Christoph Ableitinger
email: christoph.ableitinger@univie.ac.at

Astrid Anger
email: astrid.anger@univie.ac.at

Christian Dorner
email: christian.dorner@uni-graz.at

Comparison of students' and researchers' choice of significant events of math lessons

Keywords: student voice, teaching practices, significant events, learners' perspective, secondary school mathematics

Considering the learners' perspectives on classroom teaching is central in our project 'AmadEUs', in which we work together with 79 students of four school classes in Austria and 23 pre-service teachers. In a first stage, we interviewed these students about significant events in math lessons they had just attended. In this paper, we analysed what kinds of events our students have mentioned and chosen and compared them with our (researchers') choice of significant events. While different groups of students tended to choose similar events, we observed differences between students' and researchers' choices of significant events. In a qualitative analysis, we tried to identify the characters of most commonly selected events by students in contrast to the teachers' and the researchers' selections.

Introduction

Mathematics education is a complex network with several players involved, such as students, teachers, researchers and teacher educators, who prioritize different aspects of quality and therefore attach importance to different events during math lessons. Hence, one can understand and optimize teaching practice only through research that takes into account various perspectives, and particularly those of the learners as an essential part (see [START_REF] Clarke | The Learner's Perspective Study[END_REF]. Through giving students the opportunity to articulate their opinions, thoughts and feelings, valuable insights into their attitudes towards mathematics can be gained. Hence, listening to their voices has the potential to influence educators and policy makers [START_REF] Mok | What really matters to Students? A Comparison between Hong Kong and Singapore Mathematics Lessons[END_REF]. This so-called Student Voice approach and some related studies will be further discussed in the theoretical framework.

One aim of our study is to explore which events of math lessons students consider as significant. The results do not only inform us about what kinds of events students consider as important for their learning, we might also get a clue about which events actually stay in their mind and in which events they might pay more attention to. This knowledge has the potential to influence and improve teaching practices [START_REF] Huang | Matches or Discrepancies: Student Perceptions and Teacher Intentions in Chinese Mathematics Classrooms[END_REF][START_REF] Lee | Learning mathematicsletting the pupils have their say[END_REF].

By comparing the choice of students to our own (researchers') choice of significant events (by that we mean short sequences of approximately 2-5 minutes) of the math lessons, we want to detect blind spots of typical researcher's views on mathematics education on the one hand. Through contrasting the two perspectives, we expect to maintain student-specific choices of concrete situations in the learning process, which can help teachers in their prospective lesson planning [START_REF] Mcintyre | Pupil voice: Comfortable and uncomfortable learnings for teachers[END_REF]. On the other hand, it is also interesting to discover events that are significant to researchers but not to students (and vice versa). For this, our study may give hints, where and in which way crucial points of lessons need to be made explicit to students, in case they do not attach importance to these points sufficiently by themselves.

Theoretical framework

Our study takes the perspectives of students into account. Such "Student voice" approaches have been of growing interest in the past two decades. Early work that has been conducted on the voice of students (e.g. [START_REF] Mccallum | Learning: The pupil's voice[END_REF][START_REF] Pollard | What pupils say: changing policy and practice in primary education[END_REF][START_REF] Fielding | Students as Radical Agents of Change[END_REF] revealed the potential of listening to students in order to understand teaching and learning and making it more effective. In the course of this development, [START_REF] Robinson | Theorizing student voice: values and perspectives[END_REF] aimed at theorizing student voice, carving out core values of student voice work. [START_REF] Flutter | Consulting Pupils: What's in it for schools[END_REF] analysed different student voice initiatives and derived potential benefits to students, teachers and schools, including the potential to improve learning (as the participation of students might have a positive impact on their meta-cognitive development) as well as teaching (as teachers and researchers are provided with valuable insights).

As [START_REF] Allen | Pupils' perspectives on learning mathematics[END_REF] pointed out, it took some time until students have been consulted to articulate their views about mathematics teaching and learning, but meanwhile several studies have been conducted in this field (e.g. [START_REF] Taylor | Children's views of their teacher's role in helping them learn mathematics[END_REF]Sullivan, Tobias, & McDonough, 2006;[START_REF] Lee | Learning mathematicsletting the pupils have their say[END_REF][START_REF] Mcdonough | Seeking insights into young children's beliefs about mathematics and learning[END_REF]. While some studies concentrate on what students attach importance to in math lessons in general (e.g. [START_REF] Wilkie | Exploring intrinsic and extrinsic motivational aspects of middle school students' aspirations for their mathematics learning[END_REF], only few studies deal with significant events of certain experienced math lessons. Of interest is the Learner's Perspective Study (LPS), which has been designed to analyse and compare practices of mathematics classrooms around the world [START_REF] Clarke | The Learner's Perspective Study[END_REF]. Within LPS, Huang and Barlow (2013) compared important events selected by students and teachers. They examined 15 videotaped consecutive mathematics lessons of one teacher, the corresponding lesson plans, teacher interviews and 30 student interviews. During the teacher interview, he described his intentions in the lesson, the important events from his point of view as well as his actions, thoughts and feelings during the important events. The two researchers analysed the data set in order to verify whether the important events identified by students coincided with the teachers' intended important events within the lessons.

Using a slightly different research design than LPS, we seek answers to the following research questions:

What kinds of events of math lessons are significant for students and/or researchers? By what criteria do students and researchers select these events and in what way do these criteria differ from each other?

Method

Teaching units: Four school classes (79 students) who visited our University were taught by preservice teachers in the following topics: Percentage calculation (6 th grade), introduction to probability theory (10 th grade), combinatorics (11 th grade), application of the differential calculus (11 th grade). Each class was divided into two groups. These 8 lesson sequences have been videotaped.

Identifying significant events of the math lessons: Right after each lesson, interviews were conducted and recorded among all students (groups of 2-4) to ascertain events that seemed significant to them (we used the German word "wichtig"). We do not predetermine the term "wichtig" as a survey construct, because we do not aspire to infer causal relationships to other constructs. In fact, we seek to learn more about what is important to students when reflecting on their attended math lessons. We thus chose the word "wichtig" out of a group of words with a similar meaning (such as "wesentlich", "markant" or "maßgeblich", which all translate to significant or important, respectively) for its easy comprehensibility and its openness for all kinds of aspects that could be meaningful to the students. "Wichtig" expresses that some aspect is of essential relevance (Duden, 2016).

We decided to use semi-structured interviews [START_REF] Galletta | Mastering the Semi-Structured Interview and Beyond: From Research Design to Analysis and Publication[END_REF] to reveal significant events from the students' point of view. Our interview protocol consists of three segments: For the opening segment, we formulated two open-ended questions that stimulate students to talk about the math lesson they just experienced. The middle segment consists of more specific and structured questions directly related to our research that should jog the students' memories as well as help them to reflect on different aspects of the lesson. These questions cover the scope of our intended meaning of important/significant events. In the concluding segment, students have to come to an agreement about the five most important/significant events.

In addition, we (as researchers) watched the complete video material (because we did not attend the lessons) and took notes of significant events. At the end of this process, each of us had to choose the five most significant events from his/her own point of view.

In order to conduct an analysis, assistants fragmented each lesson sequence into short scenes due to certain criteria (e.g. change of teaching method, change of subject, presenting or discussing a new example, etc.). Altogether, this procedure led to 221 scenes (for all lesson sequences). After our own selection of events (the researchers' choice) had been completed, we evaluated the audio material of the student interviews in order to mark their significant events. Each scene was then labelled according to the number of selected events that coincide with that scene (at least partially). For example, the scene label 2R1S means that two (out of three) researchers and one of the three groups of students selected significant events that occured during this scene (see Figure 1 for a short extract of one lesson sequence). At this stage, we were able to analyse the data quantitatively and qualitatively respectively.

Results

Differences in significance between students and researchers

In order to measure the disparity between researchers' and students' selection, we determined the difference of the number of researchers' and students' marks for each scene. While the difference is at most 1 in 75% of all scenes (n=221), we observed 15 scenes (7%) which have been found significant by either all researchers and no group of students, or vice versa. These scenes are of great interest. 85 scenes (38%) are marked solely by either students or researchers. We must mention that 42 of these 85 scenes are labelled as significant by only one group of students or one researcher, respectively.

In more detail, 23% of all scenes are significant for at least one group of students, but not for any researcher. The majority of these scenes belong to student-centred learning phases. Furthermore, students but not researchers highlighted many events assigned to group work phases and games (due to their active participation) as well as summaries. Students have predominantly chosen events in which they had fun, events that suited them as well as events in which they could easily follow, irrespective of the professional or didactical quality.

On the other hand, 15% of all scenes are significant for at least one researcher, but not for any group of students. Such scenes often contain mathematical errors of the teachers or didactical mistakes (e.g. confusing explanations, explanations that might lead to misconceptions, etc.) and have been commonly chosen by us due to the revealed lack of (pedagogical) content knowledge of the acting teacher. A small part deals with interesting procedures of the teachers that have been either considered as appropriate or inappropriate by us.

Qualitative analysis of a particular scene

For the following scene (which all researchers and students marked as significant), we go more indepth into the corresponding justifications stated by students and researchers.

Combinatorics (11 th grade), minute 63-68

Description of the scene: The teachers use this scene to sum up the content of the lesson. The graphic shown in Figure 2 is drawn on the blackboard to help students identify combinatorial problems as either a variation, permutation or combination and to repeat the associated formulas which have been taught in this lesson. During that scene, one teacher confuses variation with combination and gets corrected by his colleague. Perspective of students: All groups of students appreciated the summary as a 'helpful overview' and identified that scene as a key point. They described the provided graphic as 'very good', 'wellstructured' and 'easy to follow'. It also helped them to solve problems they had to work on after that scene. Furthermore, they liked that the teachers helped and corrected each other and got the feeling that errors are permitted.

