1

Description of the research project

The PhD candidate will, in cooperation with the principal supervisor, prepare a preliminary project

description as an appendix to the application for admission to the PhD programme.

A final project description should be handed in within three months after admission.

Check applicable box: Preliminary project description x Final project description

PhD candidate:

Sigurd Johannes Hals

Title of thesis:

Novice teachers’ work on proof and proving

Principal supervisor:

Sikunder Ali

Date: 7 October 2017

2

1 Introduction Argumentation and proving have played important roles in the history of mathematics. Despite their

crucial roles, school mathematics often have little focus on proof, both in teaching and curriculum.

Often the trend is that proof and proving are emphasised when studying Euclidian geometry.

Reasons for working with argumentation and proof are that this will give a deeper and conceptual

understanding of the mathematical phenomenon studied; it is important in discovering the nature of

mathematics and instils argumentation and deductive reasoning.

As reflected in the curriculum chapter “Purpose” LK06 (Udir., 2013) there are two main motivations

for learning mathematics. First, the need to handling our environment, nature and society, and

secondly, development of mathematical thinking and argumentation. Argumentation and proof

comprise cultivating a deeper understanding of mathematics and training in mathematical thinking

(Zaslavsky, Nickerson, Stylianides, Kidron, & Landman, 2013).

Being a novice teacher is often a stressful and challenging experience. Hollup and Holm (2015) point

out that one-third of all graduate teachers quit their teaching jobs within the first five years of

employment. These teachers justify their decision on account of the heavy workload (Hollup & Holm,

2015). Høigaard, Giske, and Sundsli (2012) indicate that teachers’ self-efficacy and work

engagements are negatively correlated with their intention to quit and the feeling of burnout.

Besides these general struggles, the novice teachers must deal with specific challenges associated

with didactics in general and subject didactics in particular. Here I find it interesting to examine how

novice teachers handle argumentation and proof didactically and identify the challenges they face in

engaging in reasoning and proving.

My tentative research questions are

- In what learning situations do novice teachers need knowledge about argumentation and

proof?

- Which challenges do the novice teachers meet when working with argumentation and proof?

In the first question, I want to investigate the mathematical classroom situations and the textbooks

to see how they relate to argumentation and proof. I will then examine how this teaching situation

challenges the novice teachers.

The knowledge that this project might reveal will be useful for school administration and colleagues

when guiding novice teachers. It will also be useful for improving teacher training. This study will be

conducted in Norwegian schools and will focus on the Norwegian context of Norwegian school

history, school politics, curriculum, textbooks, teacher education etc.

The PhD project will be associated with the current research group at the University College of

Southeast Norway. The group’s focus is Mathematical thinking, proof and argumentation in school

subjects and the discipline of mathematics. My project is lodged in the mathematics didactic

research field, which is an interdisciplinary research domain comprising mathematics, pedagogy,

psychology, sociology and philosophy (Sriraman & English, 2010), which gives rise to multiple

interpretations and approaches to mathematics.

2 Theoretical foundation

2.1 Purpose of theory and how I use theory It is great variance in the use and intention of theory in mathematics didactic research (Niss, 2006).

Niss mention some way of utilising theories: Overhanging framework, where the theory is top-down

3

with reference to other areas than the actual research; organising a set of observations and

interpretations; Theories that provide the terminology; and last, theories that give a methodology.

Independent of how the theory is used, the choice of theory is not a simple matter. Research on

mathematics teaching and learning are not yet being presented with one coherent overreaching

theoretical framework (Godino, Batanero, & Font, 2007; Steiner, 1985). Whish entails that the

researchers are left to themselves in the process of finding and making the theoretical foundation for

the research. Since there is no uniform theory, there is also no uniform theory on how to find the

right theoretical framework. In my ongoing process of establishing my theoretical foundation, I have

tried selected theoretical perspectives that are correlating with my one understating of the learning

and teaching. The purpose of my theoretical framework giving a framework that reflects my

understanding and gives me

I intend to see my theories in three intervening levels: Grand theories, middle range theories and

local theories.

Grand theories contain learning theories, elaboration on the concept of mathematics its history and

the role of proof. This together will define the learning process and what is being learned.

Middle range theories elaborate how to interpret proof in the context of the classroom and how this

may be a key notion in teaching and learning mathematics. Since my project will look into how

teachers are learning, it will include theories on professional development.

