How student teachers engage with the making of teaching tools for mathematics pedagogy

University of Johannesburg Faculty of Education How student teachers engage with the making of teaching tools for mathematics pedagogy Refilwe Judy Ntsoane Supervisor: Professor Eli

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How to cite this thesis

Surname, Initial(s) (2012) Title of the thesis or dissertation (Doctoral Thesis / Master’s

Dissertation) Johannesburg: University of Johannesburg Available from:

http://hdl.handle.net/102000/0002 (Accessed: 22 August 2017)

University of Johannesburg

Faculty of Education

How student teachers engage with the making of teaching tools

for mathematics pedagogy

Refilwe Judy Ntsoane

Supervisor: Professor Elizabeth Henning

Co-supervisor: Professor Nadine Petersen

Date: July 2018

DECLARATION

I Refilwe Judy Ntsoane (201147390) declare that, except where indicated by

reference in the text, this is my own work Any view expressed in the dissertation other than referenced material are mine

Signature Date Place

SUMMARY

Keywords: Pedagogical content knowledge, cognitive apprenticeship, tool mediation,

teacher career path, teaching tools, makerspace, homo faber

In my practice as a foundation phase mentor teacher for student teachers, I have witnessed how hard it is for them to create effective teaching tools in a low-tech, informal ‘makerspace’ during their practicum As teacher educator at a teaching school, I was motivated to investigate this topic in order to improve my mentorship Although the design of teaching aids appeared less challenging in their practicum planning groups, the same activity posed a serious challenge for them when they had

to implement the tools individually in different schools where they continued their practicum beyond our teaching school on the campus They often came back to our school to borrow teaching tools that they had created at the school and where they worked together as a group The safe environment of the teaching school on the campus was a ‘maker’ place where they could design their tools and learn to use them

I wanted to study the students to find out how they developed this part of their service education

pre-I thus initiated this study to explore how student teachers at the university teaching school behave during their practicum session with a mentor teacher I utilised the Thomas and Brown (2009) model of design and creativity, which is viewed from the three perspectives of human activity namely “knowing” (Homo sapiens), “making”

(Homo faber) and “playing” ( and imagining) (Homo ludens) I coupled this perspective

with Collins, Brown and Holum’s (1991) typology of cognitive apprenticeship

Data for the study was collected by myself as participatory and practitioner researcher, with field notes, photos, artefacts, and individual- and focus group interviews Two groups of third year BEd foundation phase education student teachers were selected

form to take part in the study The data were collated and analysed with Atlasti software

and also manually

The results indicated that despite the student teachers’ ability to design and make creative artefacts, they found it hard to integrate their tools as mediational ‘signs’ in practice Additionally, they used teaching tools that were not always relevant to the content they were teaching Through the inductive analysis, seven themes were

identified from the collated data, highlighting that the student teachers found it hard to bridge design and practice and to teach conceptually with the tools The findings reflect

a need for an addition to the methodology modules in the teacher education programme on how to design and create teacher toolkits that will come in handy as they embark on their teaching career

Professor Nadine, thank you for your valuable guidance

To my parents: Thank you for being there in my times of need To my father, Matsobane Bennett Mphahlele, thank you for constantly reminding me of the importance of education To my mother, Raesibe Mary Mphahlele, thank you for moral support and encouragement Love you

I owe a sincere and earnest thankfulness to my boys, Nthato Ntsoane and Karabo Ntsoane Thank you for being there whenever I need you Love you

My daughter, Monthati Ntsoane Thank you for inspiring me to be a better mom and a better person I love you

