Anna Teledahl
email: anna.teledahl@oru.se

Jan Olsson
email: jan.olsson@mdh.se

Feedback for creative reasoning

Keywords: Instructional design, teaching methods, feedback

This study investigates how principles of feedback to encourage students´ creative reasoning can be used by a mathematics teacher. An experienced teacher was introduced to principles, developed in pilot studies, and was instructed to plan a lesson based on four principles for feedback. During the lesson the teacher's interactions with students were recorded and the following analysis focused on the way feedback resonated with the principles. The result indicates that providing feedback which challenges students to reason creatively is difficult and complex. There are pitfalls that originate in established classroom norms for interaction, as well as beliefs about the object of teaching, and it appears that the principles, in order to become a powerful tool, require the teacher and students, to practice using them.

Introduction

Students, who are encouraged to construct their own solutions and create arguments when solving mathematical tasks, tend to, if they are successful, learn or remember more from such activities than students who are being guided by templates and prepared examples [START_REF] Jonsson | Learning mathematics through algorithmic and creative reasoning[END_REF]. Despite the disadvantages, described in numerous research reports, of teaching mathematics by providing solution methods to tasks, such teaching is still prevalent in many classrooms, in Sweden, as well as around the world [START_REF] Boesen | The relation between types of assessment tasks and the mathematical reasoning students use[END_REF][START_REF] Hiebert | The effects of classroom mathematics teaching on students' learning[END_REF]. Teaching where students create and justify their own solutions (i.e., engaging in creative reasoning) requires different teacher-student interactions than traditional teaching. Rather than explaining which method to use as well as how and why it works the teacher must encourage students, not only to construct own solutions, but also to challenge them to justify their choice of method [START_REF] Hmelo-Silver | Scaffolding and achievement in problem-based and inquiry learning: A response to Kirschner, Sweller, and Clark[END_REF].

A teacher-student interaction aimed at supporting students' construction of solutions can be compared to feedback aimed at supporting the students' learning processes and thus relies on the active involvement of the student [START_REF] Hattie | The power of feedback[END_REF][START_REF] Nicol | Formative assessment and self-regulated learning: a model and seven principles of good feedback practice[END_REF]. Research on formative assessment and feedback often reports general guidelines on how to provide feedback in teaching but few studies present empirical results detailing how feedback can be prepared and designed in classroom situations [START_REF] Hattie | Visible learning for teachers: Maximizing impact on learning[END_REF][START_REF] Palm | A review of the impact of formative assessment on student achievement in mathematics[END_REF]. Hence there is need for a deeper understanding of how formative assessment/feedback can be implemented at a classroom level (Hirsch & Lindberg, 2015). As a preparation for this study a series of four pilot studies, involving mathematical problem solving, was conducted. The interaction between students and their teacher was studied with the aim of identifying what characterizes the kind of feedback that leads students to reason creatively. The findings can be interpreted as four basic principles: (1) find out how the students are thinking, demand that students are specific, (2) encourage the students to formulate their thinking without interrupting or interpreting, (3) challenge the students to explain why their solution is working (or not working)instead of confirming or disconfirming, and (4) encourage the students to find a way to test their solution. The present study investigates how the principles can be used by a teacher who is unfamiliar with them. Our aim is to refine the principles and to form guidelines for formative feedback that teachers can use to foster creative reasoning.

Background

Teaching that encourages students to construct and justify solutions to mathematical tasks entails a focus on reasoning. Furthermore, it is reasonable to assume that teachers' feedback may guide the character of students' reasoning. The following paragraphs will outline distinctions between different types of reasoning and feedback relevant for the didactic design addressed in this paper.

Imitative and creative reasoning

If the teacher explains a definition of a mathematical concept and then demonstrates how to solve tasks associated with this particular concept, it is possible for students to solve similar tasks by remembering the procedure without understanding the definition. [START_REF] Lithner | A research framework for creative and imitative reasoning[END_REF] found that students trying to apply memorized procedures often had difficulties when solving tasks for which there had been no recent teaching. For example, calculating 2 3  2 4 using a memorized process could mean mixing up whether the numbers should be added or multiplied. The reasoning associated with such an approach is defined as imitative (IR) [START_REF] Lithner | A research framework for creative and imitative reasoning[END_REF]. A variant of IR is AR, algorithmic reasoning which is relevant for this paper. AR entails recalling a memorized, stepwise, procedure or following procedural instructions from a teacher or textbook, that are supposed to solve a task [START_REF] Lithner | A research framework for creative and imitative reasoning[END_REF]. AR is algorithmic in the sense that it solves the associated task, but it does not require an understanding of the mathematics on which the procedure is based.