The last is local theories, giving the tools to intervene in the field, and analyse and interpret the

empery.

2.2 Grand theory As I understand, in mathematics education research grand theory are mainly concerning the

researchers’ understating of learning and the philosophy of mathematics.

I find the sociocultural understating of knowledge and learning suitable for my work.

History shows how mathematics has developed in different cultures with a variety of symbols and

style of languages. Despite these variations, the mathematical content is related and often similar.

Today we have one global community communicating and developing mathematics across borders.

This modern global mathematics is rooted in the Greek epoch where the deductive axiomatic

thinking arises. The mathematics has developed when individuals have engaged in and adopted that

times culture of mathematics. Mathematics developed through the exchange between individual and

community and process facilitated by psychological and cultural mediating tools. Lerman (2001) sees

the process of constituting mathematics as a discourse and further understands the learning process

as a discursive practice.

The sociocultural research field is not a unified and homogeneous field, but has since the young

Vygotsky died in 1934 developed in different directions. The focus is how individual knowledge

develops out of a cultural and a historical situation. From this view of inseparable unity, the object of

study is characterised by Lerman (2001) as person-in-practice-in-person, or mind-in-society-in-mind

(p. 98). Thus, to investigate students’ mathematical understanding requires looking at student-in-

mathematics-classroom-in-student, i.e. studying the students’ actions and their environment where

the main elements are the teacher’s approach, teaching material and students’ history. The view on

participation makes consequences for the view of mathematics itself. Lerman (2001) emphasises that

mathematics should be seen not as an abstract and single practice task but as a social creation of

meaning existing within a community.

4

Mercer (2004) uses the term sociocultural discourse analysis for a method that combines the

linguistic discourse analysis with the perspective that dialogue is a joint intellectual activity. The

purpose is to study how the quality of dialogue creates knowledge in the sociocultural understanding

of knowledge (Mercer, 2004).

The sociocultural perspective will have a major impact on my research. I will adopt the

methodological approach allowing me to capture dynamic interactions within the classroom and how

engagement with proof and proving in mathematics developed over time in class. In this regard, I

consider conducting interviews with students and teacher as well to observe the classroom and do

video recording. Video observation will enable in-depth analysis of the social interaction and the

common thinking.

2.3 Middle range theories In my case, middle range theories are concerning with defining and conceptualising proof and the

learning of proof. These setts should fit with resonate with the grand theories.

A simplified and standard view is that proof, and the axiom based mathematics appeared in Ancient

Greece. Thales worked with proving the known mathematics in 600 B.C.E. Three decades later,

Euclid’s Elements exemplified an axiomatic system. Since then, the ideal way of doing proving is to

state postulates followed by deductive reasoning, which provides “true” knowledge. Proof and

proving have been prominent in the development of mathematics since the Greek period. It seems

that the mathematics in Egypt, China, India and Babylon lacked focus on deductive proof and their

mathematics was, perhaps, based on empirical and intuitive thinking.

There are several aspects of proofs regarding its function. A. J. Stylianides (2007) emphasises the

importance of proofs and proving in mathematics as:

Proof is fundamental to doing and knowing mathematics – it is the basis of

mathematical understanding and essential for developing, establishing, and

communicating mathematical knowledge (p. 1, emphasis in the original).

Three conventional aspects of proofs are: 1) that the proof should verify the conjecture; 2) it should

illuminate the mathematical phenomenon and give rise to deeper understanding, although this is not

always the case; 3) it should organise concepts and axioms into a deductive system (e.g. Bell, 1976).

Furthermore, proof is a vehicle for the communication of mathematical ideas. Proof can give rise to

exploration as a process of organising and revealing new association. Proof is also an important

element in problem-solving, as the solution must be approved. One aspect seldom mentioned is the

aesthetic side of the proof. I believe that some of the proofs in school mathematics are selected not

only because they are important or simple enough but also because they are elegant. The last

mention by Reid and Knipping (2010) is self-realisation which also may be interesting from a school

perspective.

The form of the proofs remains varied, but the core of it is the deductive and logical reasoning.

Nowadays, it is common to separate between formal and semi-formal proofs. A formal proof is

where all necessary axioms and definition are stated, followed by a consequent of this that leads to a

theorem. This information could be symbolised and the process of proving can be mechanised and

could be conducted by a computer. Not all proofs may be formalised, and a formalised proof must be

understood and verified by a specialist, a process that may take years (Reid & Knipping, 2010). The

semiformal proof is less formal, as it only states what is necessary to convince a knowledgeable

person.