Perspective of researchers:

We observed several inaccuracies and errors. The teachers define the permutation solely by choosing all elements of a certain set. They do not mention whether order matters or repetition is allowed and even indicate once that the order does not matter:

Teacher:

If I cannot deduce from the task that the order matters or that a subset is chosen, then it is always a permutation.

In their graphic, combination and variation only differ in the matter of the order. It is neither mentioned that combinations are without repetition nor that repetitions are allowed in variations. By saying, "We choose k out of n elements", the teacher indicates that k cannot be greater than n in variations. A correct and thoroughly overview must include those two aspects (order and repetition) for each counting problem.

Further, the scene indicates a highly syntactical approach:

Teacher: No selection occurs; I just take all elements. I've only got an 'n'.

Student: I see. This is a permutation.

Teacher: Yes, this is a permutation, because we do not have a 'k' [...] because I simply need one number.

Without being able to verify the criteria of a permutation, students give the correct answer, knowing that the formula of permutations (out of the three presented formulas n!, n choose k and n k ) is the only one that consists of one parameter.

Comparison and discussion: While all groups of students looked upon that scene favourably, we chose that scene due to the syntactical approach and inaccuracies/errors showing a lack of (pedagogical) content knowledge. As students do not know any other counting problem apart from those three presented in that lesson, the given overview is sufficient to distinguish between them, although misconceptions might result (e.g. that k cannot be larger than n in a variation). As the whole topic is new to them, they trust in the expertise of the teachers and are not able to identify errors. Thus, they focus on the use of the summary (e.g. repeating the formulas), whereas we focus on the content quality. We agree that a summary of counting problems is a good idea, but the poor implementation has been decisive for our selection.

Discussion and conclusions

The results have demonstrated that the difference of the number of researchers' and students' marks for each scene is at most 1 in 75% of the cases, however, the reasons for choosing certain scenes are quite different, as we have shown in the above mentioned example.

While we focused on events that deploy the didactic strategies of the teachers or revealed existing or missing (pedagogical) content knowledge, the students concentrated on student-centred learning phases, summaries, quizzes and games. This emphasis on student-centred learning phases is in line with the findings of [START_REF] Huang | Matches or Discrepancies: Student Perceptions and Teacher Intentions in Chinese Mathematics Classrooms[END_REF] and [START_REF] Mok | What really matters to Students? A Comparison between Hong Kong and Singapore Mathematics Lessons[END_REF]. Their interviewed students attached importance to seatwork and group work. [START_REF] Wilkie | Exploring intrinsic and extrinsic motivational aspects of middle school students' aspirations for their mathematics learning[END_REF] illustrated that students wish among others more such phases in their math lessons. In the present study, students described teacher-centred phases as 'conventional instruction', which could be a reason for choosing student-centred events. Being active is more meaningful and therefore more memorable to them. The findings of [START_REF] Lee | Learning mathematicsletting the pupils have their say[END_REF] confirm the importance of quizzes and games for students. In their study, students explicitly demand more interactivity, games and creative tasks. The significance of such events might issue from feelings of success (discover connections, find solutions to certain problems and so on) or emotional involvement. According to [START_REF] Huang | Matches or Discrepancies: Student Perceptions and Teacher Intentions in Chinese Mathematics Classrooms[END_REF] and [START_REF] Mok | What really matters to Students? A Comparison between Hong Kong and Singapore Mathematics Lessons[END_REF], students valued review and feedback parts as important.

Those findings are also in line with our study: Students have chosen summaries as significant and valuable events (as has been shown in the qualitative analysis). However, our data reveals that the researchers did not mention summaries unless they included mistakes or highlights.

Contrary to our expectations and to the results of [START_REF] Mok | What really matters to Students? A Comparison between Hong Kong and Singapore Mathematics Lessons[END_REF], students barely talked about teacher explanations. This even holds true for some appalling explanations by the teachers that were not even mentioned by the students during the interviews, although some questions focused on teacher explanations. We conclude that the students did either not remember those scenes or did not attach any importance to them. In a certain way, their focus on methodical aspects of math lessons is understandable, due to students' less developed meta-cognitive skills and the fact that highlights during classes are kept in mind more strongly. These findings are in line with the study of [START_REF] Waldis | Der schweizerische Mathematikunterricht aus der Sicht von Schülerinnen und Schülern und in der Perspektive hochinferenter Beobachterurteile (Swiss math classes from students' perspectives and in the perspectives of highly inferential observer estimations)[END_REF], who compared student ratings with class observer ratings of math classes and found differences between these two groups when rating cognitively activating instructions and the structure of math classes.

In cases where students and researchers selected the same events, different reasons were decisive. This is similar to the result of [START_REF] Huang | Matches or Discrepancies: Student Perceptions and Teacher Intentions in Chinese Mathematics Classrooms[END_REF], who obtained differences in students' and teachers' interpretations in such match cases. We presented an example, where students described the visualisation and the associated explanation as well-structured while we as researchers valued it as confusing and identified several content-related mistakes. This may be due to a certain inability of students to detect mistakes or to think about alternative approaches to particular concepts [START_REF] Lee | Learning mathematicsletting the pupils have their say[END_REF] while still being involved in the learning processes and not being able to oversee the topic as a whole. Some statements expressed by students during the interviews hypothesise that they tend to evaluate the quality of explanations according to how easily they are able to solve corresponding tasks. This implies that their self-assessment relating to their mathematical understanding depends on (amongst other aspects) the tasks they need to solve.

For mathematics teaching practice, we can deduce that students obviously and understandably depend on the teachers' knowledge and the way content is dealt with in class. We have seen that they did not mention several inconsistencies or misleading actions of the teachers, which were obvious to us as researchers and teacher trainers, respectively. Hence, teachers cannot fully rely on students' feedback concerning content-related or didactical inappropriateness, they shouldbeside asking students for regular feedbacksee to get feedback from experienced mathematics teachers or teacher educators (e.g. through job shadowing) in order to gradually improve their teaching practice.

It is essential for the learning process that students are aware of essential content points of the lesson (e.g. phases where concept formation and the development of basic mathematical ideas are in the centre of interest), but our study reveals that they hardly mentioned them nor valued many of them as important. According to [START_REF] Huang | Matches or Discrepancies: Student Perceptions and Teacher Intentions in Chinese Mathematics Classrooms[END_REF], the agreement on the importance of such events between students and teachers is necessary for effective learning. We deduce from our results that teachers have to emphasise these essential content points on a meta-level to be sure that the students value these events appropriately. Teacher training has to do its duty at this point by calling the teachers' attention to this problem.

What we can learn from the students' choices of events and the associated justifications is that teachers indeed should make an effort to include more student-centred and methodically varied learning phases into their math lessons (as they do have the potential to motivate students and to enhance their activity as well as their attention). However, it is crucial that there has to be a strong focus on the quality of the content promoted in these phases. As we have seen in our data, games and quizzes are popular among students, but it is difficult to get any content-related feedback to these events from them on the spot, even in cases where we have chosen the same scene out of misleading or confusing aspects. Hence, entertaining elements have the potential to inspire students and to stay in mind. However, they must not cover the content and the learning goals of the lesson.

Teachers can gain useful insights into their students' learning by knowing what classroom events they perceive as important, as they are then able to adjust their teaching practice in order to catch their attention and draw it to essential content points. In addition, a lot can be learned from students' justifications of their selected events as this has the potential to revealthrough further analysiscertain predispositions students have towards math lessons in general.

Figure 1 :

 1 Figure 1: Scene labels according to selected significant events

Figure

  Figure 2: Decision tree

Acknowledgement

AmadEUs is a project carried out as part of the funding programme Sparkling Science which is supported by the Austrian Federal Ministry of Education, Science and Research.


Nick Andrews
email: nicholas.andrews@education.ox.ac.uk

Between-desk-teaching as a deliberate act of making content available: The case of Bernie teaching ratio

Keywords: Mathematics instruction, Instructional design, Student centred learning, Research methodology, Teacher education

Between-desks-teaching is recognised as a feature of mathematics classrooms across the world. Previous studies have sought to describe the range of purposes of this lesson event within a lesson and have been concerned with the orchestration of the classroom. Yet there has been little attention given to the role of between-desks-teaching beyond a single lesson, such as its contribution to how mathematical content is made available to learners over time. The current study explores what is happening mathematically when teachers and learners engage in between-desks-teaching across a topic. There is a particular focus on one case in which this lesson event features extensively. This case reveals how between-desks-teaching can have a strategic purpose and has potential to be used deliberately when teaching a topic.

Introduction

In this paper I focus on between-desk-teaching and how it can be distinctive in terms of its purpose when teaching a topic over a series of lessons. To do so, I draw on data from a wider exploratory study of four cases of classroom practice over a series of lessons. I then focus on one particular case in order to understand better the distinctive contribution of between-desks-teaching to how subject matter is transformed and connected over time in this context. This study therefore, in common with previous CERME papers [START_REF] Skott | Classroom practice and teachers' knowledge, beliefs and identity[END_REF], brings to the surface a specific act within mathematics teaching and discusses how it provides opportunities for learning.