An alternative approach to the example above, 2 3  2 4 , may be to consider what the mathematical meaning behind the expression is, i.e., 2 3 means 2  2  2, and 2 4 means 2  2  2  2. After realizing this, the next step is to put the two together, 2  2  2  2  2  2  2, which is 2 7 . If the student can express mathematical arguments for the solution she is engaged in creative mathematical reasoning (CMR). CMR is characterized by the construction or reconstruction of a solution method and the expressing of arguments for the solution method and the solution [START_REF] Lithner | A research framework for creative and imitative reasoning[END_REF].

Feedback

It is possible for students to reach correct answers to tasks without understanding the mathematical concepts involved [START_REF] Brousseau | Theory of didactical situations in mathematics[END_REF]. If a student should fail in his or her attempts to solve a mathematical task, the most obvious feedback from the teacher may be an explanation regarding how to solve the task, not to explain the mathematics it is based on. Should the student, however, be responsible for the construction of the solution method, she is helped by understanding the mathematics required by the task. In such cases, it is appropriate for the teacher to inquire into the student's thinking. Inquiring into students' thinking can be seen as part of formative feedback at the process level whereas delivering or assessing a right or wrong answer is seen as task level feedback [START_REF] Hattie | The power of feedback[END_REF]. Research indicates that 90 % of feedback in classrooms is on task-level [START_REF] Airasian | Classroom assessment[END_REF]. Feedback on process-level focuses on the underlying mathematical processes, which can support students' conceptual understanding as well as their autonomy [START_REF] Kazemi | Promoting Conceptual Thinking in Four Upper-Elementary Mathematics Classrooms[END_REF]. This type of feedback creates interactions in which students have an opportunity to develop their ability to use mathematical ideas to formulate arguments and justifications for their choices [START_REF] Michaels | Deliberative discourse idealized and realized: Accountable talk in the classroom and in civic life[END_REF].

Development of principles for feedback aimed at supporting creative reasoning

The principles for feedback were developed iteratively during a year in four pilot studies in collaboration with a teacher (see [START_REF] Olsson | Feedback to encourage creative reasoning[END_REF]. The starting point for the pilot studies was the design of appropriate tasks for CMR. Then possible solutions to the tasks, together with the potential difficulties that students will encounter, were considered. For each of these solutions and difficulties feedback aimed at supporting CMR was designed. The lessons were recorded and the interactions between the teacher and the students were analyzed with the aim of identifying the characteristics of the feedback that led students to reason creatively. The result of every analysis was used in the planning of the next lesson. After the fourth intervention, principles for feedback supporting CMR were formulated. These principles (see Introduction) were used in the present study to support the design of the intervention (to setup student activities where feedback for CMR is appropriate) and to prepare the teacher to formulate feedback for CMR.

Method

In order to test the previously developed principles for feedback a new study was set up in which a lesson was planned so that students would have the opportunity to engage in CMR. The principles were also used to prepare the teacher and help him support students' CMR through feedback. During the lessons the teacher's interactions with students were recorded. Compared to other possibilities (e.g., interviews) data was considered to be reasonably close to the thinking processes that create feedback and reasoning. Feedback is considered as both a response to students' actions and guiding their continued reasoning. The chain students' action -teacher's feedback -students' continued reasoning was the unit of analysis.

Sample

A teacher and students in 8 th grade of a Swedish elementary school volunteered to participate. The teacher is experienced in teaching mathematics through problem solving and the students were used to problem-solving activities. The teacher did not participate in the pilot studies where the principles for feedback were developed. Instead, he was introduced, during a half-day seminar, to the idea of supporting students´ CMR through feedback according to the principles. Transcripts from the pilot studies were discussed and tasks suitable for the class participating in this study were considered. The teacher considered the class to be average in terms of achievement, and a mix of students with Swedish as their native or second language. With respect to feedback, in his everyday teaching, the teacher considered it his intention not to guide students to solutions, although he did not have explicit strategies for achieving this. Neither the lesson, nor the feedback, would thus be considered something out of the ordinary and according to the teacher, the interactions were typical for his style of teaching.

Lesson plan

The teacher was instructed to plan a lesson containing problem solving. The role of the teacher was to support students in constructing solutions rather than explaining how to solve the problems. It was assumed that successfully constructing solutions, without knowing a solution method in advance, would require students to engage in CMR. The tasks were designed in line with [START_REF] Lithner | Principles for designing mathematical tasks that enhance imitative and creative reasoning[END_REF] principles: (1) creative challenge, no solution methods are available from the start and it must be reasonable for the students to construct the solution, (2) fair conceptual challenge to understand mathematical properties (e.g., representations and connections) and (3) justification challenge, is it reasonable for the particular student to justify the construction and implementation of a solution. The tasks were part of the curriculum and involved combinatorics. Task 1a asks the students to find all of the different ways to put three blocks of different colors in a row. The students were supplied with concrete materials. In task 1b another block, of yet another color, is added. The teacher was also instructed to prepare to give feedback according to the principles.