5

The idea of the content and how the proof is expressed vary in the mathematics didactic research,

while the trust in the logic and deductive reasoning is common (Reid & Knipping, 2010).

Key elements in sociocultural theory are the emphasis on the unity of individual and society the

notion that human internal and external actions are mediated by tools and signs (John-Steiner &

Mahn, 1996).

Vygotsky argued for this perspective through a generic and developmental perspective. The aspect of

communal knowledge developed is reflected in work on argumentation and proof. A. J. Stylianides

and Ball (2008) define some general principles for a proof, which fits all levels from first grade to

scholars.

(i) it uses statements accepted by the classroom community (set of accepted statements)

that are true and available without further justification;

(ii) it employs forms of reasoning (modes of argumentation) that are valid and known to, or

within the conceptual reach of, the classroom community; and

(iii) it is communicated with forms of expression (modes of argument representation) that

are appropriate and known to, or within the conceptual reach of, the classroom

community.

(p. 309)

The foregoing points show its dependence on the surrounding community. Following this reasoning,

the process of argumentation and proving should be seen and understood within the surrounding

social processes. Further, the principle of intellectual honesty, which emphasises mathematical truth

in classrooms, connects the classroom activity to the global mathematics community (A. J. Stylianides

& Ball, 2008; G. J. Stylianides, 2008).

2.3.1 Student perspective Zaslavsky et al. (2013) highlight five categories intellectual situations where the students itself may

require proof. Firstly, is the need for certainty. Often the students find certainty through empirical

examples, but when facing contradictions and cognitive conflicts, the student may ask for general

arguments. Secondly, the need for causality. When the student asks why questions, then there is a

need for understanding the underlying mechanisms which can be explored through proofs. Then

there is a need for computation, communication and structure. Zaslavsky et al. (2013) elaborate the

last point in historical and general terms. These five categories have implications for teaching.

How students’ proofs appear is examined by Balacheff (1988), who categorises students’ way of

proving in four categories; Naive empiricism, The crucial experiment, The generic example and The

thought experiment. These categories show how the students thinking may develop their form of

reasoning, from being convinced by examples advancing into the deductive reasoning. By

interpreting the Balacheff study, it is possible to see how the students have a unique and authentic

contribution to the work of mathematics. I believe this student work could be seen in relation to the

aspect of self-realisation mentioned by Reid and Knipping (2010). In this work, the voice of the

student will be valued in the classroom community and voices of students could be recognised by the

teachers. This is well illustrated by Hovik and Solem (2013), giving empirical examples on how

student argue and produces proof that could be understood through the classroom language and

thinking.

6

2.3.2 Teacher perspective - challenges and knowledge for teaching Research shows that most teachers see the role of proof as verification (Reid & Knipping, 2010, p.

79). If teachers do not see the explorative and explanatory aspects of proof as an important element

for developing a deeper understanding, then it is natural that proof and proving get sidelined in

school and curriculum. Zaslavsky et al. (2013) give three strategies that could activate the students’

need for proof through evoking uncertainty and cognitive conflicts; facilitating inquiry-based

learning; and conveying the culture of mathematics (p. 223). These concepts give ideas about what

kind of classroom culture nourish argumentation and proof.

A. J. Stylianides and Ball (2008) investigate what kind of subject matter knowledge and pedagogical

content knowledge may be needed for working on a proof, cf. Figure 1.

Figure 1. A classification of different forms of knowledge about proof for engaging students in proving. (A. J. Stylianides & Ball, 2008, p. 313)

This kind of framework will be important to understand how teachers’ knowledge of argumentation

and proving.

2.4 Local theories I have not made clear what is the local theories in my case. I think that the local theories should be

close to the actual data. Thus, it might be the toulmin-model, different kind of proving task, students’

way of representing proof. This is concepts that ale may be directly used in handling the case and the

empirical data.