Between-desk-teaching has been identified as a feature or "lesson event" of mathematics classrooms across the world (O' Keefe, Xu, & Clarke, 2006, p.73). It is a distinct event from 'classwork', which might include teacher exposition and whole-class discussion, but is a sub-set of 'seatwork', which is when "students work individually or in small groups on assigned tasks" and "talk is mostly privateteacher-student or student-student" (Stigler & Hiebert, 1999, p.67). It is this mostly private teacher-student talk, while students work on assigned tasks, which defines between-desk-teaching, although O' Keefe et al. (2006) also point to instances where this talk becomes more public. [START_REF] Inoue | Deconstructing teacher expertise for inquiry-based teaching: Looking into consensus building pedagogy in Japanese classrooms[END_REF] describe how between-deskteaching offers learners individual support when problem solving, while [START_REF] Jakonen | Professional embodiment: Walking, re-engagement of desk interactions, and provision of instruction during classroom rounds[END_REF] states that teacher actions during this lesson event are 'occasioned' in that they are sensitive to the learner's present situation. Yet between-desk-teaching is often 'repetitive' in that the teacher addresses the same issue both with multiple learners and also potentially multiple times with the same learner [START_REF] Jakonen | Professional embodiment: Walking, re-engagement of desk interactions, and provision of instruction during classroom rounds[END_REF].

O' Keefe et al. (2006) discuss both the monitoring and guiding functions of between-desk-teaching. Providing feedback, including references to learning goals [START_REF] Svanes | Teachers' instructional practices during pupils' individual seatwork in Norwegian Language arts[END_REF], or observing learners working serve a monitoring function. The guiding function can be associated with either 'extending' or 'enabling' teacher prompts [START_REF] Roche | Describing the nature and effect of teacher interactions with students during seat work on challenging tasks[END_REF]. Extending prompts are typically discussed in the literature with reference to challenge and questioning (e.g. [START_REF] Svanes | Teachers' instructional practices during pupils' individual seatwork in Norwegian Language arts[END_REF], while enabling prompts may involve telling, modelling or offering hints [START_REF] Svanes | Teachers' instructional practices during pupils' individual seatwork in Norwegian Language arts[END_REF], or referring to prior learning or suggesting alternative approaches [START_REF] Inoue | Deconstructing teacher expertise for inquiry-based teaching: Looking into consensus building pedagogy in Japanese classrooms[END_REF].

While between-desk-teaching can be analysed as a distinct lesson event, or as an event within a lesson [START_REF] Inoue | Deconstructing teacher expertise for inquiry-based teaching: Looking into consensus building pedagogy in Japanese classrooms[END_REF]), O'Keefe et al.'s (2006) study emphasises that in order to understand its purpose consideration needs to be given to the events that surround it and whether these events are classwork or seatwork. In terms of how content is made available to learners, between-deskteaching can be seen as a bridge between independent-of-the-teacher seatwork and classwork, or vice versa. When the flow is from seatwork to classwork, the purpose of between-desk-teaching is more likely to be 'monitoring student activity' (O' Keefe et al., 2006) in order to inform the precise nature and focus of the classwork to follow, whereas when the flow is from classwork to seatwork the purpose is more likely to be 'guiding student activity' and making connections between the assigned task and the whole-class teaching the preceded it [START_REF] Inoue | Deconstructing teacher expertise for inquiry-based teaching: Looking into consensus building pedagogy in Japanese classrooms[END_REF]. This emphasises how these lesson events mutually influence each other, but that there is a pedagogical choice regarding their sequencing.

The previous studies of between-desks-teaching discussed above focus particularly on the nature of teacher-learner participation during this lesson event rather than how the subject matter is being transformed and connected through these interactions. My own work however has focused on how subject matter is made available to learners over the course of teaching a topic [START_REF] Andrews | Differences in the forms of content teachers are seen to offer over time: Identifying opportunities for teacher growth[END_REF], through the actions of the teacher. I use the term 'manifestation' of a mathematical concept to refer to the form of content I observe the teacher offering learners at any moment in time, and a manifestation may be categorised as visual, technical, functional, or a combination of these. Visual manifestations often include diagrams or forms of imagery, technical manifestations often relate to methods or symbolic manipulation, and functional manifestations include worded problems and applications of methods.

In order to connect the sequencing of manifestations with opportunities for learning, I draw on a theory of growth in mathematical understanding [START_REF] Pirie | A recursive theory of mathematical understanding[END_REF][START_REF] Pirie | Growth in mathematical understanding: How can we characterise it and how can we represent it?[END_REF]. This constructivist theory proposes a progression of levels, each of which represents a reorganisation of knowledge structures, from 'primitive knowing', through 'image having' and 'formalising', to 'structuring' and beyond [START_REF] Pirie | A recursive theory of mathematical understanding[END_REF]. I associate visual, technical and functional manifestations with opportunities to focus learners' attention on image having, formalising and structuring respectively. I associate visual-functional, visual-technical and functional-technical manifestations with opportunities to shift the focus of learners' attention, captured by [START_REF] Pirie | A recursive theory of mathematical understanding[END_REF] terms 'image forming', 'property noticing' and 'observing' respectively.

In a recent paper I highlight mathematical differences between what is addressed during classwork and seatwork in each of the four cases studied [START_REF] Andrews | Are there mathematical differences in what is addressed through different modes of interaction in classrooms?[END_REF]. One pattern in practice revealed in this paper is a flow from classwork to seatwork within lessons and also across lesson series, with greater foregrounding of visual manifestations during classwork and functional manifestations during seatwork. This analysis suggests a degree of alignment in these cases between how content is made available to learners over time and the underlying progression proposed in Pirie & Kieren's theory. The current study builds on this recent paper and explores ways in which between-deskteaching might support this progression.

From the range of purposes of between-desk-teaching identified above, I focus particularly in this paper on the guiding function and references back to prior learning [START_REF] Inoue | Deconstructing teacher expertise for inquiry-based teaching: Looking into consensus building pedagogy in Japanese classrooms[END_REF]. Pirie and Kieren describe growth as "levelled but non-linear" (Pirie & Kieren, 1994, p.166) and use the term 'folding back' to describe returning to a previous level "in order to extend one's current, inadequate understanding" (Pirie & Kieren, 1994, p.173). I associate teacher references during between-deskteaching to prior learning with offering explicit opportunities for learners to 'fold back' to a previous level of understanding in order to come to understand the assigned task at hand. In such a situation the between-desk-teaching will not, in terms of manifestation, align to the assigned task but rather more closely to what has been offered previously (for example through classwork). The current study explores therefore whether during seatwork between-desk-teaching aligns with the assigned task, or is more associated with opportunities for folding back. Addressing this question over a series of lessons on a given topic rather than an individual lesson affords exploring patterns in practice, which in turn reveals if between-desk-teaching is being deployed as a strategic act for transforming and connecting the subject matter of a topic.

Methodology

A summary of the approach taken to address the current research question will be offered here. A methodological paper detailing the range of approaches taken in the wider study is being worked on separately, and this will in due course provide fuller details and justifications of methods employed. The approach is an analysis of the text of what the teacher is seen to offer learners over the course of a series of lessons on a single mathematical topic. By text here I mean the literal text of teachertalk during lessons, whether that be public or privately to individual learners, and also the tasks, video clips, simulations, physical resources and other teaching media that the teacher makes available.

A quantitative approach is taken in order to detect differences in manifestation over periods of time through applying what I call a tri-polar analysis. The relative stressing of the three elements of the triad {visual, functional, technical} in the ratio v : f : t and satisfying the condition that v + f + t = 1 can be succinctly represented by the Barycentric co-ordinate (v, f, t).

How the teacher makes content available to learners is conceptualised as having two elements: teacher-talk and given-tasks. Teacher-talk may be public (for example, during periods of classwork) or private (during seatwork). Given-tasks are only present during seatwork and are the prompts that initiate subsequent activity in a lesson rather than that activity itself. The combination of teachertalk and given-task during an individual between-desk-teaching event may serve to alter what is offered to learners when compared to what would have been available through the given-task alone. Therefore the effect of between-desk-teaching is revealed in any differences between what is offered during seatwork through considering both teacher-talk and given-task in unison to what would have been offered over the same time period through the given-task alone.

For the current study, the text of each lesson series is parsed into 30-second intervals, each of which is then qualitatively coded for manifestation. Each manifestation category is associated with a Barycentric co-ordinate. The quantitative code for teacher-talk for the interval is the mean of the Barycentric co-ordinates of all of the manifestation categories present in the teacher-talk in that interval (or (0, 0, 0) if there is no teacher-talk), and the quantitative code for given-task for the interval is determined similarly. The overall quantitative code for the interval (referred to as the interval centre) is either the co-ordinate for teacher-talk if there is no given task, the co-ordinate for given-task if there is no teacher-talk, or the mean of the teacher-talk and given-task co-ordinates. Each 30-second interval is also coded as either seatwork or classwork. The seatwork centre for the lesson series is then the mean of the Barycentric co-ordinates of all the intervals coded as seatwork, and the classwork centre is similarly defined. The given-task centre is the mean of the quantitative codes for given-task of all the intervals coded as seatwork. A statistical analysis of the visual coordinates for each interval using the Wilcoxon rank-sum test [START_REF] Wilcoxon | Individual comparisons by ranking methods[END_REF] allows the stressing of visual manifestation between classwork and seatwork, and between seatwork and given-task to be compared quantitatively (reported here using an effect size), and a similar analysis is conducted on the functional and then the technical co-ordinates. The extent of differences is represented by the effect size, and an effect size greater than .3 is treated as a material difference. By this I mean that the difference is large enough to infer systematic differences in how content is made available associated with strategic differences in teacher action.

