CHAPTER 3 THEORETICAL FRAMEWORK
Much has been written and said about the mismatch between successful algebraic problem solving and conceptual understanding of the phenomena related to the problem. It is well known that many students can solve problems without being able to explain the meaning of solutions and that they can use physics formulae but are unable to apply the underlying principles of physics to concrete situations. How do these algebraic solutions become disconnected from reality? How can such disconnectedness be remedied?
This chapter aims to shed light on these two questions. Firstly, Greeno’s model (Greeno, 1989) is discussed as a framework that explains the disconnection between algebraic problem solving and conceptual understanding. Secondly, the problem-solving strategy used in the present study is interpreted in terms of Greeno’s model to demonstrate how application of the strategy can develop both problem-solving skills and conceptual understanding.
GREENO’S MODEL FOR SCIENTIFIC PROBLEM SOLVING
In work on information processing, Greeno (1989) proposed a framework to analyse cognition relevant to scientific problem solving and reasoning. His model was called the extended semantic model and was based on four domains of knowledge, namely:
- concrete domain (physical objects and events)
- model domain (models of reality and abstractions)
- abstract domain (concepts, laws and principles)
- symbolic domain (language and algebra)
One-to-one correspondences exist between the domains. For example, the physical situation of a crane lifting a container corresponds to a force diagram in the model domain, the concepts of forces and Newton’s second law in the abstract domain and language as well as algebraic descriptions in the symbolic domain. According to Greeno, scientific problem-solving and reasoning skills involve the realization of the correspondences between these domains. Chekuri and Markle (2004) argued that although problem solving in physics usually implements algebra (operations in domain 4), a learner should always be in touch with the other three domains to consolidate concepts.
The four domains are represented in figure 3.1. The correspondences (mappings) between the domains are indicated by Φ, with subscripts c, m, a, s respectively denoting the concrete, model, abstract and symbolic domains. For example, Φsc represents the mapping between the symbolic and concrete domains. Figure 3.1 also shows a double-layered structure of each of the four domains as well as connections indicated by θ and Ψ, within domains. These features are discussed below.
Each domain consists of two layers, namely an a-layer containing independent items, or pieces of knowledge, while the associated b-layer contains structures of items, i.e. meaningful combinations, or structures of the pieces of knowledge. Using the crane example, in the concrete domain, layer 1a contains items like crane, cable, container and the action of lifting, while the combination of the crane lifting the container is a structure in layer 1b. In the model domain, arrows and a dot/circle/square are items in layer 2a, while the force diagram combining these as vertical arrows attached to the dot/circle/square, one arrow pointing up and the other pointing down, is the structure in layer 2b. In the abstract domain, the concepts of force, mass, weight and acceleration belong in layer 3a, while Newton’s second law is a structure in layer 3b. In the symbolic domain, symbols like F, m, g and a are items in the 4a layer, while the formula ΣF= ma is a structure in the 4b layer. A written problem statement about the crane lifting the container is another structure in 4b, while the independent words are items in 4a. An example from electrical circuits is given in table 3.1 to illustrate how independent items in a-layers are combined into meaningful structures in the associated b-layers.
Figure 3.1 also shows connections within each domain, indicated by θ and Ψ, with subscripts c, m, a and s to indicate the relevant domain. Connections marked θ represent relationships between independent items in a layers to form meaningful structures in the corresponding blayers. For example, a connection θm in the model domain can represent the rules for how to draw a circuit diagram from components, or the parallelogram rule to add vectors. Connections marked Ψ represent alternative ways to represent a particular structure in a particular layer, governed by sets of rules for the particular domain. For example, to associate a constant velocity with a zero resultant force is an operation Ψa in the domain of abstract structures. Another relevant example is algebraic manipulation, classified as an operation Ψs in the symbolic domain.
