluation of the internal and external representations. These physical actions take place in the interaction with the conceptual knowledge and perceptual activities. They identified that the visual reasoning activities engaged the visual knowledge to complement the perceived visual and the memory system to produce the visual information. The visual schema from the memory system guides the transformation and reorganization of the visual perception. The arrangement and relationships in the visual displayed may cause different types of activities during the visual reasoning process based on the complexity of the structure of the visuals. They concluded that visual reasoning process is an essential cognitive activity that has specific relation to any visual process and therefore, students ought to be trained on the reasoning activities through well-constructed and well-structured visual systems.

(c) Characteristics of visualisers

By the use of grounded theory methods, Alcock and Simpson (2004) developed a theory to assess students mathematical performance that resulted from their tendency to visualize or not to visualize, and their self-belief about themselves and their roles as mathematics learners. The results exhibited three major indicators to describe the

students’ pattern to visualize: 1) introduced and made use of diagrams when solving problems, 2) used gesture when explaining solutions or arguing on concepts, 3) prefer to think in diagrams rather than algebraic expressions. Subsequently, those students who were categorized as visualisers were more focused on the mathematical concepts as objects, quick-thinking for drawing initial conclusions and were more confident in their own solutions and decisions. The study led to the awareness on how students’ understandings on mathematical concepts were influenced by their learning environment. Only patterns (1) and (3) were considered in the present study.

(d) Diagram drawing sub-skills

Due to the importance of diagram drawing as heuristic strategy in solving mathematical problems, Simon (1986b) identified a set of six sub-skills that described how pre- calculus students attempted to use diagrams to solve mathematical problems: 1) represent all relevant information, 2) creating an integrated diagram that are critical to the conceptualisation of the problem, 3) labelling completely, 4) checking the accuracy of the diagram, 5) drawing multiple representations that are not critical, and 6) verbalising what is represented and what needs to be represented. He had also discovered five factors that contributed to whether students may or may not opt for diagrams to help them search for solutions: 1) their understanding on the mathematical concepts and arithmetic related to the problems, 2) their previous knowledge and skills to drawing diagrams, 3) their understanding of mathematical concepts, 4) their self- concept in mathematics, and 5) their motivation to correctly solve the mathematical problems. Feedback given to students resulted in them providing higher quality diagrams which indicate that it is a necessity for them to gain some metacognitive skills to successfully draw diagrams in their mathematical learning.

(e) Mental actions of the co-variation

Carlson’s (1998) co-variation framework incorporated five groups of mental actions that were observed when students reasoned in representing and interpreting graphical model of live operating event on concepts of rate of change. The mental actions include; 1) visualizing two variables that change simultaneously, 2) visualizing weak relationship of two variables that changes with respect to each other such as the increasing and decreasing functions, 3) visualizing specific change in one variable with respect to a specific change in the other variable, 4) visualizing continuous changes of the function over the domain, and 5) visualizing changes of rates over the domain of the function. The framework was based on multiple refinements and analysis of the co-variational reasoning can be detected to a finer degree. It can also assist to guide the structuring of teaching and learning activities.

(f) Understanding of tables and graph

Sharma (2013), on the basis of meta-analysis on various research investigating the students’ thinking claimed that students ought to start probing worry questions and able to justify their opinions on any graphical representations or relationships to data values in tables and algebraic expressions. Her study identified a broad range of ability, from no to over considerations on the contexts of mathematical education. One of her findings was that teaching students to extract information from graphs and tables was much easier as compared to assist them to mature in their questioning with how and why the need to gather and compare within and between categories and to further thinking about the data in the specific contexts. She finally provided a conceptual framework that can be used to assess information that is displayed in data representations and guide teachers and curriculum developers with firm pedagogical teaching and learning of mathematical concepts. Her framework outlined five stages of behaviours when

students dealt with graphs to solve statistical problems. The five stages were informal or idiosyncratic, consistent non-critical, consistent non-critical, consistent, early critical and advanced critical. For each of the behaviour, students were revealing the characteristics of their thinking starting from pre-structural thinking, uni-structural thinking, multi-structural thinking, relational thinking to extended abstract thinking.

(g) Level of reasoning

Yumus’s (2001) level of reasoning emphasized on the importance of transforming students’ instrumental understanding of the basics mathematical rules and concepts without referring to reasons, to more relational understanding that involved the detailed of how rules and concepts worked. The first part of the levels deals with the what and how while the latter involved the why for the what and how. The levels of reasoning include: 1) unable to produce any reasoning, 2) aware of models, known facts, properties and relationships used as basis of reasoning, but cannot produce any arguments, 3) able to provide reasons although arguments are weak, and 4) able to provide strong arguments to support reasoning.

(h) Levels of graph comprehension

The six behaviours of reading, describing, interpreting, analysing, predicting and extrapolating data stated by Friel, Curcio and Bright (2001) were based on the follow-up of two main findings. One of the findings was the results of the research carried out by Curcio (1987) on fourth and seventh grades student where she identified three levels of graph comprehension and the other one was determined by Friel and Bright (1998) on how students make sense of information on graphs. Curcio argued students’ prior knowledge on structural components of graphs do affect their ability to read and understand the mathematical information and relationships shown on graphs. It was also

identified that they struggled in responding to tasks that need higher order thinking skills, for example, when the information is not displayed on the graphs. The researchers noted that the students tended to manipulate or to interpret that proved their inconsistent understanding on the concepts. They concluded that the process of dealing with a massive data and the structural components of the graphs contribute to the ability to read and interpret graphs.