The current paper initially reports on four lesson series taught in local secondary schools in England with students aged between 13 and 16. One of the teachers, Ashley taught a series of five lessons on geometrical constructions. Bernie was at the same school as Ashley and taught a series of four lessons on ratio with one class and three lessons on linear equations with another. Courtney was at a different school and taught a series of four lessons on linear equations. In each case, the teacher determined the topic and the teaching approach. The four cases are used to address the question of whether there are mathematical differences in what the teacher makes available through betweendesk-teaching when compared to the underlying given-task alone. A closer analysis is then conducted of Bernie teaching ratio in order to understand better between-desk-teaching in this case.

Cross-case findings

Table 1 sets out the seatwork and given-task centres for the four lesson series studied. For each case the statistical significance of any differences between seatwork and given-task are given. The number (n) of 30-second intervals given over to seatwork is also given in each case.

The seatwork and given-task centres are descriptive statistics, and so comparisons between them only indicate where there are differences. It is the statistical analysis that quantifies these differences. The Barycentric co-ordinates highlight differences in each case between the seatwork and given-task centres, which indicate that during between-desk-teaching teacher-talk did not always align mathematically with the given-task. Instead this form of teacher-learner participation changed the mathematical nature of the activity. The greatest misalignment between seatwork and given-task was in the case of Bernie teaching ratio, and as this is an extreme case it has been selected for further analysis below. As this difference was material, the inference is that between-desk-teaching was being deployed as a strategic act for transforming and connecting the subject matter of a topic in this case.

The particular case of Bernie teaching ratio

The lesson series was concerned with translating part-to-part relationships into part-to-whole relationships, such as dividing quantities in a given ratio. Inspection of the statistical analysis of differences between the classwork, seatwork and given-task centres (see table 2) indicate that the visual and technical components of manifestation were stressed more during classwork. This underlying pattern in practice is triangulated by the narrative account of the lessons that follows. In the classwork event of Lesson 1 Bernie displayed the problem "share £20 in the ratio 2 to 3" on the interactive whiteboard and learners were tasked with solving it individually. The problem was presented along with twenty circles representing the twenty £1 coins that were to be shared between the two people. After allowing a brief period for individual consideration, Bernie led a whole-class sharing of ways to solve this problem. As part of this, Bernie demonstrated how the 20 coins might be shared in the ratio two to three by counting them out two, then three, then two, then three, and so on. This classwork phase of the lesson was categorised as featuring visual and technical manifestations as Bernie offered a visual representation of an enactive method of sharing the money in the given ratio. Building on Lesson 1, Bernie provided an opportunity for the whole class to recap methods for sharing quantities in a given ratio at the start of Lesson 2. The first example offered was: "share £140 in the ratio 3 to 4." Three methods were suggested by learners, the third of which was a simple integer 'scaling up' method: 4 : 3 ≡ 40 : 30 ≡ 80 : 60 and 80+60=140. The limitations of this method were revealed by the second example: "share £45 in the ratio 2 : 3 : 7." Following this, Bernie endorsed a single method, the first step of which was dividing the quantity by the total of the parts. Thus there was an emphasis placed on formal methods in this introductory classwork phase to the lesson, with technical manifestation foregrounded.

Technical and particularly functional manifestations were emphasised in the given-tasks assigned to learners during seatwork. For example, in Lesson 1 learners were provided with the opportunity to practise sharing quantities of money in given ratios, for example: "divide £20 in the ratio 4 : 1", and "divide £10 in the ratio 3 : 2." The given-task here was categorised as technical. In each of the following lessons students were set problems that required sharing in a given ratio, but increasingly these became worded, contextual problems. Thus functional manifestation was more emphasised over time.

The effect of between-desk-teaching was that overall seatwork was materially different to the assigned given-task, with visual manifestation emphasised more and functional less. This can be largely attributed to Bernie drawing on the image of 'moneybags' when between-desk-teaching in order to represent a unitary share, such as in Lesson 1:

In effect you've got 3 bags of money and 2 bags of money, which altogether is 5 bags of money.

If you share your £10 equally into those 5 bags you'll have £2 in each bag won't you? That person gets 3 bags [and] that person gets 2 bags, yes? (Bernie, Ratio Lesson 1)

Between-desk-teaching in Lesson 2 also referenced moneybags and included four separate conversations with different students over a period of 10 minutes in which Bernie offered essentially the same explanation of the solution to the same problem. It is interesting to note that while Bernie used a particular visual method for dividing in a given ratio during classwork in Lesson 1, the visual method offered when between-desk-teaching was equivalent but not identical.

Lesson 3 continued to include between-desk-teaching in which Bernie made reference to moneybags, but not Lesson 4. This suggests a progression from Lesson 3 to Lesson 4, as in Lesson 4 there was less recourse to the imagery of moneybags in Bernie's between-desk-teaching.

My reading of the between-desk-teaching actions across Lessons 2 and 3 was as a return to more familiar-to-the-learner manifestations of ratio, which I associated with offering learners explicit opportunity to 'fold back' [START_REF] Pirie | A recursive theory of mathematical understanding[END_REF]) to a secure level of understanding. In Lessons 2 and 3, this secure level of understanding was inferred to be a visual method for sharing quantities in a given ratio. Whereas in Lesson 4 Bernie was heard to return to more technical manifestations of ratio when between-desk-teaching, which I associated with offering learners explicit opportunity to fold back from an inadequate contextual understanding to a more secure understanding of a method for sharing in a given ratio.

Discussion

There are indications in this lesson series that between-desk-teaching was a lesson event foregrounded by Bernie in the context of teaching ratio to this class. Firstly, in total 52% of lesson time across the series was given over to between-desk-teaching compared to 27% to independentof-the-teacher seatwork and 21% to classwork. Secondly, I have offered examples of Bernie repeating the same explanation on multiple occasions through between-desk-teaching when an alternative choice could have been to explain once to the whole class. This was an instance of the repetition effect noted by [START_REF] Jakonen | Professional embodiment: Walking, re-engagement of desk interactions, and provision of instruction during classroom rounds[END_REF].

Furthermore I infer from the statistical analysis that between-desk-teaching was being deployed strategically as a way of making content available to learners. Non-linear opportunities were offered for growth in understanding [START_REF] Pirie | Growth in mathematical understanding: How can we characterise it and how can we represent it?[END_REF] that were sensitive to the needs of individual learners [START_REF] Jakonen | Professional embodiment: Walking, re-engagement of desk interactions, and provision of instruction during classroom rounds[END_REF]. Teacher explanation at the start of the lesson series emphasised visual methods for solving ratio problems, while technical and increasingly functional components of manifestation were emphasised in the given-task that were subsequently assigned. While the methods of problem solving offered through between-desks-teaching in Lessons 2 and 3 were not identical to the methods seen at the start of Lesson 1, there was a clear intention to offer visual representations of problems in order to facilitate their solution, and for these to remain available throughout the lesson series. By Lesson 4 it can be inferred that visual representations were no longer needed, but symbolic representations of contextual problems were being offered through between-desks-teaching by this stage. In summary, whether through conscious deliberation or out of habit Bernie's actions aligned in this context with choosing to use between-desks-teaching in order to offer learners explicit opportunities to fold back to a secure level of understanding from earlier in the lesson series.

Conclusion

Previous studies have considered between-desks-teaching as an isolated event or as an event within a lesson (e.g. [START_REF] Inoue | Deconstructing teacher expertise for inquiry-based teaching: Looking into consensus building pedagogy in Japanese classrooms[END_REF], but the current study has taken the lesson series as the unit of analysis. There is stronger evidence of Bernie's between-desks-teaching being a deliberate act because this phenomenon was explored across a series of lessons. A further affordance of this research design is that through the analysis of teacher-talk and given-task it supports making sense of the contribution of between-desk-teaching to the way content is made available to learners over time. The lesson series is a unit of time over which teachers typically plan, and while the role of between-desks-teaching at this grain size may not be an issue they (and researchers) currently attend to, it could be. The two case studies involving Bernie reported in this paper offer an ideal example of when it might be appropriate for a teacher's attention to be drawn to this. In the case of Bernie teaching ratio, as we have seen, between-desk-teaching was used extensively and strategically. Yet in the case of teaching linear equations it was not used in this strategic way at all. Such contrasts in the practice provide in my experience powerful prompts for teacher reflection.

