SECONDARY MATHEMATICS TEACHERS’ CONTENT KNOWLEDGE: THE CASE OF HEIDI Tim Rowland, Libby Jared and Anne Thwaites University of Cambridge, UK The Knowledge Quartet (KQ) is a theoretical framework for the analysis and development of mathematics teaching. It focuses attention on classroom situations when the teacher’s knowledge of mathematics and of mathematics-related professional knowledge comes to the fore. This focus for analysis and reflection is a stimulus to the enhancement of teacher knowledge and the improvement of teaching. The KQ has been developed in empirical research since 2002 in the context of elementary mathematics teaching: the purpose of this paper is to demonstrate the application of the KQ in a secondary school mathematics context. Keywords: teacher knowledge, secondary, novice teacher, Knowledge Quartet INTRODUCTION A programme of research at the University of Cambridge (SKIMA: subject knowledge in mathematics) from 2002 to the present has investigated the mathematics content knowledge of novice teachers, and the ways that this knowledge becomes visible in planning for teaching and within classroom instruction itself. Aspects of this research programme have been reported at each CERME conference since 2003 (e.g. Huckstep, Rowland & Thwaites, 2006). A significant outcome has been the identification of a framework for the observation, analysis and development of mathematics teaching, with a focus on the contribution of the teacher’s mathematical content knowledge. The framework in question, called the Knowledge Quartet, categorises events in mathematics lessons with particular reference to the subject matter being taught, and the mathematicsrelated knowledge that teachers bring to bear on their work in classrooms, as opposed to more generic features of the lesson. While Shulman’s distinction between subject matter knowledge and pedagogical knowledge underpins this ‘theory’ of mathematics teaching, the Knowledge Quartet (KQ) is more interested in the situations in which such knowledge comes into play than in categorising the different ‘kinds’ of mathematics teacher knowledge. The origins of the KQ were in observations of elementary mathematics teaching, and grounded theory methodology (Glaser and Strauss, 1967), in the context of one-year graduate elementary teacher preparation. The Knowledge Quartet According to the KQ, the knowledge and beliefs evidenced in mathematics teaching are conceived in four categories, or dimensions, named foundation, transformation, connection and contingency. The application of subject knowledge in the classroom always rests on foundation knowledge. This first category consists of knowledge and understanding of mathematics per se and of mathematics-specific pedagogy, as well as beliefs concerning the nature of mathematics, the purposes of mathematics education, and

the conditions under which students will best learn mathematics. The second category, transformation, concerns the presentation of ideas to learners in the form of analogies, illustrations, examples, explanations and demonstrations. The third category, connection, includes the sequencing of material for instruction, and an awareness of the relative cognitive demands of different topics and tasks. The final category, contingency, is the ability to make cogent, reasoned and well-informed responses to unanticipated and unplanned events. This conceptualisation of each of the four dimensions of the KQ is the synthesis of a set of codes which emerged from grounded analysis of the primary mathematics classroom data. Each dimension is composed of a small number of subcategories that we judged, after extended discussion, to be of the same or a similar nature. Table 1 shows the codes contributing to each of the four dimensions. Dimension

Contributory codes

Foundation:

awareness of purpose; adheres to textbook; concentration on procedures; identifying errors; overt display of subject knowledge; theoretical underpinning of pedagogy; use of mathematical terminology.

Transformation:

choice of examples; choice of representation; use of instructional materials; teacher demonstration (to explain a procedure).

Connection:

anticipation of complexity; decisions about sequencing; making connections between procedures; making connections between concepts; recognition of conceptual appropriateness.

Contingency:

deviation from agenda; responding to students’ ideas; use of opportunities; teacher insight during instruction.