Family and friends thank you for your unfailing support

To God be the glory

TABLE OF CONTENTS

DECLARATION i

SUMMARY ii

ACKNOWLEDGEMENTS iv

TABLE OF CONTENTS v

LIST OF TABLES viii

LIST OF FIGURES ix

CHAPTER 1 OVERVIEW OF THE STUDY 1

1.1 Introduction 1

1.2 Research question, aim and objectives 4

1.3 Research design: A case study utilizing ethnographic tools 5

1.4 Data collection and analysis 6

5 TERMINOLOGY………5

1.5.1 Student teachers 7

1.5.2 Mentor teachers 7

1.5.3 Teaching tools 7

1.5.4 Triarchic model of learning and design 7

1.5.5 Cognitive apprenticeship 7

1.7 Conclusion 8

CHAPTER 2 STUDENT TEACHERS LEARNING IN A PRACTICUM 9

2.1 Introduction: Making tools as mediational artefacts 9

2.2 Learning practice in a teaching school with a ‘dual curriculum’ 10

2.3 Teacher knowledge and PCK of foundation phase teachers 14

2.4 Learning in an apprenticeship of ‘thinking’, ‘making’ and ‘playing’ 17

2.5 Situating tool-making in practice 19

2.6 Tool-mediation in early number concept learning 21

2.7 Conclusion: Limitations of a practicum and the promise of a career 26

3.1 Introduction 28

3.2 The unit of sampling 31

3.3 Ethnographic elements of research design 31

3.3.1 Ethnography in the study of student teachers 32

3.4 Data Collection 33

3.4.1 ‘Ethnographic’ observations and the ‘emic’ perspective 34

3.4.2 Photographic data 35

3.4.3 Artefacts 36

3.4.4 Interviews 35

3.5 Reliability and validity of the study 37

3.6 Data Analysis 38

3.6.1 Computer assisted data analysis 38

3.7 Research ethics 40

3.8 Conclusion 41

CHAPTER 4 DATA OF THE STUDY 42

4.1 Introduction 42

4.2 Data collection 42

4.3 Data analysis: From codes to categories and themes 43

4.4 Outcome of the analysis 60

4.4.1 Theme 1: Inability to mediate via artefacts 60

4.4.2 Theme 2: Lack of design thinking tools 62

4.4.3 Theme 3: Limited use of artefacts in school 65

4.4.4 Theme 4: Tools as memory aids 67

4.4.5 Theme 5: Connect with life outside school 64

4.4.6 Theme 6: Limited interaction with theory 65

4.5 Conclusion: Glimpses of the data process 75

CHAPTER 5 DISCUSSION AND CONCLUSION 76

5.1 Introduction 76

5.2 Themes constructed from the data 77

5.2.1 Inability to mediate via artefacts 77

5.2.2 Lack of design thinking tools 79

5.2.3 Limited use of artefacts in schools 80

5.2.4 Limited practical work 81

5.2.5 Teaching tools as memory aids 82

5.2.6 Connection with life outside school 84

5.2.7 Limited interaction with theory 85

5.3 Readily available resources 86

5.4 Limitations 88

5.5 Conclusion 90

REFERENCES 92

ADDENDUM A: PERMISSION TO CARRY OUT RESEARCH IN SCHOOL 95

ADDENDUM B: ETHICS CLEARANCE APPROVAL 96

ADDENDUM C: PARTICIPANT CONSENT FORM 97

ADDENDUM D: OBSERVATION SCHEDULE 101

ADDENDUM E: MEMORANDUM OF AGREEMENT BETWEEN UNIVERSITY OF JOHANNESBURG AND GAUTENG DEPARTMENT OF EDUCATION 99

ADDENDUM F: EXAMPLE OF STUDENT TEACHER’S LESSON PLAN 117

ADDENDUM G: EXAMPLE OF TRANSCRIBED RAW DATA 128

ADDENDUM H: HOW DATA WAS CODED 142

LIST OF TABLES

Table 3.1: Charecteristics of the case study (adopted from Cresswel,

2011: 223 ) 32

Table 3.2: Definitions of ethnography and elements invoked in the study 35

Table 3.3: Ethicalconsiderations 43

Table 4.1: Interview and observation schedule 45

Table 4.2: schoolteachers’ demographics 46

Table 4.3: Categories derived from collapsed codes 55

Table 4.4: Final categories and themes 57

LIST OF FIGURES

Figure 1.1: Mentoring of student teachers in making teaching artefacts 4

Figure 2.1: The progression of the chapter 10

Figure 2.2: Learning in an apprenticeship at a teaching school 11

Figure 2.3: Teacher knowledge: Career pathway as levels of development (adapted from Snow et al 2005:) 16

Figure 3.1: Integrated data collection 37

Figure 3.2: Steps of analysis 42

Figure 4.1: Some of the examples of coding 41

Figure 4.2: Second level of coding 46

Figure 4.3: Conceptual pattern of the main findings 59

Figure 4.4: Backward tracing from the pattern to 1st levels of codings 60

Figure 4.5: Mathematics resource designed by student teachers 61

Figure 4.6: Some of the resources used to show the difference between 2D and 3D shapes 65

Figure 4.7: Dice as mediational artefact 68

Figure 4.8: Mediational artefact created by student teachers 71

Figure 4.9: Student teachers as knowing beings 67

Figure 4.10: Student teachers as maker of tools (Homo faber) 68

Figure 4.11: Student teachers as players (Homo ludens ) 69

CHAPTER 1 OVERVIEW OF THE STUDY

1.1 Background, problem and motivation for the study

This study is about the use and making of tools for pedagogical purposes: Teachers

of young children regularly use material teaching aids as mediational artefacts in their classrooms How they use them and with what effect is not always clear When education students learn to make and use such artefacts, they often mimic existing ones and fail to reflect on the mediational value of the artefacts It is here that the problem of this study is situated I set out to capture how university students in a foundation phase teacher education programme learn to design, make, and use these tools, arguing that it requires specialized skills to conceptualize and to produce such tools, and that it warrants a close-up inquiry I, furthermore, argue that the use of these tools requires careful planning, taking cognisance of child learning and conceptual development (Henning & Ragpot, 2015) and of pedagogical craftsmanship to foresee

the usability of the tools I thus started the study with the idea that I wanted to see how

the students go about planning and making artefacts and what they observed during their teaching practice at schools One of these schools is the teaching school where I

am a head of the foundation phase department A part of my work is to mentor (Hamilton & Riley, 1999; Gratch, 1998) student teachers It is, thus, in this capacity that

I undertook the study as a practitioner in the field

Mentor teachers in a teaching school that is affiliated to a university teacher education programme are expected to train student teachers to design and make teaching tools

as part of their student mentorship programme at the school (Hobson, Ashby, Malderez

&Tomlinson, 2001:2) Such teachers are briefed to attend to specific aspects of student apprenticeship in pedagogical content knowledge (PCK) (Sothayapetch, Lavonen & Juuti, 2013:85) In the case of making concrete/material teaching tools, mentor teachers at the research site of this study work with groups of students who make pedagogical artefacts that they can add to their teacher toolkit The work we do can

be likened to what is currently referred to as makerspaces; the spaces for pre-service

teachers in the context of this study are decidedly low-tech, using recycled materials

and natural objects, much like the teacher who was interviewed on a ‘makerspace’ blog recently:

Laser cutters, robots, 3D printers: when people talk about educational

makerspaces, images of expensive, high-tech gadgetry comes to mind In

Colleen Graves’ library, they make use of a much cheaper resource “It’s trash,”

she said “But don’t call it that.” (on-making-makerspaces-accessible-to-all/)

https://hechingerreport.org/teachers-share-tips-Hannah Arendt (Arendt, 1958) articulated the concept of Homo faber as the tool that

humans use to control the environment ‘To know’ is to use the tool (of knowledge as well) to control the environment I refer to her view because I would argue that children need to be partners in making the artefacts that are used to teach them, so that their thinking and their attraction to design and to playfully create is given some space in classrooms, instead of letting teachers make these or use ready-made tools My view

is that every classroom could be a ‘makerspace’ of some kind and children should have the opportunity to ‘control’ some of their learning by being free to express themselves

by design and by being learner-makers In other words, I would say that children need

to create these learning artefacts themselves sometimes - asking them to make something that would help others understand the topic that is in the theme of a lesson

In Chapter 5, I will come back to this issue of learners needing to be co-developers of instructional tools Young children are makers of idea tools and material, concrete tools

to teach themselves and each other (Vygotsky, 1933) I argue that it is not just the domain of teachers and traders who sell ‘educational material’ and ‘toys’ 1 I also argue that student teachers need to learn and apply this type of principle

To this end I utilised an epistemological model of design and creation, as formulated

by Thomas and Brown (2009) to situate tool making conceptually (Thomas & Brown, 2009:1-2) The construct of student teachers’ learning in a specific mentorship

programme is further framed by the cognitive apprenticeship model as described by

Collins, Brown and Holum (1991), and Brown, Collins and Duguid (1989) In these two models pre-service teacher education is seen as an apprenticeship in tool-making for learning and teaching purposes

As researcher, I was motivated to investigate the design, production and usability of innovative materials for the teaching of mathematics in the foundation phase During the early grades, children learn many mathematical concepts through concrete

manipulatives and demonstration materials In practice, as a foundation phase teacher

in a teaching school, I have witnessed much repetition (and copying) in the making of such teaching tools by student teachers I seldom come across tools that truly exemplify the pedagogical content knowledge of teachers, as formulated by Shulman (1987:5), which would include students’ knowledge of children’s early conceptual development in mathematics as, described for example, by Fritz, Ehlert and Balzer (2013:40) In a country with a limited public budget, it is necessary for teachers to be self-sufficient and to be able to improvise teaching tools Pre-service teachers can experiment with different kinds of materials, including recycled material, especially in

a teaching school set-up, where pedagogical artefact production is a requirement In this school, teachers are also encouraged to use natural objects from the school grounds as teaching aids

My interest, and with that also the framing of this research, comes from a blend of theoretical areas I use an integrated lens in my approach of this investigation: 1) I see the study as an inquiry of ’cognitive apprenticeship’ (Brown, Collins and Duguid, 1989:33; Collins et al., 1991) in learning to use and to make teaching tools as apprentices of the teaching profession and 2) I also utilise Thomas and Brown’s (2009) model of design and creation/production of such tools (See Figure 1) In this model, knowledge-making as a design and creation phenomenon is viewed from three

perspectives of human activity, namely ’knowing’ (Homo sapiens) ’making’ (Homo faber), and ’playing’ (Homo ludens) (Thomas & Brown, 2009:5) Incorporated in this

blend is also 3) the model of a teacher’s career path (Snow, Griffin & Burns, and 2005:201), to reflect on what can be realistically expected of pre-service teachers in their practicum