3 Research approach Many aspects influence teachers’ thinking and actions. Their knowledge and beliefs, students,

colleges, school administration, curriculum and society are all aspects that influence teachers’

knowledge. I will, therefore, argue that teachers aggregate their knowledge within a social

community. In my view, there is a connection between teachers’ inner processes and their

interaction with the social and non-social environment. In my view, it is not possible to understand

teachers’ cognitive processes isolated from the school environment. This understanding is

considered as a socio-cultural stance. Will I, in order to understand a social phenomenon,

comprehend the actors’ understanding. This means that social phenomena also consist of the

“phenomenon’s” understanding of itself. Cohen, Manion, and Morrison (2011) describe this

interpretativism as “‘double hermeneutic’, where people strive to interpret and operate in an already

interpreted world.” (p.31). I would characterise my paradigm as interpretive.

7

4 Research design In designing empirical research, the main concerns are how to gather data that will answer the

research questions, how to analyse the data and to ensure a reliable and valid result.

The research question inquiries about the form and content of the challenges that the newly

qualified teachers face in their work with arguments and proof. Hence, it will be appropriate to adopt

a qualitative approach (Bryman, 2012; Yin, 2009).

My object of study is the teachers’ teaching where the focus is on argumentation and proof. The unit

of analysis is how the teacher organises and manages this situation. The method I will use for data

collection is video observation and interviews. Based on the initial interviews, I will compile a

selection of teachers that I will observe. After observation, I will conduct a follow-up interview. In this

way, I could get a picture of the teachers’ assumptions and intentions; how their teaching is manifest

and, finally, get the teacher’s reflection on their own teaching.

4.1 Case study Yin (2014) uses the notion “method” when describing case study. Ashley (2017, p. 114) claims that

case study may be referred to as research strategy, a design frame or research genre, and further,

case study should not be referred to as a method because case studies might employ different

methods. Bryman (2012) characterises case study as a design. I choose to follow Bryman and use

“design” when describing case study.

A case study is an intensive and detailed analysis of one or more cases. The case may be an object or

an incident, an individual, an event, an organization or a policy. Case studies are sometimes criticized

for not being generalised, and that this is the weakness of a case study (Bryman, 2012; Wellington,

2000). However, Wellington (2000) argues that case studies can be both enlightening and insightful.

It gives researchers the opportunity to explore situations or phenomena in great depth. As I interpret

Flyvbjerg (2006), ideographic knowledge formation is one of the principles of a case study.

According to Bryman (2012), a case study may be explanatory or descriptive. A descriptive case study

aims to describe various conditions in and around the situation, event or phenomenon; the goal is

not to explain why it is as it is. In a descriptive case study, the case is not chosen because it is unique

or special, but because it may represent similar situations, and because it would create a suitable

basis for answering the relevant research question. Yin (2014) identifies five types of case studies:

critical, extreme or unique, common, revelatory, or longitudinal. All case studies will have a

combination of these elements. Common case studies seek to find situations that are considered

normal and not unique. A longitudinal study follows a trend over time and will reveal how conditions

change.

In order to reveal the variation that will be present in teaching situations, I want to have more

observation points, in the form of several teachers. Yin (2014) does not make a distinction between

methodologies in single and multiple case studies. Multiple case studies are a variant of case study

design following the idea of replication, or test repetition, as the natural sciences would call it. With

more cases, I will be increasingly able to say something about how the result is related to various

parameters, such as how the teacher’s attitudes and prior knowledge affect the various didactical

challenges in working with argumentation and proof. When selecting the different cases I must be

aware of the purpose of that particular case; whether it should stay in contrast to the other or

complement it.

8

The validity of the result will be increased with more cases. The number of cases should be in

proportion to the complexity of the phenomenon and the number of competing explanations. The

challenge of multiple case studies is that it requires more resources, and analysis is more complex

and requires a greater theoretical foundation. To meet the complexity of the classroom, I will have

more than one case. I will define my study as a descriptive, common longitudinal case study.

4.2 Triangulation To do triangulation means to find the position out of several bearings. Triangulation may be done by

means of multiple data sources, several evaluators, using different theory and lastly, using different

methods (Yin, 2014, p. 120). In a case study, this would be analogous to having multiple measures of

the same phenomenon, which would result in greater confidence in the findings. Yin (2014) explains

how the different data sources should corroborate and converge in the analyses, answering the same

research question. In contrast, different data sources are analysed separately, and then the different

conclusions are compared.

5 Methods of data collection Case studies are well suited for using multiple sources of data, and it will raise the quality of the case

study (Yin, 2014). Using multiple sources is one method of doing triangulation, making the findings

more coherent and convincing. I think this is an analogy of how humans use all senses in the

idiographic process of creating knowledge, where rich perception gives the most refined knowledge.

