Analytical framework for tasks and students’ examination scripts

In the following table, I present the categories from Morgan and Sfard’s (2016) analytical framework (for the full framework see Appendix 12.5) which I am examining in my data namely: student-author relationship, student autonomy, specialisation, logical complexity, the presence of multiple visual mediators, transitions between visual mediators, the types of actions

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demanded of students. I have added two columns indicating how these aspects of the discourse are examined in my data.

For the analysis of the tasks, I chose to focus on specific parts of the analytical framework aiming mainly to examine the routines the students are asked to engage in and the mathematical areas involved. As these provided insights into the discourses expected from the students. I did not examine the aspects of “objectification of the discourse”, the degree of specialisation and logical complexity in the tasks’ analysis. In the objectification of the discourse: The questions guiding the analysis is “To what extent does the discourse speak of properties of objects and relations between them rather than of processes” and the textual indicators are the following: “Nominalisation: use of a ‘grammatical metaphor’, converting a process (verb, e.g., rotate) into an object (noun, e.g., rotation); the use of specialised mathematical nouns such as function, sequence which encapsulate processes into an object; complexity of compound nominal groups”. I chose not to examine these aspects in the tasks but decided to examine them in students’ scripts. These aspects, of course, provide useful insight into the objectification, specialisation and logical complexity expected by the students at this first-year of their studies. Moreover, an analysis on these aspects on tasks coming from various universities can illustrate the different expectations that lecturers from various institutions can have from their students. However, for the current study, this will not be examined as these are tasks coming from one module and one institution.

88 Aspects of the

discourse

Morgan and Sfard – Questions guiding the analysis (Q) and textual indicators (TI) as reported in the tables in Morgan & Sfard (2016, pp. 106-108)

Tasks Solutions of tasks

S tud en t- au tho r rel atio ns hi

p _{Q: What kind of relationship is}

constructed between the student and a mathematical community?

TI: Use of personal pronouns (inclusive or exclusive we, other personal pronouns)

Not examined The pronoun “we” being

used in the students’ solutions

89 S tud en t A utono m y Q: In responding to an examination question, how many independent decisions is the student allowed/required to make in:

- Designing the path to follow? - Interpreting the tasks?

TI: Complexity of utterances: length of a sentence; grammatical complexity: the depth of the “nesting” of subordinate clauses and phrases; logical complexity. - Choosing the form of the

“answer”

TI: The layout: the physical size of the answer; the space provided for the work to be done on the way toward solution; format of the answer (units, precision, no. of solutions); modality of the

How many decisions is the student required to make when designing the path to follow? (The procedure of the routine; the endorsed narratives)

Instructions/directions given (or not given) to the students regarding the procedure of a routine (either given the name of the routine, a hint in brackets, implicit connection with other parts of the task)?

Instructions/directions given (or not given) to the students regarding the justification required.

90 answer (graph? algebraic

expression?)

- Choosing/constructing the mode of the response? TI: Visual mediators: Verbal; symbolic; or graphic: supplied or to be produced? S pe ci al isatio n

Q: To what extend is specialised mathematical language used?

Not examined Use of mathematical

terminology (words) which is not compatible with the mathematical discourses required to be used in the task. - Commognitive conflict Lo gi cal compl exi ty

Q: What kinds of logical relationships are present and how explicit are they? TI: the types and frequencies of conjunctions, implications, negations and quantifiers

Not examined Use of implications,

equivalences and

quantifiers illustrating problematic meaning making of the logical

relationships. -

91 Th e pr ese nce o f m ul ti pl e visu al m ed ia tors

Q: To what extent does the discourse make use of specialised mathematical modes?

TI: presence of tables, diagrams, algebraic notation etc.