Therefore, from the outset, I accepted that pre-service teachers would not be where they could be after a few years in practice, but that they would be able to form the foundations of designing and using tools for teaching In combining these three theories I argue that, as apprentices in teaching, the students can learn in the daily

practice of school how to integrate 1) knowledge - what they know about child learning,

the content of mathematics (specifically) and those artefacts which exemplify their

knowing, and 2) tool-making and design with 3) playfulness and innovative design (Thomas & Brown, 2011:121).2

Figure 1.1: Mentoring of student teachers in making teaching artefacts

This amalgamated theoretical lens incorporates, heuristically, ideas to think about how students can design teaching materials for their lessons This is an important part of the teaching practicum at the university’s teaching school - both in mentoring practice and in activities around making tools for pedagogical use In this practice, there could

be typical patterns, routines and pedagogical rituals of action, which is regarded as salient indicators of students’ apprenticeship in the mentoring situation I noticed from

my first encounters with students that they had fixed ideas and routines about ‘teaching aids’ My sense was that they had been enculturated into certain practices of tool-use

in their own school lives, in what Lortie (1975) refers to as an ‘apprenticeship of observation.’

I also realised that there could be specific discursive practices of mentoring around mathematics learning tools as contained in discussions and logbooks of students (Collins et al., 1991:1) These authors postulate that patterns of lesson preparations

2 This view of ‘play’ is not to be equated with games, or gaming and ‘gamefication’, which has a different epistemology, with an

and selection of tools may yield usable data for a study that adopts ethnographic research tools - albeit not a full educational ethnography in the traditional sense such

as the one of Henning (1992)

1.2 Research question, aim and objectives

To investigate the object of study, the research question that guided the study addresses a two-sided issue about the making of educational tools for the classroom: How do student teachers engage with the making of teaching tools?

The objectives were:

 To describe and analyse how the students exemplify the Thomas and Brown (2009) epistemological model in the tools they design and make

 To depict how students engage, with the mentor teacher’s assistance, in tool design and creation in the practicum apprenticeship

1.3 Research design: A case study with ethnographic tools

In this study, as practitioner researcher (and also participatory researcher), I utilised ethnographic tools to provide a description of student’s “way of life” (Wolcott, 1994) in

an aspect of their work during the apprenticeship programme at the school The study

is thus more of a case inquiry (Yin, 1993; Stake, 2013) with the construct as a ‘bounded system’ (Stake, 2013) rather than a full, anthropological type of ethnography The study does have ethnographic qualities, however, because the inquiry describes the way people do things in everyday practice in a specific setting and as a defined group (Wolcott, 1994; Henning, Van Rensburg & Smit, 2004; Henning, 1992, 1995) With this approach, a researcher observes and records human activity as it happens In the instance of this inquiry the student teachers were observed by an insider, thus by a practitioner researcher, or participatory observer, collecting data through everyday

interaction, with participants in their natural setting (MacMillan & Schumacher, 2006:315)

To provide an in-depth understanding of the phenomenon that is studied, a descriptive analysis is required In this analysis, from different data sources, I aimed to show how

the epistemological model (Thomas & Brown, 2009) - featured in the student’s work,

especially how patterns that relate to the phenomenon (of tool-making) could be identified Furthermore, in my role as the participatory observer and the mentor, I had the double task of ’making visible’, how he elements of cognitive apprenticeship (Collins et al., 1991)featured

1.4 Data collection and analysis

Data was collected from scheduled logbook entries of the mentor teacher and the students as well as video recordings of lessons and audio recordings of focus group discussions/interviews about lessons In addition, teaching tools of the selected sample

of participants were analysed according to criteria of the epistemological model (Thomas & Brown, 2009) I also observed mentor teachers and had informal interviews with them

The various data was analysed in different modes Data from the logbook and the group discussions and interviews were coded and categorised broadly in a grounded theory mode (Strauss & Corbin, 1998; Henning, Van Rensburg & Smith, 2004) without aiming to theorise directly from it, but just using the inductive ‘open’ and ‘axial’ coding mechanism of this analytical method The content of lessons and the artefacts and their use, were coded according to requirements/criteria for pedagogical content knowledge and tool-use in foundation phase mathematics teaching In both these analyses I was, invariably, influenced by the idea of Thomas and Brown (2009), much as I was trying to

focus-do ‘open coding’ Integration of thinking, making and playing, as described in the

Thomas and Brown (2009) model of design and creation, was thus considered throughout the analysis The video recorded lessons were analysed per episodic unit for the use of the learning tools

1.5 Terminology

Terms were used in specific relation to the context of the study and its setting

1.5.3 Teaching tools

This tern refers what is generally known as learning and teaching support material (LTSM) in the South African discourse; in this study it includes a variety of pedagogical materials used in the classroom These materials range from resources made by teachers (and rarely by learners) to commercially produced classroom resources such

as wall charts, workbooks, textbooks, e-books, readers, stationary, science kits, dictionaries, encyclopaedias and so forth Teacher-created resources refers to material created by teachers for use in their classrooms For this study the LTSM will be referred

to as ‘teaching tools’ (Department of Basic Education, 2014:7)

1.5.4 Triarchic model of learning and design

This term refers, in various guises in the study, to the Thomas and Brown (2009:5) model of design and creation/production of tools In this model, knowledge-making as

a design and creation phenomenon is viewed from three perspectives of human

activity, namely ’knowing’ (Homo sapiens) ’making’ (Homo faber), and ’playing’ (Homo

ludens)

1.5.5 Cognitive apprenticeship

This term refers to a theory of how people learn from one another through observation, imitation and modelling (Collins, Brown & Newman, 1989; Collins, Brown & Holum, 1990) and under the guidance of an expert

1.6 Components of the dissertation

In the first chapter I gave the background and motivation of the study, as well as the blended theoretical framework underpinning the study, with the overall aim and

objectives embedded and clearly stated

The second chapter gives a discussion of some of the literature that I engaged with to

investigate the unit of analysis (Linden, Trochim & Adams, 2006) of the research,

namely, student teachers’ design and creation of teaching tools in a practicum apprenticeship with a mentor teacher

The third chapter provides a description of the methods that I used during the data collection and the analysis This chapter also describes the ethical considerations pertaining to the research

The fourth chapter focuses on the data collection and analyses, with some examples

to teach a concept with the tools as mediation and unless children use the tools in a goal-directed way

CHAPTER 2 STUDENT TEACHERS LEARNING IN A

PRACTICUM

2.1 Introduction: Making tools as mediational artefacts

This chapter contains a brief discussion of the scoped literature with which I engaged during the study and which will clarify the theoretical framework as set out in the first chapter I will be dealing specifically with the socialisation of student teachers as future professionals with regard to the specific knowledge and skills they may need to start off as new teachers, especially with regard to the production and use of tools in their practicum The argument of this chapter is that student teachers need to understand that, in addition to their knowledge development, they need to learn specific methods and skills while being exposed to teacher educators’ demonstrations of practice (Joyce & Showers, 2002:2) One of these skills is to design, make and use personally made artefacts and to aim to let children also participate in the making of such learning tools – thus shifting the emphasis to children’s grasp of the notion of ‘making’ to learn and doing so playfully, or though play