In contrast to idealised problem solving that incorporates all four domains of knowledge, a popular formula-based approach flourishes amongst students. This technique is popularly known as the plug-and-chug method, described by Heller and Heller (1995). The students tend to start with a data list, matching information to symbols. This is followed by selecting a suitable formula to link the unknown symbol to the known symbols in the list. All that remains is to substitute and to solve algebraically; interpretations are rare. Van Heuvelen (1991a) called this method a formula-based approach. Greeno (1989) referred to the “insulation” of the symbolic world from the “situated nature” of problems to explain how algebraic solutions can become disconnected from the concrete situation they represent. In the classroom, students manipulate symbols to solve problems while the concrete problem situation is seldom present. This classroom reality can, therefore, lead to the belief that problems are about the symbols, rather than about the concrete situation represented by those symbols. The symbols are real marks on paper and the chalkboard, taking the place of the real objects described by the problem statement. Figure 3.2 illustrates the “insulation” of the symbolic world: A mathematical operation Ψs in the symbolic domain acquires the status of a mapping Φ’ from the symbolic to the concrete marks on the chalkboard and paper. Greeno refers to this inappropriate mapping as a “perverse” mapping from symbolic structures to symbolic entities. Algebraic solutions can therefore amount to operations on knowledge located only in the domain of symbolic knowledge, without translation to the concrete or abstract domains. Such an approach can sometimes lead to correct equations and correct numerical answers, but it does not demonstrate or develop understanding of the meaning of algebraic solutions.
THE PROBLEM SOLVING STRATEGY INTERPRETED IN TERMS OF GREENO’S MODEL
Each of Greeno’s four knowledge domains features in the problem-solving strategy (outlined in 1.5) implemented in the current study. When applying the strategy, various actions are performed. These actions can be described as operations in domains and translations from one domain to another. The word “operate” means working on structures within a domain, e.g. algebraic manipulation of an equation would be an operation in the symbolic domain while manipulation of a diagram would be an operation in the model domain. “Translate” means to switch representations, to relate knowledge from one domain to another, e.g. to draw a diagram that represents a concrete situation is a translation from the concrete to the model domain. The series of actions described in table 3.2 represents the application of the problem-solving strategy interpreted in terms of translations between the four domains of knowledge.
The actions involved in the seven steps of the problem-solving strategy thus require making multiple translations among all four domains. This forms a wealth of different kinds of associations to synthesise reality, models, concepts and symbols. At the same time algebraic manipulations within the symbolic domain become a relatively small part of problem solving. Instead, the diagram becomes the focus of attention.
While drawing the diagram, the concrete situation described in the problem statement is reconstructed as a two-dimensional map. Concepts, information and unknown quantities are grouped by location when superimposed on the diagram. In the analysis, these localized groupings guide the search for principles of physics applicable to different parts of the concrete situation. Links between different parts of the problem become visible as shared features between groupings on the diagram. From here onwards the solution proceeds algebraically, until the symbolic solution is interpreted in terms of the concrete situation, confirming the relation between the symbolic and concrete domains.
Regular practice in using the seven-step strategy for solving problems would thus amount to practice in traversing the four knowledge domains, instead of focusing on the symbolic domain. New associations can be made to develop understanding of how physics concepts are relevant to concrete situations. Existing associations can be reinforced to develop familiarity with how physics concepts are relevant to concrete situations (Alant, 2004). Consequently, the seven-step strategy is much more than a guide to constructing solutions: it is a process to develop skills to translate between knowledge domains, to integrate relevant knowledge structures from different domains, thus enhancing conceptual understanding and problem-solving skills.
DESIGN OF THE STUDY
A quasi-experimental design with a pre-test and post-tests was implemented. The treatment and control groups were situated in two geographically separate education districts. Each group consisted of 8 volunteering schools, with one participating teacher per school. This design differed from the Huffman (1997) study where each teacher had classes in the treatment as well as in the control groups. In the present study, the two groups were not informed about each other and because the districts were at opposite ends of town (separated by about 40 km / 25 miles), diffusion, contamination, rivalry and demoralization were effectively prevented. The study was conducted over a period of ten months, starting at the beginning of the academic year in January.