Procedure

During the lesson, the students worked in pairs. The tasks were presented in written text and the students were asked to present written solutions. The students were encouraged to collaborate and call the teacher if they did not understand the instructions, or if they got stuck. The teacher was wearing a recording device, which recorded his interactions with students. The recordings were transcribed into written text, with a focus on spoken language.

Analysis method

The analysis focused on the chain students' action -teacher's feedback -students' reasoning in connection teacher's feedback, and was performed through the following steps: (I). Parts of transcripts where the teacher interacts with students, and where it is considered possible to provide feedback according to the principles, were identified. (II). Parts of transcripts, where the teacher's feedback resonates with the principles for feedback, were identified. (III). It was determined whether feedback, according to the principles, supported or did not support students' CMR. (IV). In interactions, where the principles were not implemented consistently or only partly implemented, possible trajectories leading to CMR were considered and the way the principles could have supported CMR was analyzed.

The analysis was a joint effort that involved the teacher.

Results

The lesson was conducted according to plan in the sense that students got engaged in problem solving where they did not have access to a solution method for the tasks. Interactions with the teacher could be observed when a solution to a task was reached or when students asked for help. We will now present two extracts from the transcripts which can be seen as examples of interaction where the teacher can be considered to have possibilities to provide feedback according to the principles.

In the first example, which is representative of a number of interactions during the session, the student had come to a solution to task 1a:

1. Teacher:

Explain the way you are thinking.... it looks like you have some system. 2. Student:

Well.... if you start with a block.... for instance, a green one.... you can always change the order of the other two blocks.... and in this case, there are three blocks.... so, you have plus two combinations.... so, in this case it can be green-white-yellow and green-yellow-white.... so, if we have three blocks it will be three times two. 3. Teacher: OK.... you can go on to the next task

The teacher asks the student to explain her thinking (line 1) which is in line with the first principle. While the student did not know how to solve the task in advance it is reasonable to assume that the answer (line 2) represents, at least partly, CMR (constructing a solution). As a whole only the first principle is used. The student is not articulating arguments explicitly, which could have been encouraged by first inviting her to express her understanding (principle 2) and then challenge her to explain why her method results in correct answer (principle 3). The last utterance in line 2, when the student suggests a way to calculate the numbers of combinations, can be used to challenge her to justify their solution (principle 4). The teacher's comment on line 3 does not correct the student, which gives her an opportunity to find out that although the calculation is correct in connection to task 1a, it will not work in task 1b. This is an example of when the teacher's decision to refrain from explaining, gives the student an opportunity to discover the error in her conclusion. In the next task (1b) the teacher has the opportunity to challenge the student's reasoning using principles 2-4.

The second excerpt was chosen as an example where principle 3 could have been appropriate. It introduces a similar situation as example 1 and indicates the importance of the formulation of feedback. A pair of students have come to a solution on task 1a and now call for help with task 1b.

Teacher:

What did you conclude on task 1a? 2. Student:

We conclude that for every color.... if it is situated in one place.... there are two combinations for where the others could be situated.... that goes for every color situated in each place.... but if you add one block there will be four blocks and there will be three combinations for each block in each position.

Teacher:

Are you sure? 4. Student:

No.... there will be more then.

Teacher:

How many positions can this be in? 6. Student:

Then it would be.... these ones.... ok. 7. Teacher:

If there were three blocks they could be situated in.... 8. Student:

If they were three they can.... yes. 9. Teacher:

This was the prior task [1a].... wasn't it? 10. Student: Then there will be six ways for these. 11. Teacher: Yes.... if the blue one is first.... but you don't have to put the blue one first. 12. Student:

No. 13. Teacher: So, how many could it be.... how many ways is it if the yellow is first?

Student:

If the yellow is first there will be six ways. 15. Teacher: OK, so how many is it all together? 16. Student: Twenty four. 17. Teacher: OK.... could it be more ways.... are there any others that could be first? 18. Student:

No, these are the only ones you can have first.

In the planning for feedback it was considered that the reasoning connected to 1b could be continued from the solution of 1a. Therefore, on line 1 the teacher focuses the feedback to the task 1a. The question posed in line 1 may be associated to principle 1. The student´s reasoning on line 2 includes both how the solution is constructed and arguments which are components in CMR. The teacher's question on line 2 could have been in line with principle 3, but the student seems to perceive the question as an indication she is wrong. From here the teacher's feedback instead of following the principles is mainly on task-level, i.e., its purpose is to guide the students to a solution, not to support CMR. The student does not fulfil the CMR from line 2, instead, the reasoning turns into guided AR (imitating the teacher´s reasoning).