The chapter includes a discussion of apprenticeship as a form of learning in a practicum in a specific type of school In this school, the students learn to teach while the teachers at the school educate the learners in tandem (Petker, 2018) Petker has argued that this combination shows that the teachers at the school have a dual portfolio; they teach children and they teach student teachers One of the briefs of these teachers is to assist students to design and use handmade, or technologically constructed, materials While doing this, the mentor teachers aim to make the students aware of how they can build their pedagogical content knowledge (PCK), by showing their understanding of how children learn, combined with knowledge of the subject content and with classroom methods Shulman (1987) suggested that PCK includes skills of teaching with knowledge of content and knowledge of the learner

The chapter argues for a specific approach to the type of mentoring and coaching of student teachers – one that will elicit originality in the tools they make and use for their teaching, while converting the tool-use to learning activity The chapter also argues

that student teachers learn optimally in apprenticeship mode (Brown, Collins & Duguid 1989; Collins, Brown & Holum, 1991, Rogoff, 1990), while taking cognisance of a three-component model of human thinking and design (Thomas & Brown, 2009) The chapter

is rounded off by a discussion of reflective practice and the use of tools, drawing the conclusion that students are not likely to engage with a full range of opportunities in their undergraduate years They are likely to learn ‘just enough’ to cope, but will need

a longer experience in their career to design and make usable tools with good effect

as mediators of learning In Figure 2.1 the development of the chapter is set out

Figure 2.1: The progression of the chapter

2.2 Learning practice in a teaching school with a ‘dual curriculum’

The diagram below shows how student teachers encounter learning at the university teaching school (Figure 2.2) It shows students learn in what Ramsaroop (2016) and Gravett and Ramsaroop (2017) refer to as a “pedagogical laboratory” This means that the practicum at the teaching school is a place where students can freely experiment with ideas and methods and also with the use of tools to mediate children’s learning That is the ‘real life’ setting, which is a live laboratory In this study I go a bit further and look at this live laboratory from the view of three models: 1) I see the students as apprentices with a mentor, at the beginning of their career as a teacher (pre-service) and as 2) thinkers, makers and designers in their ‘human activity’ I thus see them as

Homo sapiens, Homo faber and Homo ludens in their thinking design and use of

5 Mentoring student teachers in situated practice

4 Learning in apprenticeship of 'thinking', 'making' and 'playing'

7 Conclusion:

Limitations of a practicum and the promise of a career

in which to develop

skill

materials 3) I also view them as beginners, who only learn the foundations of the profession, and who will become adaptive experts of teaching only during their years

of classroom experience (Snow, Griffin & Burns, 2005)

Figure 2.2: Learning in an apprenticeship at a teaching school

“Though the theory of a need to actively participate in reactive change is appealing, it has to be coupled with both teacher educators and student teachers embracing the

theory of becoming: (Thomas & Brown, 2009:1) Before the constant change in

learning that we are witnessing now in the 21st century, knowledge used to be static Lately, learning is different and this requires mentor teachers to look at the student teachers as participants in a productive inquiry, or, as Hickman argued: “The function

of (productive) inquiry is the production of new artefacts, including new habits” (Hickman, 2007:9)

I agree with Marais and Meier (2004) who say that practical pre-service teacher training

in South Africa needs to improve, because teacher educators do not show the commitment which is needed to positively influence the way student teachers link theory (and design) to practice These lecturers at a university in South Africa allude

to the fact that the quality of teacher educators plays an important role in initiating student teachers into the culture of good practice (Marais and Meier, 2004:221)

• The three perspectives on human activity

• Student teachers

as developers of their own mediational artefacts

3 Situated practice for preparing toolkiits for teaching

In my experience teacher educators at universities generally do not have sufficient school teaching experience at a suitable grade- or phase level themselves Elliot (2001) suggested that initial teacher training should entail a close collaboration between the schools where the student teachers do their practical teaching with the institution where they are trained This is also what Darling-Hammond (2010) advocates Mindful of the effect on student teachers’ competence of practicum sessions and how they are conducted, Gravett and Ramsaroop (2017:8) allude to the fact that student teachers need to spend time in a ‘model’ teaching environment as a remedy to unpreparedness when leaving the university with qualifications but being unable to teach and to adapt to changing conditions (see also Snow et al., 2005) They further add, referring to pre-service years, that, “(i)t is during this time that student teachers are afforded the opportunity of strengthening skills and abilities that will help them to function effectively in schools with limited resources (and to learn) how to improvise, how to make the best of the situation” (Gravett & Ramsaroop 2017:8) Focusing on student teachers’ custom designed teaching tools as an essential part of the lessons they teach at the teaching school, this “strengthening of skills and abilities” (Joyce & Showers, 2002:2) happens through their entire practicum at the school over four years - particularly during the time when they teach micro-lessons (Gravett, Petersen & Petker, 2014; Petker, 2018), when they observe mentor teachers in action

as well It is also during this time that they observe the individual child allocated to each one of them for observation over four years, to record their development One of the pedagogical principles that is stated by (Shulman,87:6) is the ‘principle of observation’, which relies on lived experiences

However, when students go to different schools, away from their home base ‘model’

school, most of the common teaching aids (mediational artefacts) that they encounter

during their teaching practicums have already been designed with specific learning

objectives in mind Students just use them almost in an ad hoc way These are seldom

analysed for their effect and purpose and are often purchased from traders in educational materials who visit schools to sell their wares In the school where this study was conducted the teachers’ artefactual toolkits were no different; yet, the student teachers were expected to learn to be innovative in their design and making of teaching aids What they encountered were ready-made commercial objects If one uses an object to clarify thinking, I would argue that a personally made object has much more currency than one made in a factory The question I pose is: How else can the

design be original? I acknowledge that there are powerful artefacts, such as blocks and that they can be used creatively, but that is not the point I am trying to make

Lego-in this study I am arguLego-ing for custom-crafted tools that show how a future teacher’s mind works in relation to teaching a specific topic I am, thus, thinking about ‘design thinking’ in a teacher education context (Schon,1987)

I argue that not only should student teachers have a good grasp of making their own teaching tools that suit their teaching preference and communication style, but they should also be able to effectively expose the learners to a variety of these mediational artefacts, suitable for the content of a lesson I would even go as far as saying that the learners should assist in developing tools In doing so student teachers will be showing

the learners to learn from the knowledge they already have, to make new knowledge

and to show it though an artefact that makes their thinking ‘visible’ (Collins et at, 1989) based on what they know and think, but also designing and making tools to help them think further, as is often exemplified in ealry grades teaching:

