International Portal of Teacher Education

Source: Journal of Mathematics Teacher Education volume 25 issue 2, pages191–215

(Reviewed by the Portal Team)

This paper considers the place of mathematical activity in Initial Teacher Education (ITE) for elementary school preservice teachers.
As a group of researchers working on ITE programmes in four English universities, the authors explore preservice teachers’ learning within the context of mathematical activities in a university-based ITE session, and as they plan and evaluate their subsequent teaching of mathematics.
Exploring the kinds of awarenesses that preservice teachers articulate within an environment that encourages deliberate retrospective analysis of their learning and implications for teaching is highly significant for understanding mechanisms that can enhance prospective teachers’ capabilities for critical pedagogical action.
They seek to address the following research question: What are the kinds of expressed awarenesses that emerge when preservice teachers engage in a process of deliberate retro-spection of their own mathematical activity and pro-spection of their teaching relating to generalisation of visual patterns?
The paper contributes a new dimension to the discussion about the focus of novice elementary school teachers’ retrospective reflection by examining how deliberate retrospective analysis of doing mathematics, and not only of teaching actions, can develop awarenesses that underlie the growth of expertise in teaching mathematics.

Methodology
The authors collected pilot data of preservice teachers’ growing pattern activities in two of their universities, and of their reflections on taking part in the activities.
Between each taught session they met as a group to discuss their interpretations of these data, recognising and respecting their differing backgrounds and perspectives (Reid 1996).
During these sessions, their focus was on identifying and understanding what helped the preservice teachers to generalise, on the order of activities and how effective their data collection tools were.
These decisions were refined to develop their methodological approach.

Research design and methods
Two universities were selected in which to focus the fieldwork: both of them provide one year postgraduate elementary Initial Teacher Education (ITE) programmes.
University One (U1) offered a mathematics specialist option within its ITE programme, and all eight preservice teachers taking this option agreed to participate in the study.
University Two (U2) ran a generalist programme, including a mathematics component for all its ITE students, and seven of these preservice teachers opted to be involved in the study.
One of the authors of this paper was a teacher educator on the U1 programme; another author was a U2 teacher educator.
In each of these programmes, a taught session on algebraic reasoning took place; this included a selection of growing pattern activities agreed by the two teacher educators.
After each class session, on the same day in most cases, the participants were interviewed individually by visiting members of the research team who had not been involved in the class.
They asked introductory questions about the participant’s mathematics background, and their approaches to the growing pattern tasks.
The participants brought jottings made in the session to inform their description of their approaches.
Semi-structured interviews provided an opportunity for participants to revisit actions and to analyse retrospectively the mathematics that they enacted, illuminating key elements from their own perspectives.
Lesson plans and subsequent lesson evaluations were collected.
A group session, where all the preservice teachers in attendance contributed to a discussion about their own lessons, was audio recorded with their permission.

Data analysis
The authors analysed data from interviews, lesson plans, lesson evaluations and the recorded follow-up university session over two phases.
In the first phase, the project team worked in pairs focusing on the mathematical actions described by each preservice teacher in the interviews, and as illustrated in their jottings.
They identified preservice teachers’ different approaches to generalising the visual patterns, and identified shifts in reasoning, with attention to recursive and functional relationships (Ferrara and Sinclair 2016).
In the second phase, the authors’ analysis foregrounded Mason’s (2010) “Working on mathematics for themselves” as a key mechanism for developing teachers’ “spection”.
In this paper, they present findings from data that were analysed across the dimensions of retro-spection and pro-spection relating to “Working on mathematics for themselves”.