Summary of results

Mathematical reasoning is considered creative (CMR) if students themselves construct or reconstruct a solution method to a problem and express arguments to support this. The two examples above indicate that students, who are encouraged to explain their thinking, express CMR. Through their explanations it seems clear that they have constructed solutions to problems, for which they did not have access to a solution method in advance. Less clear is the way students formulate arguments for their solution. In neither of the examples are students encouraged to justify or articulate arguments. The principles for feedback in this study were developed to encourage students to construct their own solution methods and formulate mathematical support for their solutions. The results indicate that it may not be an easy task, even for experienced teachers, to implement these principles.

Discussion

Creative reasoning is a powerful tool in the learning of mathematics. In mathematical problemsolving creative reasoning, where conjecturing and justifying are viewed as important parts, leads students to construct their own solutions to mathematical tasks, something that previous studies have found beneficial for their learning [START_REF] Jonsson | Learning mathematics through algorithmic and creative reasoning[END_REF]Olsson & Teledahl, 2019). How to design teaching, in particular feedback, which will encourage students to construct solutions and present support for why they are correct, is still something that we know too little about.

Teaching where students are constructing solutions while engaging in CMR does not entail a passive teacher [START_REF] Hmelo-Silver | Scaffolding and achievement in problem-based and inquiry learning: A response to Kirschner, Sweller, and Clark[END_REF]. The teacher in this study planned the activities with the intention of creating situations where feedback, in line with the principles, would be appropriate. In the first excerpt, the students have managed to solve task 1a and principle 1 is used to challenge the students to articulate their reasoning explicitly. After the students have explained their thinking, the teacher without verifying the students' solution, encouraged them to move on to task 1b. By not confirming or disconfirming the solution to the first part of the task the teacher creates an opportunity for the students to discover that their method for solving in 1a will not work in 1b. He does not challenge them to explain anything, or ask them why they think it works, but rather leaves them to test their solution in the next stage. He is thus creating an opportunity for the students to continue their CMR, and for himself to challenge the students to formulate arguments for their solution, at a later stage when they, hopefully, have made useful discoveries about their original idea. This situation leads us to realize that the teacher not only has to consider the principles for how to challenge students to explain and provide argument but also when to do so. In this interaction giving students time and space might lead them to insights that helps them to argue for their final solution. It might be challenging to teachers to be consistent in constantly providing feedback, according to the principles, but also to retain a sensitivity to when it may be a good idea to leave the students alone.

Example 2 shows, what may be an even greater challenge, to avoid explaining how to solve a task when students ask for help. Again, principle 1 seems to work well. Students express their reasoning and leave space for feedback in line with principles 2, 3, and 4. Here, the teacher may have in mind not to explain the way to solve the task, but instead of asking the students to explain he uses questions that, step by step, guides the students towards a solution. On the surface, it might seem like he is sticking to the principles, given that he is using questions instead of statements and hints, but the questions can be seen as a way to implicitly provide the solution. The responsibility for the mathematical reasoning shifts to the teacher and the students miss an opportunity not only to develop their ability to provide mathematical arguments but also to retain their autonomy [START_REF] Kazemi | Promoting Conceptual Thinking in Four Upper-Elementary Mathematics Classrooms[END_REF][START_REF] Michaels | Deliberative discourse idealized and realized: Accountable talk in the classroom and in civic life[END_REF]. It may be tempting for the teacher to explain or to guide the students to a solution, not only because it saves time when the teachers is stressed, but also because it produces a result, a solution. In this way the product, rather than the process, becomes the main object. A solution to a problem, even if it was obtained by imitating the teacher (AR according to [START_REF] Lithner | A research framework for creative and imitative reasoning[END_REF] is considered more valuable than CMR, which may not even result in a complete solution. To focus feedback on the underlying thinking process that will solve the task is most likely difficult and it places high demands on both teachers and students.

Since the collaborative teacher in the pilot studies was involved in developing the principles for feedback she became gradually aware of how to use them and what they would lead to. In addition, while she was positive to the idea of students learning through CMR, she also practiced using the principles between the four interventions, which were planned in collaboration with the authors. The teacher, who participated in the current study, was also positive to teaching according to the principles. The difference was that he was introduced to the concept through a few hours of instructions. The two examples of interaction, shown above, highlight how important it is to carefully formulate your questions. In every classroom there is an established pattern for teacherstudent interaction, and in order for such patterns to develop towards CMR teachers and students need, not only to practice using the guidelines, but also to reflect on their current patterns of interaction.