Table 1 : The seatwork and given-task centres of the four lesson series

 1

	Case		Seatwork		Given-task
		n		410
	Ashley teaching	Centre	(0.24, 0.27, 0.49)	(0.24, 0.32, 0.44)
	geometric constructions
		Difference	**
		n		301
	Bernie teaching ratio	Centre	(0.06, 0.29, 0.65)	(0.00, 0.39, 0.61)
		Difference	MAT
		n		208
	Bernie teaching linear	Centre	(0.20, 0.02, 0.78)	(0.18, 0.02, 0.80)
	equations
		Difference	NS
		N		270
	Courtney teaching linear	Centre	(0.17, 0.09, 0.74)	(0.15, 0.12, 0.73)
	equations
		Difference	NS

N.B. Difference indicators show whether differences between the centres were material (MAT)

, statistically significant at the 1% level (**), or not statistically significant at the 1% level

(NS)

Table 2 : The classwork, seatwork and given-task centres for Bernie's lesson series on ratio

 2

	Case		Classwork	Seatwork	Given-task
		n	82	301	301
	Bernie teaching	Centre	(0.14, 0.11, 0.75) (0.06, 0.29, 0.65) (0.00, 0.39, 0.61)
	ratio
		Difference	MAT	MAT
	N.B. Difference indicators show whether differences between the centres were material (MAT), statistically
	significant at the 1% level (**), or not statistically significant at the 1% level (NS)


Kristin Krogh

Yvonne Grimeland

Teachers' learning

Keywords: Learning goals, identifying mathematical ideas, revised Bloom's Taxonomy

come

Introduction

To plan lessons with the intended learning outcome (in this paper called a "(learning) goal") as the point of departure is an old tradition. Tyler's influential model for planning [START_REF] Tyler | Basic principles of curriculum and instruction[END_REF] is described by [START_REF] John | Lesson planning and the student teacher: re-thinking the dominant model[END_REF] as linear, meaning that you start with identifying a goal for the lesson, and then choose activities, assessment, and other elements that fit the goal. More dynamic models (see e.g. [START_REF] John | Lesson planning and the student teacher: re-thinking the dominant model[END_REF]) also include specifying goals. In one framework for learning how to teach [START_REF] Hiebert | Preparing Teachers to Learn from Teaching[END_REF], specifying the learning goals for an instructional episode is listed as the first item of the framework. Moreover, learning goals are closely linked with theory on assessment for learning, where one principle is that students should understand the learning intentions, including an understanding of what would count as success criteria [START_REF] Hodgen | Mathematics inside the black box: Assessment for learning in the mathematics classroom (Black box assessment for learning series)[END_REF]. Thus, goals are typically not only a part of a teacher's planning process, but also present in the classroom. For instance, it is common in Norwegian schools that pupils get a goal for the week/day/lesson in every subject (Vikan & Buland, 2014). Another influence on the use of learning goals in Norway is that the education system is managed by objectives. The practice of using learning goals in teaching is probably similar to what is found in other countries where the education system is managed by objectives, e.g. Denmark (see [START_REF] Hansen | På vej mod en målbevidst, målstyret kompetenceorienteret matematikundervisning[END_REF]. In Denmark, a study has shown that visible learning goals implicitly lead to a focus on skills and performance instead of mathematical competence [START_REF] Hansen | På vej mod en målbevidst, målstyret kompetenceorienteret matematikundervisning[END_REF]. A Norwegian Master's thesis investigating the mathematical competences present in five 8 th grade teachers' learning goals (as they appeared in teaching) found that most of the goals had a short-time horizon and mostly focused on procedures, rather than on understanding [START_REF] Selling | Målsättningar i matematiken studie av fem högstadielärares målsättningar i matematikundervisning[END_REF].

In mathematics teaching, deciding upon a goal is closely connected with identifying a particular mathematical topic to discuss [START_REF] Superfine | Planning for Mathematics Instruction: A Model of Experienced Teachers' Planning Processes in the Context of a Reform Mathematics Curriculum[END_REF]. This identifying process can be linked with Askew's term "precision" [START_REF] Askew | Mathematical discipline knowledge requirements for prospective primary teachers, and the structure and teaching approaches of programs designed to develop that knowledge[END_REF]. Being precise, including about the mathematical knowledge at stake, is one of the key elements in his framework for mathematics subject knowledge for teaching. It is suggested in a study by [START_REF] Sullivan | Processes and priorities in planning mathematics teaching[END_REF] that teachers tend to concentrate on activities and general aspects of pedagogy rather than on content objectives when planning mathematics lessons. In the study, Australian teachers were asked "what is the most important idea that you will focus on for [the next mathematical topic you plan to teach]?". Only half of the teachers in the study could describe the important ideas with a level of precision "likely to assist them in their planning" (Sullivan et al., 2012, p.478). In the conclusion to the study, the authors write that "it seems that teachers may need support in articulating the 'important ideas' on which they will focus and the sequencing and interrelationship of these ideas" (Sullivan et al., 2012, p. 478). Our study is based on the assumption that identifying mathematical ideas, or relationships between them, is intrinsic to producing adequate learning goals. It is therefore of interest to gain more knowledge about teachers' identification of ideas underpinning the goals. To this end, our research questions are:

What characterises teachers' mathematical learning goals for mathematics lessons?

What characterises teachers' identified mathematical ideas behind the learning goals?

With the above stated assumption in mind, there is a hypothetical outcome where the two research questions gain similar results (a mathematical learning goal may be referring to a mathematical idea). However, for the first question, we aimed to analyse the learning goals given explicitly to students, and there could be pedagogical reasons not to give the mathematical idea to students in advance. Thus, the second research question will give more insight into the teachers' practices regarding learning goals. In order to answer the research questions, we designed a questionnaire, which was answered anonymously by 51 teachers from primary and lower secondary school. In the subsequent analysis, we used two frameworks: The revised Bloom's Taxonomy [START_REF] Anderson | A Taxonomy for learning, teaching, and assessing: A revision of Bloom's Taxonomy of educational objectives[END_REF]) and a framework from the already mentioned paper by [START_REF] Sullivan | Processes and priorities in planning mathematics teaching[END_REF].

As the study is set in Norway, we provide the reader with some contextual background. In Norway, the non-linear "didactic relation model" for lesson planning [START_REF] Bjørndal | Nye veier i didaktikken? En innføring i didaktiske emner og begreper (Pedagogisk perspektiv)[END_REF] has become part of the Norwegian "pedagogical canon", as it is used in a number of textbooks in teacher education [START_REF] Engelsen | Kan laering planlegges? Arbeid med laereplanerhva, hvordan[END_REF]. In this model, six (originally five) elements are shown as mutually interplaying, with "goal" being one of them [START_REF] Bjørndal | Nye veier i didaktikken? En innføring i didaktiske emner og begreper (Pedagogisk perspektiv)[END_REF]. However, although goals are not the typical starting point for lesson planning in Norway, they are an important part of teachers' practices. More precisely, "[g]oals are visible in a variety of ways; in the weekly plans, when teachers write goals on the blackboard and when they discuss goals orally in class" (Vikan & Buland, 2014, p. 17). The current curriculum, implemented in 2006 (Ministry of Education and Research, 2016), is written in terms of goals. For all school subjects, there is a subject curriculum, which consists of a list (ordered by year) of so-called competence aims, and the students are assessed by whether they have achieved these competence aims. 1 As an example, the curriculum for mathematics lists that after year 4 (aged 9-10), a student should be able to "make estimates of and find numbers by means of counting in one's head, using counting aids and written notes, making estimates by calculating with simple numbers, and assessing answers" (Ministry of Education and Research, 2010). The competence aims are further decomposed, ultimately by the teacher(s) when planning a lesson.

Theoretical frameworks

One of the most well-known models for analysing goals is Bloom's Taxonomy [START_REF] Bloom | Taxonomy of educational objectives: The classification of educational goals[END_REF]. We build our analysis of the learning goals on the revised Bloom's Taxonomy [START_REF] Anderson | A Taxonomy for learning, teaching, and assessing: A revision of Bloom's Taxonomy of educational objectives[END_REF], in which the original taxonomy has become a two-dimensional model, with one dimension focusing on the cognitive aspect of the goal, the other on the knowledge at stake. Unlike the original taxonomy, the revised version does not stress a rigid hierarchical structurethat is, a goal in a given category in either dimension is not necessarily "better" or "more demanding" than a goal in the previous category in that dimension. On the other hand, the revised taxonomy is meant as a tool to, among other things, "help educators consider the panorama of possibilities in education" (Anderson & Kratwohl, 2001, p. 35). As such, it is most common among educators, but it is also used in research (see e.g. [START_REF] Hansen | Teachers' mathematical discussions of the body mass index[END_REF]. We chose the revised Bloom's Taxonomy because it would give an overview of the learning goals, suggesting some characteristic features and some features that do not appear among the goals.

We explain here how we have interpreted the categories of the revised Bloom's Taxonomy. The four categories of the knowledge dimension are interpreted as follows: Factual knowledge means terminology and conventions. Conceptual knowledge consists of mathematical concepts, ideas, and models. Procedural knowledge includes everything related to doing something [START_REF] Anderson | A Taxonomy for learning, teaching, and assessing: A revision of Bloom's Taxonomy of educational objectives[END_REF]: computing, algorithms, strategies. Metacognitive knowledge is knowledge about cognition, including meta-knowledge of strategies. Moreover, the six categories of the cognitive process dimension are interpreted as follows: Remember is about memorising things (like definitions, terminology, or the steps of an algorithm). The Understand category is for goals including verbs like "learn", "know", "explain" and others listed by [START_REF] Anderson | A Taxonomy for learning, teaching, and assessing: A revision of Bloom's Taxonomy of educational objectives[END_REF] (in our data, it is hard to identify the category in other ways). The Apply category is for doing something of a non-algorithmic nature. Note that this differs slightly from Anderson and Krathwohl's description, as we would classify performing a given algorithm to a standard problem as Remember procedural knowledge rather than Apply procedural knowledge. Analysing means breaking something down to look at relationships, while Evaluate is to test or detect something based on given criteria. Finally, Create means to produce something new, like a hypothesis or an argument.

To analyse the teachers' stated mathematical ideas, we use a framework introduced by [START_REF] Sullivan | Processes and priorities in planning mathematics teaching[END_REF]. The framework consists of five categories, which we have interpreted as follows: For A mathematical idea within the topic, the mathematical idea/relationship is clearly specified. For the second category, An element of a hypothetical learning trajectory, the mathematical idea/relationship is underlying the response, but not explicitly statedinstead, the focus is on the students' discoveries or activities. The third category, A sub-topic of the larger mathematical topic, is characterised by keyword-style answers. The two final categories have self-explanatory titles: A statement which could apply to most other topics and not specifically an important idea tied to the particular topic and A statement which describes aspects of pedagogy and does not particularly address content. We did not intend to connect the two frameworks. Instead, it was our aim to treat the two research questions separately, and then use the second to shed light on the first, as the goals are our main topic of interest.

