4.1.1 Zenobia’s epistemology of teaching/learning in tutorials
I interpreted Zenobia’s views for teaching/learning in tutorials from my analysis of interview data, where Zenobia explained to me her underlying considerations and thinking for what she did in observations. I found consistency between what Zenobia did in various observations and what she said in distinct discussions with me. This finding indicates that Zenobia drew on her views for teaching/learning in order to act in tutorials. In other words, these views form her thinking and perception for her teaching actions; and as such, they form her epistemology of teaching/learning in tutorials.
4.1.1.1 Zenobia’s views of small group tutorials and her role as a tutor
I discussed with Zenobia about SGT1 and SGT8 after the end of my last observations of her SGTs. In that discussion, I asked Zenobia why she always starts the tutorials with a discussion with the students about their welfare and their suggestions for group work. Zenobia stressed the pastoral aspect of the SGT setting where the tutor offers week-to-week care to the students about their well-being at the University. Within this setting, she viewed her role as tutor to be about checking that the students’ welfare is OK every week. She also distinguished the tutor from “just another person testing them all the time” by saying that the tutor should not only care pastorally about the students, but should also be “on their side”.
In the same discussion about SGT1 and SGT8, my next question to Zenobia was why during the tutorial she used humour and valued the students, for example, by telling them “good job”. Her response gave me insight into what she meant by a tutor being on the students’ side and, in that case, why she viewed “that the students work harder if they feel like you are on their side and you care about them.” In her response, Zenobia stressed her communication with students which enabled them “to be comfortable enough in that group, to feel safe in that group, not to mind admitting
what they know, what they don’t know, what they want help with or whatever”. So in order to have students who “speak” in the tutorial and “work harder”, she viewed her role as breaking down the barrier of the member of staff and being on the students’ side.
I think it is just if they feel comfortable with me, if they know that I am their friend and I am their advocate, then they are more likely to be willing to admit that they don’t understand something or to ask a question; than, if they are intimidated and see me as the lecturer, the member of staff. So, I am trying to break down that barrier so that we can communicate more effectively.
Excerpt 1_Discussion about SGT1 and SGT8 The section Creating students’ positive feelings of this chapter, Section 4.2.2.2, provides evidence of my analysis and interpretations about what Zenobia does in her tutorials to care pastorally about the students and to be “on their side”. In my question why after the pastoral discussion with the students, she talks with the students and together they select a small number of tasks or one task to tackle in the SGT, she said:
Almost all the instruction that they [students] get involves people [lecturers] having decided ahead of time useful examples to show them and going through them for them. And I feel that at least once a week they need a chance for them to direct it and to really check each step that they have really understood what it’s going on.
Excerpt 2_Discussion about SGT1 and SGT8 In Excerpt 2, Zenobia talked about lecture teaching where lecturers choose and use examples which they demonstrate on the board. They usually choose examples before the lecture and demonstrate without being able to check the students’ meaning making. My interpretation from Excerpt 2 is that, contrary to lectures, Zenobia’s views for the small group tutorial is to let students select the task. Her view of her role as tutor is then to check at “each step that they have really understood”. Zenobia’s declaration about students who may “have really understood” is illuminated in the
Furthermore, a first insight into the way she lets students select tasks and thus “direct” the tutorial is in the next section about her views of what teaching is (Section 4.1.1.2). A full analysis is in the section Selecting tasks of this chapter (Section 4.2.2.1).
To conclude, her views of the SGTs include pastoral care for students; and students who “speak” and choose as a group the tasks in the tutorial time. In these SGTs, she views a threefold role as a tutor: to check that the students’ welfare is OK every week; to break down the barrier of the member of staff thus to be on the students’ side; and to check at each step of the task solution that the students have really understood. 4.1.1.2 Zenobia’s views of what teaching is and its connection with mathematical research
In Zenobia’s tutorials, I usually observed that the students discussed with each other and agreed about the selection of a few tasks with which they faced difficulties. Then, they agreed with Zenobia to solve one or two of these tasks in the tutorial. Zenobia discussed with the students each step of the process of the solution of the tasks; in particular, she prompted them to elicit the steps. In discussion with her after SGT2, I asked her why she does not present solutions of tasks on the board in a lecturing format. She responded that she does so only in case her “prodding doesn’t result of anything” and she gives “the next step and the next step”. However, she considered that the students come to the SGT with difficulties in tasks from the lecture material. Her view was that another lecturing hour in which the tutor guides the students through the mathematics might result into the same difficulties for students. In contrast, a tutor who discusses with the students and checks at each step that the students have made sense has the potential to resolve the difficulties. Zenobia also started to explain to me her thinking about what she discusses with the students in her effort to elicit the solutions of tasks.
[W]hen I work with students and sort of recognise where they struggle, I have thought, well, when I’ve got a new concept I struggle with how do I tackle it and that has informed the way that I teach now. Like I read something in a paper and I am like ‘Well I have no idea what that means, can I think of a single example that fits that? What’s the simplest possible example I can think of that fits this or what’s the simplest possible example that doesn’t fit to this? Excerpt 3_Discussion after SGT2 Excerpt 3 is crucial for Zenobia’s design of teaching with regard to the way of working with the mathematics to resolve the students’ difficulties. She declared that the way she works with the mathematics in her research to enable herself to make sense of a new concept has informed the way she works with the mathematics in her teaching to enable the students to make mathematical sense. My analysis of data indicates that various mathematical heuristics are central to this way of working. In the section Decoding the mathematics and encoding the mathematics of this chapter, Section 4.2.2.5, I draw on data to provide insight into the way of working with the mathematics and the nature of the connection between Zenobia’s teaching and mathematical research.