Steven Watson
email: sw10014@cam.ac.uk

Revisiting teacher decision making in the mathematics classroom: a multidisciplinary approach

The purpose of this paper is to consider the role of mathematics teachers' thinking and decision making in the classroom. This has been a somewhat neglected area of research since the mid-1980s, but I will argue here that understanding the nature of teachers' thinking and decision making in lessons is important in understanding practice and can inform approaches to initial teacher education and professional development. While mathematics teachers' knowledge and beliefs are important, the decisions they make and the actions they implement in the lesson influence the learning environment, culture and interpretation of tasks and activities. I draw on my own empirical research along with a multidisciplinary account based on developments in cognitive psychology, neuroscience and ontology (e.g., posthumanism) to provide a theoretical account of teacher thinking and decision making and how this influences practice.

Keywords: Teachers' decision making, teaching practices, teaching methods, teachers' knowledge, teachers' beliefs.

There was considerable research into teachers' thinking and decision making during the 1970s and 1980s, but since the mid-1980s this was rather displaced by a surge of interest in teachers' knowledge [START_REF] Borko | Teachers' decision making: from Alan J. Bishop to today[END_REF]. It is not entirely clear why there was a change in direction, but it is likely that the seminal American Educational Research Association presidential speech by Lee Shulman in 1986 heralded widespread interest in teachers' pedagogic content knowledge [START_REF] Shulman | Those who understand: Knowledge growth in teaching[END_REF]. Since then researchers have been more interested in what teachers know in terms of discipline-based pedagogical knowledge than they have been in teachers' thinking. This is not to say, of course, there have been no recent contributions to the field of teacher decision making. Notably, Alan Schoenfeld's book How We Think (2011) directly addresses teacher decision making and thinking in the classroom and builds on the earlier research. But there remains further opportunity to revisit and extend the existing research using recent thinking from neuroscience and psychology. My aim here is to argue why, while acknowledging the important contribution of research into teachers' knowledge and beliefs, research into teacher thinking is necessary in order to gain a better understanding of the nature of classroom practice.

The background to this is mine and colleagues' recent research [START_REF] Watson | A phenomenological study of a novice mathematics teacher's instructional decision making[END_REF] into how teachers implement ambitious teaching approaches [START_REF] Stylianides | The role of instructional engineering in reducing the uncertainties of ambitious teaching[END_REF] in post-16 (A Level) mathematics classrooms. An intense idiographic inquiry led us to consider how teachers make decisions in the classroom and the nature of the underlying thinking and reasoning processes. We found we could not easily explain, like many researchers before us, why even in the context of supported teaching reforms, a teacher almost naturally tends to a traditional teacher-centred approach in the classroom. And when the teacher is encouraged and supported in implementing ambitious teaching and in giving mathematical authority/authorship [START_REF] Povey | Learners as authors in the mathematics classroom[END_REF] to the students they seem compelled to retain authority/ authorship. This is not by any means a criticism of teachers, but it is undoubtedly a phenomenon that warrants exploration and a search for further understanding. Our research, building on past research into teacher thinking and recent developments in cognitive psychology, neuroscience and ontological accounts such as posthumanism, suggests that in-the-moment decision making [START_REF] Schoenfeld | How we think: A theory of goal-oriented decision making and its educational applications[END_REF] in the classroom has an important role in this.

It is first necessary to distinguish between teachers' thinking in the lesson as they interact with students and the thinking and decision making that takes place in the planning of and reflecting on lessons. I adopt here the following categorization of teacher thinking developed by [START_REF] Clark | Teachers' thought processes[END_REF]:

•
Teacher planning (preactive and postactive thought); • Teacher interactive thoughts and decisions; • Teacher theories and beliefs.

There are qualitative differences between the decisions made in the moment in the lesson and the teacher's thinking about the lesson before and after the lesson. Decisions in the classroom have to be made quickly in a complex and demanding environment. But both interactive, in-the-moment decisions and planning are underscored by a teacher's theories, knowledge and beliefs. The knowledge and perspectives in memory provide the resources that contribute to reasoning processes. The turn to pedagogic content knowledge in the late 1980s tends to privilege the role of teachers' theories and beliefs, I argue here (and following our empirical research) it is the momentary decisions that influence the character of the lesson. That is, in-the-moment decisions can lead to a lesson becoming more traditional and teacher-centred even though the teacher may have the knowledge of and hold beliefs in reform-oriented student-centred approaches (or 'ambitious' teaching)1 .

Traditional teacher-centred teaching typically involves teacher exposition or demonstration followed by student practice on routine graduated exercises and might conclude with a whole-class review. In contrast, student-centred 2 practice involves students in collaborative dialogic problemsolving, inquiry-based or investigative activities in which they construct meaning and conceptual understanding (or 'relational' rather than 'procedural' understanding, see, [START_REF] Skemp | Relational understanding and instrumental understanding[END_REF] where the mathematical authority and authorship [START_REF] Povey | Learners as authors in the mathematics classroom[END_REF] is transferred to students. This has been given a catch-all characterization of 'ambitious' teaching [START_REF] Stylianides | The role of instructional engineering in reducing the uncertainties of ambitious teaching[END_REF].