In case of representational pictures, one type of artefact that has

received considerable attention is pictures What is our notion of the

artefacts picture of a dog that distinguishes it from related kinds such

as a picture of a cat? Just as with chairs, clocks and paws, there is no

physical property that all and only pictures of dogs possess Some are

tiny black-and-white sketches; others are huge abstract drawings

Some look like dogs (in the sense that one might even mistake them

for actual dogs in the dim light) others are diffuse smears of colour that

do not readily call dogs to mind (Bloom, 1996:6)

Learning to make meaningful artefacts to add to their toolkit, student teachers should pay attention to the school curriculum and the university teacher education programme

at the same time My sense is that this is likely to be hard for them

It is an issue to consider in that the teaching school hosts these two curriculums, running concurrently The core function of the teachers at the teaching school is to ensure the effective delivery of both curriculums (Department of Basic Education, 2011) Teachers teach children and, at the same time, they have to teach students how to teach (Petker, 2018) This partnership incorporates, among other programmes,

service learning which, according to Petersen and Petker (2014:130) “allows the teacher education curriculum planners to make optimal use of central organising pedagogical stance of the teacher education programme, which in these institutions is knowledge of how children learn and develop” A part of this is to learn to make usable materials to augment their teaching, especially in the teaching of abstract concepts, such as numbers and patterns in mathematics during their practicum sessions So, in the end a mentor teacher engages in much additional work and must try to ignite students’ creativity in making of tools, while they are in practicum

Practicum sessions play an important role as a bridge between the university curriculum (theory) and the school curriculum practice (Gravett & Ramsaroop, 2017) According to Kaasila and Lauriala, (2010:854), who are authors who write about this topic in a Finnish context, student teachers operate as adult professionals during practicum sessions and this puts them in a dual-purpose position, where their task is

to study and learn simultaneously about the school learners and the content that they have to teach One of the things they learn for this purpose is how to make material tools that can mediate their teaching Students should be alert to what the learners have to know and at the same time also what they have to do in their university curriculum Their work is also dual

For student teachers to be experienced in both curricula they need to observe a master (mentor teacher) in action, while also learning form a lecturer at the university about design thinking and design principles for cognitive development The mentor teacher, however, is the one who serves as example Cain (2009) suggests that mentoring means that a class teacher (in the school curriculum) entrusts her learners to the student teachers as she takes the role of scaffolding the student teacher teaching In this dual role a mentor teacher trains student and builds relationships with them, making the student teachers feel welcomed, accepted and supported (Cain, 2009:54) during the four years of B.Ed studies

My claim is that mentor teachers are able to guide students They are assumed to be ideal human resources that need to model good teaching practice (Korthagen, 2004; Petker, 2018) Yet, they are not trained as teacher educators, especially in the important area of teaching tools, and that their own practice needs to reflect ‘thinking’,

‘making’ and ‘creativity of design’ In my own practice as a teacher, I generally do not exemplify this fine-grained practice

Mentor teachers are expected to engage in the apprentice programme, helping to develop the student teacher’s skills in making teacher toolkits, and at the same time they also to model competence and pedagogical creativity (Hamilton & Riley, 1999) Novice teachers depend on their modelling of how to implement the curriculum, attending, first and foremost, to the teaching of their own learners (Petker, 2018); they have little time to attend to the needs of the students’ creativity and PCK in its entirety Gradually student teachers will come to develop the different types and knowledge and skills that they need in the profession Their pre-service years only serve as the beginning

2.3 Teacher knowledge and PCK of foundation phase teachers

As an experienced foundation phase teacher, who is at the same time also a teacher educator, I attempt to instil the following idea in the students: If they can build the beginnings of teacher knowledge as proposed by Shulman (1987), they can create their own teaching materials and help children to ‘see’ abstract notions

Shulman (1987) developed a typology of different categories of teacher knowledge, and describes them as follows: general pedagogical knowledge, knowledge of curriculum, pedagogical content knowledge, knowledge of learner’s characteristics, knowledge of educational context and of educational ends/aims The most important one, for me, is pedagogical content knowledge, as it represents the distinctive body of knowledge for teaching; it is a representation of the blended content, pedagogy and distinguishes the specific understanding of the school teacher (Shulman, 1987) My argument is that for student teachers to acquire this important knowledge they need mentor teachers’ ‘guidance in the teaching and learning environment with regard to making user-friendly tools With the closing of teacher colleges in South Africa, there has been a shift away from the abundance of ‘tool-kitting’ Gravett and Ramsaroop (2017) refer to the complaint that some scholars have about the diminishment of this type of ‘hands-on’ training and the advancement of situated learning Engaging novice

teachers in situated learning programmes, which are located in schools, rather than in

universities, are often presumed to provide more ‘situated’ than knowledge learned in the lecture room (Kennedy, 1999:75) By the same token, I am not sure if ‘situated

cognition’ is perhaps overrated as a source of PCK and of general pedagogical knowledge (GPK)

Given the knowledge types as proposed by Shulman, I argue that for student teachers

to acquire these different knowledge(s), the contribution of teacher educators from the university is essential in the apprenticeship programme which runs concurrently with their practicum sessions in the classroom environment In other words, the integration

of the two programmes should be optimal A study by Lim, Cock, Lock and Brook (2009:27) refers to the fact that there are many teacher educators who are practicing old-style teaching approaches that stem from their own ‘apprenticeship of observation’ (Lortie,1975) In such practices, the message that student teachers receive is sometimes that artefactual tools are illustrations, and even ‘decorations’, but not the powerful mediators that they can be The student teachers that I have encountered do not come with the knowledge that tools are to be used by individual learners and that each child uses them differently The tools are not simply illustrations or demonstrations for a whole class This reminds me of decorative classrooms that have many posters, mostly because they have been purchased from the sales travellers who visit schools I believe that students should learn in practice how tools can be used (and also made) by individual learners Apprentices in teaching need to know why some tools are useful and others not

Hill, Rowan and Ball (2005:403) argue that the onus lies with mentor teachers who need to demonstrate the idea that such specific professional knowledge can make a difference to the quality of instruction and that they have to model it for the student teachers in their care during practicum This can be the reason why Gravett, Petersen and Petker (2014:7) explain that for a teacher to become a teacher educator the teacher needs to take on a different role than only that of educating children These authors believe that this can be achieved if the mentor teacher undergoes proper training within a coherent programme, designed with the aim of developing the necessary skills for this (Gravett, Petersen and Petker, 2014:7) I agree with these authors that mentor teachers need the development offered by experts to ensure that their demonstration lessons reflect the relevant knowledge In my opinion, the university prepares the teacher educators well and equips them to actively engage with the student teachers in the practicum, which aims to bridge the theory taught at the

university with the practice taught at the teaching school (Gravett & Ramsaroop, 2017) The pre-service years, according to Snow et al (2005) are at the beginning of a career Most of the learning will be of an immediate type, also known as ‘declarative knowledge’ To some extent, students many progress with specific skills for specific lesson types, using suitable materials In the Snow et al (2005) model this would be known as ‘situated, can-do’ knowledge In Figure 2.3 the knowledge that comes from experience in the ‘real’ world are included in the rest of the model

Figure 2.3: Teacher knowledge: Career pathway as levels of development (adapted from Snow et al

2005)

