Multiple representations and cognitive load: words, arrows, and colours when solving algebraic problems

WORDS, ARROWS AND COLOURS

WHEN SOLVING ALGEBRAIC PROBLEMS

Amina Brey

Submitted in fulfilment of the requirements for the degree of

Magister Educationis

in the Faculty of Education

at the

Nelson Mandela Metropolitan University

Port Elizabeth

South Africa

Supervisor: Professor Paul Webb

PO Box 77000

Nelson Mandela Metropolitan University Port Elizabeth

6013

Enquiries: Postgraduate Examination Officer

DECLARATION BY CANDIDATE

NAME : AMINA BREY

STUDENT NUMBER : 208072174

QUALIFICATION : MAGISTER EDUCATIONIS

TITLE OF PROJECT: MULTIPLE REPRESENTATIONS AND COGNITIVE

LOAD: WORDS, ARROWS AND COLOURS WHEN SOLVING ALGEBRAIC PROBLEMS

DECLARATION:

In accordance with Rule G4.6.3., I hereby declare that the above-mentioned treatise/dissertation/thesis is my own work and that it has not previously been submitted for assessment to another University or for another qualification.

SIGNATURE : _____________________

i

ACKNOWLEDGEMENTS

To my Almighty Allah, I give thanks. It is only through His abundant love, mercy and guidance that I had the ability to complete this study.

To my supervisor, Professor Paul Webb, thank you for having faith in me and accepting me as your student. Thank you for your patience, encouragement, wise counsel, academic integrity, valuable insight and endless guidance through this beautiful journey.

To Mrs Carol Poisat and Mrs Marina Ward, for your encouraging words, continued support and help at all times, and my friends Zaakiyah Gangat and Kholisa Papu, for your assistance and friendship.

To the principals, teachers and learners who participated in this study, a special thanks.

To my loving parents, Cassiem and Farida, for their continuous prayers, support, help, interest, patience, encouragement, sacrifice and confidence in me. I never doubt for a second that I would be here without all that they have done for me and continue to do for me.

To my caring siblings, Muhammad, Haajar and Zulaikha, for their endless support, love and prayers over the years and through this journey.

To my beautiful son Muhammad Naasir, for your patience, understanding and laughs when I needed it most. You are my blessing and joy.

ii

ABSTRACT

This study investigates the possible effects that access to selected multiple representations (words, arrows and colours) have in terms of cognitive load and learner achievement when presented with algebraic problems at grade nine level. The presentation of multiple representations (the intervention) was intended to decrease extraneous cognitive load, manage the intrinsic cognitive load (algebraic problems) and optimise germane cognition (schema acquisition and automation). An explanatory sequential mixed-method design was employed with six hundred and seventy three learners in four secondary schools. Quantitative data were generated via pre-, intervention and post-tests/questionnaires, while qualitative data were obtained from open-ended questions in the pre-, intervention, and post-tests/questionnaires, eight learner focus group interviews (n = 32), and four semi-structured, open-ended teacher interviews. Statistically and practically significant improvement in mean test scores from the pre- to intervention test scores in all schools was noted. No statistically and practically significant improvement was noted in further post-tests except for post-test 2 which employed more challenging problems (statistically significant decrease with a small practical effect). Learners expressed their preference for arrows, followed by colours and then words as effective representations. Teacher generated qualitative data suggests that they realise the importance of using multiple representations as an instructional strategy and implicitly understand the notion of cognitive load. The findings, when considered in the light of literature on cognitive load, suggest that a reduction in extraneous cognitive load by using a more effective instructional design (multiple representations) frees working memory

iii

capacity which can then be devoted to the intrinsic cognitive load (algebraic problems) and thereby increase germane cognition (schema acquisition and automation).

Key words: Cognitive load, multiple representations, learner achievement, working memory capacity, schema acquisition and schema automation.