From the perspective of cognitive psychology, human reasoning and decision making are characterized as either unconscious and intuitive Type 1 processes or rational and conscious deliberative Type 2 processes [START_REF] Evans | Dual-process theories of higher cognition: advancing the debate[END_REF]. Intuitive unconscious reasoning is rapid and is employed in complex and demanding situations where decisions have to be made quickly [START_REF] Johnson-Laird | How we reason[END_REF]. Conscious reasoning uses working memory to construct mental models and assess possibilities prior to making a decision and taking action. Research into teacher decision making in the 1970s and 1980s acknowledged the distinction between unconscious (Type 1) and conscious (Type 2) reasoning and recognized that classroom demands require intuitive Type 1 decisions [START_REF] Clark | Teachers' thought processes[END_REF][START_REF] Shavelson | Research on teachers' pedagogical thoughts, judgments, decisions, and behavior[END_REF]. At the same time, Clark and Peterson recognized deliberative aspects of teachers' interactive decision making in the classroom. [START_REF] Shavelson | Research on teachers' pedagogical thoughts, judgments, decisions, and behavior[END_REF] suggested teachers draw on existing routines and sub-routines to simplify their interactive decision making.

…teachers' interactive teaching may be characterized as carrying out well-established routines.

In carrying out the routine, the teacher monitors the classroom, seeking cues, such as student participation, for determining whether the routine is proceeding as planned. This monitoring is probably automatic as long as the cues are within an acceptable tolerance. However, if the teacher judges the cue to be outside tolerance (e.g., student out-of-seat behavior during discussion), the teacher has to decide if immediate action is called for. If so, the teacher has to decide if a routine is available for handling the problem. The teacher may take action based on a routine developed from previous experiences (Shavelson & Stern, 1981, p. 483).

This implies a heuristic approach to decision making that is not entirely intuitive and involves conscious decisions using rehearsed actions and pedagogic routines. The nature of teacher decision making is somewhere between Type 1 and Type 2 reasoning.

Recent research on human reasoning goes beyond a Type 1 and Type 2 dichotomy and offers an intermediate type of reasoning. [START_REF] Stanovich | The complexity of developmental predictions from dual process models[END_REF] propose an algorithmic system. This is a Type 2 conscious process but makes use of heuristics and preestablished routines and processes to reduce the demand on working memory. Algorithmic reasoning draws on knowledge which is "…tightly compiled due to overlearning and practice" (Stanovich et al., 2011, p. 107) and is based on social and cultural knowledge acquired through participation [START_REF] Stanovich | The complexity of developmental predictions from dual process models[END_REF].

In our research [START_REF] Watson | A phenomenological study of a novice mathematics teacher's instructional decision making[END_REF], most of the decision making in the classroom was based on algorithmic reasoning. For example, during the lesson we observed the teacher deciding to stop the class and explain a specific aspect of the mathematics to the whole class. He explained in a stimulated recall interview later that he made the decision to take this action during the lesson. He said that he felt that there were enough student misunderstandings to warrant some remedial action. This heuristic, he said, was learnt during his initial teacher education programme. He explained how he observed that if a teacher found that there were three or more students with a similar question, then it could be assumed that more students were likely to have the same difficulty, then they would stop the class and offer an explanation to the whole class. Based on his explanation of this vicarious learning process, it appeared likely that this heuristic had been supported with some justification from a mentor or experienced teacher. This is an example of algorithmic reasoning, it is a simplified (and almost intuitive) judgement of the situation and the retrieval of rules, procedures and strategies [START_REF] Stanovich | The complexity of developmental predictions from dual process models[END_REF].

Mathematics teaching and teaching more generally is replete with pedagogical routines and strategies. Indeed, Alexander's (2001) analysis of primary classrooms observed complex hierarchical and temporal systems of routines and rituals. Leinhardt characterizes teachers' knowledge as "…highly proceduralized and automatic and in which a highly efficient collection of heuristics exist for the solution of specific problems in teaching" (Leinhardt, 1988, p. 146).

As a secondary mathematics teacher, my own practice featured automated, proceduralized routines and heuristics. For example, I had a routine for teaching how to solve simultaneous equations. It was highly procedural, I would refer to it as a 'recipe'. And as an experienced mathematics teacher, I did not have to think, in the moment, about the pedagogical process in too much detail. I had learnt it from other teachers and used it on many occasions. If students were having difficulty solving simultaneous equations I could quickly stop the class and invoke this routine. I knew that if students were becoming anxious (and potentially disruptive) I could quickly and authoritatively provide this well-rehearsed explanation and give the students some routine practice questions, and this would provide reassurance and calm the situation.

