The Knowledge Quartet: a Framework for Developing Mathematics Teachers’ Content Knowledge
Anne Thwaites, Tim Rowland and Peter Huckstep
University of Cambridge
Paper presented at the European Conference on Educational Research, University College Dublin, 7-10 September 2005
This paper is a continuation and extension of our research into the mathematics content knowledge of prospective primary school teachers. In our earlier work, a grounded theory approach to data analysis led to the identification of a ‘Knowledge Quartet’, with four broad dimensions which we are calling foundation, transformation, connection and contingency. Teachers’ subject content knowledge could be evidenced in practice through these dimensions. This most recent development applies and builds on this theoretical framework. In this paper we give a case study of a trainee who has been invited to reflect upon her lesson a few hours after teaching it. In talking freely about her lesson and also by responding to an interviewer who had previously viewed and analysed a video recording of the lesson, further insights were gained by both the trainee and the researchers.
The process of audit and remediation of subject knowledge within primary Initial Teacher Training (ITT) became a high profile issue following the introduction of government requirements for ITT (DfEE, 1998), and provoked a body of UK research on prospective primary teachers’ mathematics subject knowledge (e.g. Rowland, Martyn, Barber and Heal, 2000; Goulding and Suggate, 2001; Jones and Mooney, 2002; Sanders and Morris, 2001, Morris, 2001; Goulding, Rowland and Barber, 2002). However, the balance of advice has gradually shifted away from assessment of this knowledge, by some form of ‘testing’, towards seeking evidence for trainees’ subject knowledge in the act of teaching i.e. within the school-based placements (TTA, 2002a, b).
The focus of the case study reported in this paper is, in the first instance, on the ways a trainee’s mathematics content knowledge may be seen to be cashed out in practical teaching during a school-based placement. In this respect, it is similar to previous work. (Rowland, Huckstep and Thwaites, 2003a) However, what is distinctive about this particular study is that some of the inferences made about the trainee’s mathematics content knowledge are informed by the reflective responses that take place in an interview with the trainee during the same day that her lesson had been taught. This process of discussion and reflection is at heart of the learning relationship between trainee teacher and mentor (university and school-based) and the earlier work has been used as a basis of guidance to mentors. The new study provides another perspective.
Context and purpose
In the UK, most trainee teachers follow a one-year, postgraduate course leading to a Postgraduate Certificate in Education (PGCE) in a university education department. Over half of the PGCE year is spent working in schools under the guidance of a school-based mentor. Placement lesson observation is normally followed by a review meeting between a school-based teacher-mentor and the student-teacher (‘trainee’, in the terminology of recent official UK documentation). On occasion, a university-based tutor will participate in the observation and the review. Research shows that such meetings typically focus heavily on organisational features of the lesson, with very little attention to mathematical aspects of mathematics lessons (Brown, McNamara, Jones and Hanley, 1999; Strong and Baron, 2004). The purpose of the research reported in this paper was to develop an empirically-based conceptual framework for the discussion of the role of trainees’ mathematics SMK and PCK, in the context of lessons taught on the school-based placements. Such a framework would need to capture a number of important ideas and factors about content knowledge within a small number of conceptual categories.
This study, as in the previous one, is of a trainee’s final professional placement during the last of term of a one-year PGCE course. From earlier grounded theory research (e.g. Rowland, Huckstep and Thwaites, 2003a), critical episodes in trainees’ teaching were identified. In these, subject matter knowledge (SMK) and pedagogical content knowledge (PCK) could be articulated in terms of one or more of 18 coded descriptions such as adherence to textbook and recognition of conceptual appropriateness. But it soon became apparent that 18 codes were too unwieldy for classroom observation. The resolution we have termed the ‘Knowledge Quartet’; the 18 categories are grouped into four broad, superordinate units – the dimensions or ‘members’ of the quartet. (Rowland, Huckstep, and Thwaites, 2003b) We have named these units foundation, transformation, connection and contingency.