iv

LIST OF ABBREVIATIONS

ANA Annual National Assessment ANOVA Analysis of Variance

LoLT Language of Learning and Teaching

NCTM National Council of Teachers of Mathematics

v

TABLE OF CONTENTS

CHAPTER ONE

INTRODUCTION

1. INTRODUCTION ... 1

2. COGNITIVE LOAD THEORY ... 2

3. MULTIPLE REPRESENTATIONS ... 3

4. PROBLEM STATEMENT ... 4

5. RESEARCH OBJECTIVES ... 5

6. RESEARCH QUESTIONS ... 6

7. DESIGN AND METHODOLOGY ... 7

7.1 Sample and setting ... 8

7.2 Research instruments ... 8

7.3 Data collection ... 9

7.4 Data analysis ... 9

7.5 Reliability & Validity ... 10

8. ETHICS... 11

vi

CHAPTER TWO

LITERATURE REVIEW

1. INTRODUCTION ... 13

2. COGNITIVE LOAD THEORY ... 14

2.1 Human cognitive architecture ... 15

2.2 The role of memory in cognition ... 17

2.2.1 Sensory memory ... 18

2.2.2 Working memory (short-term memory) ... 19

2.2.3 Long-term memory ... 20

2.3 Sources of cognitive load during learning ... 21

2.3.1 Intrinsic cognitive load ... 22

2.3.2 Extraneous cognitive load ... 24

2.3.3 Germane cognitive load ... 28

2.4 Adding sources of cognitive load (intrinsic, extraneous and germane) ... 29

2.5 Measuring cognitive load ... 31

2.6 Cognitive load theory and instructional design ... 32

3. MULTIPLE REPRESENTATIONS ... 33

3.1 Mathematical representations ... 33

3.2 Design of multiple representation systems ... 35

3.3 Classification of representations ... 36

3.4 Visual representations ... 38

vii

3.6 A concern regarding multiple representations ... 41

3.7 Importance of multiple representations in learning (cognition) and teaching ... 41

4. FRAMING THIS STUDY ... 43

5. CHAPTER SUMMARY ... 43

CHAPTER THREE

RESEARCH DESIGN AND METHODOLOGY

2.5 Matching paradigms, methods and tools ... 51

3. MIXED-METHODS ... 52

3.1 Rationale for using a mixed-method approach ... 56

3.2 Challenges to the mixed-methods approach ... 57

3.3 Mixed-method designs ... 58

3.4 Mixed-method procedures ... 59

4. RESEARCH DESIGN ... 61

5. METHODOLOGY ... 65

viii

5.2 Research instruments ... 69

5.2.1 Questionnaires and pre-, intervention and post-tests ... 69

5.2.2 Learner focus group interviews and semi-structured, open-ended teacher interviews ... 71

5.3 Data generation ... 73

5.4 Data analysis ... 74

5.5 Reliability and validity/trustworthiness ... 78

5.6 Ethical issues ... 80 6. CHAPTER SUMMARY ... 81

CHAPTER FOUR

RESULTS

3.3. Home language distribution ... 85

3.4. Distribution of learners’ - first, second or third time in grade 9 ... 86

3.5 Distribution of learners’ mathematics marks at the end of grade 8 ... 87

4. QUANTITATIVE RESULTS: PRE-, INTERVENTION AND POST-TESTS/QUESTIONNAIRES ... 87

ix

4.1 Overall results from the four participating schools ... 89

4.1.1 Average test results ... 89

4.1.2 Test results per school ... 90

4.1.3 Comparison between schools by test ... 93

4.1.4 Differences in terms of gender ... 95

4.1.5 Differences in terms of age ... 95

4.1.6 Differences in terms of home language ... 98

4.1.7 Learners’ representation preferences ... 98

5. QUESTIONNAIRE RESULTS ... 101

5.1 Tests ... 101

5.1.1 Pre-Test ... 101

5.1.2 Intervention test ... 102

5.1.3 Post-Test ... 106

5.2 Learner focus group interviews ... 108

5.3 Teacher interviews ... 112

5.3.1 Teachers’ understandings and views pertaining to mathematics and cognitive load ... 113

5.3.2 Teachers’ understandings and views on multiple representations ... 117

5.3.3 Teachers’ views on the pre-, intervention and post-tests... 120

x

CHAPTER FIVE

DISCUSSION, CONCLUSIONS AND RECOMMENDATIONS

1. INTRODUCTION ... 124

2. MULTIPLE REPRESENTATIONS AND LEARNER ACHIEVEMENT ... 125

3. LEARNERS’ BELIEFS REGARDING MULTIPLE REPRESENTATIONS AND COGNITIVE LOAD... 129

4. LEARNERS’ MULTIPLE REPRESENTATIONAL PREFERENCES ... 133

5. DIFFERENCES IN TERMS OF GENDER, AGE AND HOME LANGUAGE ... 136

6. TEACHERS’ UNDERSTANDINGS AND BELIEFS AROUND COGNITIVE LOAD AND MULTIPLE REPRESENTATIONS ... 138

7. MAIN FINDINGS ... 144

8. LIMITATIONS OF THE STUDY... 147

9. RECOMMENDATIONS ... 147

9.1 Recommendations for further research ... 148

9.2 Recommendations for curriculum and materials developers, subject advisors and teachers ... 149

10. CONCLUSION ... 149

xi

LIST OF APPENDICES

APPENDIX A: PRE-TEST... 168

APPENDIX B: INTERVENTION ... 170

APPENDIX C: POST-TEST ... 172

APPENDIX D: TEACHER INTERVIEW QUESTIONS ... 173

APPENDIX E: APPROVAL TO CONDUCT RESEARCH ... 174

APPENDIX F: LETTER OF INVITATION TO SCHOOL PRINCIPALS / SCHOOL PRINCIPAL CONSENT FORM ... 175

APPENDIX G: TEACHERS’ INFORMATION AND INFORMED CONSENT FORM . 178 APPENDIX H: ETHICS APROVAL ... 181

xii

LIST OF FIGURES

Figure 2.1: Pictorial representation of Atkinson and Shiffrin “modal model” which

consists of three distinct memory types, namely, the sensory memory, the working memory (short-term memory) and the long-term memory (Cooper, 1998, p. 4) ... 18 Figure 2.2: The picture above is an example of an image that illustrates the sensory

memory retention. If one were to shut one’s eyes whilst looking at the above image, the image will still remain somewhere in the mind for a split second before disappearing (sensory memories disappear very quickly) (Cooper, 1998, p. 4-5) ... 19 Figure 2.3: Visual representations illustrating (i) a low intrinsic cognitive load with a high

extraneous cognitive load, (ii) a high intrinsic cognitive load with a high extraneous cognitive load, and (ii) a high intrinsic cognitive load with a low extraneous cognitive load. The thick, bold arrows indicate the mental resources required (working memory capacity) by learners. The two thick, bold arrows (i) and (iii) that extend beyond the image indicates learning has occurred whereas the thick, bold arrow (ii) which falls short indicates learning has been hampered (Cooper, 1998, p. 14-15) ... 30 Figure 2.4: A pictorial representation of the relationship between internal and external

representations in developing a child’s understanding of the concept of numeracy (Pape & Tchoshanov, 2001, p. 119) ... 38 Figure 3.1: The framework for the explanatory sequential mixed-methods design depicting

the sequence researchers adhere to (Creswell & Plano Clark, 2011, p. 69) . 59 Figure 4.1: Graphical representation of the distribution of learners by participating

xiii

Figure 4.2: Average test results in terms of learner achievement from the participating schools as a percentage (n = 673)………... 90

Figure 4.3: Test results in terms of learner achievement for each of the schools as a percentage (n = 673)………... 91

Figure 4.4: A comparison between schools for each of the tests as a percentage (n = 673)……….... 93

Figure 4.5: Differences in terms of gender for each of the tests in terms of learner achievement from the four participating schools as a percentage (n = 669 out of 673; four learners did not indicate their gender on the tests/questionnaires)……… 96

Figure 4.6: Differences in terms of age for each of the tests in terms of learner achievement from the four participating schools as a percentage (n = 516 out of 673; one hundred and fifty seven learners did not indicate their age on the tests/questionnaires)………..………...…...… 97

Figure 4.7: Differences in terms of home language for each of the tests in terms of learner achievement from the four participating schools as a percentage (n = 652 out of 673; nine learners did not indicate their home language on the tests/questionnaires and 12 learners fell into the other languages category; English; n = 464, Afrikaans: n = 93, and isiXhosa: n = 95)………..… 99

Figure 4.8: Learners preferences for the various representations (words, arrows, colours) expressed as a percentage of those who responded)……….… 100

xiv

LIST OF TABLES

Table 3.1:

A table depicting three paradigms (positivist, interpretivist and pragmatic) with their methods (quantitative or qualitative) and data collection tools (Adapted from Mackenzie & Knipe, 2006) ... 52 Table 3.2:

A comparison between quantitative and qualitative research approaches (Adapted from du Plessis & Majam, 2010) ... 54 Table 3.3

Sequential design of the study in terms of activities, products, data generated and data analysis ... 64 Table 3.4:

A comparison between quantitative and qualitative research approaches (Adapted from Creswell & Plano Clark, 2011, p. 205-206) ... 75 Table 4.1

Gender distribution of learners per school as a number and percentage (n = 669 out of 673; four learners did not indicate their gender on the tests/questionnaires) ... 84 Table 4.2

Age distribution of learners per school as a number and percentage (n = 516 out of 673; one hundred and fifty seven learners did not indicate their age on the tests/questionnaires) ... 85 Table 4.3