Yin (2014) accentuates six types of data sources: documentation, archival record, interview, direct

observation, participant observation, and physical artefact. I will in this text focus on my main

methods interviews, direct observation and textbook analysis. Other sources such as documentation

(student works, schoolbooks, official curriculum, local curriculum, task collection, lesson plan, with

more) and physical artefact (concretes and manipulatives) may be relevant to the study, but this will

happen informally.

The interview will reveal teachers’ knowledge, attitudes and interpretations of the activities in the

classroom. The observation will give me access to the content of the classroom activities and enable

me to interpret them. The dataset will make it possible to analyse classrooms discourse and how the

teacher is making use of these.

I will interview 10 teachers selected in the region. Based on this first interview I will select four

teachers that I will observe in practice. After the observation, I will do a follow-up interview. This

process will take place in the teachers’ first and second semester. The teachers that will be observed

will be selected based on their motivation and attitudes towards mathematics.

5.1 Interviews Interviewing is a common method in case studies and is suitable for probing people’s own lifeworld.

Teachers’ descriptions of classroom events would be coloured by their own awareness and

interpretation. Using interviews will provide me with insight into the teachers’ perspectives and

knowledge on working with arguments and proof (Kvale, 1997). Bryman (2012) characterises three

types of interview, where the structured interview with its fixed form and rigid questioning is mainly

used in quantitative research. Qualitative researchers commonly use semi-structured and

unstructured interviews. The flexibility makes it possible to get deep insight into the interviewees’

lifeworld. These interviews, which are more like conversations, make it possible to follow aspects

that the researcher finds relevant. To make my multiple case study coherent, I have to assure that all

interviews shed light on how teachers’ challenges with argumentation and proofs are related to their

beliefs, knowledge and ambition.

9

As I see it, semi-structured is more appropriate when I have theoretical concepts that I want to put in

relation to the interviewee’s lifeworld, which is the case in my initial interview. The intention with

the follow up interview is to reveal the teacher’s own reflection on the conducted teaching. Hence,

the follow up interview will be unstructured.

Table 1. The focus in data collecting.

Initial interview Participant Observation Follow up interview

Semi-structured Video observation Unstructured

Topics that will be elaborated

Attitudes towards mathematics

Knowledge of teaching, including MKT (Mathematical Knowledge of Teaching)

Ambitions

Their working methods and strategies

Focus of observation

Content

Form

Action in the moment

Topics that will be elaborated

Interpretation of the happenings

Evaluating the learning process.

The initial interview has two purposes: viz. It should provide knowledge that will form the basis for

the selection of teachers I will observe, secondly, it will provide information on the teachers’ own

attitudes, knowledge and ambitions. This information will be part of the data that will answer the

research question. For this to be possible, when providing such a foundation it will be necessary to

use theory to create an Interview Guide.

The evaluating interview will take place shortly after observation. This interview will reveal the

teacher’s experience of, and reflection on, the current teaching situation. This would be what Yin

(2009) describes as an in-depth interview, where there I will focus on teacher’s understanding and

thoughts. I do consider using stimulated recall interview.

5.2 Observation To elaborate on the novice teachers’ work, I have to observe their actions and environment.

Alternatives are either participant observation or direct observation (Yin, 2014). Participant

observation would involve my taking part in their task and management of teaching. Thus, I would

experience the situation at close hand, but at the same time, make big changes to the case. Direct

observation would mean that I observe in the classroom. To have someone sitting in the classroom is

not unusual for the teacher and students. In the classroom, I would consider it better to be a passive

observer than going “native” in this situation. As I am video recording, I will have to choose where to

focus the camera, either on a student, groups of students, whole class or teacher. This will depend on

the form of teaching. The main purpose is to capture how teachers are using mediating tools in

orchestrating the classrooms discourse with the purpose of developing action in proving. Video

observation has the advantage of recalling the situation to make a detailed analysis though generates

a vast amount of data and subsequent increased workload.

5.3 Textbook analysis I want to conduct a thorough analysis of mathematics textbooks, teacher books and curriculum

documents in common use in Norway. The purpose is to get an understanding of the positioning of

proof and proving in mathematics in Norwegian Curriculum documents and how this positioning is

then translated within mathematics textbooks in schools in Norway. This analysis will enable me to

locate the actions and choices of the novice mathematics teachers with whom I want to work with.

