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## Mathematical Resources activated

An unidentifiable mathematical resource is activated when the student, upon reading the question writes the equation in step 1. This resource is similar to the ones described in previous solutions (see section 6.2 and 6.3) and therefore designated unidentified resource 1.

In step 2 the student notice from the diagram the relation that; current I3 breaks into I1 and I2 and writes is mathematically as I3= I1+ I2. Step 2 could not be immediately derived from step 1, which is an incorrect mathematical expression of the Kirchhoff‟s rule around a closed loop. Step 2 thus involves activation of reasoning primitives that I shall call sum of parts is whole. Sum of parts is whole is an intuitive sense of physical mechanism (reasoning primitive) with the abstract notion that a whole can be divided into its individual parts. The same reasoning could be used for a river breaking into two streams, to say the water in the two streams is the same as the water in the river, for example. Even though the student is not aware that step 2 is the correct solution, the equation was motivated by a sense of physical mechanism; where students use a form of intuitive knowledge about physical phenomena and processes (Tuminaro, 2004; p. 45).

Formal interpretive devices are used in the rearrangement of variables from step 2 to step 3.

Neither intuitive mathematics knowledge resources nor do symbolic forms appear to be activated in this student‟s solution.

### Awareness of ESM domains

Despite the question being presented in the model domain, student H1 starts on the abstract domain by writing the incorrect Kirchhoff‟s rule. The student awareness of the model domain helps him to come up with the expression I3 = I1 +I2 in step 2. While step 2 could be interpreted as indication of the model domain – demonstrating understanding of the relationship of the three currents from the diagram, it also could appear accidental. Step 3 indicates awareness of the symbolic domain, and is further proof of the accidental nature of step 2 as the student appears to be working in reverse.

Step 1 gives a mistaken mathematical expression for current around a loop. It is therefore ESM layer b since an expression is a relation, or a comparison. Since step 2 is not motivated by step 1, it can only be due to the awareness of first; the currents I3, I2 and I1 as separate entities (ESM layer a) and then their combined relation (ESM layer b). Step 3 is an awareness of ESM layer b resulting from step 2.

**Use of units, variables and constants**

In student H1’s solution to this problem, the following was observed;

• No units are used

• No variables are substituted

• No constant are substituted

Since no units, variables or constant were used or substituted in this solution, there is no order of use or substitution to be discussed.

#### Study summary

The use of mathematics in physics must be understood for the role its serves. In students‟ learning of physics, it is even more important that students understand why they use mathematics the way they do. Do students‟ use mathematics so that mathematics helps them understand the physics, or is students‟ “efficient” use of mathematics when solving physics problems simply an indication of their understanding of the subject – mathematics? Is students‟ use of mathematics in physics much like solving a puzzle? More so, does students‟ use of mathematics in the physics topic of electricity bring out any unique approaches or notable types of understanding?

A comprehensive coverage of the relevant literature indicated that students‟ effective use of mathematics in physics is still a contentious issue (see sections 2.1; 2.2; 2.3; 2.4 & 2.5). There are those researchers who reckon that students do badly in physics because students do not have requisite mathematical preparedness (Ayene et al., 2012). Others argue that even if students did have the required level of mathematics, the issue of transfer of knowledge across different domains is really the problem (Basson, 2002; Redish, 2005). Researchers have described how mathematics and physics are ontologically different types of knowledge, which also require different epistemological energies (Pettersson & Scheja, 2008). These differences include descriptions of knowledge as procedural as opposed to conceptual, or objective as opposed to subjective. Inevitably, these varying and at times conflicting descriptions also affect the way students perceive mathematics in physics.

Salaam (2007) and Quale (2011) both maintain that students‟ perceptions with regard to their understanding of the purpose of problem solving are polarized. This, they say results from their view of physics as objective knowledge, and mathematics as subjective knowledge. Physics is viewed as representing real physical objects, while mathematics relates to human imagination. Quale (2011) recommends for some kind of middle ground between the positions of realism (physics) and relativism (mathematics), since he says even “so called” objective objects are perceived by the human mind.