4.1.1.3 Zenobia’s views of what making sense of mathematics is
Observational data from SGT10 sheds light into Zenobia’s views of what making sense of mathematics is; and thus into her declaration in Excerpt 2 about checking whether students “have really understood”. In this tutorial, the following discussion between Zenobia and the students took place:
Zenobia: Do you guys find the way that I use examples to extract your understanding of the definitions and then work back again useful? How do you guys like to understand the definitions? How do you go about understanding a definition?
St: I use ‘The exercise teaches the theory’. The theory doesn’t teach the exercise.
Zenobia: Right, yeah. Exactly. So, doing it in an example is what makes you actually understand what is going on. It’s not that you understand the definition and then the exercise is straightforward. Which is exactly how we design them, right? We do design them to give you context in which to understand.
Excerpt 4_SGT10 Observation In Excerpt 4, Zenobia referred to her use of examples of a concept that is difficult for students. She said that she uses examples to elicit the students’ sense making of the definition of a concept and then works to solve the task. The student’s perspective was that work with examples of a concept is possible to promote their sense making of the definition of the concept. My interpretation is that at this point of the discussion, Zenobia shared with the students her view of what making sense of mathematics is, she said:
“doing it in an example is what makes you actually understand”.
In Excerpt 3, she declared that in her research she devises examples to figure out what works and what does not work for the new concept in order to make sense of it. From Excerpts 3 and 4, I interpret that her view of ‘what making sense of mathematics is’ is informed by her own mathematical research practice. Concluding the above discussion with students, Zenobia stressed a difference between her view of mathematical sense making and views of mathematical sense making in the design of lecture teaching: lecturers (“we” in Excerpt 4) design the presentation of the definitions in a way to give students “context in which to understand” so that “the exercise is straightforward”. However, evidence in Excerpt 4 indicates that within the latter design the student does not consider that the exercise is straightforward.
4.1.2 Zenobia’s epistemology of mathematics
Zenobia continued the discussion with the students, from which Excerpt 4 is a part, by sharing her views on the nature of mathematics. Before starting to talk about the nature of mathematics, she explained to the students the Platonic ideals with the metaphor of “person”.
What is a – what we call – “person”? What is our vision of “person”? So, Plato would have said that there was some sort of Platonic ideal of person and that we recognise anything on Earth as representing a flawed version of that Platonic ideal of “person”. And that’s how we come to recognise that somebody is a person. But at the other end, the more modern version – I don’t remember who said this – is that it’s a cultural consensus to lob a set of objects together and give them a label. But that label is prone to change its meaning as our experience of what those… You know, that we construct a generalisation of “person”, but that generalisation of “person” is not fixed, because if we – for instance – had never met a female person, because we’re mathematicians and we don’t know any women, and then suddenly we meet a female person, it’s not that she’s not a person because she doesn’t fit the ideal of “person” which in our brains is a male person. It’s that – suddenly – the concept of “person” has to change to accommodate a wider set of “person” than we previously had had, right?
Excerpt 5_SGT10 Observation In this excerpt, Zenobia started to explain her epistemology of mathematics according to which a mathematical object is a Platonic ideal; and people come to make sense of it by familiarising themselves with the cultural consensus of their time about the object. In this discussion with the students, Zenobia’s use of humour is evident in her reference to the cultural group of mathematicians. She said that despite the consensus of “person” as the male person in the cultural group of mathematicians, the female person also belongs in the Platonic ideal of person; resulting into accommodating a wider consensus of “person” in that cultural group. My analysis from Excerpt 5
referred to the history of mathematics to explain to the students her view on the nature of mathematics.
In mathematics, we do rigorously define things. And so, it’s a situation where – from the set of examples that we have – we’ve come up with an ideal idea, and then we can actually rigorously then check that something is in that or not. And what happens in mathematics is that if we see a more general version of things that doesn’t fit that, but that still has some things in common with it, then we create a new definition that’s more general. We come up with new definitions any time we recognise that there are some sets of structures that have some relevance. But it really does emerge out of the examples. And if you look at the history of mathematics, it’s not that people have had the idea of a function. It’s that they’ve had lots of examples of functions and they’ve tried to distil what the critical characteristics of a function are. Does that make sense? So, I think it’s a very natural way to think about the relationship between examples and theories – it’s that we don’t define definitions just off the tops of our heads. We define them because they capture a behaviour we see in examples that have interesting kinds of properties.
Excerpt 6_SGT10 Observation Excerpt 6, with reference to the history of the development of mathematics, is key for the connection between Zenobia’s reading of the history of mathematics, her views on making mathematical sense, her views on researching mathematics and her views on teaching mathematics. She said that mathematicians come up with a consensus of a mathematical object from sets of relevant structures, or properties, distilled out of examples. That consensus forms a definition of the mathematical object which can be accommodated at a later stage to better describe this ideal object.
Considering my analysis of Excerpts 3-6 into Zenobia’s views, it seems to me that in the case of Zenobia’s teaching a path of informing from her views on the history of mathematics, to her views on the sense-making of mathematics, to her views on conducting her own research in mathematics, to her views on the teaching of mathematics is revealed. In the next sections, I draw on data to unpack Zenobia’s
views with regard to her teaching practice which I distinguish in my analysis into strategies and tools for teaching.