Home language distribution of learners per school as a number and percentage (n = 664 out of 673; nine learners did not indicate their home language on the tests/questionnaires) ... 85

xv Table 4.4

Distribution of learners first, second or third time in grade 9 as a number and percentage (n = 667 out of 673; six learners did not indicate whether it was their first, second or third time in grade 9 on the tests/questionnaires) ... 86 Table 4.5

Distribution of learners’ mathematics marks into categories at the end of grade 8 per school as a number and percentage (n = 666 out of 673; seven learners did not indicate their grade 8 percentage on the tests/questionnaires) ... 87 Table 4.6

Inferential statistics indicating the change in score from pre-test to intervention, intervention to post-test 1, and post-test 1 to post-test 2 ... 92

Table 4.7

Distribution of learners in terms of achievement who fell within the four categories (0-40%, 40-60%, 60-80% and 80-100% per school and per test)... 94 Table 4.8

Learners’ responses as to whether they found the six algebraic problems in the pre-test difficult to complete or not, per school as a number and percentage (n = 669 out of 673; four learners did not indicate whether they found it difficult or not on the tests/questionnaires) 101 Table 4.9

Learners’ responses as to whether they found the six algebraic problems in the intervention test difficult to complete or not, per school as a number and percentage (n = 669 out of 673; four learners did not indicate whether they found it difficult or not on the tests/questionnaires) ... 103 Table 4.10

Learners’ responses as to whether they found the eight algebraic problems in the post-test difficult to complete or not, per school as a number and percentage (n = 667 out of 673; six learners did not indicate whether they found it difficult or not on the tests/questionnaires . 106 Table: 4.11

xvi Table 5.1:

Summary of the main findings of the study in terms of achievement; reduction of cognitive load; learner preferences; variables in gender, age and home language; and teachers’ understandings of multiple representations and cognitive load ... 146

1

CHAPTER ONE

INTRODUCTION

1. INTRODUCTION

Mathematics has long been seen as a subject that is difficult to understand (Ali & Reid, 2012) and in which the attainment of educational goals are often not met (Mwakapenda, 2004). Studies also reveal that many learners perceive mathematics as abstract and struggle with the concepts (Ali & Reid, 2012; Brown, Brown, & Bibby, 2008). Researchers such as Raiker (2002) point out that an important objective of any mathematics lesson is the acquisition of mathematical concepts, and most curricula worldwide concur with the expectations of the South African curriculum that the teaching and learning of mathematics is supposed to enable learners to “develop deep conceptual understandings in order to make sense of mathematics” (Department of Education, 2002, p. 5). However, the mental activities required for acquiring deep conceptual understanding may impose a cognitive load which exceeds an individual’s currently available working memory capacity. In such cases learners resort to the memorisation of procedures and facts, while understanding becomes a casualty of the process (Ali & Reid, 2012).

Instructional strategies and designs to assist learners to develop conceptual understanding and lower the cognitive load associated with mathematics are continuously being researched. de Jong (2010, p. 105) recommends the “design of such instructional systems which optimize the use of the working memory capacity and avoid excess cognitive load”. According to Ainsworth (2006), an instructional strategy gaining much attention is the usage of multiple representations in the teaching and learning of mathematics. Strategies

2

which focus on learning by using more than one representation are underpinned by the idea that “two representations are better than one” (Ainsworth, 2006, p. 183). The notions of cognitive load and multiple representations in mathematics are the theoretical constructs which frame this study and are touched on briefly below.

The overall aim of this study is to explore whether selected multiple representations (words, arrows and colours) affect achievement when solving algebraic problems (products of monomials and binomials/trinomials) by grade nine level learners in selected South African schools. Their preferences for the different types of representations offered, why they preferred those representations, and whether they believe that the representations lower cognitive load (make the algebraic problems easier to understand and solve) are explored.

2. COGNITIVE LOAD THEORY

de Jong (2010, p. 105) explains cognitive load theory as follows; “the basic idea of cognitive load theory is that cognitive capacity in working memory is limited, so that if a learning task requires too much capacity, learning will be hampered. The recommended remedy is to design instructional systems that optimize the use of working memory capacity and avoid cognitive overload”. As mathematics comprises of symbols, pictures, words, numbers, notations, terminologies, definitions, concepts, theorems, as well as graphical and abstract representations, and a number of these aspects can be found in a single mathematics lesson, acquiring comprehension and understanding may be intimidating and overwhelming for learners (Orr & Schutte, 1992).

Cognitive load theory provides some guidelines for increasing learner performance. It aims at designing instructions that do not overburden a learner’s working memory capacity

3

(Chandler & Sweller, 1992; Sweller, van Merrienboer, & Paas, 1998). Three types of cognitive load are distinguished within cognitive load theory (Ayres, 2006a; de Jong, 2010):

 Intrinsic cognitive load, which relates to the difficulty of the subject matter,  Extraneous cognitive load, which refers to the way that instructional material

and information is presented externally, and

 Germane cognitive load, which is concerned with the construction and formation of schemas.

An educator’s task is to consider ways in which to make complex tasks easier to learn and facilitate learning in such a way that the cognitive load is kept at a manageable level (Ayres, 2006a). In this study these three aspects (intrinsic, extraneous and germane cognitive load) will be taken into consideration when investigating the effects of selected multiple representations on learner achievement, perceptions of effect on cognitive load, and learners’ preferences for selected multiple representations to the tasks undertaken.

3. MULTIPLE REPRESENTATIONS

Research in mathematics, physics and biology education indicates that the design and usage of multiple representations is gaining recognition (Ainsworth, 2006) and that multiple representational learning environments are now commonplace in schools (Ainsworth, Bibby, & Wood, 2002). Ainsworth (1999) maintains that the use of more than one representation is more likely to capture a learner’s interest and therefore plays an important part of effective learning. Multiple representations can also aid teachers when they trying to make abstract concepts more concrete (Pape & Tchoshanov, 2001). Kaput (1992) proposed that multiple representations allows learners to perceive complex ideas in a new way and apply them more

4

effectively. In these ways multiple representations support cognitive processes in learning and help develop deeper understanding of the subject being taught.

These findings have been taken up by the National Council of Teachers of Mathematics (NCTM, 2000, p. 67) where it is stated, “Representations should be treated as essential elements in supporting students’ understanding of mathematical concepts and relationships”. However, research has also shown that while affording learners an opportunity to interact with an appropriate representation enhances performance (Ainsworth, 2006), learners have different learning styles and they should be presented with a choice of representations, from which they can choose the representation that best suits their learning style (Dunn & Dunn, 1993). In this study three aspects of representations are used, namely words (explanation), arrows and colours.