Q: How are multiple visual mediators incorporated into the discourse? TI: Provided in the text or to be produced by the student; Linguistic, visual and/or spatial relationships between modes

Visual mediators (algebraic notation) Visual mediators (diagrams, graphs, algebraic notation) present in the students’ solution

92 Transi ti on s be tw ee n visu al m ed iato rs

Q: What transformations need to be made between different modes? TI: The presence of or demand for two or more modes of communicating “equivalent” information, e.g. an equation formed from a word problem; a unit of text that involves table, graph and algebraic expressions corresponding to the same function

Q: How are transformation indicated in the discourse?

TI: provided in the text or to be produced by the student; explicit linguistic or visual links between modes

Not applicable for the tasks Visual mediators used in the solutions. Examining the links between the modes (text and graphs)

Use of visual mediators (graphs and algebraic notation) which are not compatible with the mathematical discourses required to be used in the task. - Commognitive conflict

93 Th e t ype s o f ac ti on s de m an de d o f s tud en ts

Q: What areas of mathematics are involved?

TI: topics

Q: What are the characteristics of the routine procedures?

TI: algorithmic or heuristic? Complexity, explicitly hinted at?

Determining the mathematical discourses involved in the task

Examining the routines:

Characterising them as rituals, recall, substantiation or construction (based on the lecturer’s solutions and comments)

Explicit directions on the procedures of the routines (this is also examined at the Student Autonomy)

Determining the

mathematical discourses involved in the student’s solution.

Use of procedures which are not compatible with the mathematical discourses required to be used in the task. - Commognitive conflict

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4.6 Ethical Considerations

Prior to collecting any data, I applied for ethical approval from my department’s ethics committee. Then I gained permission from the Head of the Mathematics department. To gain this permission, I send an e-mail including an information sheet for my study. As Boeije comments the researcher has the

“obligation to outline fully the nature of the data collection and the purpose for which the data will be used to the people or community being studied in a style and a language that they can understand” (Boeije, 2009, p. 45).

This was done through the information sheets given to all the participants of the study. These included the: aims of my study, participants involvement in the various stages of data collection, and the use of data for the thesis and publications (see information sheet in Appendix 12.1). Additionally, I used consent forms to receive written consent from the participants to audio- record the interviews and provide the document data (see consent forms in Appendix 12.2). The lecturers of the three modules agreed for me to make a short introduction to my research at the end of a lecture and they forwarded an invitation electronically from me to their students regarding my research. It was always made clear that participation in the study is completely voluntary and that it would not have any effect with their studies. Also, the students and lecturers who participated in the study were informed that they could drop out of the study if they wished to (Boeije, 2009, p. 45). Moreover, they had the right to contact my supervisor or the Head of the department in case they did not feel comfortable with their involvement in the study. Two of the student participants decided that they only wished to take part in the first interview and thus a second interview was not conducted (B2 and B8). Particularly for the participation in the second interview, where students were asked to solve past papers, I made clear that I would not offer any support to them. All the interviews took place on the university premises, and the participants were offered juice and cakes during the interview. There was no other form of payment to the participants of the study. Prior to observing any lectures, the lecturer had signed a consent form and informed the students about my research and my presence in the lectures.

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The raw data (interviews, interview transcripts, observation notes and document data) was locked up physically, in a locked cabinet, and electronically, in a password secured folder. Adhering to confidentiality and anonymity, I anonymised the data using aliases. For the students who participated in the interviews, I used a letter and a number. Similarly, for the lecturers of each of the module and for the students’ examination scripts I used numbers. Finally, the university where the study was conducted was not named and when the name of the university appeared in a task, I concealed this information. As Stake points out

“In social research the dangers are almost never physical. They are mental. They are the dangers of exposure, humiliation, embarrassment, loss of respect and self-respect, loss of standing at work or in the group” (Stake, 2010, p. 206)

All the forms of data prior to analysis were only shared with the supervisors. The analysed data was presented in conferences of the Mathematics Education community: British Society for Research into Learning Mathematics (BSRLM), European Congress on Research in Mathematics Education (CERME) and International Network for Didactic Research on University Mathematics (INDRUM). It was also shared in meetings with the Research in Mathematics Education (RME) group of the university.

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