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Problem solving in the Revised National Curriculum

Statement 45 The RNCS (DoE 2002:5) expounds that one of the seven critical outcomes for Outcomes-based Education in South Africa “…envisages learners who are able to identify and solve problems and make decisions using critical and creative thinking”. The Statement about Mathematics as learning area declares that one of the unique features of learning and teaching Mathematics is problem solving. The learning of mathematics through problem solving has a strong real-life connection and is evident in the curricula of different countries. In the RNCS in South Africa (DoE 2002:43,71) the real-life connection is stressed in the Assessment Standards for Mathematics, stating that problems should be solved in context, including contexts that may be used to build awareness of other learning areas, as well as human rights, social, economic and environmental issues.

**The contextualisation of problem solving in the modeling**

**The contextualisation of problem solving in the modeling**

process Problem solving took on a deeper dimension when it was contextualised in the modeling process. Modeling and the use of models was mainly used in fields of Applied Mathematics; it is only in the past decade or so that models and modeling became the lens through which the construction of knowledge was studied in Mathematics Education (Lesh et al 2002; Dossey et al 2002; Lesh & Doerr 2000; Gravemeijer et al 2000). The word model in connection with problem solving is used by Carpenter, Fennema, Franke, Levi and Empson (1999:55) who describe problem solving as modeling. Counting and direct modeling strategies are cited as specific examples of the fundamental principle of modeling and should be seen “as attempts to model problems rather than as a collection of distinct strategies”.

**MENTAL MODELS**

**MENTAL MODELS**

The relationship between models and the representation thereof surfaced in the discussion of models and modeling. The notion of models being internal conceptual systems that are expressed in external notation systems and thereby becoming accessible to others, were discussed. A question that ensues is about the nature of mental representations and the role they play in mathematical understanding. David Bartholomew as quoted in Wild and Pfannkuch (1999:223) connect mental models with our interpretation of everyday experiences: We all depend on models to interpret our everyday experiences. We interpret what we see in terms of mental models constructed on past experience and education. They are constructs that we use to understand the pattern of our experiences.

**Statistics in school curricula**

Statistics are in most school curricula considered as a branch of mathematics dealing with collection, analysis, interpretation and representation of data. Statistics and probability are sometimes referred to as stochastics (Reading & Pegg 1995:140; Truran 1997:538; Truran, Greer & Truran 2001:258; Shaughnessy & Watson 2003:192). As mentioned in 2.2, the term stochastic refer to uncertainty, as opposed to the term deterministic, which refers to certainty. The term stochastics is not used in school curricula and even the word statistics does not appear in most national curricula, but rather the terms data handling and probability.

** Data modeling**

Model development as learning was discussed in 2.1. This connection between model development and learning is also pointed out by Doerr and English (2003:111), but with specific reference to data modeling contexts. Lehrer and Schauble (2000:52) define data modeling as a multicomponential process of posing questions; developing attributes of phenomena; measuring and structuring these attributes; and then composing, refining, and displaying models of their relations.

**TABLE OF CONTENTS
**

- CHAPTER 1: RESEARCH DESIGN
- 1.1 INTRODUCTION AND ORIENTATION
- 1.2 MOTIVATION FOR THE STUDY
- 1.3 RESEARCH FOCUS
- 1.4 AIM AND OBJECTIVES OF THE RESEARCH
- 1.5 RESEARCH DESIGN
- 1.5.1 Research methodology
- 1.5.2 Qualitative research
- 1.5.3 Research principles
- 1.5.3.1 Legal principles
- 1.5.3.2 Ethical principles
- 1.5.3.3 Philosophical principles
- 1.5.3.4 Procedural principles

- 1.5.4 Criteria for data generation
- 1.5.5 Phases of the research
- 1.5.5.1 Phase One: Literature study
- 1.5.5.2 Phase Two: Empirical study
- 1.5.5.3 Phase Three: The analysis and interpretation of findings

- 1.6 VALUE OF THE RESEARCH
- 1.7 AN OVERVIEW OF THE RESEARCH

- CHAPTER 2: MODELING AND PROBLEM SOLVING IN MATHEMATICS AND STATISTICS EDUCATION
- 2.1 INTRODUCTION
- 2.2 MODELING IN MATHEMATICS EDUCATION
- 2.2.1 Definitions of modeling
- 2.2.2 The process of modeling
- 2.2.3 Emergent models
- 2.2.4 Modeling in the school curriculum

- CHAPTER 3: REPRESENTATION IN MATHEMATICAL AND STATISTICAL MODELING AND PROBLEM SOLVING
- 3.1 INTRODUCTION
- 3.2 DIFFERENT PERSPECTIVES INFLUENCING THE CONCEPT OF REPRESENTATION
- 3.3 CONCEPTS IN A THEORY OF REPRESENTATION
- 3.3.1 Representational systems
- 3.3.1.1 Characters, configurations and structures
- 3.3.1.2 Symbolic relationships
- 3.3.1.3 Ambiguity in representation
- 3.3.2 Internal and external representation
- 3.3.2.1 Internal representation and representational systems
- 3.3.2.2 External representation and representational systems
- 3.3.3 The role of context and content in representation
- 3.3.4 Representational fluency

- 3.4 THE RELATIONSHIP BETWEEN MODELING, PROBLEM SOLVING AND REPRESENTATION IN MATHEMATICS
- 3.5 REPRESENTATION IN THE MATHEMATICS CURRICULUM
- 3.6 DATA REPRESENTATION

- CHAPTER 4: METHOD OF RESEARCH
- 4.1 INTRODUCTION
- 4.2 REVISITING THE RESEARCH QUESTION
- 4.3 STUDY POPULATION AND SAMPLE
- 4.3.1 Study population
- 4.3.2 Sample
- 4.4 INSTRUMENT
- 4.5 VALIDITY AND RELIABILITY OF THE INSTRUMENT
- 4.6 DATA SOURCES AND DATA GENERATION
- 4.6.1 Data sources
- 4.6.2 Data generation
- 4.7 DATA ANALYSIS
- 4.7.1 Data arrangement types
- 4.7.2 Representational types
- 4.7.3 SOLO levels of representation
- 4.7.4 Coding reliability check
- 4.8 SUMMARY

- CHAPTER 5: ANALYSIS OF RESULTS

- CHAPTER 6: FINDINGS AND RECOMMENDATIONS

DOCTOR OF EDUCATION Project Topics