4. PROBLEM STATEMENT

Learners’ underachievement in mathematics and science education has been documented for decades both internationally (Driver, Guesne, & Tiebergien, 1995) and nationally (de Lange, 1981). The poor performance of South African schools when compared to both developed and developing countries is also well known (Fleisch, 2008; Taylor, 2008). National achievement in the Trends in International Mathematics and Science Study (TIMMS) tests has been disappointing (Howie, 2004). South Africa has had the lowest (or near lowest, depending on which year) performance in mathematics and science of the 50 participating countries (Reddy, 2006). Within South Africa, the Eastern Cape Province and Limpopo rated as the lowest performing provinces (Reddy, 2006).

The South African Annual National Assessment (ANA) requires all schools to conduct the same grade-specific Language and Mathematics tests for Grades 1 to 6 and Grade

5

9 (Department of Education, 2012). The 2012 ANAs reveal that performance in mathematics decreases from Grade 1 (68%) to Grade 9 (13%). During my personal experience in my grade 9 mathematics lessons at a secondary school in the Eastern Cape over a year, learners were often heard saying, “I don’t understand”, “maths is too difficult” or “there are too many things to remember in maths.” The worryingly low average mark of 13% at Grade 9 level (Department of Education, 2012), as well as the fact that the algebraic problems (researched in this study) form the basis of algebra motivated this study which aims at exploring whether access to mathematical multiple representations when presented with algebraic problems is perceived to reduce the cognitive load and lead to better mathematical achievement in terms of the problems presented at grade nine level.

5. RESEARCH OBJECTIVES

Primary Objective

The primary objective of this study is to investigate the possible effects that access to selected multiple representations have in terms of cognitive load and learner achievement when learners are presented with algebraic problems at grade 9 level.

Secondary Objectives

The secondary objectives of the study are:

 To determine learner achievement before and after introduction to the selected multiple representations

 To investigate learners’ perceptions on the effect multiple representations have on the cognitive load associated with the problems presented

 To investigate learner preferences in terms of the selected multiple representations presented

6

 To investigate whether factors such as gender, age, and home language, are associated with learner achievement when presented with the selected multiple representations

 To investigate teachers’ understandings and beliefs around cognitive load and the use of multiple representations in mathematics education.

6. RESEARCH QUESTIONS

There are a number of sub-questions that have to be answered in order to adequately answer the principal question asked in this study. These questions, and the principal question, are posed below:

Principal research question:

 Does access to selected multiple representations in algebraic problems lower the cognitive load and enhance learner achievement at grade 9 level?

Sub-Questions:

The sub-questions that are required to be answered to answer the principal research question are:

 Does the provision of the selected mathematical multiple representations result in improved achievement by learners?

 Do the learners believe that mathematical multiple representations reduced the cognitive load by making the problems more accessible and understandable?

 Do the learners have preferences in terms of the representations presented?

 Are there any differences in responses in terms of gender, age, and home language on learner achievement when presented with the selected multiple representations?

7

 What are the teachers’ understandings and beliefs around cognitive load and the use of multiple representations in mathematics education?

7. DESIGN AND METHODOLOGY

The research was conducted within an interpretivist paradigm using a mixed method approach within a pre-post-test quasi-experimental design. Literature on multiple representations and cognitive load was researched and used to design a questionnaire (mostly quantitative data) of three parts (pre-test, intervention and post-test). The questionnaire included demographic details, algebraic problems, worked examples exemplified by three representations (arrows, words and colours), and close ended questions (Appendices A, B & C).

The design is quasi-experimental in that it involves pre- and post-testing with an intervention in between (presentation of selected representations). The design is not ‘experimental’ in its true form as it lacks a control group and random selection of the participants, issues which were not deemed necessary for the type of information that was sought. The three sets of algebraic problems were designed in such a way that the pre-test determined learners’ baseline knowledge. The introduction of multiple representations related to the problems to be solved formed the intervention and a third set of algebraic problems in the post-test was be used to determine whether there were any changes in terms of achievement after the intervention. All of the problems presented, although similar, are unique. Qualitative data was generated through learner focus group and teacher interviews. The focus group and teacher interview questions were designed to match the test/questionnaire and probe areas where further clarification was required.

8

7.1 Sample and setting

The research was conducted with grade 9 learners from four secondary schools situated within Port Elizabeth in the Eastern Cape, South Africa, in which the language of learning and teaching (LoLT) is English. While the majority of schools in the Eastern Cape are described as ‘dysfunctional (Fleisch, 2008), functional government schools with a diverse learner population were chosen for this study (purposive sampling) as such schools offer a better chance of successfully determining the factors associated with multiple representations and cognitive load in mathematics than in schools where there may be a number of confounding factors such as learning and teaching taking place in a second language (a common situation in South Africa in general, and in the Eastern Cape in particular), administrative dysfunctionality, teachers who are not qualified to teach the subject, etc.

As noted above, the participants who completed the test/questionnaire were not randomly sampled (whole classes were used), nor was there a control group as the main aims were to determine responses and possible differences between responses in terms of gender, age, home language, in terms of cognitive load, learner preferences for selected multiple representations and learner achievement. Four teachers who are currently the Grade 9 head teachers at the four participating schools were interviewed. These teachers were chosen because of their seniority and experience. The learners’ ages ranged between 13-16 years.

7.2 Research instruments

The research instruments included a questionnaire which provided both quantitative and qualitative data and teacher interview and learner focus groups protocols which enabled the generation of qualitative data. The questionnaires included a biographical section, three sets of algebraic problems as well as structured questions which required both closed and open ended responses. The semi-structured, open-ended interview and focus group questions

9

were informed by the data generated by the questionnaire and allowed further probing of issues which arose from these data.

7.3 Data collection

Quantitative data were gathered through pen and paper pre-, intervention and post-tests. Teachers from the participating schools were tasked with the completion of these questionnaires by the learners. The researcher availed herself during this time and answered any queries that arose. The purpose of the pre-test was to determine learners’ baseline knowledge before the intervention in terms of multiple representations whilst the purpose of the post-test was to determine whether any changes occurred in terms of achievement after the intervention. These questionnaires were completed within a single class period (approximately 30 minutes)

Learner focus groups and semi-structured, open-ended teacher interviews were voice recorded and the teachers were tasked with questioning the learners in their focus groups. In actuality, the teachers requested that the researcher conduct the interview, which she did. All participants were asked the same questions in the same order and all of the interviews were conducted in English.

7.4 Data analysis

The questionnaires (quantitative data) were scored and entered into an excel sheet and then sent to the Nelson Mandela Metrolpolitan University’s Statistical Services. The data generated from the sample of 673 learners was normally distributed which enabled parametric analyses to be undertaken such as t-tests and analysis of variance (ANOVA). Non-parametric measures were made of the distribution of scores within the schools using χ2

.