It is widely acknowledged in the literature that a major preoccupation for all teachers is the management of the classroom and the management of student behaviour [START_REF] Martin | Teacher efficacy in student engagement, instructional management, student stressors, and burnout: A theoretical model using in-class variables to predict teachers' intent-to-leave[END_REF]. Shavelson and Stern explain the main aim of the teacher is to "maintain the flow of activity" (Shavelson & Stern, 1981, p. 484) and that decisions are made when there is an indication of a "potential problem or unexpected event" (p. 484). At these points teachers become "aware of reality" and their attention focuses on student behaviour [START_REF] Shavelson | Research on teachers' pedagogical thoughts, judgments, decisions, and behavior[END_REF]. Therefore, what prompts a teacher to make an in-the-moment or interactive decision is strongly influenced by the affective system, that is the senses, the autonomic system and physiology. While Schoenfeld offers a highly rationalistic account of goal-oriented in-the-moment decision making based on his previous research on mathematical problem solving, he acknowledges the importance of affect. He advocates for future research in decision making to work toward a "rapprochement of the physiological and psychological" (Schoenfeld, 2011, p. 195).

Indeed, this rapprochement can be observed in posthuman accounts of practice and potentially provides a compelling interdisciplinary account of thought, experience, affect and behaviour in the context of classroom practice. A posthuman3 critique of teacher thinking represent a departure from the Cartesian mind/body duality [START_REF] Strom | Teaching as assemblage: negotiating learning and practice in the first year of teaching[END_REF]; posthumanism emphasizes the physical, the material, the experiential and the embodied nature of cognition [START_REF] Braidotti | The posthuman[END_REF]. The posthuman classroom is not populated by discrete individuals but is "…an assemblage of multiple human and non-human elements that are all connected and work together to jointly produce teaching and learning activity" (Strom, Mills, Abrams, & Dacey, 2018, p. 144). A feeling or sensitivity to the environment is processed by the amygdala, which is reciprocally connected with the brain cortex and the thalamus. The amygdala is often associated with 'fight or flight' responses. Effectively it helps us decide whether a response should be intuitive and reactive or conscious and deliberative. Importantly it draws on episodic memories which have affective and emotional signatures to quickly make sense of the situation [START_REF] Markowitsch | Amygdala in action: Relaying biological and social significance to autobiographical memory[END_REF]. In contemporary liberal society human responses are much more nuanced than simply 'fight or flight': in any critical situation then the amygdala is involved in the following processes, do we have to react quickly or intuitively with deeply embedded culturally and evolutionarily compiled behaviours? Is there time to invoke socially learned and culturally embedded heuristics, strategies, routines and processes (the algorithmic mind)? Or is there time to reflect and construct and mentally evaluate alternative hypothetical courses of action and use logical processes? (see, for example, [START_REF] Janak | From circuits to behaviour in the amygdala[END_REF][START_REF] Markowitsch | Amygdala in action: Relaying biological and social significance to autobiographical memory[END_REF][START_REF] Pessoa | A network model of the emotional brain[END_REF][START_REF] Phelps | Emotion and cognition: insights from studies of the human amygdala[END_REF].

Much human behaviour is intuitive, we respond to the context with culturally and evolutionarily compiled behaviours, like, for example, the fight or flight response. Algorithmic responses are invoked where simple decisions and well-practised responses are required, and these have been the focus of the research I have referred to in this paper [START_REF] Watson | A phenomenological study of a novice mathematics teacher's instructional decision making[END_REF]. Reflective Type 2 thinking might take place in the planning and evaluation of lessons, or in the work of research and scholarship, for example.

While it might be argued that the goal of teachers' decisions is to optimize student learning (see [START_REF] Schoenfeld | How we think: A theory of goal-oriented decision making and its educational applications[END_REF] and this might be possible in the planning and design of lessons, tasks and activities, in interactive decision making, in the classroom, decisions have to be more immediate. The primary aim, in the moment, is to manage the social situation, as Cuban observes:

… teachers have learned to ration their time and energy to cope with conflicting and multiple societal and political demands by using certain teaching practices that have proved over time to be simple, resilient, and efficient solutions in dealing with large numbers in a small space for extended periods of time (Cuban, 2009, pp. 10-11).