10

Moreover, this analysis of textbook regarding proof and proving will also allow me to see how the

novice teachers see and interpret the roles of proof and proving as part of their dispositions. Further,

in what ways their understandings and interpretations can then impact the organization of teaching

and learning of situations within their classrooms.

6 Analytical approaches Case study analysis addresses big challenges to researcher’s empirical thinking. Despite all support

facilities, the researcher has to find the answers with his bare mind. Yin (2014) describes three

general principles of analysis. Firstly, conceptualising and understanding with the use of already

existing theory. Second, is ground up, where the researcher kneads the data to develop a theoretical

framework based on the collected data. Last, is when the findings are presented by narratives.

Additional to this is the critical discussion of the competing understandings.

The analysis will probe the data to find an answer to my main research question: “Which challenges

do the novice teachers meet when working with argumentation and proof?”. I have the intention to

have focus in two directions; firstly, what do the teachers define as challenging, and secondly, what

do I see as challenging in teachers’ work. These two aspects are important in their own ways. I want

to use these two perspectives to create an understanding of the challenges the teachers face.

My raw data will be video files of classroom observation and audio files from the interviews. I believe

that the first step should be to sort out video sequences that reveal teachers’ work on argumentation

and proof. In the transcribed interview, I will search for sequences where the teachers reflect on

their actual teaching containing argumentation and proof, and sequences where the teachers have

reflections that are more general concerning argumentation and proof.

To analyse the video, I need a tool that measures how the teacher handles the situation. Research

shows that higher qualities of teacher and of classroom environment increase pupil performance

(Kilday & Kinzie, 2008). In recent years, tools have been developed to measure the quality of

classroom activity (Kilday & Kinzie, 2008). I want to use the tool Mathematical Quality of Instruction,

(MQI), developed by Hill et al. (2008). It measures (1) richness of mathematics, (2) working with

student and Mathematics, (3) errors and imprecision, and (4) student participation in meaning-

making and reasoning (Borko, Jacobs, Koellner, & Swackhamer, 2015, p. 54). All four points’

measurements could be linked to evidence and argument. MQI is not developed with a focus on

argumentation and proof. Thus, I have to consider if the tool needs modification in order to handle

this narrow focus.

11

To analyse these two different perspectives, I will have two different analytical frameworks. As

teaching is particularly a complex situation, it may be appropriate to use the activity theory as a

framework. Activity Theory is an interdisciplinary framework used to analyse and understand various

human activities, including how mathematics is taught (Monaghan, 2016; Page & Clark, 2010).

7 Trustworthiness and ethical considerations One way of increasing the internal validity of analysis and interpretation of the data is to use more

than one researcher in part of the analysis. As this observation requires certain skills, it is important

to do some training in advance (Hambleton, 2017; Yin, 2014).

To gain construct validity, I must scrutinise the conduct of my research and be sure that my

investigation adheres to the concept and research questions. I have elaborated how I will establish a

reasonable understanding of the main concepts. My text does not show a great concern for internal

validity. In further work, I must ensure that my interpretation is adequate and that it supports true

conclusions. To enhance reliability, I will make the research transparent so that others could test the

research. In the coding and analysis, I will cooperate with others that will check the accuracy.

7.1.1 Ethical considerations Bryman (2012) describes four groups of ethical principles that are pertinent when conducting social

research: 1) Whether there is any harm to participants 2) Whether there is lack of informed consent

3) Whether there is an invasion of privacy 4) Whether deception is involved. The first three principles

concern ethical considerations towards people and cohorts. In my case study, I will examine the

teachers’ challenges, which can be regarded as a critical view of their teaching. I need to be honest

about my research focus and make sure they trust my intentions. My aim is not to investigate their

weaknesses, rather my aim is to investigate the challenges they meet in their profession as novice

teachers. I will make sure that they feel safe and do not feel that they are being put in a bad light. I

will make sure that the participants get all the relevant information they need to decide whether

they will participate in the study or not. The fourth principle relates to how I conduct and report my

study. I believe that good research is research that tells an honest story.

Interview: general reflection

Video observation

Interview: reflection

own practice

Understanding of novice teachers’ challenges

in the work on argumentation and proof

Figure 2. Two different perspectives answering the research questions.