The interviews and focus group data were inspected, categorised and analysed thematically according to the Tesch method (Creswell, 2005). The Tesch method involves

10

careful reading of the transcriptions several times, making note of key ideas that emerge. A code word or phrase is then assigned to quotes. Similar codes are then categorised into main themes and sub-themes (Creswell, 2005). The interview and learner focus group questions afforded insight into the teachers and learners thoughts, understandings and views regarding the teaching and learning of algebraic problems using selected multiple representations, their perceptions on the effect multiple representations have on the cognitive load, and their preferences in terms of the selected multiple representations.

7.5 Reliability & Validity

The reliability of qualitative data refers to the trustworthiness, credibility and accuracy of the findings and interpretations (Golafshani, 2003). Creswell (2005) proposes triangulation to validate research findings. Triangulation of data generated was achieved by corroborating the different data collection methods intended, namely tests/questionnaires, focus groups and teacher interviews.

The reliability of quantitative data means that the results/scores obtained from the instrument administered are accurate, stable and consistent (Creswell, 2005; Struwig & Stead, 2001). It also suggests that the same or nearly the same results will be achieved if the same instrument is repeated at different times (Creswell, 2005) and is usually indicated by statistical analysis. The statistical reliability in this study was calculated using the Cronbach



The validity of data relies largely on the instruments used, such that the instrument measures what it set out to measure and the researcher is then able to draw conclusions from the data generated (Creswell, 2005; Struwig & Stead, 2001). The data generating instrument used in this study consisted of standard tests of problems comprising three sets of algebraic

11

problems from school text books and past examination papers. As such, the examples were considered to have construct and face validity.

8. ETHICS

Written permission was obtained from the Eastern Cape Education Department to undertake this research (Appendix E). Informed consent was solicited from the principals and teachers from the participating schools (Appendix F). Informed consent was also obtained from the teachers who participated in the interviews (Appendix G). The research plan was submitted to the Nelson Mandela Metropolitan University faculty of Education Human Ethics Committee and was approved (H13-EDU-ERE-009) (Appendix H).

9. OUTLINE OF THE STUDY

This study is described in six chapters. Chapter one provides a general introduction and orientation of the study and introduces the issues of cognitive load and multiple representations. In terms of background, chapter one also notes the status of mathematics and grade nine learners’ mathematical achievement within a South African context. The research problem is formulated, why the study was conducted is explained, the research question, design and methodology, ethical issues are presented, and an outline of the study is described.

Chapter two provides a literature review and theoretical framework for this study focusing on issues of cognitive load theory, multiple representations as well as the role that multiple representations can play in lowering the cognitive load associated with mathematics, particularly algebraic problems. Chapter three explains the methodological framework and methods which were adopted when collecting and analysing the data. The sample, data

12

gathering instruments, as well as issues pertaining to validity and reliability of the research process and procedures are also described.

Chapter four focuses on the results obtained from the study. These results are discussed in chapter five in light of the literature review in chapter two. The main conclusions drawn from this study, their implications, and recommendations for further research are also argued in this chapter.

13

CHAPTER TWO

LITERATURE REVIEW

1. INTRODUCTION

This chapter provides an overview of two main issues, namely, cognitive load theory and multiple representations, which form the theoretical framework of this study. Relevant literature pertaining to cognitive load theory and multiple representations are reviewed and discussed in detail. Topics relating to cognitive load theory such as human cognitive architecture, the role of memory in cognition, sensory memory, working memory and long-term memory are considered. Sources of cognitive load during learning, namely intrinsic cognitive load, extraneous cognitive load, and germane cognitive load are interrogated. Relationships between cognitive load theory and instructional design are explored. The additive nature of cognitive load is recognised and the measurement of cognitive load is examined. Issues pertaining to multiple representations which include mathematical representations, design and classification of representations, visual and verbal representations, concerns regarding representations, as well as the importance of representations in teaching and learning are highlighted. Finally, it is noted that these theoretical frameworks not only provide an explanatory purpose, but shape the conceptual framework within which this study is designed.

14

2. COGNITIVE LOAD THEORY

Cognitive Science is a discipline which deals with the mental processes of learning, memory, and problem solving (Cooper, 1998). It is from this field of research that cognitive load theory (CLT) originated in the early 1980s (van Merriënboer & Sweller, 2005) and underwent substantial development by researchers from around the world in the 1990s. Internationally, it continues to play a central role in educational and psychological research (Paas, Renkl, & Sweller, 2003).

According to Sweller et al. (1998) cognitive load theory is based on cognitive theories of human cognitive architecture. Some aspects of human cognitive architecture include working memory, long-term memory, schema automation and schema construction. Cognitive load theory holds that human cognitive architecture consists of a limited working memory with partly independent processing units for visual and audio information, which interacts with an unlimited long-term memory. The working memory processes and constructs information which is stored in the long-term memory in the form of cognitive schemas (Beers, Boshuizen, Kirschner, Gijselaers, & Westendorp, 2008).

Ayres (2006a) clarifies that the basic tenet of cognitive load theory is that interactions between working memory and long-term memory play a significant role in learning. Cognitive load theory provides guidelines to assist in the presentation of information in such a manner that encourages learner activities so that their intellectual performance is optimised (Sweller et al., 1998). In short, cognitive load theory relates working memory characteristics and the design of instructional systems and assert that learning is hampered when the working memory capacity is exceeded (de Jong, 2010).

15

In order to understand cognitive load theory, human cognitive architecture and the role that memory plays in cognition (learning, memory and problem solving) is discussed below.

2.1 Human cognitive architecture

Torcasio and Sweller (2010) contend that there are two kinds of knowledge; biologically primary knowledge and biologically secondary knowledge. Biologically primary knowledge is specific knowledge that humans acquire which is not explicitly taught. Examples of these are learning to listen and speak which are acquired by mere immersion in society. In contrast, biologically secondary knowledge is explicitly taught and this knowledge is acquired through instruction. Examples of these include learning to read and write.

Geary (2007) maintains that human cognitive architecture applies to biologically secondary knowledge. Leahy and Sweller (2008), Sweller (2009) and Torcasio and Sweller (2010) explain that human cognition is a natural information processing system and this system is based on five principles which provide a base for cognitive load theory. These principles include:

(i) Long-term memory and information store principle: Human activity is driven by a huge amount of information stored in the long-term memory. A primary aim of instruction is the accumulation of knowledge in the long-term memory.

(ii) Schema acquisition and the borrowing and reorganising principle: Most of the information in the long-term memory is borrowed from other people; by imitating them, listening to what they say or reading what they write. This process is assumed to be constructive because before the information is stored, the learner transforms and reorganises the information presented with previous information in the long-term

16

memory into new schemas. Hence, this schema acquisition and the borrowing and reorganising principle do not create new information.

(iii) Problem solving and the randomness of genesis principle: Novel information is created during problem solving because when a learner is presented with a problem he/she has to randomly generate problem solving moves and then test these moves for effectiveness. The effective moves are then retained in the long-term memory and the ineffective moves are discarded.