The algorithmic mind quickly accesses routines and procedures that are learnt through participation and provide well-rehearsed and culturally-embedded approaches to respond to situations in the classroom. The mainstay of these established cultural scripts [START_REF] Stigler | The teaching gap: best ideas from the world's teachers for improving education in the classroom[END_REF] correspond to traditional teacher-centred approaches. Moments of doubt, pressure and uncertainty in a classroom are likely to result in an almost automatic response with teacher-centred routines, as we found in our research [START_REF] Watson | A phenomenological study of a novice mathematics teacher's instructional decision making[END_REF]. This shifts the mathematical authority/ authorship back to the teacher. Research into mathematics teachers' beliefs has shown that there is frequently a misalignment between teachers' espoused and enacted beliefs [START_REF] Thompson | The relationship of teachers' conceptions of mathematics and mathematics teaching to instructional practice[END_REF][START_REF] Wilson | Mathematics teacher change and development[END_REF], teachers may profess beliefs in reform-oriented student-centred ambitious teaching, yet their teaching is largely traditional and teacher-centred. Cuban refers to teacher-centred progressivism [START_REF] Cuban | Hugging the middle: How teachers teach in an era of testing and accountability[END_REF], where there are many surface features of reform and student-centred practice but fundamentally the pedagogy is teacher-centredthe mathematical authority/ authorship remains with the teacher. The classic illustration of this is Cohen's case of Mrs Oublier [START_REF] Cohen | A revolution in one classroom: the case of Mrs[END_REF]. The difference between espoused and enacted beliefs, and the manifestation of teacher-centred progressivism can be explained by considering the momentary interactive decisions and algorithmic reasoning that leads to teachers implementing culturally established and embedded teacher-centred routines and scripts, as [START_REF] Leinhardt | Situated knowledge and expertise in teaching[END_REF] observes:

[The] resistance to change on the part of the teacher should not be perceived as a form of stubborn ignorance or authoritarian rigidity but as a response to consistency of the total situation and a desire to continue to employ expert-like solutions (Leinhardt, 1988, p. 146).

The "expert-like solutions" represent the momentary intuitive judgements that draw on the teacher's established knowledgethe essence of practical reason in Aristotelian terms4 .

In this paper, I have considered the nature of teachers' decision making in the classroom. Drawing on multidisciplinary approach, I have argued how important momentary interactive decisions are in the classroom, how they can shape the character of the lesson and influence the nature of students' learning experiences. It seems as important as ever to research this decision making further, to better understand the affective dimensions, the culturally embedded routines and strategies that teachers acquire through participation in the professional community (and vicariously as a student, see, [START_REF] Lortie | Schoolteacher[END_REF]Stigler & Hiebert, 1999, for example). An improved understanding of mathematics teachers' interactive decisions would also inform the design of reform-oriented professional development and initial teacher education.

I suggest the main implication for future research is to give greater attention to teachers' decision making from a multidisciplinary perspective while using techniques such as stimulated recall. What I have presented here prepares the way for a model of teacher thinking that integrates cognition, affect and the social setting. Further research could develop and test the model.

The implicit assumption here is that the traditional teacher-centred teaching is a default and largely prevalent approach to practicethis is certainly underpinned by evidence from the Office for Standards in Education (Ofsted) in England (Ofsted,

2008, 2012). I contend this to be a justifiable claim for high school and secondary mathematics teaching practice in Europe and the US (see, for example,[START_REF] Cuban | How teachers taught: Constancy and change in American classrooms, 1890-1990[END_REF][START_REF] Stigler | The teaching gap: best ideas from the world's teachers for improving education in the classroom[END_REF].2 The characterisation of traditional teacher-centred practice and student-centred practice draws on Larry Cuban's historical analysis of classrooms in the USA[START_REF] Cuban | How teachers taught: Constancy and change in American classrooms, 1890-1990[END_REF].

Posthumanism is not a rejection of humanity but a counter to humanism, as an anthropocentric project and a challenge to Enlightenment rationality that privileges the mind as a conscious and rational system as the primary mechanism for solving individual and collective problems.

The Nicomachean Ethics[START_REF]The Nicomachean ethics[END_REF] alludes to effective practical wisdom in terms of a harmony of conscious reasoning, intuition and affect.

Acknowledgements

I would like to pay tribute to the late Professor Malcolm Swan, my PhD supervisor, with whom I did not always agree but always respected his profound insight into mathematics education. It was within the Shell Centre milieu that this line of thinking was born. I also thank Dr Matthew Inglis (Mathematics Education Centre, Loughborough University) who introduced me to the ideas of Type 1 and 2 reasoning, the work of Philip Johnson-Laird and Jonathan Evans on human reasoning from a cognitive perspective. Finally, to my colleagues in the Mathematics Education Research Group and the Culture, Politics and Global Justice cluster in the Faculty of Education, University of Cambridge who have provided helpful formative criticism from hugely diverse disciplinary, philosophical and theoretical perspectives.