GROWING MATHEMATICAL UNDERSTANDING: LAYERED OBSERVATIONS

Jo Towers University of Calgary towers@ucalgary.ca

Lyndon Martin University of British Columbia lyndon.martin@ubc.ca

Susan Pirie University of British Columbia susan.pirie@ubc.ca

Abstract

This paper discusses the use of the Pirie-Kieren Dynamical Theory for the Growth of Mathematical Understanding. In particular it illustrates and draws upon two recently extended elements of the theory, those of ‘folding back’ and ‘teacher interventions’. From an initial analysis drawing upon the Theory, significant instances of ‘teacher interventions’ and ‘folding back’ are identified and focused upon in greater detail. We suggest that by analysing these points in the pathway of growth a fuller, thicker, and more complete account can be developed of the ways in which the understanding of learners grows. This analysis illustrates the power of the Pirie-Kieren Theory as a theory for the growth of mathematical understanding, and illuminates the way in which different elements of the theory can be combined to provide a rich, multi-layered account of what is observed.

The Pirie-Kieren Dynamical Theory for the Growth of Mathematical Understanding

The Pirie-Kieren theory has been fully presented and discussed in a number of previous PME meetings (see, for example, Pirie & Kieren, 1992; Martin, Pirie & Kieren, 1996; Martin & Pirie, 1998) and many of its features have been set out and elaborated there and elsewhere. It will therefore only be briefly reviewed here. This theory provides a way of considering understanding that recognises and emphasises the interdependence of all the participants in an environment. It shares the enactivist view of learning and understanding as an interactive process (Davis, 1996; Varela, Thompson & Rosch, 1991) and is based in a belief that although understanding is still the creation of the learner, the classroom, curriculum, other students, and actions of the teacher occasion such understanding, which can be seen as co-emergent with the space in which it was created (Varela, Thompson & Rosch, 1991). The theory, developed to offer a language for, and way of observing, the dynamical growth of mathematical understanding, contains eight potential levels for understanding. These are named Primitive Knowing, Image Making, Image Having, Property Noticing, Formalising, Observing, Structuring and Inventising.

A diagrammatic representation or model is provided by eight nested circles (see Figure 1). This nesting and the associated traces of students’ understanding illustrate the fact that growth in understanding need be neither linear nor mono-directional. Growth occurs through a continual movement back and forth through the levels of knowing, as the individual reflects on and reconstructs his or her current and previous knowledge. When faced with a problem at any level that is not immediately solvable, an individual will need to return to an inner layer of understanding. This shift to working at an inner layer of understanding actions is termed folding back and enables the learner to make use of current outer layer knowing to inform inner understanding acts, which in turn enable further outer layer understanding. The theory suggests that for understanding to grow and develop, folding back is an intrinsic and necessary part of the process. This element of the theory was explored and expanded by Martin (1999) who developed a framework for describing in detail an act of folding back. Included in this framework is the development of the notion of folding back to collect (Martin, Pirie & Kieren, 1996). An act of collecting involves the retrieval of an earlier understanding without the modification of the earlier understanding. Instead, there is the action of reviewing it in the context of the present problem.

The second recently extended element of the Pirie-Kieren theory on which we will focus is the notion of teacher interventions. Understanding what occasions the growth of mathematical understanding requires an understanding of the coemergence of teachers’ and students’ actions and understandings in classrooms. In studying these interactions and co-evolving understandings, Towers (1998a, 1998b) has developed a number of intervention themes that describe teachers’ actions-in-the-moment. The intervention themes identified (showing and telling, leading, shepherding, checking, reinforcing, inviting, clue-giving, managing, enculturating, blocking, modelling, praising, rug-pulling, retreating, and anticipating) acknowledge that the growth of students’ mathematical understanding is dependent on, but not determined by, teaching (Davis, 1996).

Layering a Story

The Pirie-Kieren theory provides a way to analyse the mathematical actions of a learner or a group of learners and can describe and account for the way that their mathematical understanding grows and develops. Through the use of a technique known as “mapping”, a pathway of growth can be traced out on the model diagram, indicating the ways in which the understanding actions of the learners shift within the layers of the theory (see Figure 1).

We will focus particularly in this paper on two learners who were Grade 9 students in a middle-ability grouping taught by Towers. The session was concerned with finding the area of any segment of any circle. The prompt offered to the students was in the form of a diagram of a circle with a sector and segment drawn as shown in Figure 2, but with no dimensions offered. The students in question established the need to find the area of the sector of the circle first, and then proposed to subtract the area of the enclosed triangle. This is the point (Image Having) at which the mapping seen in Figure 1 starts:

Figure 1: The Pirie-Kieren model for the dynamical growth of mathematical understanding

