Mental Representations of Diagrams, Views about Diagrams, and Problem Solving (original) (raw)

Diagrams and problem solving

The use of diagrams as external aids to facilitate cognitive abilities is not new. This paper looks into cognitive studies for insight into when, why and how diagrams are effective in problem solving. A case study examines the use of diagrammatic representations as thinking tools and tools for communicating information. The purpose is to examine whether diagramming could be used as a design method, as part of the designer's creative process.

THE USE OF DIAGRAMS IN SOLVING NON ROUTINE PROBLEMS

2004

This paper explores the role of diagrams in a specific problem solving process. Two types of tests were administered to 194, 12 year old students, each of which consisted of six non-routine problems that could be solved with the use of a diagram. In Test A students were asked to respond to the problems in any way they whished whereas in Test B problems were accompanied by diagrams and students were asked to solve these problems with the use of the specific diagrams presented. The results revealed that there was no statistical significant difference between the two tests. The result also revealed that it was not the same group of students that were successful in the two tests.

Constructing mental diagrams during problem-solving in mathematics

Pythagoras

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 fir...

Does the Use of Diagrams as Communication Tools Result in their Internalization as Personal Tools for Problem Solving?

Proceedings of the 30th Annual Conference of the …, 2008

Although diagrams are considered as effective personal tools for solving problems, applied research in education has identified a widespread problem: that students lack spontaneity in diagram use. One way to address this problem was reported by : their findings indicate the effectiveness of using peer instruction to enhance students' spontaneous use of diagrams. However, it was not clear from their study whether actual interaction is necessary, and whether formulation of explanations in using diagrams to solve problems would in itself be sufficient. The present study sought to clarify the role of communication in enhancing the spontaneous use of diagrams, and involved 5 days of experimental classes for 59 participants in the 8 th grade. Two conditions were used: one where participants really interacted with each other in peer instruction sessions (the experimental condition), and another where the participants formulated explanations but were not involved in peer instruction interactions. At post test, both quantity and quality of diagrams spontaneously produced by participants in the experimental condition were higher than those in the control condition, suggesting that the communication process involved in actual interactions with peers is a critical factor. This result supports the notion that using diagrams as communication tools results in their internalization as personal tools for problem solving.

Using diagrams as tools for the solution of non-routine mathematical problems

Educational Studies in Mathematics, 2009

The Mathematics education community has long recognized the importance of diagrams in the solution of mathematical problems. Particularly, it is stated that diagrams facilitate the solution of mathematical problems because they represent problems’ structure and information (Novick & Hurley, 2001; Diezmann, 2005). Novick and Hurley were the first to introduce three well-defined types of diagrams, that is, network, hierarchy, and matrix, which represent different problematic situations. In the present study, we investigated the effects of these types of diagrams in non-routine mathematical problem solving by contrasting students’ abilities to solve problems with and without the presence of diagrams. Structural equation modeling affirmed the existence of two first-order factors indicating the differential effects of the problems’ representation, i.e., text with diagrams and without diagrams, and a second-order factor representing general non-routine problem solving ability in mathematics. Implicative analysis showed the influence of the presence of diagrams in the problems’ hierarchical ordering. Furthermore, results provided support for other studies (e.g. Diezman & English, 2001) which documented some students’ difficulties to use diagrams efficiently for the solution of problems. We discuss the findings and provide suggestions for the efficient use of diagrams in the problem solving situation.

What can we learn by comparing students' diagram-construction processes with the mathematical conceptions inferred from their explanations with completed diagrams?

Despite recent research interest in student-created diagrams, little research has systematically investigated students' diagram-construction processes, meaning the order and manner in which students create markings as they physically generate diagrams. In this study, we characterize the various processes students use to create diagrams that represent a quadratic motion situation involving increasing speed, and we explore how these diagram-construction processes are related to students' conceptions of speed as inferred from their explanations with their completed diagrams. Previous literature suggests contrasting predictions regarding whether or not students' diagramconstruction processes are closely related (from our perspective as researchers) to students' inferred conceptions. We see the study as having value for research and practice by raising new questions related to diagram-construction processes, pointing to the potential formative assessment value of attending to diagram-construction processes, and demonstrating the need for the development of theory to explain the relationships identified by this study.

A good diagram is valuable despite the choice of a mathematical approach to problem solving

2013 Physics Education Research Conference Proceedings, 2014

Drawing appropriate diagrams is a useful problem solving heuristic that can transform a problem into a representation that is easier to exploit for solving the problem. A major focus while helping introductory physics students learn problem solving is to help them appreciate that drawing diagrams facilitates problem solution. We conducted an investigation in which 118 students in an algebra-based introductory physics course were subjected to two different interventions during the problem solving in recitation quizzes throughout the semester. Here, we discuss the problem solving performance of students in different intervention groups for two problems involving standing waves in tubes, one which was given in a quiz and the other in a midterm exam. These problems can be solved using two different methods, one involving a diagrammatic representation and the other involving mostly mathematical manipulation of equations. In the quiz, students were either (1) asked to solve the problem in which a partial diagram was provided or (2) explicitly asked to draw a diagram. A comparison group was not given any instruction regarding diagrams. Students in group (1), who were given the partial diagram, could not use that partial diagram by itself to solve the problem. The partial diagram was simply intended as a hint for students to complete the diagram and follow the diagrammatic approach. However, we find an opposite effect, namely, that students given this diagram were less likely to draw productive diagrams and performed worse than students in the other groups. Moreover, we find that students who drew a productive diagram performed better than those who did not draw a productive diagram even if they primarily used a mathematical approach. Interviews with individual students who were asked to solve the problem provided further insight.