The teachers in the treatment group were introduced to the structured problem-solving strategy at an afternoon workshop early in the year. During this workshop the strategy was explicitly described and demonstrated, using sample problems taken from final grade 12 departmental examinations. At their schools the teachers explicitly explained the strategy to the students. Thereafter they applied the strategy in the classroom when solving sample problems and whenever assisting students. The students were required to apply the strategy not only when working on their own, but also in the social context of classroom sample problems and group work. Care was taken to ensure that the teachers knew exactly what was expected of them. During the first semester, three teacher workshops of 90 minutes each were held before starting new topics. Typical problems were discussed during these workshops, giving teachers the opportunity to practise using the problem-solving strategy for particular topics. The researcher provided no written solutions to problems in order to encourage teacher participation. The teachers were, therefore, actively engaged, contributing to the construction of solutions during the workshops. Similarly, the students were expected to be active participants in the classroom situation.
The control group teachers and students were informed that the purpose of the project was to investigate students’ problem-solving skills. No mention was made of a treatment group using a problem-solving strategy.
The study was designed to be non-disruptive: the way in which problems were solved in the treatment group was the only change from an ordinary school situation. The problem-solving strategy was applied and practised while solving classroom and homework problems that would form part of the ordinary routine of learning physics by doing problems. The ordinary grade 12 syllabus and the school’s textbooks were used. No extra classes for students were required. Tests were structured like ordinary 30-minute classroom tests consisting of typical exam problems. For each syllabus topic, a problem collection was given to the teachers of both groups to ensure that the students had the same exposure to problems. The researcher did not provide solutions to these problems to either group in order to prevent rote learning of model solutions in either group and to provide opportunities for each group to practise problem solving.
Participating in the study could benefit teachers as well as students from both groups. Students and teachers were exposed to the correct test and marking standard. Teachers did not have to set or mark tests from January to June, as this was done by the researcher in order to ensure a uniform standard. Problem collections were provided for all the topics. The fixed test dates provided a time schedule to enable teachers to finish the syllabus. The treatment group had the additional benefit of learning and implementing the problem-solving strategy.
Treatment fidelity was checked by means of questionnaires, students’ written solutions and videotaping students while solving problems. Sustained classroom observation was not considered an option, as the researcher was employed as a full time teacher at the time of data collection. Besides, the presence of an observer may have caused distractions, or it could have been a threat to the teacher’s authority, or it could have encouraged a special effort by the teacher and class. Whatever the effect of the presence of the observer, it could lead to a biased interpretation of results.
Chapter 1 Introduction
1.1 Research questions
1.2 The South African context
1.4 The role of problem solving in physics
1.5 The structured problem solving strategy
1.6 Outline of the study
1.8 Organisation of the thesis
Chapter 2 Literature survey
2.1 Problem solving in physics by individuals
2.2 Conceptual understanding of physics
2.3 Instructional strategies
2.3.1 Structured problem solving
2.3.2 Explicit problem solving
2.3.3 Structured problem solving and cooperative groups
2.3.4 The classroom based study of Huffman
2.3.5 Modelling Instruction
2.3.6 Overview, Case Study Physics
2.3.7 Qualitative strategy writing
2.4 Implications for science education in South Africa
Chapter 3 Theoretical framework
3.1 Greeno’s model for scientific problem solving
3.2 The problem solving strategy interpreted in terms of Greeno’s model
Chapter 4 Research methodology
4.1 Design of the study
4.3 Collection of data
4.4 Data analysis
4.6 Validity of the design
4.6.1 Internal validity
4.6.2 External validity
Chapter 5 Results: quantitative
5.1 Sample profile
5.2 Test scores
5.3 Pass rate
5.4 Quantitative evidence of reliability and validity
5.5 Interactions with the treatment
5.6 Quantitative description of conceptualizing
5.7 Chapter summary
Chapter 6 Results: qualitative
6.1 Solutions maps
6.1.1 Work and energy
6.1.4 Solutions maps interpreted in terms of a conceptual approach
6.1.5 Section summary
6.2 Case studies: video session
6.2.1 The accelerating charge
6.2.2 The pendulum collision problem
6.2.3 Section summary
6.3.1 Treatment group’s questionnaire
6.3.2 Video group’s questionnaire
6.3.3 Teachers’ questionnaire
6.3.4 Section summary
6.4 Chapter summary
Chapter 7 Synthesis
7.1 Interpretation of results
7.1.2 The co-development of conceptual understanding and problem solving skills
7.1.3 The development of a conceptual approach to problem solving
7.2 Significance of the study
7.2.1 Social knowledge construction
7.2.2 Second language development
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