**Table of contents :**

Declaration

Abstract

Acknowledgements

List of Tables

List of Figures

List of Abbreviations

Language Editing Certificate

Turn it In Report

Turn it in Receipt

**Chapter 1 Introduction and Background **

1.1 Introduction

1.2 Study context

1.3 Problem Statement

1.4 Rationale for the Study

1.5 Objectives and Research Questions

1.6 Operational definition of key terms

1.7 Dissertation overview

**Chapter 2 Literature Review **

2.1 Introduction

2.2 Contrasting mathematics and physics

2.3 Problem solving in physics

2.3.1 Use of Algorithms and Heuristics in Problem Solving

2.3.2 Multi – Step Strategy

2.3.3 Group Work

2.4 The dichotomy in students‟ use of mathematics in physics

2.4.1 Mathematics as indispensable in students‟ learning of physics

2.4.2 Mathematics as a barrier in students‟ learning of physics

2.5 Students‟ learning outcomes on electricity as a topic in physics

2.5.1 Students use of mathematics in the topic of electricity

2.5.2 Students conceptual understanding of electricity

2.5.3 Students‟ misconceptions of electricity

2.6 MPERG and related studies on students‟ use of mathematics in physics

2.6.1 Students‟ interpretation of constants and variables

2.6.2 Semantic analysis

2.6.3 Mathematics – Physics entanglement

2.7 Mathematical thinking in physics

2.7.1 Mathematical Resources

2.7.2 Epistemic Games and Frames

2.8 Summary of the observations

**Chapter 3 Conceptual Framework **

3.1 Introduction

3.2 General Systems Theory (GST

3.3 Extended Semantic Model (ESM)

3.4 Some relevant approaches for students‟ use of mathematics in physics

3.4.1 Integration approach

3.4.2 Modeling approach

3.5 Design of the conceptual framework

3.5.1 Electricity layer

3.5.2 Mathematical Resources layer

3.5.3 MATHRICITY

3.6 Application of MATHRICITY through analysis of a typical first year electricity question

3.7 Chapter summary

**Chapter 4 Research Method **

4.1 Introduction

4.2 Research Design

4.3 Instruments

4.3.1 Expectation survey

4.3.2 Test scripts

4.3.3 Focus group semi – structured interviews

4.4 Validity and Reliability of the Instruments

4.4.1 Validity and Trustworthiness of SERMP

4.4.2 Validity and Trustworthiness of the focus group interview

4.4.3 Validity and Trustworthiness of the test Scripts

4.4.4 Pilot study

4.5 Participants

4.6 Analytical Framework

4.6.1 Survey and interview analysis

4.6.2 Scripts analysis

4.6.3 Integrating all the analyses

4.7 Ethics

4.8 Summary

**Chapter 5 Students‟ Expectations on the Use of Mathematics in Physics **

5.1 Introduction

5.2 Students‟ response to the SERMP

5.3. Emergent responses from the SERMP

5.3.1 Students agree

5.3.2 Students are neutral

5.3.3 Students disagree

5.3.4 Summary of emergent responses

5.4. Students‟ Epistemological Frames

5.5. Epistemological frame: What students think it takes to learn physics

5.5.1 Use of equations in learning physics

5.5.2 Memorization in learning physics

5.5.3 Conceptualization in learning physics

5.6. Epistemological frame: What Students think about the Use of Mathematics in Physics

5.6.1. The meaning of mathematical answers

5.6.2. Relationship between mathematics and physics

5.7 Summary

**Chapter 6 Students‟ test scripts **

6.1. Introduction

6.2 Analysis of students‟ work on Electric Force

6.2.1 Analysis of Student V1‟s work on Q1A1

6.2.2 Analysis of Student M3 on Q1A1

6.2.3 Analysis of Student M5‟swork on Q1A1

6.2.4 Summary of the three students‟ work on Q1A1

6.3 Analysis of students work on Electric Field question

6.3.1 Analysis of student V1‟s work on Q1B2a

6.3.2 Analysis of student M4 on Q1B2a

6.3.3 Analysis of student M5‟s work on Q1B2a

6.3.4 Summary of the three students‟ (V1, M4, M5) work on Q1B2a

6.4 Analysis of students‟ work on Electric Circuit question

6.4.1 Analysis of Student M6 on Q2B2

6.4.2 Analysis of Student H1‟s work on Q2B2

6.4.3 Analysis of student H2‟s work on Q2B2

6.4.4 Summary of the three students‟ (M6, H1, H2) work on Q2B2

6.5 Chapter Summary

**Chapter 7 Summary of Study and Findings **

7.1. Study summary

7.2 Discussion of findings

7.2.1 What they think it takes to learn physics

7.2.2 What students think about the use of mathematics in physics

7.2.3 How students used mathematics in the physics topic of electricity

7.2.4 Updated MATHRICITY

7.3 Conclusion

7.3.1 Students Expectations

7.3.2 Mathematical Approaches

7.3.3 Types of understanding

7.4 Limitations of the study

7. 5 Implications and further studies

**REFERENCES **

**APPENDICES**

Appendix A: MPEX

Appendix B: VASS

Appendix C: EBAPS

Appendix D: SERMP

Appendix E: CONSENT FORM

Appendix F: UNISA Ethics Clearance

Appendix G: University of Botswana Research Permission letter

Appendix H: Interviews

Appendix I: Instructor Solutions to Test Questions

Appendix J: Students Use of Units, Variables and Constants

Appendix K: Students‟ Solutions to Questions