Findings and discussion
The authors have presented three themes of expressed awarenesses relating to the teaching and learning of mathematical reasoning with a focus on growing, visual patterns: awareness of “how you see” a pattern, awareness of the difficulty translating “what you see” in a pattern into a mathematical expression of generalisation, and awareness of the role that resources may play in helping learners to “see” the pattern.
Awareness of “how you see” a pattern was strongly embedded in preservice teachers’ noticing of what changes and what remains the same in the visual patterns when working towards identifying the general rule.
In deliberate retrospective analysis of this approach to pattern generalisation and pro-spection of their teaching, preservice teachers articulated the importance of making distinctions between different elements of the visual pattern in order to establish meaningful relations between visual and symbolic representations.
The value of making distinctions appears as a key learning feature in the current study, in preservice teachers’ deliberate retrospective analysis of their own mathematics activity and their prospective approach to teaching.
The authors identified examples that illustrate how deliberate retrospective analysis and articulation of their own mathematics actions and struggles can sensitise preservice teachers to the difficulties that their students may face in the classroom.
This kind of sensitivity was demonstrated in preservice teachers’ anticipation of potential difficulties that their students may face and their planning for pedagogical action.
On this basis, the authors argue that teacher education environments that encourage a process of prompted retro-spection of the preservice teachers’ own learning, and pro-spection of teaching, can support and enhance their ability to respond rather than react.
To respond, according to Mason (2010 p. 37), is “to make an intentional, conscious, considered choice of action”, which he considers to be rare, as “we usually react”.
This notion relates to creating stepping stones to continued learning from the enactivist perspective, for the purpose of enabling preservice teachers to maintain an “on-going alertness” and become “aware of their own awareness” (Brown and Coles 2012, p. 223), thus taking the middle path between the extremes of spontaneous, unreflective action and rational calculation and deliberateness that can characterise the actions of novices (Varela 1999).
The notion of “seeing” a pattern emerges strongly in the data and connects the three themes of expressed awarenesses that the authors have described and exemplified.
This is consistent with the enactivist notion of embodied cognition, whereby “perception consists of perceptually guided action” (Varela 1999, p. 4), as well as previous evidence that has supported the importance of visualisation in pattern generalisation (e.g. Vale et al. 2018).
When working with the patterns, many of the participants chose to use drawing, shading, colouring and physical resources to support their “seeing” of elements of the structure. Preservice teachers’ verbal reports of their drawing actions provided an interesting insight into action as a visible aspect of their “embodied (enacted) understandings” (Davis 1995, p. 4), both in cases where the visual structure of the pattern was discerned leading to generalisation , as well as in cases of difficulty engaging with the visual elements of the pattern.
This study focused on preservice elementary teachers’ deliberate retrospective analysis on their own processes and experiences of doing mathematics for themselves during university-based taught sessions.
The authors explored the kinds of awarenesses that they articulated and the connections they then made to their future teaching, thereby tracing threads between retro-spection of learning and pro-spection of teaching.
Their mathematical focus on generalisation of visual growing patterns proved to be an area which many found challenging.
Their findings support their contention that attempting to generalise visual growing patterns can enable preservice teachers to sensitise themselves to students’ struggles.
They argue that learning experiences that trigger preservice teachers’ awarenesses and sensitivity to their own, individual, and possibly differing, embodied processes of pattern generalisation need to constitute an important component of teacher education programmes, in order to prepare teachers who will be sensitive and astute to the individually embodied processes of knowing manifested by the learners in their classrooms.

References
Brown, L., & Coles, A. (2012). Developing ‘deliberate analysis’ for learning mathematics and for mathematics teacher education: How the enactive approach to cognition frames reflection. Educational Studies in Mathematics, 80(1–2), 217–231.
Davis, B. (1995). Why teach mathematics? Mathematics education and enactivist theory. For the Learning of Mathematics, 15(2), 2–9.
Ferrara, F., & Sinclair, N. (2016). An early algebra approach to pattern generalisation: Actualising the virtual through words, gestures and toilet paper. Educational Studies in Mathematics, 92(1), 1–19.
Mason, J. (2010). Attention and intention in learning about teaching through teaching. In R. Leikin & R. Zazkis (Eds.), Learning through teaching mathematics: Development of teachers’ knowledge and expertise in practice (pp. 23–47). New York: Springer.
Reid, D. (1996). Enactivism as a methodology. In L. Puig & A. Gutiérrez (Eds.), Proceedings of the 20th annual conference of the international group for the psychology of mathematics education (pp. 203–210). Valencia: PME.
Vale, I., Pimentel, T., & Barbosa, A. (2018). In N. Amado, S. Carreira, & K. Jones (Eds.), Broadening the scope of research on mathematical problem solving: A focus on technology, creativity and affect (pp. 243–272). Cham, CH: Springer.
Varela, F. (1999). Ethical know-how: Action, wisdom, and cognition. Stanford: Stanford University Press.
https://www.heartoftheart.org/wp-content/uploads/2017/08/Varela-F.-J.-19.... Accessed 15 October 2020.