Method

The research questions require data in the form of teachers' mathematical learning goals, together with information on what the teachers regard as the mathematical idea underpinning the learning goals. To obtain this, we designed an anonymous questionnaire for teachers, introduced by the request "base your answers on the next mathematics lesson you give". The questionnaire and the quoted responses are translated from Norwegian by the authors. The questions relevant for the research questions treated in this paper are Q2: "What is the content-related learning goal of the lesson?"; Q4: "Is there a particular mathematical relationship or idea within the learning goal you want to especially focus on in the lesson? If yes, what?"; Q6: "Do you as the teacher have any other content-related or pedagogical goals for the lesson, that are not made known to the students? If yes, what?" Additionally, we asked about the overall topic of the lesson, the age of the students, and whether the goal would be made explicitly known to the class.

We piloted a first version of the questionnaire among a group of in-service teachers taking a course given by the first author. In the discussion with the pilot respondents, we were made aware that the terms "mathematical relationship/idea" could be hard to distinguish from "content-related goal". As suggested by the pilot respondents, we changed the order of some questions, intending to make this distinction clearer. Yet, we chose to keep the terms used, as this would be in line with the terms used in the theoretical frameworks of the study. Nevertheless, this suggested confusion could indicate that the validity of the study is not optimal, as we will comment further upon in the discussion.

In-service teachers supervising school placement of student teachers were chosen as the cohort because we had easy access to them via the university's digital learning management system (LMS). In general, supervisors have not taken more mathematics courses than other teachers have. The respondents are called "teachers" throughout this paper. We posted information of the study in the digital LMS, and additionally issued individual messages to all teachers, with a link to the digital questionnaire. Later we sent two reminders to all the teachers. In this period, we obtained 29 digital answers. To get more data, the second author visited a meeting for supervisors hosted by the university the following term, distributing paper copies of the questionnaire (as well as link to the digital version). At the meeting, 22 completed questionnaire sheets were handed in. Thus, the data in this study consists of 51 questionnaire answers (29 digital and 22 on paper). It makes sense to assume that most teachers answering in the first round had access to their actual planning notes, while most teachers answering in the second round did not. Yet, we have treated the answers from both rounds as one data set, because the teachers at the meeting had the option of answering digitally later, which they chose not to do (so we can assume they considered that they remembered the details sufficiently). Some teachers might have answered the questionnaire twice (once in each round); this has not been taken into consideration during analysis, as we cannot identify them in the data material. The teachers work in primary or lower secondary school, and the majority (37 teachers) teach grades 4-7. One teacher in the cohort did not state a goal when answering Q2; thus, 50 goals were analysed.

After collecting all the data, the first author coded the replies to Q2 and Q4 with the revised Bloom's Taxonomy and Sullivan's framework for mathematical ideas, respectively. Then the coding was discussed with the second author. We used the revised Bloom's Taxonomy as follows: First, we coded by the knowledge dimension, subsequently by the cognitive process dimension. Not all goals were easy to reliably classify by the framework. For instance, it is sometimes unclear whether a goal belongs to the Remember or the Understand category, in particular for goals in the Procedural knowledge category. In these situations, we used the replies to additional questions (in particular Q4) to look for indicators on how to classify the goal. We also intended to use Sullivan's framework to analyse goals given as replies to Q6, but here the replies were (with very few exceptions) either blank, or belonging to the last of the five categories (general aspects of pedagogy). Thus, we concluded that this question did not shed light on the research question, and we include no analysis of the replies.

Because of the methodological weaknesses in the study, we need to be careful when addressing the data. Nevertheless, we can view the results as preliminary results, initiating discussions and indicating directions of future research.

Results

The complete results of the analysis are presented in Table 1 (Q2) andTable 2 ( Q4). From the analysis of Q2 (the goals), the most prominent findings are a focus on understanding and a lack of higher-level goals. Regarding Q4 (the mathematical ideas), the analysis reveals a vague/missing description of mathematical ideas. All teachers except two state that they would present the goals to their students before the class.

The cognitive process dimension

A focus on understanding

Of the 50 goals that were analysed, 27 goalsmore than half of the goalswere found to belong to the Understand category. Examples include "understand fractions as a part of a set" (Understanding conceptual knowledge; the knowledge at stake can be identified as the set model for fractions) and "The students should learn addition with regrouping" (Understanding procedural knowledge; the respondent states in reply to Q6 that no algorithm will be given, but that the work builds a fundament for a later understanding of the standard algorithm). In contrast, there are few goals identified in the Remember category.

A lack of higher-level goals

By "higher-level", we mean the three upper categories in the cognitive dimension; Analyse, Evaluate and Create. No goals were found to fit within these categories. The "highest" category we found examples of in our analysis were C3, Applying conceptual knowledge. In this category, any method mentioned for the given procedure is subject to the students' exploration, or there is no method given at all. One example, with a given method, is "that the students should get experience with using strategy with 'friendly numbers' in computing". With a particular idea highlighted in the reply to Q4, this goal could belong to category C4, but no such idea is stated.

Category Number of replies

1. A mathematical idea within the topic 8

2. An element of a hypothetical learning trajectory 8

3. A sub-topic of the larger mathematical topic 7 4. A statement which could apply to most other topics and not specifically an important idea tied to the particular topic 12 5. A statement which describes aspects of pedagogy and does not particularly address content 3

Missing 11

Uncategorised 1

Vague/missing description of mathematical ideas

As Table 2 shows, only 16 of the 50 analysed replies fall into the two top categories in Sullivan et al.'s framework. Eleven of the teachers do not state a mathematical idea at all. Examples of category 1 and 2 include "the relationship between division and multiplication" and "understanding that decimal numbers consist of parts that are less than one whole" (category 1); "strategy thinking how to find fourths and eights -focus on halving" and "converting to correct unit of measurement before computing" (category 2). Examples from category 3 are "friendly numbers" and "the decimal number system". In category 4, we find "focus on exactness and using a sketch figure", in category 5 "more than one way leading to the target". The uncategorised reply is "show a way to do it ['it' is division by one digit numbers]", where we do not know whether the "show" regards the teacher or the students.

Table 1 : Revised Bloom's Taxonomy table. Number of goals in each intersected category

 1

	The knowledge dimension	1. Remember	2. Understand	3. Apply	4. Analyse	5. Evaluate 6. Create
	A. Factual knowledge	2
	B. Conceptual knowledge	1	10	2
	C. Procedural knowledge	6	13	6
	D. Metacognitive knowledge		4
	Other	6

Table 2 : Sullivan et al.'s framework. Number of replies in each category

 2

Regulations of the Education Act,

2006, §3-3. Available at https://lovdata.no/dokument/SF/forskrift/ 2006-06-23-724?q=forskrift%20til%20opplaeringslova

Discussion

We can assume from the preliminary findings that many teachers intend to teach for understanding, judging by their use of verbs like "understand", "learn", "see" and so on (the verbs attributed to the Understand category in Bloom's Taxonomy). This is in contrast to Selling's findings, where the goals were dominated by a focus on procedures and not understanding [START_REF] Selling | Målsättningar i matematiken studie av fem högstadielärares målsättningar i matematikundervisning[END_REF]. One reason for this difference can be that we have analysed learning goals from the teachers' lesson planning, while Selling analysed goals that appearedexplicitly or implicitlyduring lessons. Nevertheless, the teachers' focus on students' understanding is an uplifting finding. As terms like "understand" and "learn" are ambiguously used in pedagogics and mathematics education research, more information is needed in order to a more detailed picture of the teachers' use of such words. Yet, it is tempting -from the researchers' point of view -to connect the idea of "understanding" to a focus on the mathematical ideas underpinning what is to be understood. Here, we note an intriguing discrepancy between the emphasis on understanding in the learning goals and the vague/missing description of mathematical ideas.

When discussing the teachers' seemingly disappointing reports on mathematical ideas, it is necessary to return to the methodological problem we encountered when wording Q4 in the questionnaire. Because we wanted to elicit information about the teachers' view on mathematical ideas, we ended up using a termmathematical ideathat seemed unfamiliar and confusing to the pilot respondents. As mathematicians, we may have assumed that the meaning of the term is evident. However, a brief search in literature reveals that very few papers using the term (or similar terms, like "big idea") actually define it. An exception is the in the German-speaking tradition, where mathematics educators use "fundamental ideas" to guide their mathematics teaching (see [START_REF] Vohns | Fundamental Ideas as a Guiding Category in Mathematics Education-Early Understandings, Developments in German-Speaking Countries and Relations to Subject Matter Didactics[END_REF], but this isfrom our experiencenot familiar to the Norwegian mathematics education community. Interestingly, we find traces of a similar problem in Sullivan et al. 's paper (2012). In their study, they asked the teachers "what is the most important idea that you will focus on for that topic", assuming that the teachers would interpret "idea" as "mathematical idea". When the responses showed that this was not always the case, the authors conclude that "[t]he fact that the important idea in the teaching of a forthcoming mathematical topic in the mind of a teacher might not be mathematical is revealing in itself" (Sullivan et al., 2012, p. 468).

Following this discussion, we infer that we cannot conclude whether we have found a problem with the teachers' level of precision (cf. [START_REF] Askew | Mathematical discipline knowledge requirements for prospective primary teachers, and the structure and teaching approaches of programs designed to develop that knowledge[END_REF], with the teachers' own mathematical competence, or with the use of a term that is understood differently by the researchers and by the teachers. To ensure the validity of further studies, such terms should be avoided. Other approaches (like interviews) can help to distinguish the teachers' mathematical focus when writing learning goals.