(iv) Working memory and the narrow limits of change principle: The working memory, which processes new information, is limited in capacity and duration. It can only deal with small amounts of novel information when generating a hypothesis or testing its effectiveness at a given time.

(v) Environmental organising and linking principle: The previous four principles permit information processing within its environment. However, this fifth principle links the environment to the term memory by linking the working memory to the long-term memory. The working memory has no limits when dealing with previously organised information from the long-term memory. Huge amounts of schematic information from the long-term memory can be transferred to the working memory in order to generate actions required by complex environments.

(Leahy & Sweller, 2008; Sweller, 2009; Torcasio & Sweller, 2010).

The five principles of human cognitive architecture mentioned above which provide the base for cognitive load theory are encapsulated in the following sentence by Leahy and Sweller (2008, p. 274) “the dual nature of working memory, very limited when dealing with novel information from the environment, but unlimited when dealing with organised

17

information from long-term memory, is central to cognitive load theory”. Research based on this human cognitive architecture shows that the function of learning is to ultimately store information in the long-term memory. If nothing has been stored in the long-term memory, nothing has been learnt (Torcasio & Sweller, 2010).

2.2 The role of memory in cognition

According to Roediger (2008), the field of memory theory is so complex that even experts cannot easily summarise it. Memory theory clarifies how the human memory works and how new information is absorbed (Miller, 2011). In the late 1960s, Atkinson and Shiffrin developed the ‘modal model’ which theorised that the memory consists of three distinct memory types (modes), namely, the sensory memory, the working memory (short-term memory), and the long-term memory. Each mode has a separate function with its own characteristics and limitations, but work in concert to process information (Kirschner, Sweller, & Clark, 2006). The Atkinson and Shiffrin ‘modal model’ of memory, which links these functions, is depicted in figure 2.1.

18

Figure 2.1: Pictorial representation of Atkinson and Shiffrin “modal model” which consists

of three distinct memory types, namely, the sensory memory, the working memory (short-term memory) and the long-term memory (Cooper, 1998, p. 4).

2.2.1 Sensory memory

The sensory mode of memory deals with incoming stimuli from the human sense organs. It holds information for a very short time, approximately 1-5 seconds for visual and auditory information. Sensory memories disappear very quickly (Miller, 2011). (Cooper, 1998) elucidates this by means of the following explanation for the picture below. If one were to shut one’s eyes whilst looking at figure 2.2, you will still notice an image of the picture

19

remaining for a split second somewhere in the mind. This demonstrates the working of the sensory memory that is responsible for visual perceptions.

Figure 2.2: The picture above is an example of an image that illustrates the sensory memory retention. If one were to shut one’s eyes whilst looking at the above image, the image will still remain somewhere in the mind for a split second before disappearing (sensory memories disappear very quickly) (Cooper, 1998, p. 4-5).

2.2.2 Working memory (short-term memory)

Working memory is also known as short-term memory and can be equated with consciousness. It enables humans to think, to reason and to solve problems (Cooper, 1998). This mode stores small amounts of information for a very short duration (de Jong, 2010). Working memory is limited. It can hold no more than five to nine elements of information and actively process no more than two to four elements simultaneously. It is only able to deal with information for a few seconds and within twenty seconds all information is lost unless it is refreshed by rehearsal (van Merriënboer & Sweller, 2010). Sweller et al. (1998) found that this mode of memory is responsible for the processing of information by identifying, comprehending, organising, classifying, combining, comparing, or contrasting. Novel

20

information is processed in the working memory in order to construct and automate long-term cognitive schemas in the long-long-term memory (Torcasio & Sweller, 2010). Cowan (2008) adds that working memory is also used to plan and carry out behaviours such as solving an arithmetic problem without a pen and paper or baking a cake without making the mistake of adding the same ingredient twice.

According to Friso-van den Bos, van der Ven, Kroesbergen, and van Luit (2013), Baddeley and Hitch (1974) proposed a three-component model of working memory, each with its own limited capacity. These include:

(i) the phonological loop which processes verbal information. Spoken words enter the phonological store directly whereas written words are first converted from a visual code into an articulatory code and then transferred to the phonological loop,

(ii) the visuo-spatial sketchpad which processes, maintains and manipulates visual and spatial information. It deals with characteristics of objects like shape and colour and is responsible for picture processing, and

(iii) the central executive which monitors and coordinates the phonological loop and the visuo-spatial sketchpad and then links them to long-term memory.

Schüler, Scheiter, and van Genuchten (2011).

2.2.3 Long-term memory

Long-term memory has the ability to store unlimited amounts of information permanently (Cowan, 2008). The information stored here does not only consist of small, isolated facts but also includes large, complex interactions and procedures (Sweller et al., 1998). Examples of these include; our names, date of birth, how to read, how to write, and

21

anything else that we ‘know’ (Cooper, 1998). Miller (2011) maintains that the information in this mode cannot be used to perform a task unless it is activated. It must first be retrieved and reloaded into the working memory in order to carry out a specific task. The working memory has no limitations when dealing with information that has been retrieved from the long-term memory.

van Merriënboer and Sweller (2005) confirm that the knowledge stored in the long-term memory is stored in the form of cognitive schemata. These schemata organise, categorise and store information according to how it will be used (van Merriënboer & Sweller, 2010). Paas et al. (2003. p. 2) define schemata as “cognitive constructs that in- corporate multiple elements of information into a single element with a specific function”.

Constructed schemata may become automated if they are repeatedly applied and will lead to a reduced working memory load (van Merriënboer & Sweller, 2005). Sweller et al. (1998) clarify automation by providing the example of adults who read with minimal conscious effort (with minimal working memory load) because they do not have to process each letter which makes up the prose being read. This is because the procedures involved in reading the letters were automated in childhood. In contrast, a young learner just learning to read must consciously process each letter and this poses a working memory load. In summary, schema construction aids in the storage and organisation of information in long-term memory and reduces working memory load. Both schema acquisition and schema automation are essential for effective learning (Kirschner, 2002).

2.3 Sources of cognitive load during learning

Ayres (2006b) and de Jong (2010) state that cognitive load theory researchers have identified three sources of cognitive load that occur during learning, namely, intrinsic

22

cognitive load, extraneous cognitive load and germane cognitive load. Each of these loads is discussed in more detail below.

2.3.1 Intrinsic cognitive load

According to Sweller (2010, p. 124) “intrinsic cognitive load is concerned with the natural complexity of information that must be understood and material that must be learned, unencumbered by instructional issues such as how the information should be presented or in what activities learners should engage to maximise learning”. Intrinsic cognitive load considers the prior knowledge a learner brings to the task (Beers et al., 2008). van Merriënboer and Sweller (2010) add that intrinsic cognitive load is fixed and is dependent on the nature of what is to be learned (e.g. simplification) or by the act of learning itself.