For this new part of the study, a trainee was again videotaped teaching a lesson by one member of the research team. Soon afterwards the team met to view the tape and to identify some key episodes in the lesson using the codes developed in the earlier work. It is important to note that our approach to analysing the lesson was not grounded theory, as before, since we now viewed the lesson through the theoretical lens of the Knowledge Quartet. The procedure from this point marked a further development in both methodology and application. Later, one team member met with the trainee to view the videotape and to discuss some of the episodes. An audio recording was made of this discussion, and a full transcription of the recording was made later. In this stimulated recall, the trainee was encouraged to reflect on the lesson. In the first part of the ‘interview’ most of the videotape was shown and the trainee invited to ‘think aloud’ about any aspect of the mathematical content of the lesson. (Unsurprisingly, the re-run of their lesson provoked comments on other matters - the context, the children, and so on). The interviewer then drew the trainees’ attention, one at a time, to key issues that had been identified by the team in their earlier analysis using the Knowledge Quartet, and invited the trainee to comment and offer their own perspective on the relevant episodes. We aimed to complete the three stages (videotaping the lesson, team reviewing the lesson, discussion with the trainee) in a short time span. In the case considered in this paper, the whole process occurred within one day.
The Knowledge Quartet
The brief conceptualisation of the knowledge quartet that now follows draws on the extensive range of data (24 lessons) collected from the study so far. Some aspects of the characterisation below will emerge from our consideration of the lesson that we have singled out for attention later in this paper.
Our first unit is rooted in the foundation of the trainees’ theoretical background and beliefs. Both empirical and theoretical considerations suggest that the other three units flow from it. Our conceptualisation of this category includes trainees’ beliefs, knowledge and understanding – gained both in their ‘personal’ education and in their learning in the academy, in preparation for their role in the classroom. The key, defining feature of the category under immediate consideration is its propositional form (Shulman, 1986). We take the view that the possession of such knowledge informs pedagogical choices and strategies in a fundamental way. The key components of this theoretical background are: knowledge and understanding of mathematics per se; knowledge of significant tracts of the literature and thinking which has resulted from systematic enquiry into the teaching and learning of mathematics; and espoused beliefs about mathematics, including beliefs about how and why it is learnt. (Bramall and White, 2000)
The second of these four categories shifts attention from the acquisition of knowledge to knowledge-in-action as demonstrated both in planning to teach and in the act of teaching itself. At the heart of this category, and acknowledged in the particular way that we name it, is Shulman’s observation that the knowledge base for teaching is distinguished by " … the capacity of a teacher to transform the content knowledge he or she possesses into forms that are pedagogically powerful". (1987, p. 15) Our second category, unlike the first, picks out behaviour that is directed towards a pupil (or a group of pupils), which follows from deliberation and judgement. The trainees’ choice and use of examples has emerged as a rich vein for reflection and critique. This includes the use of examples to assist concept formation, to demonstrate procedures, and the selection of exercise examples for student activity.
The next category unites certain choices and decisions that are made for learning discrete parts of mathematical content. It concerns the coherence of the planning or teaching displayed across an episode, lesson or series of lessons. Mathematics is notable for its coherence as a body of knowledge and as a field of enquiry, and the cement that holds it together is reason. In recent, influential work (Askew, Brown, Rhodes, William and Johnson, 1997) five of the six case study teachers, found to be highly effective, gave evidence of a ‘connectionist’ orientation. Our conception of coherence includes the sequencing of topics of instruction within and between lessons, including the ordering of tasks and exercises. To a significant extent, these reflect deliberations and choices entailing not only knowledge of structural connections within mathematics itself, but also awareness of the relative cognitive demands of different topics and tasks.
The fourth member of the quartet is distinguished both from the possession of a theoretical background, on the one hand, and from the planned deliberation and judgement involved in making learning meaningful and connected for pupils, on the other. Our final category concerns classroom events that are almost impossible to plan for. In commonplace language it is the ability to ‘think on one’s feet’; it is about contingent action and is thus knowledge-in-interaction. The two constituent components of this category that arise from the data are the readiness to respond to children’s ideas and a consequent preparedness, when appropriate, to deviate from an agenda set out when the lesson was prepared. When a child articulates an idea, this points to the nature of their knowledge construction, which may or may not be quite what the teacher intended or anticipated.
Sonia had completed a joint honours degree in Religious Studies and Education. Although she had passed the required GCSE mathematics examination she joined the PGCE course with concerns about her own mathematical knowledge and confidence. Her lesson is with a Year 4 class (pupil age 8-9). She begins with a short exploration of shape, introducing the learning outcome of the lesson - that pupils will be able to "... make and describe repeating patterns which involve translations and/or reflections". Then she moves on to a numerical task, as an ‘oral and mental starter’(1), before returning to the work on shape. We shall outline and discuss three episodes within the lesson.