When it comes to the third finding, the lack of higher-level goals, there is also a potential explanation due to the same issue. The teachers' replies to Q2 were partly analysed based on their reply to Q4. Thus, it is possible that some goals were misjudged due to our use of the term "mathematical idea". At the same time, many goals were easy to analyse without the need to consider Q4. Based on this, the preliminary findings indicate the same trend that has been found in Denmark, i.e. that the focus on visible learning goals can lead to more focus on skills and less on competences [START_REF] Hansen | På vej mod en målbevidst, målstyret kompetenceorienteret matematikundervisning[END_REF].


Fay Baldry
email: fb128@le.ac.uk

A comparison of the treatment of mathematical errors arising from teacher-initiated and student-initiated interactions

Keywords: Classroom norms, errors, mathematics education, misconceptions

This paper explores teachers' treatment of errors as they occur both within and outside of Initiate-Response-Evaluate (e.g., Cazden, 1988) interactions. Data are drawn from a classroom-based video study in England, involving three participating secondary mathematics teachers and six classes. This paper focusses on one of those teachers, whose lessons contained the widest range of treatment of errors. This included facilitating extended peer-to-peer turn-taking and bald negative evaluation of student contributions; the later appeared to be atypical, both in relation to the other teachers in this study and the wider literature. Analysis showed, for this teacher, indirect evaluation of errors was associated with teacher-initiated interactions, whereas bald negative evaluation was associated with student-led exchanges. Here, consideration is given to the interactional patterns related to the treatment of errors, and the potential impact on learning opportunities for students.

Introduction

Patterns of classroom talk vary, but [START_REF] Lefstein | Classroom discourse: The promise and complexity of dialogic practice[END_REF], amongst other, report that in Anglo-American schools whole-class talk is typically controlled by the teacher. Moreover, a large proportion of interactions are in Initiate-Response-Evaluate (IRE) format, that is to say these are initiated by the teacher asking a question, which is followed by a student response and concluded with another teacher-turn, in which the mathematical validity of the student contribution is evaluated [START_REF] Cazden | Classroom discourse: The language of teaching and learning[END_REF]. The role and treatment of mathematical errors has been reported in previous studies (e.g., [START_REF] Steuer | Dealing with errors in mathematics classrooms: Structure and relevance of perceived error climate[END_REF], but the ubiquitous nature of IRE interactional patterns means that many of these discussions draw on evidence from this type of turntaking exchange. Other types of interaction patterns, such as talk initiated or led by students, are less common in many classrooms [START_REF] Edwards-Groves | Classroom talk: Understanding dialogue, pedagogy and practice: PETAA Newtown[END_REF], but a move away from univocal teacher talk has been associated with enhancing learning opportunities [START_REF] Alexander | Towards dialogic teaching : rethinking classroom talk[END_REF][START_REF] Edwards-Groves | Classroom talk: Understanding dialogue, pedagogy and practice: PETAA Newtown[END_REF]. This paper focusses on one secondary school teacher in England and the shifts in turn-taking patterns that are seen when he teaches two classes. In particular, this study analyses the treatment of errors as they arise in IRE exchanges and student-initiated turns, with the aim to explore the leverage of errors to develop productive talk in different types of interaction.

Theoretical Framework

The use of 'error' varies between studies; some use the term to indicate anything treated as such by the teacher (e.g., [START_REF] Ingram | Handling errors as they arise in whole-class interactions[END_REF], whereas others use 'error' when they identify an issue with a mathematical contribution (e.g., [START_REF] Heemsoth | Secondary school students learning from reflections on the rationale behind self-made errors: A field experiment[END_REF], although it is supposed there is considerable overlap between these two conditions. Whilst research has highlighted the role errors can play in providing learning opportunities [START_REF] Bray | A collective case study of the influence of teachers' beliefs and knowledge on error-handling practices during class discussion of mathematics[END_REF], there is evidence a productive use of errors is not a routine part of many teachers' pedagogical practice [START_REF] Ding | Teacher interventions in cooperative-learning mathematics classes[END_REF]. The potential conflict between errors being seen as learning opportunities and the notion that errors should be avoided has been discussed. For instance, [START_REF] Swan | Dealing with misconceptions in mathematics[END_REF] argues some teachers avoid errors because they believe students might be confused by the discussion and erroneously remember the error rather than the correct solution. The notion that errors are embarrassing has been used to justify the avoidance of errors and to explain why both teacher and students take mitigating actions when they do occur [START_REF] Pekrun | Emotions at school[END_REF]. However, work by [START_REF] Steuer | Dealing with errors in mathematics classrooms: Structure and relevance of perceived error climate[END_REF] has shown that the classroom climate and students own goal orientations have a significant effect on the affective reaction of students to errors.

Questions permeate most aspects of whole-class talk and hence are often the precursor to mathematical errors. In many classrooms, IRE sequences dominate whole-class interactions, being described as the 'default' setting for many teachers [START_REF] Resnick | How (well structured) talk builds the mind[END_REF]. Whilst questions can play a variety of roles, from developing student understanding through to a tool for drawing attention to mathematically important features [START_REF] Mason | Asking mathematical questions mathematically[END_REF], within IRE interactional sequences the questions asked are often more limited in scope [START_REF] Edwards-Groves | Classroom talk: Understanding dialogue, pedagogy and practice: PETAA Newtown[END_REF]. Teachers frequently ask questions that have a single expected response, with a shared understanding that the teacher knows the answer and that they will evaluate the accuracy of the student's response in the third turn. Within this type of turn-taking, correct solutions are usually acknowledged immediately by the teacher, whereas errors are rarely directly acknowledged as such [START_REF] Ingram | Handling errors as they arise in whole-class interactions[END_REF]. When errors do occur in student responses, the negative evaluation in the third turn is often communicated indirectly, through teachers' pedagogical actions, such as redirecting the question to another student, asking follow-up questions or other actions to draw attention to the nature of the problem. Indeed, the pedagogical moves taken to avoiding direct evaluation are often directed towards allowing students to undertake additional mathematical work in subsequent turns, including self or peer correction of the error [START_REF] Ingram | Handling errors as they arise in whole-class interactions[END_REF]. However, whilst there are examples of learning opportunities being created after an error has been identified in an IRE sequence, there is also evidence that teachers tend to reduce the mathematical load in subsequent turns and may 'funnel' students towards particular solutions [START_REF] Jones | Teachers' Interpretations of Effective Whole-class Interactive Teaching in Secondary Mathematics Classrooms[END_REF], thereby limiting the learning opportunities made available. From a conversation analysis perspective, the avoidance of direct negative evaluation would imply that errors are embarrassing and should be avoided [START_REF] Seedhouse | The case of the missing "no": The relationship between pedagogy and interaction[END_REF], but [START_REF] Steuer | Dealing with errors in mathematics classrooms: Structure and relevance of perceived error climate[END_REF] argue that the lesson context is more complex, with students' engagement shaped by their reading of the error climate of the classroom.

This study seeks to contribute to the understanding of the treatment of errors through the exploration of their occurrence in different patterns of whole-class interactions.

Methodology

This paper draws on data from a classroom-based video study in England. As part of a qualitative case study in the interpretative tradition, eighteen secondary mathematics lessons have been observed and recorded by this author. Three teachers participated and for each teacher two classes with different attainment profiles were observed. Lessons were recorded with two static video cameras and lesson artifacts, such as students' work, were collected. Semi-structured interviews were conducted with the teachers before and after each lesson and lesson planning documents were obtained. Audio data from each lesson was transcribed and collated with copies of mathematics written or projected in the shared space of the class whiteboards. The initial coding of the transcripts classified activities as either mathematically relevant or not, with episodes in the former category being subject to further scrutiny. As part of the wider study, the Orchestration of Mathematics Framework (OMF) has been developed to capture teachers' pedagogical moves and the potential impact this has on the mathematics made available to learners [START_REF] Baldry | Analysing a teacher's orchestration of mathematics in a 'typical' classroom[END_REF]. The further analysis of the mathematically relevant episodes involved an iterative process of coding interactions against this conceptual framework and cross-referencing with other data sources, such as teacher interviews and student work.

A key element of the analysis was the nature of interaction patterns in class-level talk, including the treatment of errors, although it is acknowledged that the interpretation of interactions is predicated on understanding the classroom culture [START_REF] Larsson | Exploring a framework for classroom culture: a case study of the interaction patterns in mathematical whole-class discussions[END_REF]. The notion of classroom norms [START_REF] Cobb | Participating in Classroom Mathematical Practices[END_REF] has been drawn on to establish whether interactions are typical or atypical, and thereby allow inferences beyond the particular occurrence, but this is limited to what is discernible at a whole-class level. This paper focusses on one teacher, Sam, and four lessons from two classes, selected when the analysis highlighted distinct patterns of interaction in relation to errors. These have been chosen because they do appear to be atypical, both in relation to the other two teachers in the study and the wider literature. Here, 'errors' are taken to be students' contributions that include a mathematically incorrect statement, but these occurrences were also all treated as errors by Sam. The practices of one teacher cannot be generalized to others, and indeed it could be argued that the analysis of a limited number of lessons in a study of this size may not provide a sufficiently representative picture to allow comparisons [START_REF] Staub | Mathematics classroom cultures: Methodological and theoretical issues[END_REF]. However, I believe that a consideration of these more atypical behaviours has the potential to contribute to our wider understanding of the relationships between interactional patterns of whole-class talk, the treatment of errors and the learning opportunities for students.