Figure 2: Finding the area of a segment of a circle

The predominant impression of Towers’ teaching during this lesson is of a teacher who continually wants to encourage the pupils to move from working with the specific to thinking in more general ways. In doing this, she intervenes to help the pupils realise some of the limitations of their current understandings, and to consider how they need to broaden their existing images for the concept. This nudging towards a deeper understanding is an intervention style that Towers (1998a, 1998b) has named “shepherding”. A close analysis of the video of this lesson reveals that the teacher invites the students to explain their thinking in order to have a place from which she hopes to occasion their growth of understanding (for example she uses phrases such as “Tell me what you think you are doing…”). This intervention, occurring at a point at which the students are Image Having (Figure 1) but with an incomplete and partially incorrect image of the problem, enables the teacher to engage in an interaction which prompts the students to fold back to Image Making to build a thicker understanding that recognises the need for trigonometry. She challenges the students’ method (as they initially assume that the angle of intersection of the radius of the circle and the chord is a right angle), and offers a new perspective (prompting them to consider the chord as the ‘base’ of the enclosed triangle) when the students had been struggling for some time working with one of the radii of the circle as the base of their triangle. Rather than halt the flow of the moment with an explanation of why it would be better to use the chord of the circle as a base for the triangle, though, she simply smiles when one of the students recognises the value of the suggestion (evident in the student’s sudden recognition that she might “chop the triangle in half” thereby creating a right-angle), and instead prompts the students to access their own Primitive Knowing about trigonometry to help them continue (reminding them that “we’ve done something like that very recently, haven’t we?”) Such teaching requires a teacher to be willing to refrain from telling (at least immediately), and be aware that to move forward, these students need to fold back.

When we tried to consider in detail the folding back actions of the learners at this point, analysing the video suggests that this is not a simple automatic process, and that the learners are not able to instantly apply trigonometry to solve the new problem. They know what piece of their Primitive Knowing is needed and why they need it. They are able to fold back and begin to collect from their Primitive Knowing the formalisings and images for trigonometry that they need to use in the current problem. They know that they know what is needed, and yet their understanding is not sufficiently structured for the automatic recall of the appropriate piece of usable knowledge. It is this lack of ability to simply instantly recall that we call collecting. There is however, no act of modification of the earlier understanding, instead there is the action of finding and reviewing it in the context of the present problem. It is this reviewing that gives the ‘thickening’ effect of folding back. The students are able to re-collect the role of angle in trigonometry and especially the way in which dropping a perpendicular to the base from the apex of the triangle creates a right angle and halves the angle at the apex. They are then able to state what they think is the necessary trigonometric formula. However, they are not totally confident with this as an answer, thus confirming that the formula was not immediately accessible to them for automatic use. They deliberately choose to use an exercise book as an aide-memoir to confirm their collecting and quickly find a similar example that supports the formula they have stated. Throughout this episode they have to work on collecting small fragments of their understanding. There is a sense of the learners knowing and being aware that they have the necessary understandings but that they are just not immediately accessible.

Observations and Conclusions

The micro-analyses presented above of one small classroom incident have been offered as an illustration of a way of getting at the complexity of growing mathematical understanding. Although the mere noting that folding back and teacher interventions play a part in this episode begins to tell us something about the way that the understanding of these two students is growing, the subsequent more detailed study of these aspects of the theory provides a greater insight into the process through which this growth occurs. The power of the Pirie-Kieren theory is that it is a theory for, not a theory of, the growth of mathematical understanding, and as such it is validated by its usefulness to someone seeking to make sense of the growing mathematical understanding of learners. This paper illustrates how the theory might be useful both from the perspectives of teaching and of learning, through offering to an observer two different yet complementary lenses within the Pirie-Kieren theory through which to view particular elements of the process of coming to understand.

The choice of which lens to use in a particular situation is of course determined by the needs of the observer. The extended aspects of the Pirie-Kieren theory, folding back and teacher interventions, provide a way of layering an account of the developing understanding of learners and an observer may choose to make use of both of these elements, only one, or indeed neither – depending on what they want to observe and to account for. This paper has aimed to briefly illustrate the multi-layered, flexible and dynamic character of the Pirie-Kieren theory and, more specifically, the ways in which it can be used to look closely at growing mathematical understanding as it is seen to emerge.

References

Davis, B. (1996). Teaching mathematics: Toward a sound alternative. New York: Garland.

Martin, L. (1999). The nature of the folding back phenomenon within the Pirie-Kieren Theory for the growth of mathematical understanding and the associated implications for teachers and learners of mathematics. Unpublished doctoral dissertation, Oxford University.

Martin, L., & Pirie, S. (1998). She says we’ve got to think back: Effective folding back for growth in mathematical understanding. In A. Olivier & K. Newstead (Eds.), Proceedings of the twenty-second annual meeting of the International Group for the Psychology of Mathematics Education, 4 (pp. 255-262). Stellenbosch, South Africa.

Martin, L., Pirie, S., & Kieren, T. (1996). Folding back to collect: Knowing you know what you need to know. In L. Puig, & A. Gutierrez (Eds.), Proceedings of the twentieth annual meeting of the International Group for the Psychology of Mathematics Education, 3 (pp. 241-248). Valencia, Spain.

Pirie, S., & Kieren, T. (1992). The answer determines the question - interventions and the growth of mathematical understanding. In W. Geeslin & K. Graham (Eds.), Proceedings of the sixteenth annual meeting of the International Group for the Psychology of Mathematics Education, 2 (pp. 1-8). New Hampshire.

Towers, J. (1998a). Teachers’ interventions and the growth of students’ mathematical understanding. Unpublished doctoral dissertation, University of British Columbia.

Towers, J. (1998b). Telling tales. Journal of Curriculum Theorizing, 14 (3), 29-35.

Varela, F., Thompson, E., & Rosch, E. (1991). The embodied mind: Cognitive science and human experience. Cambridge, MA: MIT Press.