Paas et al. (2003) argue that the main generator of intrinsic cognitive load is element interactivity. In other words, in order to determine the level of intrinsic cognitive load that a particular task or knowledge level may pose, the level of element interactivity must first be determined. Learning materials differ in their levels of element interactivity and thus in their intrinsic load. An element refers to “anything that has been or needs to be learned, most frequently a schema, such as a concept or procedure” (Sweller, 2010, p. 124). Ayres (2006b) suggest that element interactivity refers to the way in which individual elements of a task interact with each other.

Tasks may either contain a small number of interactive elements (low interactivity) or a large number of interactive elements (high interactivity) (de Jong, 2010). Low interactivity tasks consist of single, simple elements and can be learnt in isolation. They can be learnt serially rather than simultaneously and require a relatively low working memory load. In contrast, high interactivity tasks consist of individual elements which can only be well understood in relation to other elements. Several elements in the working memory must be

23

manipulated simultaneously and therefore requires a higher working memory load. Consequently, a task is termed difficult not because of the number of elements the task contains, but because of the number of elements that must be simultaneously assimilated (Ayres, 2006b; Sweller et al., 1998).

Pollock, Chandler and Sweller (2002) and Sweller et al. (1998) distinguish between low and high interactivity by providing the examples of vocabulary learning and grammatical syntax. When learning vocabulary, words can be learnt independently of each other as an instance of low element interactivity, which imposes a low intrinsic cognitive load because it is easy to understand and learn. When learning grammatical syntax, an instance of high element interactivity, elements cannot be learnt independently because the elements interact. This imposes a high intrinsic cognitive load because it is not easy to understand and learn.

Certain learning tasks, such as mathematical tasks tend to be high in element interactivity (Sweller et al., 1998). In a more recent paper, Sweller (2010) explains by means of the following algebraic equation (a + b)/c = d where a learner is asked to solve for a. Each symbol in the equation acts as an element. The learner is required to process each element simultaneously for the equation to be understood. Some learners may find the intrinsic cognitive load overwhelming and resort to memorisation at the expense of mathematical understanding. However, it is not possible to gauge the levels of element interactivity by merely analysing the learning material. High element interactivity for one learner might pose as low element interactivity for another more experienced learner who has already constructed or automated the schema (Sweller et al., 1998).

Apart from element interactivity, another important premise governing intrinsic cognitive load is that it cannot be changed by instructional designs. As mentioned earlier “intrinsic cognitive load is … unencumbered by instructional issues” (Sweller, 2010, p. 124).

24

Both arguments by Ayres (2006a, p. 389) “intrinsic cognitive load is fixed and innate to the task” and Paas et al. (2003, p. 1) “Different materials differ in their levels of element interactivity and thus intrinsic cognitive load, and they cannot be altered by instructional manipulations” concur with Sweller (2010).

Recently a somewhat different stance has been taken by others who claim that intrinsic cognitive load can be managed. Four instructional designs have been noted; van Merrienboer, Kirschner, and Kester (2003) suggest sequencing the material in a simple-to complex order so as to reduce the complexity of the material. Pollock et al. (2002) and Ayres (2006b) recommend introducing isolated elements of information that could be processed serially and then the integrated task. Gerjets, Scheiter, and Catrambone (2004) advocate the part-whole sequencing modular approach where solutions are subdivided into smaller units. van Merriënboer, Kester, and Paas (2006) also propose the whole-part approach where the learning material is presented in its full complexity from the start, but the learners’ attention is focused on particular subsets of interacting elements.

Sweller et al. (1998) infer that element interactivity, learners’ expertise, and instructional design all contribute to the level of intrinsic cognitive load. From a learner’s perspective, intrinsic cognitive load is directly related to, and can be added to, extraneous cognitive load.

2.3.2 Extraneous cognitive load

According to Sweller et al. (1998, p. 262) “extraneous cognitive load provides the core of cognitive load theory and is under the direct control of instructional designers”. de Jong (2010, p. 108) adds “extraneous cognitive load is cognitive load that is evoked by the instructional material and does not directly contribute to learning”. The above definitions are encapsulated by van Merriënboer and Sweller (2005, p. 150) “extraneous cognitive load … is

25

load that is not necessary for learning (i.e., schema construction and automation) and that can be altered by instructional interventions”.

These definitions clarify that extraneous cognitive load is the load imposed by non-optimal designed instructional materials (Sweller, 2010) which could have been avoided with a different design (de Jong, 2010). Some instructional designs involve cognitive activities that involve a heavy working memory load at the expense of understanding. Based on the five principles of cognitive load theory, Leahy and Sweller (2011, p. 944) deduce that “information should be presented explicitly to learners in a manner that reduces any unnecessary load on working memory (extraneous cognitive load) and instead directs working memory resources to information that is intrinsic to the task at hand (intrinsic cognitive load) in order to increase the amount of information transferred to long-term memory”. The basic idea is to reduce the extraneous cognitive load by designing instructional materials optimally in order to free the working memory (Bannert, 2002). The working memory can then be devoted to schema acquisition and schema automation (Leahy & Sweller, 2008).

Based on this idea, cognitive load theorists focus on identifying instructional designs that create unnecessary working memory load and devise instructional designs that are effective or efficient (Paas et al., 2003). Sweller et al. (1998) reviewed six effects which have been shown to reduce extraneous cognitive load by assisting learners to process information in working memory for storage in long-term memory. These include: goal-free, worked example, completion problem, split-attention, modality, and redundancy effect (van Merriënboer & Sweller, 2005). For the purpose of this study, the worked example and redundancy effects will be discussed.

26

Worked example effect

Cooper (1998, p. 20) provides a concise explanation of worked examples; “worked examples are presented to students to show them directly, step by step, the procedures required to solve different problem types. Worked examples contain explicit information that equates to schemas and automation”.

The worked example effect allows learners to devote available working memory capacity to carefully studying solution procedures. This method reduces the extraneous cognitive load because learners study solutions rather than generate them (van Merriënboer & Sweller, 2010). Worked examples facilitate and foster learning and transfer when compared to problem-solving, and has also been found to be more efficient in terms of time and mental effort (van Gog, Paas, & Sweller, 2010). Sweller and Cooper's (1985) early studies found that learners who studied worked examples outperformed those who learned by solving problems. Worked examples also enable learners to induce generalised solutions or schemas (Sweller et al., 1998). Cooper (1998, p. 20) maintains that “worked examples promote the acquisition of knowledge and skills required to: identify problems as being of a particular type, recall the steps (in sequence) needed to solve each particular type, and perform each step without error”.

