Constructing mental diagrams during problem-solving in mathematics

Abstract

In mathematics, problem-solving can be considered to be one of the most important skills students need to develop, because it allows them to deal with increasingly intricate mathematical and real-life issues. Often, teachers attempt to try to link a problem with a drawn diagram or picture. Despite these diagrams, whether given or constructed, the student still individually engages in a private discourse about the problem and its solution. These discourses are strongly influenced by their a priori knowledge and the given information in the problem itself. This article explores first-year pre-service teachers’ mental problem-solving skills. The emphasis was not on whether they solved the problems, but rather on their natural instincts during the problem-solving process. The research shows that some students were naturally drawn to construct mental images during the problem-solving process while others were content to simply leave the question blank. The data were collected from 35 first-year volunteer students attending a second semester geometry module. The data were collected using task sheets on Google Forms and interviews, which were based on responses to the questions. An interpretive qualitative analysis was conducted in order to produce deeper meaning (insight). The findings point to the fact that teachers could try to influence how students think during the problem-solving process by encouraging them to engage with mental images.

Unveiling problem-solving strategies of pre-service mathematics teachers: A visual and discursive exploration

“…Her approach was effective, and her answer was similar to those of participants who had used algebraic techniques. These findings align with Mudaly's (2021) study, emphasizing the importance of encouraging students to use mental imagery during problem-solving.…”

“…His narrative also concurred with Mudaly and Rampersad's (2010) concept of visualization processes that are rooted within internalization and externalization activity. Additionally, PMT4's writing down of information diagrammatically and adding information onto the diagram to make sense of the question correlates with several other findings on the interpretation of how students comprehend questions (Diezmann, 2000;Mudaly, 2021;Mudaly & Reddy, 2016). Diezmann (2000) explains that writing information down in a diagram (i.e., creating an iconic visual mediator) is a process of translation that involves decoding linguistic information and encoding visual information in a spiral, iterative manner-as Mudaly (2021) outlines.…”

“…This suggests that, depending on the difficulty of a mathematics problem, mental visualization can be used to solve the problem without relying on iconic visual mediators. This was demonstrated in an empirical study conducted by Mudaly (2021), which found that students were able to organize data and develop strategies for solving mathematics problems using only internally visualized diagrams. This approach requires consciously reflecting on one's own thinking during problemsolving, which Sfard (2007) refers to as "intrapersonal communication".…”

“…Mathematical problem-solving is a crucial aspect of learning and teaching mathematics. In particular, visualization and discourse are two essential components of the problem-solving process (Mudaly, 2021). Visualization techniques assist learners to understand a problem, explore potential solutions, and develop new insights (Samosa et al, 2021).…”

An analysis of pre-service mathematics teachers’ geometric thinking and classroom discourse using a commognitive lens.

“…PSTs carry several unresolved misconceptions from previous learning, which accounts for their difficulties in learning at higher level education. For example, a study conducted by Mudaly (2021) involved 51 undergraduate mathematics students. Figure 2.1 was one of the tasks used in the study, and the people who took part were asked to write down the value of the angle formed at A.…”