Table 1: The Knowledge Quartet – dimensions and contributory codes

Further details are given e.g. in, Huckstep, Rowland & Thwaites (2006), and in Rowland, Turner, Thwaites & Huckstep (2009). We emphasise that the conceptualisation of the KQ has been refined, and the constituent codes enhanced, in an iterative response since 2003 to additional classroom data, in the process of application. This paper is best understood in the context of that process of theory evolution. Rationale From time to time questions have arisen about the adequacy and relevance of the KQ to analyses of mathematics teaching in secondary schools, and even to subjects other than mathematics. While we could not comment on the second of these questions, we believed that it would be meaningful and productive to test the application of the KQ to mathematics teaching beyond the primary years, and began to do so systematically in 2010. We began unsure about how well the KQ might ‘fit’ secondary teaching – and the work of trainee secondary mathematics teachers in particular – on account of certain characteristics of both the teachers and the subject matter being taught, when compared with their primary mathematics counterparts. In particular, these secondary trainees are all specialist mathematics teachers, with evidence of recent success in their own study of mathematics, and their teaching is supported by mathematics specialists throughout their

practicum placements. By contrast, generalist primary teachers, who have typically specialised in the arts and humanities in their own education, often lack confidence in their own mathematical ability (e.g. Green & Ollerton, 1999). From the mathematical point of view, the subject matter under consideration in secondary classrooms becomes significantly more abstract and complex than that in the primary school. As Potari and her colleagues indicated at CERME5, “teachers’ knowledge in upper secondary or higher education has a special meaning as the mathematical knowledge becomes more multifaceted and the integration of mathematics and pedagogy is more difficult to be achieved” (Potari et al., 2007, p. 1955). The purpose of this paper is to test, and to illustrate, the application of the KQ as an analytical and developmental tool in the context of novice secondary mathematics teaching. One could expect that the secondary context could necessitate annexing additional codes to those which emerged earlier in the analysis of primary mathematics teaching (Table 1). METHODS UK teacher education context In the UK, the majority of pre-service secondary teacher education takes place under the auspices of university education departments. Trainee mathematics teachers are required to have at least half of their undergraduate study in mathematics, and expected to have achieved at least an “upper second” class bachelor’s degree, before following a one-year, full-time course leading to a Postgraduate Certificate in Education (PGCE). In order to achieve a good theory/practice balance in the PGCE, for the last 20 years the programme has been conceived as a ‘partnership’ between the university and several collaborating schools, and two-thirds of the 36-week course is spent working in two schools under the guidance of mathematics specialist, school-based mentors. Participants The project participants were three volunteer trainee teachers from one secondary mathematics PGCE cohort at our university. Their professional placements were in different schools, all within a half-hour commute from the university. Data Collection The trainee participant in each school taught two ‘project’ lessons to the same class. These lessons took place in May, towards the end of the trainees’ second school-based placement, which had begun in January of the same year, so that the participants were familiar with their schools and with the pupils in their classes. One or two members of the research team (the authors) observed and videotaped each lesson. One tripod-mounted camera, operated manually, was placed at the rear of the classroom. Sound recording was via a radio microphone worn by the trainee-teacher. Trainees were asked to provide a copy of their lesson plan, for reference in later analysis. As soon as possible after the lesson, the research team met to undertake preliminary analysis of the videotaped lesson, and to identify some key episodes in it by reference to the KQ framework. These were fragments, typically 5-10 minutes long, in which two or more of the four KQ dimensions were