In this view of a teacher’s career it is unlikely that students will progress further than level 2, and in some cases up to level 3 It is thus to be expected that third year students, such as the participants in this study, are likely to function on level 2

2.4 Learning in an apprenticeship of ‘thinking’, ‘making’ and

‘playing’

The idea of apprenticeship in learning to teach is aimed at ensuring the smooth

transition by the student teacher from learning about teaching to learning to teach This

means that learning to teach in one special teaching school should be transferable to other schools as well The work done by Lave and Wenger (1991), Brown, Collins and Duguid (1989) and Collins, Brown, & Holum (1991) inspired many researchers to investigate the notion of apprenticeship and the establishment of communities of

practice They were some of the first authors to put forward a proposal of cognitive apprenticeship as a model of instruction Their argument was based on their

5 Reflective, organised, analysed knowledge

1 Decalrative

knowledge

2 Situated, can do procedure knowledge

3 Stable procedural knowledge

4 Expert, adaptive knowledge

observations of successful learning situations; the researchers then searched for samples of effective work and made an analysis of the shared structures in subjects like language and mathematics lessons, which were presented in creative modes and with authentic methods, with learners seen as apprentices who witness the ‘master teacher.’ In this study the student teachers are the learners and the mentor teacher is the ‘master’

In this adapted model the mentor teacher serves as the model-in-action and her responsibilities are to provide support, assistance, and guidance and to assess the work of the pre-service teacher In doing so, Grossman (1991) suggests ways to counteract the legacy of the ‘apprenticeship of observation’ (Lortie, 1975 Borg,2004:274), which comes from students’ years at school More importantly, the mentor teacher’s work is to serve as an example to counter the legacy of default mechanisms of pedagogy (Darling-Hammond et al., 2005; John, 2013; Grossman & McDonald, 2008) Cognitive apprenticeship, according to Brown, Collins, and Holum (1991), is a view of education (for teacher professionals in this instance) by which the apprentice sees the procedure of the activity, with its faults and loopholes, and gets the opportunity to engage practically In the practical engagement the master (expert)

is constantly demonstrating, supporting and evaluating the apprentice and making the procedures of the activity visible This means that the mentor also shows typical problems and how to address them and, I would add, refer to what the students remember from their own school days

It is in this almost ‘microscopic’ analysis that the apprenticeship is made ‘visible’ Dennen and Burner (2008:427) agree and further explain that “the concept of cognitive apprenticeship is learning through guided experience on cognitive and metacognitive, rather than physical, skills and processes” With this said, I argue that student teachers need to be enculturated into a practice that gives them the opportunity to start bridging the gap between book learning and learning by doing Herrington & Oliver (1995:7) explain; “pre-empted situated learning by proposing that bridging apprenticeship be designed to bridge the gap between theoretical learning in a formal instruction of the classroom and the real-life application of the knowledge in the work environment”

The lesson plan of an apprentice

Given that pre-service teachers need to be skilled so that they can create and design their own teaching toolkit that will assist them as they embark on their foundation phase teaching career, the university equips the pre-service teachers with theory about the components of lesson planning This includes teaching tools as assistive and mediating devices that support teaching The practical part of this type of lesson, which

is the implementation of its design, is then observed by the pre-service teachers at the teaching school, where the in-service teachers are expected to guide the learning According to Collins, Brown, and Newman (1989), cognitive apprenticeship differs from traditional apprenticeship in that the former helps the apprentice to adapt to and apply knowledge and skills according to the requirements of diverse settings (Collins, Brown

& Newman 1989:5) Unlike an apprentice in a trade, for example, the demands for learning to be adaptive are many, because individual children differ much Darling-Hammond (2005) and Snow et al (2005) provide models of learning as a pre-service teacher and how experience in schools provide the scaffolding for further advancement

in a teacher’s career Their models consider that a teacher will adapt her teaching as she gains experience and also more knowledge from new research

Self-monitoring

During observation in the practicum, the apprentice develops self-monitoring skills and learns to integrate skills and knowledge, while trying to improve and, in the process, developing a ‘theoretical model’ for herself Learning problem solving depends mostly

on information sourced from regular ‘textbook’ patterns, as opposed to practically learning problem-solving strategies When encountering problems that fall outside of the ‘textbook’ pattern, student teachers do not know what to do In some cases, student teachers do have the necessary resources at their disposal but they do not know how

to effectively utilise them because they have not observed anyone demonstrate how the resources are used Often, students also do not make use of available resources because they do not have examples or models of the required processes, (Collins Brown & Newman, 1989:7) Santoro (2009:43) explains that “the retrieval of a craft apprenticeship model of teacher training means that pre-service teachers will spend more time in schools learning the craft of teaching and less time in the universities engaged with theory”

I would argue that cognitive apprenticeship is a suitable model for preparing student teachers for their entry into the real world of teaching The knowledgeable and experienced person demonstrates and takes the apprentice step by step through the procedure of the task, with much of the learning taking place while the apprentice observes the ‘master’ Secondly, the ‘master’ scaffolds (gives support to) the

‘apprentice’ and, depending on how the learner understands the task, the master can assist with the entire task In a case where there is evidence of the student teacher understanding the specific aspect which is being demonstrated, the master slowly

‘fades’ support – thus affording the apprentice more accountability

2.5 Situating tool-making in practice

As mentoring ‘fades’, and students are able to design and implement their own tools, they enter the phase of mentoring where they are no longer coached The students have developed from novices who had theoretical knowledge and some memory of their “scholarship of observation” (Lortie, 1975) to a budding practitioner with trust in

their own judgment and creativity in tool-making and tool-use According to Cain,

(2009:54) mentoring is, therefore, a developmental practice in which an expert, or a

more experienced person, models good practice to a less skilled person with the aim

of supporting and developing the protégé The act of mentoring is carried out in the process of a thoughtful, helpful and caring relationship with the goal of ‘fading’ support

in mind Ashby, Hobson, Malderez and Tomlinson (2009:207) agree that mentoring can also be defined “as the one-to-one support of a novice or less experienced practitioner (mentee) by a more experienced practitioner (mentor)” who withdraws gradually

Mentor teachers serve as a human resource to student teachers throughout their year training in their B.Ed studies at the university where this study was conducted The mentor teachers help the student teachers to develop skills that should ensure the smooth transition from being a student teacher to becoming a teacher Du Plessis (2011:31) explains that mentor teachers in schools have a major influence on student teacher’s professional development The comments about the support received from mentors constitute different categories, which are the practical and immediate knowledge that the mentor teachers can offer, and the relationship that forms over the course of the apprenticeship period The relationship is thus both professional and

four-personal, because trust is at the core In the Finnish teacher education system trust is valued as much as it is valued in schools (Niemi, 2011; Sahlberg, 2010; Lavonen, 2013)