Sonia’s differentiated oral and mental starter concerns finding complements in 100 and 1000. The three pairs of examples she uses are:
82 + ? = 100 35 + ? = 100 63 + ? = 100
820 + ? = 1000 350 + ? = 1000 630 + ? = 1000
A valuable insight into Sonia’s ability to undertake subject knowledge transformation comes from her response - firstly instantaneous, then reflective - to the interviewer’s question about the choice of examples that she uses for this activity.
Teachers often structure their examples to achieve optimal learning uptake. Sometimes this might be simply a matter of making them progressively more demanding in some way as their pupils display success. But this raises the question of what counts as one complement in 100 being more demanding than another. More specifically, does Sonia have explicit (or implicit) decision criteria for her choices? Whereas in our earlier work we were obliged to make inferences from our observations, in the present case we were in a position to consult to Sonia on the matter. In fact, her first response was consistent with our own inference:
Interviewer: You know when you did these … something add something equals…
Interviewer: … 100 and 1000 and so forth, and the examples that you chose were 82 …
Sonia: Completely random.
Sonia … there was whatever came into my head
It is tempting to suppose that since Sonia’s ‘choices’ involved no apparent deliberation, they must have been arbitrary. Yet on further questioning, she was able to account for what simply seemed to have come straight to her:
Interviewer: … Sometimes there’s a choice, when you’re giving examples, sometimes … students or teachers have a particular reason for doing it. In your case these were just sort of…
Sonia: What were they? There might have been a reason
Interviewer: 82, 35
Sonia: 35 because it was a smaller … was an actually smaller number, I remember the reason for that one.
Interviewer: So you had a smaller number after the …
Sonia: Yeah, after the big number. And then I made sure that that the … the last digit of the 63 was a different last digit to the other two
Interviewer: Why did you have the smaller one … in the middle?
Sonia: Don’t know. I just thought, this, I’d have a smaller number, like a substantially smaller number than 82
Sonia: The units … the ten was intentional but the unit was random, in that case. And in the last one, 63, the … ten was random but the unit was intentional.
Interviewer: Right, so there’s some … thinking behind it.
Through this discussion, a rationale for Sonia’s choice of examples has been teased out. She gives the impression that, although with help she is able to articulate her rationale, she was unaware of it in the moment (in the classroom) or until she was asked to talk about it. Ideally the examples that teachers use for pedagogical purposes should be chosen deliberately, and with care. This type of discussion can be helpful in reflecting on practice and making explicit those decisions in planning, both actual and potential, that can affect the quality of children’s learning experiences.
Perhaps the most interesting episode arises when Sonia dwells on the pupils’ solutions to 63 + ? = 100. Since she is presented with three answers 37, 27 and 47 the way is open for contingent action on her part.
It is to Sonia’s credit that she does not immediately pronounce the correct answer but decides to invite a volunteer to discuss his method publicly. Matt firstly finds the complement in 10 of the 3 of 63, saying "If you do 3 add 7 that makes 10". It is at this point that Sonia prompts him by asking "Where have we got to?". There is something ambivalent in this utterance. In saying it, Sonia could simply be drawing the pupil back onto the task by asking him how much of the problem had been solved. On the other hand, the "where" could be a tacit way of suggesting a place in a specific mathematical sequence suggesting that she is guiding Matt into a sequential (or ‘whole number’) method. Either way he takes the cue, and increasing the 63 by 7 writes 70 + 30 = 100. Sonia then ties things together asking "What have we added on?". She rings the 7 and the 30, asks the class what 30 and 7 make and finally draws out the answer to her original question: 63 + 37 = 100.
In the earlier analysis of the videotape, the research team had made a conjecture (about supporting a sequential calculation strategy) concerning the intention of the question "Where have we got to?". This was tested in the interview, with illuminating result:
Interviewer: Yes, and he said, … 3 add 7 that’s 10, so, … you … referring back to what you said earlier on, you make it up to a nice number.
Interviewer: … and you said, "Where have we got to?"
Interviewer: … and he said 70
Sonia: Yeah … I think, was it 63, was the number?
Interviewer: Yes, but he said three add 7 is 10 … were you trying to get him to do it in sequence, then?
Sonia: I thought that was what he was going to do, so I was just hoping he was, and tried to push him in that direction.
Interviewer: Yes, so "Where have you got to?" is just the right sort of prompt there …
In this interchange Sonia confirms that her intention was to draw out a sequential process from Matt, even though his reference to 3, 7 and 10 may have derived from an intended split-tens strategy. This may also have helped to clarify the thinking of those children who gave answers of 27 and 47.