particularly salient, according to our preliminary analysis. Then, again with minimum delay, one team member met with the trainee to view a selection, typically two, of these episodes1 from the lesson and to discuss them, in the spirit of stimulated-recall (Calderhead, 1981). These interview-discussions addressed some of the issues that had come to light in the earlier KQ-structured preliminary analysis of the lesson. An audio recording was made of this discussion, to be transcribed later. In some cases the observation, preliminary analysis and stimulated-recall interview all took place on the same day. In the case of Heidi, the trainee featured in this paper, the delay between observation and interview was nearer 20 days for both lessons, on account of her prior commitments and those of the researchers. Data analysis The data analysis consisted mainly of fine-grained analyses of each of the lessons, both before and after the stimulated-recall interview, against the theoretical framework of the KQ. In this sense, in contrast with the earlier SKIMA research, analysis was primarily theory-driven as opposed to data-driven. Initially, we identified in the video-taped lessons aspects of trainees’ actions in the classroom that could be construed to be informed by their mathematics content knowledge (including their pedagogical content knowledge). In addition, when possible, our interpretation of the trainees’ mathematical and pedagogical purposes and intentions was further assisted by reference to their lesson plans and the postlesson interviews. These actions were, where possible, coded in accordance with the KQ and its 20 constituent codes, thereby testing the adequacy of the theoretical framework, developed in the context of elementary mathematics teaching, in this secondary school context. Therefore, this research phase entailed ‘theoretical sampling’ (Glaser and Strauss, 1967), whereby the application of a theory has the potential to expose its shortcomings, laying it open to refinement, modification and possible improvement. THE CASE OF HEIDI We come now to our case study. Heidi was, in many respects, a ‘typical’ secondary PGCE student, having come to the course direct from undergraduate study in Mathematics and Statistics at a well-regarded UK university. Her placement school was state-funded, for pupils aged 11-16. The school was ‘comprehensive’, providing for some 1400 pupils across the attainment range. In keeping with almost all secondary schools in England, pupils were ‘setted’ by attainment in mathematics, with 10 or 11 sets in most years. We offer Heidi’s lesson as a ‘case’ in the following sense: it is used to illustrate, and to test in the secondary context, how the KQ can be used to identify, for discussion, matters that arise from lesson observation, and to structure reflection on the lesson. Heidi’s lesson This was Heidi’s second videotaped lesson. Her class was one of two parallel ‘top’ mathematics sets in Year 8 (pupil age 12-13), and these pupils would be expected to be successful both now and in the high-stakes public examinations in the years ahead. The observation notes record that there were 30 pupils in the class, 17 boys and 13 girls. They

were seated at tables facing an interactive white board (IWB)2 located at the front of the room. Heidi stood alongside the IWB for the whole of this lesson, and the video camera was trained on her and/or the board. The objectives stated in Heidi’s lesson plan were to “Go over questions from their most recent test, and then introduce direct proportion”. As soon as the pupils were settled at their tables, Heidi returned test papers to them, from a previous lesson, and proceeded to review selected test questions with the whole class. These questions included two on simultaneous linear equations. Several pupils offered solution methods, and these were noted on the IWB. Nearly 30 minutes of the 45-minute lesson had elapsed before Heidi moved on to the topic of direct proportion. She began by displaying images of three similar cuboids on the IWB: she explained that the cuboids were boxes, produced in the same factory, and that the dimensions were in the same proportions. The linear scale factor between the first and second cuboids was 2 [Heidi writes x2], and the third was three times the linear dimensions of second [x3]. Heidi identified one rectangular face, and asked what would happen to the area of this face as the dimensions increased. They calculated the areas, and three pupils made various conjectures about the relationship between them. The third of these said “I think it is that number [the linear scale factor] squared”. Heidi then introduced two straightforward direct proportion word problems, such as “6 tubes of toothpaste have a mass of 900g. What is the mass of 10 tubes?” Different approaches were offered and discussed. THE KNOWLEDGE QUARTET: HEIDI’S LESSON Earlier, we introduced the four dimensions of the KQ, and gave a general account of the characteristics of each of them. We now offer our interpretation of some ways in which we have observed, or inferred foundation, transformation, connection and contingency (but not in that order) in Heidi’s second videotaped lesson. It will become apparent that many moments or episodes within a lesson can be understood in terms of two or more of the four dimensions. We also draw upon her lesson plan, and upon her contributions to the postlesson, stimulated-recall discussion with a researcher (Anne, in this case). This discussion had homed in on two fragments3 of the lesson that had been selected at the preliminary analysis session a few days earlier. The first of these fragments was Heidi’s review of the first of the two test questions on simultaneous equations; the second was the introduction of the proportion topic using the IWB-images of the three cuboids. Transformation Heidi had little or no influence regarding the choice of examples (a key component of this KQ dimension) in her test review, since the test had been set by a colleague. However, the stimulated-recall interview gave an opportunity and a motive for her to reflect on the test items. There had been two questions (7 and 8) on simultaneous equations, and the related pairs of equations were Q7: 2x + 3y = 16, 2x + 5y = 20 Q8: 3b – 2c = 30, 2b +5c = 1 In response to an interview question, Heidi thought the sequencing appropriate. In particular (regarding Q7) she said “They could do it the way it was”, seeming to refer to