This relationship with the mentor teacher can be further nurtured by opening the lines

of communication and engaging student teachers in professional ‘math talk’, when students learn the discourse of maths pedagogy Classroom discussions with the Such discussions could be talk that “moves, that engages students in discourse” (Lisa, 2015:15) Mentor teachers can also encourage talk by checking whether the student teacher understands her own practice by discussing some of the issues This can, for example, be about reasons for using and making certain tools When students are encouraged to listen, and then to rephrase what the other student teachers have said

in the student teacher discussion groups, it can encourage them Maphosa (2007) states that, as a mentor teacher, one needs to know and understand the university requirements and standards that speak to guidelines in supporting student teachers, while also acknowledging the requirements of the DBE and the DHET

The mentor teacher needs to mediate such group discussions and encourage students

to reflect critically on the lessons (De Garcia 2015; Petersen, Loukomies, Lavonen and Henning (2018) This requires the mentor to arrange the physical space in such a way that it is conducive to discussion by, for example, arranging student teachers to sit in

a circle and listen to peer talk Visual aids could also be employed by the mentor to prompt critical discussions However, as a mentor teacher one needs to be very careful how feedback is given to student teachers By giving effective, constructive feedback that is not judgmental, the student teachers develop trust in their mentor

Taking into account the fact that student teachers have not yet gained enough experience in teaching, it is understandable that when they enter the classroom they still have insufficient knowledge of pedagogical content and of effective classroom management Expert mentor teachers combine support with understanding and employ a variety of techniques For example, where possible, feedback has to be given

in private and should encourage self-assessment

According to Bird (2012: 20) the education fraternity, as compared to other fields, has two systems of mentoring, which invokes both personal and professional ‘bonding’, as

I have mentioned before These are known as the ‘informal buddy’ and the ‘formal comprehensive’ systems In the ‘buddy’ system, mentors do not get compensation for their mentoring whilst in the formal comprehensive system mentors are carefully selected whereupon they engage in scheduled training and thereafter they are compensated for the formal service they provide Rowley (1999) and Gratch (1998) explain that a good mentor is characterised by commitment and has understanding for novice teachers Sahlberg (2010) argues that the success of Finnish education is dependent on its teachers, all of whom are now educated in programmes with a practice school with mentor teachers Petker (2018) describes the process, accentuating that these teachers are the crux of the practicum at Finnish practice schools that are affiliated to universities I would like to make the point that, while we try to emulate the Finnish model, it is hard for mentor teachers to do this without having been trained as teacher educators We are not fully involved in the university teacher education programmes and often have to make judgments, based only on our own understanding In the mentoring of student teachers who are preparing to teach, grade

1 learners number concepts it can be particularly daunting

2.6 Tool-mediation in early number concept learning

While students are still in the pre-service phase of their career, they also learn to think critically about their teaching – they learn to reflect and make it part of their overall brief

as a teacher Anderson et al (2008:14) propose that reflective action occurs when an individual, be it a mentor or a student teacher, willingly and constantly takes part in self-appraisal and development Among other things, the participant needs to be flexible and rigorous with the aim of raising the standard of their professional proficiency Reflection in action and upon action (Lavonen, 2013; Loukomies, & Petersen, Lavonen, 2018) is one of the main tenets of teacher education in Finland

As the participant engages deeper with the process of learning to teach, it is important that they be given the opportunity to actively discuss and think about what they have learned in the practicum work According to Hobson et al (2008) reflective teaching takes place when the teacher is conscious of the aims and consequences of her teaching whilst keeping in mind the technical skill and efficiency This, according to these authors (ibid) helps teachers to create proper development frameworks within

which teaching and learning can take place A reflective practitioner, as first defined by Schön (1983, 1987: 7) is someone who learns “by doing and developing the ability for continued learning and problem solving throughout the professional's career” For pre-service teacher education, Adler (1991) argues that this type of learning by doing and discussion is crucial

Experienced teachers now embrace reflective practice as this ultimately improves teaching experience for both the mentor teacher and the student teacher Rodger (2002:230) defines this type of learning in practice as the “reconstruction or reorganization of experience which adds to the meaning of experience and which increases [one’s] ability to direct the course of subsequent experience Reflection is that process of reconstruction and reorganizing of experience.” He further explains that,

we can say that a reflective teacher does not merely seek solutions

nor does he or she do things the same way every day without an

awareness of both the source and the impact of his or her actions

of mentors to skill the student teachers in designing and creating their own toolkits during their teaching practicums is an important part of the work for which I am responsible and the practitioner research that must accompany it Ferraro (2000:5) argues that it is not easy to bring the idea home to students that a tool is a mediating artefact – that it becomes a channel through which they are teaching a concept, a process or factual information

Teachers in the foundation phase mathematics classroom often rely on concrete objects for effective lesson presentation, assuming that this type of playful engagement with manipulatives is the assured bridge to understanding number concepts and

number relations In studies on tool-use in childhood, however, many questions have arisen From the later work of Piaget (Piaget et al.,1978) it is evident that the use of concrete tools is not guaranteed to be the bridge to understanding Piaget et al (1978: Preface) argue that the manufacture of artificial tools has not been the origin of

intelligence and that Homo faber “may long remain a mechanic whole ignoring mechanics” (referring to Essertier, 1927) The authors continue, by citing Essertier (1927: 35): “Knowledge is contained in tools But if we look more closely it is not from tools that it is deduced, but from the intelligence as such.” This assertion form Jean Piaget, poses a dilemma for the use of manipulatives and tools in general – hence my argument that teachers should know as much as possible about child P(cognitive) development especially, so that they can make research-based (scientific) judgments about which tools to use and when learners should rather make the tools I have seen how children make their thinking visible with clay and with drawings, for instance Many teachers think that the concrete objects in themselves have teaching (and learning) power and often do not realise that these are just tools and that they have to

be used as mediators by the teacher (or peers) who give them mediational power and

also ‘making’ power if both students and learners work in a makerspace and not only

a teaching and learning space To ensure that effective learning takes place, it is important to create (and co-create with learners) materials that will be utilised in enhancing learning for young learners by making them ‘come alive’, designed and used critically-reflectively and from a solid scientific base Atieno (2014:15) has the following to say about this:

Learning is a complex activity that involves interplay of learner’s motivation, physical facilities, teaching resources, and skills of teaching and curriculum demands The availability of teaching tools therefore enhances the effectiveness

of schools as they are bases resources that bring about good academic performance in the students (Atieno, 2014:15)

In the teaching of mathematics in the early grades, it is common to use concrete objects as a first step in learning However, it is at the same time also a challenging activity and needs much thought, especially considering what Piaget et al (1978) say

about Homo faber and commenting on the thinking and the playing child as well

The thinking, making and playing child

Lancy (2017) gives an account of children’s tool-use through the ages In an aptly titled article, he argues along the same lines as Essertier (1927) and Piaget et al

(1978), proposing that Homo Faber juvenalis, as tool makers/users has learned though

imitative play and tool-making He concludes that this is how they imitated and played

since the evolvement of Homo sapiens, while discovering their environment and its

artefacts, intersecting with their social and cognitive development Importantly, Vygotsky (1933), in an analysis of play as ‘leading’ developmental modality, concludes similarly In the classic model of human activity, he introduced the theory of tools/artefacts and signs (such as language) that mediate all activity of humans (Vygotsky, 1978) In Figure 2.4 I suggest that a learning-, tool-using/making child engages in the playful, imaginative activity of making thinking visible, or, as Vygotsky (1978, 1933) put it, ‘externalising’ thinking