12

8 Publication plan I am writing a collection of articles.

Article Preliminary title Targeted journal or conference

A Analysis of the need for proof and proving in

Norwegian textbooks

Conference paper PME July 2018

B Analysis of the need for proof and proving in

Norwegian textbooks

NOMAD

C How does the teacher succeed in evoking the

need of proof, and overcoming the weakness

of textbooks

Educational Studies in Mathematics

D What is supporting the novice teachers’

pedagogical development in the work of

argumentation and proof?

Journal for Research in Mathematics Education

E ? Conference paper ICME 14 (2020) Shanghai,

8.1 Some possible ideas for publications

• Analysis of the need for proof and proving in Norwegian textbooks. • What kind of challenges do the newly educated teachers face working on proof • What is supporting the novice teachers’ pedagogical development in the work of

argumentation and proof? • What challenges the novice teachers’ beliefs about argumentation and proof? • How do challenges vary with different beliefs and values? • What characterises the novice teachers’ work on argumentation and proof? • How does the teacher succeed in evoking the need of proof, and overcoming the weakness

of textbooks? • Is there a need for proof or is it a demand in the classrooms? When do we see students

demanding proofs? • Novice teachers are exploring student need for proof. • The learning progression of the novice teachers • The teachers’ own experience of working in the mathematical classroom • Examining the teachers’ experiences of the mathematical classroom • How do the novice teachers work to solve their challenges?

9 Progress plan

Activities (courses, articles A, B, C…, lab-work, experiments,

midterm evaluation, stays at other institutions, thesis

submission, etc.)

Year:201

7

Year:201

8

Year:201

9

Year:202

0

Year:202

1

I II I II I II I II I II

Theory of Science from a Perspective of Mathematics

Education PhD course at UiA X

Pedagogical resources and teaching processes PEDRES HSN X

PME July 2018 x

Article A x

UV9112 - Interaction Analysis: Methodological Perspectives

on Learning and Communication (uke 5) at HSN x

13

Kick-off seminar x

Pilot study x

First data collection x

First-year seminar x

Second data collection x

Article B x

Article C x

ICME 14 (2020) Shanghai, China 2. Pape x

Article D x

90%-seminar X

Dismutase x

14

10 References

Ashley, L. D. (2017). Case study research. In R. Coe, M. Waring, L. V. Hedges, & J. Arthur (Eds.), Research Methods and Methodologies in Education (2nd Edition ed., pp. 114-120). London: SAGE.

Balacheff, N. (1988). Aspects of proof in pupils’ practice of school mathematics. Mathematics, teachers and children, 216, 235.

Bell, A. W. (1976). A Study of Pupils' Proof-Explanation in Mathematical Situation. Educational Studies in Mathematics, 7, 23-40.

Borko, H., Jacobs, J., Koellner, K., & Swackhamer, L. E. (2015). Mathematics Professional Development - Improving Teaching Using the Problem-Solving Cycle and Leadership Preparation Models (Vol. 1). New York Teachers College Press.

Bryman, A. (2012). Social research methods (4th ed. ed.). Oxford: Oxford University Press. Cohen, L., Manion, L., & Morrison, K. (2011). Research Methods in Education (7 ed.). New York:

Routledge. Flyvbjerg, B. (2006). Five Misunderstandings About Case-Study Research. Qualitative Inquiry, 12(2),

219-245. doi:10.1177/1077800405284363 Godino, J., Batanero, C., & Font, V. (2007). The onto-semiotic approach to research in mathematics

education. The International Journal on Mathematics Education, 39(1), 127-135. doi:10.1007/s11858-006-0004-1

Hambleton, R. K. (2017). Measurement and validity. In R. Coe, M. Waring, L. V. Hedges, & J. Arthur (Eds.), Research Methods and Methodologies in Education (2nd Edition ed., pp. 234-240). London: SAGE.