the fact that one variable (x) could be eliminated by subtraction, without the need for scaling either equation. In fact, the pupils’ response to Heidi’s invitation to offer solution methods suggested that this opportunity was not recognised, or not welcomed. The first volunteer, Matthew, proposed multiplying the first equation by 10, and the second by 6, suggesting a desire on his part to eliminate y, not x. (We shall consider Heidi’s response to this under Contingency). Heidi was able to explain this in her answer to Anne’s question “What if the y-coefficients were the same”. Heidi’s first response was “That would be less difficult because they tended to want to get rid of the y. I don’t know why”. In fact, in this lesson segment, when eliminating one variable by adding or subtracting two equations, Heidi reminds the class several times about a ‘rule’, namely: if the signs are the same, then subtract; if they are different then add. Heidi suggests, later in the interview, that the pupils tend to want to make the y-coefficients equal, as Matthew did, because their signs are explicit in both equations. This can be seen, in both Q7 and Q8, where the coefficient of the first variable is positive in both equations, and the sign left implicit, whereas + or – is explicit in the coefficient of the second variable. This insight of Heidi’s is typical of the way that focused reflection on the disciplinary content of mathematics teaching, structured by the KQ, has been found to provoke valuable insights on how to improve it (Turner & Rowland, 2010). Heidi’s observation is that restricting the xcoefficients to positive values (and emphasising the ‘rule’) has somehow imposed unintended limitations on student solution methods, with a preference for eliminating y even when “they could do it the way it was” by eliminating x. Turning to choice of representation, and Heidi’s introduction of the direct proportion topic, we note that we misinterpreted Heidi’s use of the three cuboids in our preliminary lesson analysis. Her lesson plan included: “Discussion point: What happens to the area of the rectangular face as the dimensions increase? What happens to the volumes of the cuboids as the dimensions increase?”. We took this to mean that she intended to investigate the relationship between linear scale factor (between similar figures) and the scale factors for area and volume. In the event she was drawn into this topic, but this had not been her intention, and the subsequent word problems make this clear. In the event, there is discussion of the area of one rectangular face of the cuboid, and how its area increases as the cuboids grow larger: there is not time to consider the volumes. When probed about her choice of context for the introduction of the direct proportion topic, Heidi said that she had chosen the cuboids because it was “a nice visual” which contrasted with the “wordy” presentation of the other problems. In fact she drew on her IWB expertise by unveiling the images of the cuboids, one by one, as if drawing back electronic curtains. She responded to appreciative ‘Wow’s from the pupils with modesty, saying “It’s not all that impressive is it?” In the interview when asked whether she agreed that she could have done the work on area comparison with rectangles, she replied “You’re absolutely right, rectangles would be enough … but I did like my box factory”. Here we see an example of trainees’ propensity to choose representations in mathematics teaching on the basis of their superficial attractiveness at the expense of their mathematical relevance (Turner, 2008). In