Figure 2.4 The activity of making and/or using an artefact to make thinking visible

I keep in mind that the tool itself is not the source of knowledge, but the way in which the tool is used or made by a child is the learning opportunity This means that teachers need to be examples of good tool use and tool design, because children learn by imitation of activity I have to also add that student teachers, as apprentices, likewise learn from mentor teachers I show, diagrammatically, that a learner who uses a tool

is also a thinking person and when imagination is activated, there is an element of play

MEDIATORS:Tools, artefacts signs, language

OUTCOME:

An artefact that exempifies thinking Learning through using an artefact

Through teaching tools, teachers have to afford the learners the proper opportunities

to engage creatively and critically with the lesson topic, but also to see the opportunities for ‘ludic’ symbols (Piaget, 1951) to be activated, or play that is symbolic and in which imaging and imagination is foregrounded Here Piaget argues that practical make-believe (for fun and enjoyment) is substituted by symbolic make-believe in which play

is actually quite serious In other words, the imagination takes over and the play is not only for pleasure but to imaginatively solve a problem (Piaget, 1951: 555) In spontaneous children’s play this is evident when children ‘play out’ frustration In the classroom this natural ability can be harnessed for learning and solving problems

Teachers and materials

In arguing for a view that artefacts are powerful, but not all powerful, teachers need to consider multiple factors in making materials and in using them According to Tomlinson (2002), developing teaching material is a twofold process, and I would add that with the teaching of numeracy in the early years this is a crucial thought, because teachers have to have some understanding of numeracy development Firstly, tools can cater for the practical component where the teacher applies her creativity, embedded in the curriculum, to design and produce a tool that will assist her to deliver content in a way that the learners can understand at their own level On the other hand, the teacher personally develops understanding and how to explain Chanda, Phiri, Nkosha and Tambulukani (2001) further add that the student teacher’s support material that is self-made, can act as substitutes for the material that is ready-made and expensive Furthermore, designing and creating one’s own teaching tools means that one is given the opportunity to be independent and not to rely on commercially available tools In addition to the above mentioned, the teacher engages in the construction and assessment and reworking of these materials which she will utilize for her lesson presentation in the classroom (Chanda et al., 2001:13)

The materials referred to in my research include anything that can be utilized to assist

in clarifying challenging concepts during teaching Taking into account that children on average take about six months to develop an understanding of the cardinal value of the number ‘6’ (Spelke and Kinzler, 2007), the tools used to exemplify this cardinality have to be designed carefully, considering the use of pictorial representation and three-

dimensional objects Teachers have to consider learners’ working memory and their ability to manage their attention (Henning and Ragpot, 2014) It is important that when one thinks about teaching resources one needs to bear in mind the fact that they are not teaching ‘methods’ on their own but that they are there to serve as assistive and mediating devices that can be adapted when needed These tools range from auditory tools to visual ones, which can be presented in print form as well The use of sound (musical or otherwise) assists in understanding difference and proportion Visual materials can be displayed for as long as they are functional Rogoff (1990) argued that children, when learning by example from the people around them, would mimic or copy the use of artefacts until they have become foundational

According to Tomlinson (2011:2), “an important factor in changing attitudes to material development has been the realisation that an effective way of helping teachers to understand and apply theories of learning- and to achieve personal and professional development- is to provide monitored experience of the process of developing materials” Previously, material development was treated as part of practice in which teaching materials were presented as ‘teaching techniques’ rather than seeing them is mediating learning Chanda et al (2001:16) say that the “teaching and learning material should be adapted to the understanding and level of the class” (2001:26) In the making of materials for early grades mathematics learning it is important to know enough about child cognitive development to design appropriate tools

According to Conley and Wise (2011), educational media can also be seen or thought

of as tools which are mediational artefacts, along with playful use of other artefacts As for magnitude, excellence of teaching aids - it depends on the context as prescribed

by the Curriculum Assessment and Policy Statement for foundation phase in which mathematics teachers are encouraged to use manipulatives wisely in order to ensure active involvement of the learners (Department of Basic Education, 2011) Young children need to learn how to manipulate concrete material, and in the process to learn through play I have already mentioned that the blocks or other manipulatives are in themselves not a guarantee for engagement and understanding These teaching tools serve as assistive devices for learners to engage in critical and creative thinking to solve problems (Conley et al.,2011: 278) It is, however, important to keep in mind that

there is a difference between demonstration- and illustration materials and teaching tools that can be handled, or manipulated, by the individual learner

The 2013 Diagnostic Report of the annual national assessments of foundation phase mathematics and the 2014 Framework of Improvement (Department of Basic Education, 2014) further state that some of the concepts in geometry and data handling can be effectively taught and consolidated with the support of teaching tools which, with proper training and adequate teacher pedagogical content knowledge, pre-service teachers can be skilled to create and design According to the Annual National Assessment (Department of Basic Education, 2013:10) mathematics teaching can be

a fun filled activity if mathematics teachers can ensure that their lesson plans are accompanied by relevant age-appropriate materials I would argue that teachers need

to understand how children learn and develop concepts so that they use developmentally appropriate materials In the work I do with students I use several conceptual tools, one of which is a model of number concept development (Fritz et al 2013)

Greeno, Collins and Resnick (1996:26) point out that one of the issues to consider is that material development be directed by theory, practice, curriculum and that it should contribute to teacher development, teacher needs, learner requirements, and beliefs According to Greeno at al (1996:16) the process of constructing understanding in mathematics entails learners’ interaction with concrete materials which exemplify mathematics concepts It also enhances social interaction when learners discuss their understanding of concepts and systems It is important that learners be exposed to both of these aspects – material and social – in the maths classroom

The teacher’s developing consciousness and refining her classroom teaching materials will assist in the facilitating of learning in all subjects With regard to the

teaching of mathematics Aunio and Rasanen (2016: 687) argue,

Teaching learners basic operational signs help the learners to better

understand mathematics When learners are taught place value they

grow their knowledge of the number symbols and the different values

they represent because of the value of the position they are placed on

this concept is clarified to the learners when using the base ten blocks,

that emphasize the oneness of one to the learners (Aunio & Rasanen,

2.7 Conclusion: Limitations of a practicum and the promise of a

at best, be introduced to the making of tools and, as Darling-Hammond et al (2005) and Snow et al (2005) have shown, pre-service teacher education has limitations In the data anlysis of this study these limitations will be pointed out At the very least, though, in the practicum of the teachers there is ample opportunity to see what works and what does not work However, the learners cannot be a testground – they cannot

be giunea pigs for the students

I agree with Petker (2018) that the school is, in the first instance, a place of learning for the children Mentor teachers who help students to design teaching materials have

a repsonsibility to ensure that the tools are functional Students have to also learn to

be accountable in their thinking, making and designing of teaching materials According to Maphosa (2007, 297) there are some important features of teaching practice in teacher education that should be emphasized in teacher preparation These two features are lesson planning and lesson presentation Mentors should be very