Visual consideration in problem solving

In some aspects of pre-university and university mathematical teachings, visual considerations are naturally prominent. Be it on the board or the use of electronic technology, educators would put some thought on the layout of the presentation such as the fonts and sizes of the words and the quantity and quality of information on each page, so that students are able to see everything that are supposed to be delivered. Specifically in mathematics, visual works usually involve graphs or other types of mathematical diagrams to enhance students’ ability to generate reasoning on mathematical concepts and relationships rather than manipulating symbols and expressions.

In 2014, Anderson-Pence, Moyer-Packenham, Westenskow, Shumway and Jordan took some efforts to restructure the relationships between the usage of visual tools and the students’ written worked solutions. The two open-ended tasks were distributed to 371 students, one with diagram for them to refer to and another one, a word problem for them to sketch diagrams or graphs to help the solution process, to trace the patterns of solutions and errors performed. Students were found to lack flexibility in the reading and interpreting graphs either from those provided in the tasks or the ones that they had to construct by themselves. It was also detected that their exposure to various types of mathematical representations influenced their choice of solution methods and hence their understanding of related concepts.

The use of diagrams which are visual in nature is regarded as one of the most effective ways to encourage students to strategize their method of solving mathematical problems (Ainsworth & Loizou, 2003; Cheng, 2004; Mayer, 2003; Stern, Aprea & Ebner, 2003). Nevertheless Uesaka, Manalo and Ichikawa (2007, 2010) had identified that students were reluctant to use diagrams when solving mathematical word problems. They were unaware of the diagrams’ efficacy when dealing with word problems on real life situations. In their series of studies on the area, Uesaka and Manalo (2011) identified factors related to the students’ lack of urge to use diagrams. In their first experiment on 125 Japanese students, they identified that students were prone to use diagrams on problems that require more mental efforts as compared to problems involving length or distance measurements. The National Curriculum of New Zealand (2007) stressed on the importance of both teaching students to understand diagrams and the use of diagrams as communication tools. Therefore, in their second experiment, they made a comparison between the same Japanese students and 323 New Zealand students. The tasks were translated to English language for the New Zealand student. As

expected, a significantly higher proportion of the New Zealand students exhibited their preference to the use of diagrams when solving mathematical word problems.

Many researchers and educators highlighted the importance of visual reasoning in the teaching and learning of differential calculus and had proposed that there were a lot to cultivate in the topics (Presmeg, 2006). Kannemeyer (2005) noted that teachers emphasized on the completion of the syllabus through the typical process on recurring problems instead of stressing on the handling of application or non-routine problems. Visuals such as diagrams, graphs or other representations serve both as tools for solving problems and communication purposes. Therefore designing suitable tasks or real life problems so as to promote the use of visual in solving the problems is vital (Doerr & English, 2006). Francisco & Maher (2005) carried out a study on the nature and types of visual tasks that should be used for classroom purposes and determined that the more complex the task was, the more cognitive efforts and reasoning skills that were required from the students.

In 2004, Leung and Chan’s students, Kevin, experienced a visual process of understanding the global features of graphs of functions through the manipulation of local parts using the zooming capabilities of graphing software, the graphic calculator. He was able to view the whole continuous and separated portions of graphs together in one screen. The zooming function allows him to scrutinize visually the situation of separated curves that led him to his own idea of law of continuity. This allowed him to combine all his prior knowledge on the visual information to explain his understanding on functions through graphing.

Teacher’s knowledge on the subject content has a large effect on how students learn and grasp concepts. In Singapore, Toh (2009) gathered information on 27 new (less than five years experienced) in-service mathematics teachers from various secondary schools. He adopted Amit and Vinner’s (1990) model using a questionnaire

to stimulate the teachers’ knowledge in calculus. Two, out of the seven tasks, were graphed-based. The tasks were mostly dealing with the definition and images of various concepts in derivative and calculus essential for the secondary levels. Among the mistakes that they had performed were: 1) failure in grasping the essential principles to solve problems, 2) did not recognize the discontinuity of the graphs of functions, 3) did not manage to identify the correct values of limits, and 4) unable to link the concepts to the tasks. He identified that most of the pre-service teachers did not possess strong or convincing concepts images related to the derivative concepts. They would generally favour the procedural understanding in handling the tasks.