Worked examples have been found to facilitate learning in many subject areas. Studies were conducted by Sweller and Cooper in the 1980’s in the field of mathematics. Learners studied the use of worked examples as a substitute for conventional problem solving in learning algebra. The worked examples improved schema construction as well as the ability to solve new algebra problems (Sweller et al., 1998). Paas and van Merrienboer (1994) conducted an experiment in geometry solving in which learners only studied worked examples. This experiment also yielded lower extraneous cognitive load scores, better

27

schema construction and higher transfer performance. More recently, a study conducted by Schwonke et al. (2009) compared the effectiveness of worked examples with problem solving. Their results confirmed the robustness of the worked example effect. It was also found to develop both procedural and conceptual understanding.

The use of worked examples has its disadvantages. Smith, Ward, and Schumacher (1993) argue that excessive use of worked examples may inhibit learners from generating new, creative solutions to problems. van Merriënboer and Sweller (2010) also found that worked examples may be redundant for more knowledgeable learners and so impose an extraneous cognitive load. It is also difficult to design good worked examples and, as noted above, they may convey redundant information that does not contribute to schema construction (Paas & van Merrienboer, 1994).

Redundancy effect

The redundancy effect occurs when learners are presented with unnecessary, additional information to the task at hand (Sweller, 2010). The same information is presented in two different forms, namely, a diagram with text or listening and reading, with the one form being redundant (Torcasio & Sweller, 2010). In redundancy, either the diagram or text, or, the listening or reading, are fully intelligible in isolation because it provides all the information the learner requires (Sweller et al., 1998).

Chandler and Sweller (1991, p. 319) provide the example of a diagram shown to learners indicating the flow of blood through the heart, lungs and rest of the body. Statements are presented together with the diagram, for example, “Blood from the lungs flow into the left atrium”. The diagram includes arrows indicating that blood flows from the lungs into the left atrium. In this case of redundancy, the diagram is fully intelligible in isolation proving the redundancy of the statement.

28

According to Torcasio and Sweller (2010) and Sweller et al., (1998), the redundancy effect interferes with learning because learners are required to unnecessarily process multiple sources of information which could have been understood in isolation. This imposes a heavy working memory load (extraneous cognitive load). The working memory tries to deal with the statements and how they interact with the diagram, implying that working memory resources are unavailable to deal with the intrinsic cognitive load and so the germane cognitive load is low. Eliminating the statements means eliminating the extraneous cognitive load, which in turn decreases the working memory load and facilitates learning (germane cognitive load) (Sweller, 2010). Sweller et al., (1998) recommend that the redundancy effect be carefully considered by instructional designers.

2.3.3 Germane cognitive load

The third source of cognitive load is germane cognitive load. Intrinsic cognitive load refers to the material itself, extraneous cognitive load refers to the instructional designs and germane cognitive load refers to the processes connected to learning. Extraneous cognitive load interferes with learning, but germane cognitive load enhances learning (Paas et al., 2003).

A low intrinsic cognitive load imposed by the instructional materials and a low extraneous cognitive load due to appropriate instructional procedures will result in an unused amount of working memory capacity (Sweller et al., 1998). This unused working memory capacity is devoted to schema construction and automation (Bannert, 2002; Leahy & Sweller, 2008). de Jong (2010) clarifies that schema construction involves many processes such as interpreting, classifying, inferring, differentiating, exemplifying, and organising. The load imposed by these processes is termed germane cognitive load.

29

Germane cognitive load is closely linked to long-term memory, which was discussed earlier. As this study only considers the issue of the intrinsic cognitive load of the problems chosen, and the extraneous cognitive load of the instructional design, deeper issues pertaining to germane cognitive load are not presented in this chapter.

2.4 Adding sources of cognitive load (intrinsic, extraneous and germane)

Cognitive load theory assumes that the sources of cognitive load are additive (Ayres, 2006b; de Jong, 2010). Paas et al. (2003, p. 2) argue “Intrinsic, extraneous, and germane cognitive loads are additive in that, together, the total load cannot exceed the working memory resources available if learning is to occur. The relations between the three forms of cognitive load are asymmetric”.

The additivity between intrinsic and extraneous cognitive load has been mentioned by Sweller et al. (1998, p. 263) who state, “Intrinsic cognitive load due to element interactivity and extraneous cognitive load due to instructional design are additive”. van Merriënboer and Sweller (2005, p. 150) note that “Extraneous cognitive load and intrinsic cognitive load are additive”, providing support for Sweller et al.’s (1998) statement.

Cooper (1998, p. 14-15) outlines the additivity of intrinsic and extraneous cognitive load as is shown in figure 2.3:

(i) If the intrinsic cognitive load is low (low element interactivity) and the extraneous cognitive load is high, then working memory capacity is freed and learning (schema acquisition and automation) will occur.

(ii) If the intrinsic cognitive load is high (high element interactivity) and the extraneous cognitive load is also high, then working memory capacity is overloaded and learning (schema acquisition and automation) will fail.

30

(iii) If the intrinsic cognitive load is high (high element interactivity) and the extraneous cognitive load is low, then working memory capacity is freed and learning (schema acquisition and automation) will occur.

(i)

(ii)

(iii)

Figure 2.3: Visual representations illustrating (i) a low intrinsic cognitive load with a high extraneous cognitive load, (ii) a high intrinsic cognitive load with a high extraneous cognitive load, and (ii) a high intrinsic cognitive load with a low extraneous cognitive load. The thick, bold arrows indicate the mental resources required (working memory capacity) by learners. The two thick, bold arrows (i) and (iii) that extend beyond the image indicates learning has occurred whereas the thick, bold arrow (ii) which falls short indicates learning has been hampered (Cooper, 1998, p. 14-15).

In short, a reduction in extraneous cognitive load by using a more effective instructional design will free the working memory capacity which can be devoted to the intrinsic cognitive load and thereby increase the germane cognitive load (schema acquisition and automation) (Paas et al., 2003).

31

2.5 Measuring cognitive load

According to de Jong (2010), literature reveals that there are no direct measures of cognitive load. However, the level of cognitive load can be determined from the results on knowledge post-tests. Mayer, Hegarty, Mayer, and Campbell (2005) believe that low results from knowledge post-tests suggest a high cognitive load. This indirect measurement of cognitive load is found in the following statement by Stull and Mayer (2007, p. 808) who claim “Although we do not have direct measures of generative and extraneous processing during learning in these studies, we use transfer test performance as an indirect measure. In short, higher transfer test performance is an indication of less extraneous processing and more generative processing during learning’’.

de Jong (2010) further adds that authors have expressed the need for direct cognitive load measurement. This is also evident in the words of Mayer, Mautone, and Prothero (2002, p. 180) who argue ‘‘admittedly, our argument for cognitive load would have been more compelling if we had included direct measures of cognitive load…’’. Three techniques have been used to measure cognitive load and these are briefly described below:

 Self-reporting: Thi