Hill, H. C., Blunk, M. L., Charalambous, C. Y., Lewis, J. M., Phelps, G. C., Sleep, L., & Ball, D. L. (2008). Mathematical Knowledge for Teaching and the Mathematical Quality of Instruction: An Exploratory Study. Cognition and Instruction, 26(4), 430-511. doi:10.1080/07370000802177235

Hollup, K., & Holm, M. S. (2015). Tre grunner til at lærere slutter. Bedre Skole(4). Hoover, M., Mosvold, R., Ball, D. L., & Lai, Y. (2016). Making Progress on Mathematical Knowledge for

Teaching. The Mathematics Enthusiast, 13(1). Hovik, E. K., & Solem, I. H. (2013). Arguemetasjon, begrunnelse og beivs på barnetrinnet. Paper

presented at the FoU i praksis 2012. Høigaard, R., Giske, R., & Sundsli, K. (2012). Newly qualified teachers’ work engagement and teacher

efficacy influences on job satisfaction, burnout, and the intention to quit. European Journal of Teacher Education, 35(3), 347-357. doi:10.1080/02619768.2011.633993

John-Steiner, V., & Mahn, H. (1996). Sociocultural Approaches to Learning and Development-A Vygotskian Framework. Educational Psychologist, 31(3/4), 191-206.

Kilday, C. R., & Kinzie, M. B. (2008). An Analysis of Instruments that Measure the Quality of Mathematics Teaching in Early Childhood. Early Childhood Education Journal, 36(4), 365-372. doi:10.1007/s10643-008-0286-8

Knuth, E. J. (2002). Secondary School Mathematics Teachers' Conceptions of Proof. Journal for Research in Mathematics Education, 33(5), 379-405. doi:10.2307/4149959

Kvale, S. (1997). Den kvalitativa forskningsintervjun. Lund: Studentlitteratur. Lerman, S. (2001). Cultural, Discursive Psychology: A Sociocultural Approach to Studying the Teaching

and Learning of Mathematics. Educational Studies in Mathematics, 46(1), 87-113. doi:10.1023/a:1014031004832

Mercer, N. (2004). Sociocultural discourse analysis: analysing classroom talk as a social mode of thinking. Journal of Applied Linguistics, 1, 137-168.

15

Monaghan, J. (2016). Activity Theoretic Approaches In J. Monaghan, L. Trouche, & J. M. Borwein (Eds.), Tools and Mathematics New York: Springer.

Niss, M. (2006). The concept and role of theory in mathematics education. Roskilde: IMFUFA, Roskilde University, Danmark.

Page, S., & Clark, J. (2010). Experiences of Learning and Teaching Mathematics: Using Activity Theory to Understand Tensions in Practice. Paper presented at the Shaping the future of mathematics education: Proceedings of the 33rd annual conference of the Mathematics Education Research Group of Australasia.

Reid, D. A., & Knipping, C. (2010). Proof in Mathematics Education: Research, Learning and Teaching. Rotterdam: Sense.

Sriraman, B., & English, L. (2010). Surveying Theories and Philosophies of Mathematics Education. In B. Sriraman & L. English (Eds.), Theories of Mathematics Education (pp. 7-32): Springer Berlin Heidelberg.

Steiner, H. G. (1985). Theory of Mathematics Educations (TME): an Introduction. For the Learning of Mathematics, 5(2), 11-17.

Stylianides, A. J. (2007). The Notion of Proof in the Context of Elementary School Mathematics. Educational Studies in Mathematics, 65, 1-20. doi:10.1007/s10649-006-9038-0

Stylianides, A. J., & Ball, D. L. (2008). Understanding and describing mathematical knowledge for teaching: knowledge about proof for engaging students in the activity of proving. Math Teacher Educ, 11, 307-332. doi:10.1007/s10857-008-9077-9

Stylianides, G. J. (2008). Analytical framework of reasoning-and-proving. For the Learning of Mathematics, 28(1), 8.

Stylianides, G. J., Stylianides, A. J., & Weber, K. (2017). Reserach on the Teaching and Learning of Proof: Taking Stock and Movinig Forwoard Mathematical Processes and Content.

Udir. (2013). Læreplan i matematikk fellesfag. udir.no. Wellington, J. (2000). Educational research: Contemporary issues and practical approaches. London:

Continuum. Yin, R. K. (2009). Case study research: design and methods (4th ed. Vol. vol. 5). Thousand Oaks, Calif:

Sage. Yin, R. K. (2014). Case study research: design and methods (5th ed.). Los Angeles, Calif: SAGE. Zaslavsky, O., Nickerson, S. D., Stylianides, A. J., Kidron, I., & Landman, G. W.-. (2013). The Need for

Proof and Proving: Mathematical and Pedagogical Perspectives. In G. Hanna & M. d. Villiers (Eds.), Proof and Proving in Mathematics Education (pp. 215-229). New York: Springer.