this instance, the preference for these ‘visuals’ took Heidi into mathematical territory for which she was not mathematically prepared (see Contingency). Contingency Analysis of this dimension of the KQ in Heidi’s lesson intertwines with the component of foundation concerned with teachers’ beliefs about mathematics and mathematics teaching. Here, we begin by taking up the story of Matthew’s suggestion to solve Q7 by multiplying the first equation by 10, and the second by 6. Heidi responded to Matthew (responding to students’ ideas) with “Excellent, you could do that”, and talks it through (without writing on the board), saying that Matthew is trying to “make the number in front of the y, which we call the coefficient, the same”, so that both will be 30. She does point out that “You wouldn’t have to do quite as much timesing as that, quite big numbers, if you didn’t want to”, and there might be “other multiples” which could be used. Given Heidi’s earlier comment that, with the equal x-coefficients, “they could do it the way it was”, Anne asked her why she had “run with” Matthew’s suggestion4. Heidi replied “Because it would work. You’re trying to find the lowest common denominator but it would work. Like adding fractions, it would work with any common multiple. I didn’t want him to think he was wrong”. This kind of openness to pupils’ suggestions, and ability to anticipate where they would lead, was very characteristic of Heidi’s teaching, and several examples of it can be found in our data, although there are occasions when her prior expectations blinker her reading of events as they unfold. Space limitations prevent a detailed analysis of the class discussion which followed Heidi’s introduction of the three cuboids. Suffice, here, to say that the pupils calculated (in cm2) the areas of the rectangles with sides (respectively) 2x3, 4x6, 12x18 (all cm) viz. 6, 24, 216 (in cm2). Heidi had annotated x2, x3, as we noted earlier. The first pupil contribution about the relationship between the areas conjectured that cubes (unspecified) were involved. Heidi acknowledged this suggestion, but set it aside. Now, it just so happens that the third area is the cube of the first (63=216). This is, admittedly, a coincidence, and we are in no position to know whether it is what the pupil had discovered. A second pupil suggested that the relationships were “timesed by 4 and timesed by 6”. Heidi made it clear that she was not checking these calculations numerically (“I’m going to take your word for that”), recorded this second proposal on the IWB (writing x4 and x6) and said “So two times what this has been timesed by [pointing to the linear scale factors]. Good observations”. This seemed to be the end of the matter, until a third pupil, Lucas, said “I think it is that number squared”. Heidi paused, then changed the second factor to x9 and emphasised the squares. Now, this length/area relationship in similar figures was not what Heidi had set out to teach, and it became clear at the interview that Heidi (unlike Lucas) did not know in advance about “that number squared”. In the interview, the discussion proceeded: Anne:

Then you go on to areas. They give a range of options. Now, you take all these responses and give value to all of them. But this was different, in that two of these responses were not correct.

Heidi:

I want to take everyone’s ideas on board. When you do put something on the board they correct each other rather than me being the authority. In that case I had a bit of a brain freeze, I hadn’t worked out how many times 24 goes into 216, but they’re used to me putting up everything.

We see here, paradoxically, a situation – by no means the only one – in this secondary teaching data in which some subject-matter in the school curriculum lies outside the scope of the content knowledge of the trainee at that moment in time. This should come as no great surprise. For all their university education in mathematics, and their knowledge of topics such as analysis, abstract algebra and statistics, there remain facts from the secondary curriculum that they will have had no good reason to revisit since they left school. This is no disgrace, and they will have cause to remember soon enough. What is significant, however, exemplified by Heidi but more-or-less absent in our observations of primary mathematics classrooms, is a teacher with the confidence to negotiate and make sense of mathematical situations such as this (the length/area relationship) ‘on the fly’, as they arise. We noted earlier Heidi’s response to Matthew: “Excellent, you could do that”, in a situation when his method differed from the one she had in mind. She used versions of this formula (praise, followed by an implied caution) on two other occasions in the lesson. In an earlier test item, it was given that £36 is 75% of some quantity (x). Heidi had in mind solutions such as x= 36x