The choices of shapes Sonia selects to transform in the next stage of her lesson reveals some shortcomings in her foundation knowledge. In particular, she does not appear to realise that the internal properties of a transformed figure can mask certain effects of a transformation, particularly reflection.
With the learning objective of pattern-making in place, she tries to establish that when a shape is transformed, a second shape is generated which is "the same" as the first. This is somewhat confusing, because if her notion of a transformation is a movement, when using objects, there will not be two shapes. The moved object will become the image of the transformation but the domain shape (the pre-image or ‘argument’ of the transformation) will no longer exist. Of course, this dilemma does not arise if the transformation is seen, not as a movement, but as a relationship between pairs of points (and in consequence between pairs of shapes) in the plane.
The shapes (including a circle and a rectangle) that she ‘chooses’ to use for demonstration are both have a high degree of symmetry, and for this reason they do not reveal a change of orientation under the transformations of reflection and translation. This points to some weak transformational thinking (in the sense of the knowledge quartet!) on Sonia’s part. The circle is spectacularly ineffective in conveying the particular properties of a translation. If a circle C has translation image C’, then C’ is also the image of C under a reflection or a rotation. An astute pupil presents her with an opportunity for contingent action. This pupil perceives the pedagogical inadequacy of these symmetrical shapes, and offers "If you reflect it with an L shape it wouldn’t turn out the same". This time, whilst Sonia endorses the pupil’s response she makes no attempt to enact his suggestion publicly. However, when questioned later the same day Sonia readily saw this as a missed opportunity:
Interviewer: … it’s about the boy who did the … who asked for the L-shape
Interviewer: The shapes that you chose were a rectangle
Interviewer: … and a circle, which … have got a certain amount of regularity
Interviewer: … if you flip the … the rectangle, the same … but the L-shape … hasn’t got any symmetry in it, if you like
Interviewer: Emm, so did you, were you aware of that, or just …
Sonia: [laughs] I took random shapes off a pile [laughs] um, yes.
Interviewer: Well, you can see the boy’s point …
Sonia: Oh yes, definitely.
Interviewer: It’s quite a good reply.
Sonia: If I were to do it again, I would …
Interviewer: It would be striking what has happened to the shape if itself it didn’t have any symmetry
Sonia: It would be much easier for them to see.
So here we hope that the discussion has helped Sonia extend her understanding of these transformations, and how particular example shapes can be used to demonstrate the essence of a specific transformation more effectively than others. By reflecting on selected aspects of the mathematical content of her teaching, Sonia is identifying areas of her mathematics content knowledge - both SMK and PCK - where there is scope for development. She is also becoming aware of areas where she had good instincts which she might now incorporate into rational decision-making.
The knowledge quartet has given us a framework for the analysis and review of trainees’ lessons from a mathematics content perspective. This enables a focus on the mathematics rather than more generic issues such as organisation and management of the learning. The new layer to our work has added the trainee teachers’ reflection on their lesson, assisted by viewing a recording of the lesson. We hope that this will help to develop further our understanding of the trainees’ own subject knowledge but also the ways in which we (as mentors as well as researchers) can engage them in reflective dialogue which supports and builds upon their existing subject knowledge.
Our purpose in incorporating the interview with the trainee into our procedure was in part to add to our insights on the lesson. Where previously we were limited to our own inferences and conjectures, we now had the perspective of the principal actor on his or her actions in the classroom. This is not to claim that the trainee’s understanding of what s/he did and said was in some sense ‘objective’, but it did offer the chance to test the compatibility of their understanding with ours. A second purpose of the interview was to support the trainee’s knowledge development by reflection (in discussion with a relatively knowledgeable other) on mathematical aspects of the lesson. This might be thought of as a laboratory version of the usual post-lesson review with a mentor. In this way, our procedure can be seen as exemplifying what Skott (2005) calls the ‘theoretical loop’ now arising from the dialectical relationship between theory and practice.
Theoretical constructs to conceive of the role of the teacher as well as practical suggestions for teaching and for supporting teacher development are now based on studies of the interactions of mathematics classrooms. (Skott, 2005, p. 1)
1. Conforming to the English National Numeracy Strategy (NNS) guidance (DfEE, 1999), Sonia segments the lesson into three distinctive and readily-identifiable phases: the mental and oral starter, the main activity and the concluding plenary.
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This document was added to the Education-Line database on 11 November 2005