Findings

Across all lessons, whole class-level talk represented about one third of the lesson time, and IRE interactional patterns accounted for at least two thirds of this talk. Most questions asked by Sam had a limited range of appropriate mathematical responses and about 20 percent were 'simple', that is to say students should have been able recall the answer immediately and without effort. About 75 percent of IRE exchanges were positively evaluated, a rate that was relatively stable across the lessons. The majority of the remaining exchanges were coded as containing negatively evaluations, although there were a few ambiguous cases. The interactional patterns in IRE exchanges across both classes were very similar. For example, when discussing the rules of indices, the following IRE interactions occurred, but similar exchanges were regular occurrences in all lessons: Most questions asked by Sam had a limited range of appropriate mathematical responses (e.g., line 53) and when the questions were more ambiguous his reaction in the third turn indicated a particular response was sought (e.g., line 32 and 34). Acceptable answers were indicated in two main ways. First, they were often explicitly acknowledged, with expressions such as "yep", or by the inclusion of a positive indication of quality, such as "excellent" (e.g., line 34). Second, Sam regularly repeated a student's response (e.g. line 55), and the analysis indicated there was a taken-as-shared understanding that this also indicated an acceptable response. Negative evaluations occurred when students' responses contained mathematical errors (e.g., line 56), and when there was no error per se but the response was deficient in some way (e.g., line 31). For example, in line 33, the student recognises where his response does not meet Sam's requirements and self-corrects. In line 56, a student offers a mathematically incorrect response; Sam paused before offering some indication as to location of the error and then nominated another student who had their hand up to answer.

One notable shift away from IRE sequences was when there were sequential student turns. Whilst Sam sometimes nominated students to answer questions, it was also common practice for students self-nominate by calling out. In both classes, there were instances where different solutions were given before Sam responded. There did, however, appear to be differences in how the interactions were controlled by Sam when both errors and a correct solution were present. In particular, in the higher attaining class there were eight occasions where there were extended peer-to-peer exchanges. For example, when Sam asked for the value of in the following occurred: No it isn't its one guys ok its one 254 Sam:

Er vote then (.) vote for zero

Extract 2: Class A multiple student contributions

There were no discernable responses from Sam after a number of students had offered " " (lines 244 & 245) or the correct reply of "one" (line 246). The students continued to self-nominate by calling out, although the students were not necessarily responding to the immediately preceding turn (e.g., line 249). After Sam called for a vote for zero or one, started in line 254, he reverted to IRE sequences, where he focused the interactions on the mathematically correct answer by asking about situations related to , where was a range of natural numbers; some of these questions would have been simple for students to answer, such as "four divided by four".

In the lower attaining class, there were twelve occasions when two or three students self-nominated and called out answers that included an error before Sam replied. For example: In these cases, all the students appeared to be responding to Sam's original question. In the nine occasions where responses include both an error and a correct response (e.g., lines 309 & 310), Sam positively evaluated the correct response and offered no discernable reaction to the error. The remaining three occurrences only contained errors and were followed up by further IRE exchanges.

The other notable variation was the explicit negative evaluation of errors. In whole-class discussions, there were twelve occurrences of Sam using a bald "no" in class B, the lower attaining class, with one in the higher attaining class. In all bar one case an element of correction was included in the turn containing negative evaluation and the student did not take the subsequent turn. Three quarters of occurrences arose outside of IRE interactions, including the one case from class A when students were offering prepared explanations. Towards the end of the discussion about the following exchange occurred in class B:

168 Sam: If I tried to do (..) two squared plus three to the power of four that'll (.) sorry times three to the power four (.) is two times two times (.) three times three times three times three (. Sir can't you do five to the something like three add two equals five and you do five to the power whatever 170 Sam:

No because look we can't do to the power five or the power six because we've got these threes and these twos and they're different numbers and you can't combine them (..)

Extract 4: Bald negative evaluation

In line 168, Sam concluded his turn with a statement rather than a clearly defined question, so the comment made in line 169 was coded as student-initiated. Sam took the following turn (line 170); he immediately gave a bald "no" and proceeded to offer an explanation that appeared to be directed at correcting the student's comment. The student did not take the subsequent turn.

Discussion

In common with many other studies (e.g., [START_REF] Resnick | How (well structured) talk builds the mind[END_REF], the IRE sequence was the dominant form of whole-class interaction for both classes. Moreover, within these IRE exchanges, Sam's treatment of errors typically conformed to the well-reported approach of indirect negative evaluation in the third turn (e.g., Ingram et al., 2013). These structures allowed Sam to shape the focus of the interactions and provided opportunities for students to undertake further mathematical work to correct errors. The focus on the correct solution and the use of simpler questions after errors could be seen as reducing or removing the mathematical learning opportunities; alternatively, considered as a whole, they could be seen as serving to highlight the underlying mathematical structure. In these situations, the responsibility to correct errors lies at least partially with the students, but the arbiter of mathematical correctness resides with Sam.

When other types of whole-class talk were considered, different interactional patterns emerged, including differences between classes. In both classes there were instances where contradictory solutions were called out by students in response to a question from Sam. In class A, a regular occurrence was for peer-to-peer exchanges to continue beyond multiple replies to Sam's original question and extended sequences of peer-to-peer turns were formed. The established norm was that if Sam did not respond verbally when the evaluative third turn would have otherwise of occurred, then peer-to-peer turns were allowed. Whilst some later students turns appeared to be responding to Sam's original question, and many were simply making statements of their own position, students also offered justifications or engaged with other students' comments. For example, in Extract 2, line 247, the third student added a justification for "one", even though in this case the reasoning was faulty. Sam eventually took a turn and control of the interactions (e.g. extract 2, line 254); these peer-to-peer exchanges were all followed by IRE sequences, which focused the interactions on the correct solution; errors made in the peer-to-peer exchanges were rarely mentioned or interrogated. For instance, after the vote that followed extract 2, Sam constructed an IRE sequence around , where was a range of natural numbers, but the potential origins of zero were not discussed, such as attending to the exponents of or thinking . In these student-led exchanges, students had the opportunity to consider alternative solutions and some took the opportunity to articulate their reasoning, argue for their position or to challenge others; a shift in accountability, albeit briefly. However, the resolution of the debate was conducted through the more typical IRE exchanges with the final arbitration residing with Sam.

When class B is considered, there did appear to be similar potential starting points that had led to the peer-to-peer turns in class A. For example, extract 3 was one of nine occasions where both incorrect and correct responses were offered in response to Sam's question, but none of these extended to further peer-to-peer turn taking. Instead, Sam took control of the exchanges at an earlier stage and drew immediate attention to the correct answer through positive evaluation.

The explicit negative evaluations of students' contributions were predominantly associated with class B and student-initiated comments. As in extract 4, they occurred during whole-class talk when the students self-nominated to take the next turn and without a direct question being asked by Sam. In a previous study, [START_REF] Ingram | Handling errors as they arise in whole-class interactions[END_REF] found explicit negative evaluations by the teacher, and the associated corrections, occurred when there was an issue with communication of ideas rather than with the substance of the answer, although these appeared to relate to IRE sequences. In contrast, in this study explicit negative evaluation were associated with key conceptual issues, and occurred outside of IRE exchanges. For instance, in extract 4, the bald "no" followed on from a student misapplying a rule of indices by attempting to combine powers with different bases. The corrections associated with the "no" allowed Sam to redirect the focus of the interactions back to his chosen approach within one turn, but removed the opportunity for students to correct their own or their peer's error. One possible explanation for the correlation between bald negative evaluations and student-initiated comments is that these, by their nature, have not been planned for in the same manner as responses to questions within IRE sequences As such, they were not an integral part of Sam's pedagogical goals; the bald evaluations and corrections allowed a swifter realignment with those goals. From a conversation analysis perspective, a bald evaluation would carry the message that making an error is not embarrassing. However, the benefit of this potential acceptance of errors would have to be considered alongside the management of the pedagogical trajectory away from student-initiated comments, which could be read by students as devaluing their contributions.

Conclusion

This paper seeks to complement existing research on the treatment of errors, in particular in relation to errors that occur outside of IRE sequences associated with more student-led talk. Here, within the extended peer-to-peer exchanges, students had the opportunity to consider and challenge alternative solutions. However, for this teacher at least, the resolution of those questions relied on the more common IRE exchanges, thereby revalidating the teacher as the arbiter of correctness. The potential tensions between the interactional message that errors are to be avoided and the positive role they can take in learning have been discussed in previous studies. Here, Sam's use of bald negative evaluations has raised an alternative question; the possibility of a positive interactional message that errors are a natural part of talk, set against the potential loss of learning opportunities and diminishment of the value of student contributions. It is hoped that by raising awareness of these interactional structures, teachers could make informed choices about the different routes available to their treatment of errors and the potential impact on student learning.

Positive and negative evaluation in IRE sequences

	⁞
	53 Sam:	So two the power five times two to the power four (.) what's that
	54 Student 2:	Two to the power nine
	55 Sam:	Two to the power nine (.) OK (.) what is (.) three to the power four times
		two to the power three (…)
	56 Student 3:	Is it six to the power seven
	57 Sam:	(..) what we can't do (.) really easily (..) we just can't add those powers this
		time (..) Tom
		Extract 1:
	30 Sam:	What does two to the power four mean
	31 Student 1:	Sixteen
	32 Sam:	What does it mean
	33 Student 1:	Oh it means two times two times two times two
	34 Sam:	Excellent (.) good