100 , but Adrian suggested finding a third of 36 and adding it to 75

36. Heidi responded “That’s a perfectly acceptable way, Adrian. Yes, you can do it that way …”. Later, in the dog-walker proportionality problem (a dog walker walks 7 dogs in 2 days, how many dogs in 10 days; 5 days?) a pupil suggests a unitary approach (how many dogs in one day) some way into the discussion. Heidi responds “Lovely, you can do that”, and in fact she subsequently emphasises the x3.5 scaling. These responses suggest a tension in Heidi’s mind, when responding to students’ ideas, between acknowledging and valuing flexible, idiosyncratic solution methods and promoting standard methods that will ‘work’ for them now, and in the high-stakes tests to come. This tension is probably not lost on her pupils either, and is conveyed in her language, in the modal ambiguity between possibility and permission (‘can’ and ‘could’). Foundation This lesson does raise a few issues about Heidi’s content knowledge that might be brought to her attention, and some of them were raised in the interview. Briefly, these include: her use of mathematical terminology, which is either very careful and correct (e.g. ‘coefficient’), or quite the opposite (e.g. ‘timesing’); her lack of fluency and efficiency in mental calculation, such that she did not question the suggestion that 6x24=216 herself in the cuboids situation: on occasion it appeared that she was puzzled by some of the pupils’ mental calculations; thirdly, she was not aware of the length/area/volume scale-factor relationships referred to earlier.

But, after many hours spent scrutinising the recording of this lesson, and that of the postlesson interview, our lasting impression relates to the beliefs component of the Foundation dimension. In particular Heidi’s beliefs about her role in the classroom in bringing pupils’ ideas and solution strategies into the light, even – as we remarked earlier – when she believed that ‘her way’ would, in some sense, be better. As she told Anne, “I want to take everyone’s ideas on board. When you do put something on the board they correct each other rather than me being the authority”. Her perception of this aspect of her role, as teacher, and the possibility of the pupils themselves contributing to pupil learning, is resonant of various constructivist and fallibilist manifestos. Balacheff, for example, advised that “[the] transfer of the responsibility for truth from teacher to pupils must occur in order to allow the construction of meaning” (1990, p. 259), and identified classroom discussion as a context in which this transfer can take place. Heidi constantly assists this ‘letting go’ by acknowledging pupils’ suggestions, and making them available for scrutiny by writing them on the board. Occasionally she finds herself in deep water as a consequence, but she never seems to doubt her [mathematical] ability to stay afloat. Connection We identified a few events in this lesson under connection. For example, Heidi’s introduction to direct proportionality with the cuboids seemed quite unrelated to the word problems which followed. In any case, the rather diverse objectives for the lesson were likely to make it somewhat ‘bitty’, and, again with space considerations in mind, we omit further analysis of this KQ dimension from the present narrative. CONCLUSION The purpose of this phase of our SKIMA research was to test the ‘fit’ of the Knowledge Quartet to secondary mathematics teaching. The indicative analysis of Heidi’s second lesson in this paper indicates the potential of KQ as an analytical and (potentially) developmental tool in the context of novice secondary mathematics teaching. In this lesson, there were no moments or events, in which Heidi’s mathematical content knowledge became a significant and/or influential factor in the proceedings, that could not be accommodated by one or more of the four dimensions of the KQ and the existing 20 codes. There are, however, such events in the data, as a whole, that may cause us later to want to supplement the codes within existing KQ dimensions. For example, the existing four transformation codes might not adequately capture the kinds of extended explanations, or the imaginative task design, that we saw in some other lessons. In other cases, such as Heidi’s commitment to valuing student ideas and conjectures, the difference between these data and those from our novice primary teachers is more one of degree than of kind, from a KQ perspective. In our role as mathematics teacher educators, our analysis (as researchers) of the six lessons taught by these three volunteer participants now encourages us to pilot the use of the KQ as a developmental framework for the observation and review of lessons taught by secondary PGCE trainees during their school-based placements. This, in turn, will create yet more opportunities for testing and refining the KQ in the field.

Notes 1

A DVD of the full lesson was given to the trainee soon afterwards, as a token of our appreciation, but their reflections on viewing this DVD in their own time are not part of our data.

2

Interactive whiteboards, with associated projection technology, are now more-or-less universal in secondary classrooms in England.

3

The two corresponding video clips were each about four and a half minutes.

4

Our earlier KQ research had identified three kinds of responses to unexpected ideas and suggestions from pupils: to ignore, to acknowledge but put aside, and to acknowledge and incorporate.

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